tag:blogger.com,1999:blog-29452406192909906042024-03-18T17:41:43.903+05:30Miraculous world of NumbersMiraculous World of Numbers is an eBook on Mathematical Fun. It has so many creative ideas to develop the skills & to increase the Thinking power. Along with this eBook I would like to encourage Students & Parents how they can participate to build the entire world with powerful thinking level. My Blogs will definitely encourage students & parents in this direction. Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.comBlogger171125tag:blogger.com,1999:blog-2945240619290990604.post-77331253745168969082024-03-16T12:36:00.000+05:302024-03-16T12:36:17.832+05:30171-NCERT-10-7-Coordinate-geometry - Ex- 7.2<h2 style="clear: both;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><span style="color: #0400ff;"></span></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 7.2</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 7 Coordinate geometry</span></span></div></span></span></h2><h2 style="clear: both;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2024/01/NcertMathsSolution.10.CoordinateGeometry.1.html" rel="nofollow" target="_blank"><span style="color: #0400ff;"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ </span><span style="color: #0400ff;">NCERT-10-7-Coordinate-geometry - Ex- 7.1</span></a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 7.2</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>Q 1. Find the coordinates of the point which divides the join of </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(–1, 7) and (4, –3) in the </b></span><b>ratio 2 : 3.</b></span></div></blockquote><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhxiTdnPtwXlA-f0Rx9CS0gFO5vmB_IkawLuBriS5UeBK3gDW0aZFgiU5-uI4FJZ6BgpJWbS5JFbyCHx95k8bc784GFhC7po5mxE9aO5v_dszFm2sCoqNszkG8ewOYlAxyvP4uh_gl81Jnm79-rx8CBLnRkF6baT99NNuKQHgFvlHDXN24C9Or913/s515/3.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="82" data-original-width="515" height="51" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhxiTdnPtwXlA-f0Rx9CS0gFO5vmB_IkawLuBriS5UeBK3gDW0aZFgiU5-uI4FJZ6BgpJWbS5JFbyCHx95k8bc784GFhC7po5mxE9aO5v_dszFm2sCoqNszkG8ewOYlAxyvP4uh_gl81Jnm79-rx8CBLnRkF6baT99NNuKQHgFvlHDXN24C9Or913/s320/3.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">– 1</span><span style="font-family: arial; font-size: medium;">, 7) and </span><span style="font-family: arial; font-size: medium;">B(4</span><span style="font-family: arial; font-size: medium;">, </span><span style="font-family: arial;">– 3</span><span style="font-family: arial; font-size: medium;">) in</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">the ratio 2 : 3, so,</span></blockquote></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-weight: normal;">a) first we will find x-coordinate</span> </div></div></span></h3></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><h3><span style="font-family: arial; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-weight: 400; white-space: normal;">x = ((2) (4)</span><sub style="font-weight: 400; white-space: normal;"> </sub><span style="font-weight: 400; white-space: normal;">+ (3) (</span><span style="font-weight: 400; white-space: normal;">– 1</span><span style="font-weight: 400; white-space: normal;">))/(2 + 3)</span></div></div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = (8</span><sub> </sub><span>–</span><span> 3</span><span>)/(5)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = (5</span><span style="font-family: arial;">)/(5)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 1</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">b) now we will find y-coordinate</span> </div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><h3><span style="font-family: arial; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-family: arial; font-size: medium; font-weight: normal;">y = (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)</span></div></span></h3></div><h3 style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; font-weight: normal;"><span style="white-space: normal;">y = ((2) (</span><span style="white-space: normal;">–</span><span style="white-space: normal;"> 3</span><span style="white-space: normal;">)</span><sub style="white-space: normal;"> </sub><span style="white-space: normal;">+ (3) (</span><span style="white-space: normal;">7</span><span style="white-space: normal;">))/(2 + 3)</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>y = (</span><span>–</span><span> 6</span><sub> </sub><span>+</span><span> 21</span><span>)/(5)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = (15</span><span style="font-family: arial;">)/(5)</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote><span style="font-family: arial; font-size: medium;">y = 3</span></blockquote></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) So, the point P(1, 3) </span><span style="font-family: arial;">divides the segment joining the points A(</span><span style="font-family: arial;">– 1</span><span style="font-family: arial;">, 7) and </span><span style="font-family: arial;">B(4</span><span style="font-family: arial;">, </span><span style="font-family: arial;">– 3</span><span style="font-family: arial;">)</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">in </span><span style="font-family: arial;">the ratio 2 : 3.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 2. Find the coordinates of the points of trisection of the line segment</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>joining (4, –1) </b></span><b>and (–2, –3).</b></span></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBjjQ0gJCnaAPHQJSav732mEWNTxds6WfEb6GjEr_GS_QkcfCHySwx93cGiHSWOxDDXjZTawktPumqcFEqDx6JtJvDUcjj-t2A0mxNT0AMMHMNx4pGMN9IG03U5nSqes6eQvcmjjCidU_K8PEsSTKUY4dDvfM1ctwPqj2BPGZVoZT9dHDaXQWGcUUE/s556/4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="80" data-original-width="556" height="52" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBjjQ0gJCnaAPHQJSav732mEWNTxds6WfEb6GjEr_GS_QkcfCHySwx93cGiHSWOxDDXjZTawktPumqcFEqDx6JtJvDUcjj-t2A0mxNT0AMMHMNx4pGMN9IG03U5nSqes6eQvcmjjCidU_K8PEsSTKUY4dDvfM1ctwPqj2BPGZVoZT9dHDaXQWGcUUE/w363-h52/4.png" width="363" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) The point P(</span><span style="font-family: arial;">x</span><sub>1</sub><span style="font-family: arial;">, y</span><sub>1</sub><span style="font-family: arial;">) divides the segment joining the points </span></div></span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="font-family: arial; font-size: medium;">A(4, </span><span style="font-family: arial;">– 1</span><span style="font-family: arial; font-size: medium;">) and </span><span style="font-family: arial; font-size: medium;">B(</span><span style="font-family: arial;">– 2</span><span style="font-family: arial; font-size: medium;">, </span><span style="font-family: arial;">– 3</span><span style="font-family: arial; font-size: medium;">) in </span><span style="font-family: arial;">the ratio 1 : 2, so,</span></div></span></div></h3></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">a) first we will find x-coordinate</span> <span style="font-weight: normal;">of point P(x<sub>1</sub>, y<sub>1</sub><span style="font-family: arial;">)</span></span></div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; text-align: left; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">1 </sub><span style="font-family: arial;">= (mx</span><sub>2 </sub><span style="font-family: arial;">+ nx</span><sub>3</sub><span style="font-family: arial;">)/(m + n)</span></blockquote></div></span></h3></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">x<sub style="font-weight: 400;">1</sub></span> = ((1) (</span><span style="font-family: arial;">– 2</span><span style="font-family: arial;">)<sub style="font-weight: 400;"> </sub><span style="font-weight: 400;">+ (2) (</span><span style="font-weight: 400;">4</span><span style="font-weight: 400;">))/(1 + 2)</span></span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">1</sub><span> = (</span><span>– 2</span><sub> </sub><span>+</span><span> 8</span><span>)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">1</sub><span> = (6</span><span>)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">1</sub><span> = 2</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">b) now we will find y-coordinate</span> </div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">y<sub>1</sub></span> = (my<sub style="white-space: pre-wrap;">2 </sub><span style="white-space: pre-wrap;">+ ny</span><sub style="white-space: pre-wrap;">3</sub><span style="white-space: pre-wrap;">)/(m + n)</span></span></span></div></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-weight: normal;"><span style="white-space: pre-wrap;"><span>y<sub>1</sub></span><span style="white-space: normal;"> =</span></span> ((1) (– 3)<sub> </sub>+ (2) (</span>– 1))/(</span><span style="font-family: arial;">1 + 2</span><span style="font-family: arial;">)</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>1</sub></span> = (– 3<sub> </sub></span><span style="font-family: arial;">– 2</span><span style="font-family: arial;">)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>1</sub></span> = (</span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 5</span><span style="font-family: arial;">)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>1</sub></span> = </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 5</span><span style="font-family: arial;">/3</span></span></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) So, the coordinates of the point P(</span><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">1</sub><span style="white-space: pre-wrap;">, y</span><sub style="white-space: pre-wrap;">1</sub><span>) </span><span>is P(2, </span><span>–</span><span> 5</span><span>/3)</span></span></div><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">3) The point Q(</span><span style="font-family: arial;">x</span><sub>2</sub><span style="font-family: arial;">, y</span><sub>2</sub><span style="font-family: arial;">) divides the segment joining the points </span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium;">A(4, </span><span style="font-family: arial;">– 1</span><span style="font-family: arial; font-size: medium;">) and </span><span style="font-family: arial; font-size: medium;">B(</span><span style="font-family: arial;">– 2</span><span style="font-family: arial; font-size: medium;">, </span><span style="font-family: arial;">– 3</span><span style="font-family: arial; font-size: medium;">) in </span><span style="font-family: arial;">the ratio 2 : 1, so,</span></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">a) first we will find x-coordinate</span> <span style="font-weight: normal;">of point </span><span style="font-family: arial; font-weight: 400;">Q(</span><span style="font-weight: 400;">x</span><sub style="font-weight: 400;">2</sub><span style="font-weight: 400;">, y</span><sub style="font-weight: 400;">2</sub><span style="font-family: arial; font-weight: 400;">)</span></div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">2 </sub><span style="font-family: arial;">= (mx</span><sub>1 </sub><span style="font-family: arial;">+ nx</span><sub>3</sub><span style="font-family: arial;">)/(m + n)</span></blockquote></div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">x<sub>2</sub></span> = ((2) (</span><span style="font-family: arial;">– 2</span><span style="font-family: arial;">)<sub> </sub>+ (1) (4))/(2 + 1)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">2</sub><span> = (</span><span>– 4</span><sub> </sub><span>+</span><span> 4</span><span>)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">2</sub><span> = (0</span><span>)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">2</sub><span> = 0</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">b) now we will find y-coordinate</span> </div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">y<sub>2</sub></span> = (my<sub style="white-space: pre-wrap;">1 </sub><span style="white-space: pre-wrap;">+ ny</span><sub style="white-space: pre-wrap;">3</sub><span style="white-space: pre-wrap;">)/(m + n)</span></span></div></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;"><span>y<sub>2</sub></span><span style="white-space: normal;"> =</span></span> ((2) (– 3)<sub> </sub>+ (1) (– 1))/(</span><span style="font-family: arial;">2 + 1</span><span style="font-family: arial;">)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>2</sub></span> = (– 6<sub> </sub></span><span style="font-family: arial;">– 1</span><span style="font-family: arial;">)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>2</sub></span> = (</span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 7</span><span style="font-family: arial;">)/(3)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">y<sub>2</sub></span> = </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 7</span><span style="font-family: arial;">/3</span></span></blockquote></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) So, the coordinates of the point Q(</span><span style="white-space: pre-wrap;">x</span><sub style="white-space: pre-wrap;">2</sub><span style="white-space: pre-wrap;">, y</span><sub style="white-space: pre-wrap;">2</sub><span>) </span><span>is P(0, </span><span>–</span><span> 7</span><span>/3).</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: justify;"><span style="font-family: arial; font-size: medium;"><span><b>Q 3. To conduct Sports Day activities, in </b></span><b>your rectangular-shaped school</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: justify;"><span style="font-family: arial; font-size: medium;"><span><b>ground ABCD, lines have been </b></span><b>drawn with chalk powder at a </b><b>distance of 1m each. 100 flower pots </b><b>have been placed at a distance of 1m </b><b>from each other along AD, as shown </b><b>in the following fig., Niharika runs </b><b>1/4 th the </b><b>distance AD on the 2nd line and </b><b>posts a green flag. Preet runs </b><b>1/5 th the distance AD on the eighth line </b><b>and posts a red flag. What is the </b><b>distance between both flags? If </b><b>Rashmi has to post a blue flag halfway between the line segment </b><b>joining the two flags, where should she post her flag?</b></span></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCZ66lOxkQSykRqpapvHIPkrx_d_YgFJG7Fk4VFVtAHC_HcpZ8uTTQbQr_Tgkf4R8CAlF_DGfCNnulrIVfr3xCO3CiTmULV8OxlqyLe8CT3NwDbEJZnRS4BHAsYi8yOD8AKA_Ae3p_Zk5t1mVoDSj-M3LVhga-fIpZBffJkHU5Vd-aKdqBGvjcnF60/s561/5.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="561" data-original-width="509" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCZ66lOxkQSykRqpapvHIPkrx_d_YgFJG7Fk4VFVtAHC_HcpZ8uTTQbQr_Tgkf4R8CAlF_DGfCNnulrIVfr3xCO3CiTmULV8OxlqyLe8CT3NwDbEJZnRS4BHAsYi8yOD8AKA_Ae3p_Zk5t1mVoDSj-M3LVhga-fIpZBffJkHU5Vd-aKdqBGvjcnF60/s320/5.png" width="290" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) According to the problem, the total distance of AD is 100 m.</span></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">2) Niharika covers 1/4th of the distance of AD. i.e. (1/4)(100) = 25 on 2nd line. So</span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">Niharika posts a green flag at the coordinates P(2, 25).</span></div></h3></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) In the same way, <span style="white-space: pre-wrap;">Preet covers 1/5th of the distance of AD. i.e. (1/5)(100) = 20</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">on 8th line. So </span></span><span style="font-family: arial; white-space: pre-wrap;">Preet posts a red flag at the coordinates Q(8, 20).</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) so using the distance formula, we can find the distance between two flags as</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">follows:</span></div></blockquote><span style="font-family: arial; font-size: medium;"><span>5) The coordinates of P and Q are P(2, 25) and Q(8, 20), so here</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) </span><span>x</span><sub>1</sub><span> = 2<br /></span><span>b) </span><span>y</span><sub>1</sub><span> = 25</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>c) </span><span>x</span><sub>2</sub><span> = 8</span><br /><span>d) </span><span>y</span><sub>2</sub><span> = 20</span> </span></div></blockquote><span style="font-family: arial; font-size: medium;"><span>6) We know that:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(PQ) = </span><span><b>√</b>[(</span><span><span style="line-height: 17.12px;"><span>x<sub>1</sub> – x<sub>2</sub></span></span><span>)</span></span><sup>2</sup><span> +</span><span> </span><span>(</span><span>y<sub>1</sub> – y<sub>2</sub>)</span><sup>2</sup><span>]</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(</span><span>PQ</span><span>) = </span><span><b>√</b>[(2</span><span> – 8)</span><sup>2</sup><span> +</span><span> </span><span>(25</span><span> – 20)</span><sup>2</sup><span>]</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(</span><span>PQ</span><span>) = </span><span><b>√</b>[(</span><span>– 6)</span><sup>2</sup><span> +</span><span> </span><span>(5</span><span>)</span><sup>2</sup><span>]</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(</span><span>PQ</span><span>) = </span><span><b>√</b>[36</span><span> +</span><span> 25</span><span>]</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-size: medium;"><span style="font-family: arial;">(</span><span style="font-family: arial;">PQ</span><span style="font-family: arial;">) = </span><span style="font-family: arial;"><b>√</b>61</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>7) The distance between the two flags is </span><b>√</b><span>61 m.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) As Rashmi puts the blue flag in the middle of the green and red flag, i.e.,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>R(x, y) = ((</span><span>x</span><sub>1</sub><span> </span><span>+ x</span><sub>2</sub><span>)/2, </span><span>(</span><span>y</span><sub>1</sub><span> </span><span>+ y</span><sub>2</sub><span>)/2)</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">R(x, y) = ((2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">+ 8</span><span style="font-family: arial;">)/2, </span><span style="font-family: arial;">(25</span><span style="font-family: arial;"> </span><span style="font-family: arial;">+ 20</span><span style="font-family: arial;">)/2)</span></span></div><span style="font-size: medium;"><span style="font-family: arial;">R(x, y) = ((10</span><span style="font-family: arial;">)/2, </span><span style="font-family: arial;">(45</span><span style="font-family: arial;">)/2)<br /></span><span style="font-family: arial;">R(x, y) = (5</span><span style="font-family: arial;">, 22.5</span><span style="font-family: arial;">)</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #333333;">9) Therefore, Rashmi should post her blue flag at 22.5m on the 5th line.</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 4. Find the ratio in which the line segment joining the points (– 3, 10) </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>and (6, – 8) is divided </b></span><b>by (– 1, 6).</b> </span></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3></div></div><div><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">1) Let the point P(</span><span style="font-family: arial;">– 1, 6) divides </span><span style="font-family: arial; white-space: pre-wrap;">segment A(<span style="white-space: normal;">– 3</span>, </span><span style="font-family: arial;">10) B(6, </span><span style="font-family: arial;">– 8) in the ratio k : 1, so</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">using section formula, we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = (mx</span><sub>2 </sub><span>+ nx</span><sub>1</sub><span>)/(m + n)<br /></span><span>(6</span><span>k </span><span>– 3</span><span>)/(k + 1) = </span><span>– 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(6</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 3</span><span style="font-family: arial;">) = </span><span style="font-family: arial;">– 1(k + 1)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(6</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 3</span><span style="font-family: arial;">) = </span><span style="font-family: arial;">– k </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(6</span><span style="font-family: arial;">k </span><span style="font-family: arial;">+ k</span><span style="font-family: arial;">) = </span><span style="font-family: arial;">– 1 </span><span style="font-family: arial;">+</span><span style="font-family: arial;"> 3</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7k</span><span style="font-family: arial;"> </span><span style="font-family: arial;">= 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">k = 2/7</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) The ratio is 2:7.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 5. Find the ratio in which the line segment joining A(1, – 5) and B(– 4, 5) is divided by the </b></span><b>x-axis. Also, find the coordinates of the point of division.</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3></div></div><div><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">1) Let the line </span><span style="font-family: arial; white-space: pre-wrap;">segment joining the points A(1, <span style="white-space: normal;">– 5</span></span><span style="font-family: arial;">) and B(</span><span style="font-family: arial;">– 4, 5) get divided by the</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x-axis in the ratio k : 1.</span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) We know that the y-coordinate of every point on the x-axis is 0, so first we will</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">find the </span><span style="font-family: arial;">y-coordinate. </span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) Using the section formula, we have,</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (my</span><sub>2 </sub><span>+ ny</span><sub>1</sub><span>)/(m + n)</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = (5</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 5</span><span style="font-family: arial;">)/(k + 1)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">0 = (5</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 5</span><span style="font-family: arial;">)/(k + 1)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">0 </span><span style="font-family: arial;">(k + 1) </span><span style="font-family: arial;">= (5</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 5</span><span style="font-family: arial;">)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(5</span><span style="font-family: arial;">k </span><span style="font-family: arial;">– 5</span><span style="font-family: arial;">)</span><span style="font-family: arial;"> = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5k</span><span style="font-family: arial;"> </span><span style="font-family: arial;">= 5</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">k = </span><span style="font-family: arial;">5/5</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>k = 1</span> </span></div></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) The x-axis divides the line segment </span><span style="font-family: arial; white-space: pre-wrap;">joining the points A(1, <span style="white-space: normal;">– 5</span></span><span style="font-family: arial;">) and B(</span><span style="font-family: arial;">– 4, 5) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1:1 ratio</span><span style="font-family: arial;">.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Now we will find the x-coordinate point of division with the ratio 1:1</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">Using the section formula, we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = (mx</span><sub>2 </sub><span>+ nx</span><sub>1</sub><span>)/(m + n)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = (x</span><sub>2 </sub><span>+ x</span><sub>1</sub><span>)/(1 + 1)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = (</span><span>– 4</span><sub> </sub><span>+ 1</span><span>)/(2)<br /></span><span>x = (</span><span>– 3</span><span>)/(2)<br /></span><span>x = </span><span>– 3</span><span>/2</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) So the coordinates of the point of division are (– 3/2, 0).</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 6. If (1, 2), (4, y), (x, 6), and (3, 5) are the vertices of a parallelogram taken in order, find </b></span><b>x and y.</b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQrh3QpZExxly6CflHfnZyyTxSA_Cvq-YZFsPV_b8mVMt1mg1U-P4ZrGoKQEmw4pLdcbJn-wJwzDbdRbWjN--TkiOZVoxrlobhkirngODhqduRzSPSCLJS2yrDM4V67lnsnrZXVG6ieXHk7N43Q6dxwNj0NLHCVPisB1ymTpXQsMe2jfcZuxml7d2s/s704/6.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="345" data-original-width="704" height="157" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQrh3QpZExxly6CflHfnZyyTxSA_Cvq-YZFsPV_b8mVMt1mg1U-P4ZrGoKQEmw4pLdcbJn-wJwzDbdRbWjN--TkiOZVoxrlobhkirngODhqduRzSPSCLJS2yrDM4V67lnsnrZXVG6ieXHk7N43Q6dxwNj0NLHCVPisB1ymTpXQsMe2jfcZuxml7d2s/s320/6.png" width="320" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">1) As ABCD is the parallelogram, the diagonals bisect each other.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">2) So point O is the midpoint of diagonal AC and diagonal BD.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">3) Now we will find the mid-point of AC</span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">Mid-point of AC = </span><span style="font-family: arial; font-weight: 400; white-space: pre-wrap;">((1 + x)/2, (2 + 6)/2)</span></span></div></h3></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">Mid-point of AC = </span><span style="font-family: arial; white-space: pre-wrap;">((1 + x)/2, (8)/2)<br /></span><span style="font-family: arial; white-space: pre-wrap;">Mid-point of AC = </span><span style="font-family: arial; white-space: pre-wrap;">((1 + x)/2, 4) --------- equation 1</span></span></blockquote><div><h3><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">4) Now we will find the mid-point of BD</span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-size: medium;"><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">Mid-point of BD = </span><span style="font-family: arial; font-weight: 400; white-space: pre-wrap;">((3 + 4)/2, (5 + y)/2)</span></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">Mid-point of BD = </span><span style="font-family: arial; white-space: pre-wrap;">((7)/2, (y + 5)/2)<br /></span><span style="font-family: arial; white-space: pre-wrap;">Mid-point of BD = </span><span style="font-family: arial; white-space: pre-wrap;">(7/2, </span><span style="font-family: arial; white-space: pre-wrap;">(y + 5)/2</span><span style="font-family: arial; white-space: pre-wrap;">) --------- equation 2</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) From equations 1 and 2, we will get the x-coordinate,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">(1 + x)/2 = 7/2</span></div><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">(1 + x) = 7<br /></span><span style="font-family: arial; white-space: pre-wrap;">x = 7 </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 1</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">x = 6</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 3</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) From equations 1 and 2, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">we will get the y-coordinate,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">(y + 5)/2</span><span style="font-family: arial; white-space: pre-wrap;"> = 4</span><span style="font-family: arial; white-space: pre-wrap;"><br /></span><span style="font-family: arial; white-space: pre-wrap;">(y + 5)</span><span style="font-family: arial; white-space: pre-wrap;"> = 4 (2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">(y + 5)</span><span style="font-family: arial; white-space: pre-wrap;"> = 8</span></span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">y = 8 </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 5</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">y = 3</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 4</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) From equations 3 and 4, x = 6 and y = 3.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 7. Find the coordinates of point A, where AB is the diameter of a circle whose center is </b></span><b>(2, – 3) and B is (1, 4).</b></span></div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3NHAVEeBNDRTuKjPWgJbS_dbhCNtRWxi1LTMlnZ0vB6kXqmwzh5oCYAj3mkFz1ydyM6BEwRAed1a5UDBhmH00SqypNbULAUUAnrMvrYCgFltlnaxXDWM3iI_HBJl4YQLJSy7_JAu3Imr6NC7C84pamqGgI-7KB0dPYEGW55VR2SSXzOp5mhmqJCmV/s450/7.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="298" data-original-width="450" height="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3NHAVEeBNDRTuKjPWgJbS_dbhCNtRWxi1LTMlnZ0vB6kXqmwzh5oCYAj3mkFz1ydyM6BEwRAed1a5UDBhmH00SqypNbULAUUAnrMvrYCgFltlnaxXDWM3iI_HBJl4YQLJSy7_JAu3Imr6NC7C84pamqGgI-7KB0dPYEGW55VR2SSXzOp5mhmqJCmV/w264-h175/7.png" width="264" /></a></div></span></h3><h3><div><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">1) Let the coordinates of point A be (x, y).</span></div><div><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">2) AB is the diameter of the circle with center O(2, </span><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;">– 3).</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">3) So, point O</span><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">(2, </span><span style="font-weight: normal;"><span style="font-family: arial;">– 3) is the mid-point of the segment joining the points </span></span></span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><h3><div style="text-align: left;"><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;">A(x, y) and B(1, 4).</span></span></div></h3></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) So, using the mid-point form, we will find the x-coordinate of point A,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x + 1)/2 = 2</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(x + 1) = 2 (2)<br /></span><span style="font-family: arial;">(x + 1) = 4<br /></span><span style="font-family: arial;">x = 4 </span><span style="font-family: arial;">– 1</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 3</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) So, using the mid-point form, we will find the y-coordinate of point A,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(y + 4)/2 = </span><span style="font-family: arial;">– 3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(y + 4) = </span><span style="font-family: arial;">– 3</span><span style="font-family: arial;"> (2)<br /></span><span style="font-family: arial;">(</span><span style="font-family: arial;">y + 4</span><span style="font-family: arial;">) = </span><span style="font-family: arial;">– 6</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">y = </span><span style="font-family: arial;">– 6</span><span style="font-family: arial;"> </span><span style="font-family: arial;">– 4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = </span><span style="font-family: arial;">– 10</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 2</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">6) From equations 1 and 2, x = 3 and y = </span><span style="font-family: arial;">– 10</span><span style="font-family: arial;">. So, the coordinates of </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">A are A(3, </span><span style="font-family: arial;">– 10).</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 8. If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that </b></span><b>AP = (3/7) AB and P lies on the line segment AB.</b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheENkJuSwWgrcpi0-jZpB2oRDa4XIKnYkASeGWe-qMl7HLtc2qdwhcbiV88MyOU6h2v_S11Sr1PXUUyhfmYtE8UIg-rjaBU9PuYJ2yKNaOUSnCfq2wnFsowPigyncqJ2_7MkJt324jt0PbmfgtuKg-9Jov-P45IGycf0ROC8dRuDYZxMuZk2L8Yx3B/s503/8.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="76" data-original-width="503" height="48" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheENkJuSwWgrcpi0-jZpB2oRDa4XIKnYkASeGWe-qMl7HLtc2qdwhcbiV88MyOU6h2v_S11Sr1PXUUyhfmYtE8UIg-rjaBU9PuYJ2yKNaOUSnCfq2wnFsowPigyncqJ2_7MkJt324jt0PbmfgtuKg-9Jov-P45IGycf0ROC8dRuDYZxMuZk2L8Yx3B/s320/8.png" width="320" /></a></div></span></h3><h3><div><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">1) Let the coordinates of point P be (x, y).</span></div><div><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">2) Point P divides AB such that, AP = (3/7) AB, so,</span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><div style="text-align: left;"><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;">(3/7) AB = AP</span></span></div></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(AB)/(AP) = 7/3<br /></span><span style="font-family: arial;">(AB - AP)/(AP) = (7 - 3)/3<br /></span><span style="font-family: arial;">(PB)/(AP) = (4)/3<br /></span><span style="font-family: arial;">(AP)/(PB) = 3/4</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">3) Here, point P</span><span style="font-family: arial; white-space: pre-wrap;">(x, </span><span style="font-family: arial;">y) divides the segment joining point A(-2, -2) and B(2, -4), So,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">we now, will find x coordinate</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = (3 (2) + 4 (- 2))/(3 + 4)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = (6 - 8)/(7)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (- 2)/(7)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = - 2/7</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">we now, will find y coordinate</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = (3 (- 4) + 4 (- 2))/(3 + 4)<br /></span><span style="font-family: arial;">y = (- 12 - 8)/(7)<br /></span><span style="font-family: arial;">y = (- 20)/(7)<br /></span><span style="font-family: arial;">y = - 20/7</span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 2</span></span></blockquote></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">4) From equations 1 and 2, x = </span><span style="font-family: arial;">- 2/7</span><span style="font-family: arial;"> and y = </span><span style="font-family: arial;">- 20/7</span><span style="font-family: arial;">. So, the coordinates of </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">P are P(- 2/7, </span><span style="font-family: arial;">- 20/7).</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 9. Find the coordinates of the points which divide the line segment joining A(– 2, 2) and </b></span><b>B(2, 8) into four equal parts.</b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s594/2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="594" height="58" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigkuFQddjrKoDHw8BI0l3X2dT6vVPRol7ETGIpTwHSZiEYc0ziwVHEPXGJLuRXbowh3r2eOZV-U6VQV7xpdBsiXLuuLgYILw0VmifZVa-gbawK_xOn27pxsoGbSe1wiqNgURt06SBpf75PRAKIjefM7azvYZRTrIUYiCz45v2VGcp26DO7rP4G2VSg/s320/2.png" width="320" /></a></div></span></h3><div><span style="font-size: medium;"><span style="font-family: arial;">1) The point P(x, y) divides the segment joining the points A(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) and </span><span style="font-family: arial;">B(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) in</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the ratio m:n, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = (m</span><span>y</span><sub>2 </sub><span>+ n</span><span>y</span><sub>1</sub><span>)/(m + n)</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;">2) In general P(x, y) = ((mx<sub>2 </sub>+ nx<sub>1</sub>)/(m + n), (my<sub>2 </sub>+ ny<sub>1</sub>)/(m + n)) </span><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM303jSlCi6tySFenjgZ93c_erJ-Q4iIXE1E0KpDA3rua0OAZuKsW4k2eSHAwqsdU9KJQ_JlU7jZzOwpdefsoIKrw-M9UN6xU_1euNVlCdN1zYJzyadU6Uf4V1laqhit06-rwIWU6e0wxQKMQv3VDXnKIlUgxKM59bvQV50aezUvm4bExcaUnOyXyV/s937/9.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="95" data-original-width="937" height="47" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM303jSlCi6tySFenjgZ93c_erJ-Q4iIXE1E0KpDA3rua0OAZuKsW4k2eSHAwqsdU9KJQ_JlU7jZzOwpdefsoIKrw-M9UN6xU_1euNVlCdN1zYJzyadU6Uf4V1laqhit06-rwIWU6e0wxQKMQv3VDXnKIlUgxKM59bvQV50aezUvm4bExcaUnOyXyV/w472-h47/9.png" width="472" /></a></div></span></h3></div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">1) The point Q(</span><span style="white-space: normal;">x</span><sub style="white-space: normal;">2</sub><span style="white-space: normal;">, y</span><sub style="white-space: normal;">2</sub><span style="font-family: arial; white-space: normal;">) the mid-point of the segment joining the points </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium; white-space: normal;">A(- 2, 2), B(2, 8), so the coordinates of point Q will be:</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">Q(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) = ((- 2 + 2)/2, (2 + 8)/2)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">Q(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) = ((0)/2, (10)/2)<br /></span><span style="font-family: arial;">Q(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>) = (0, 5) -------- equation 1</span></span></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">2) The point P(</span><span style="white-space: normal;">x</span><sub style="white-space: normal;">1</sub><span style="white-space: normal;">, y</span><sub style="white-space: normal;">1</sub><span style="font-family: arial; white-space: normal;">) the mid-point of the segment joining the points </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium; white-space: normal;">A(- 2, 2), Q(0, 5), so the coordinates of point Q will be:</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">P(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>) = ((- 2 + 0)/2, (2 + 5)/2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">P(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>)</span><span style="font-family: arial;"> = ((- 2)/2, (7)/2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">P(</span><span style="font-family: arial;">x<sub>1</sub>, y<sub>1</sub>)</span><span style="font-family: arial;"> = (- 1, 7/2) -------- equation 2</span></span></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">3) The point R(</span><span style="white-space: normal;">x</span><sub style="white-space: normal;">3</sub><span style="white-space: normal;">, y</span><sub style="white-space: normal;">3</sub><span style="font-family: arial; white-space: normal;">) the mid-point of the segment joining the points </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium; white-space: normal;">Q(0, 5), B(2, 8), so the coordinates of point Q will be:</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>R(</span><span>x</span><sub>3</sub><span>, y</span><sub>3</sub><span>)</span><span> = ((0 + 2)/2, (5 + 8)/2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>R(</span><span>x</span><sub>3</sub><span>, y</span><sub>3</sub><span>)</span><span> = ((2)/2, (13)/2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>R(</span><span>x</span><sub>3</sub><span>, y</span><sub>3</sub><span>)</span><span> = (1, 13/2) -------- equation 3</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>4) From equations 1, 2, and 3, </span><span>the coordinates of points P, Q, and R are as follows.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>P(</span><span>x</span><sub>1</sub><span>, y</span><sub>1</sub><span>) = P</span><span>(- 1, 7/2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">Q(</span><span style="font-family: arial;">x<sub>2</sub>, y<sub>2</sub>)</span><span style="font-family: arial;"> = Q</span><span style="font-family: arial;">(</span><span style="font-family: arial;">0, 5</span><span style="font-family: arial;">)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">R(</span><span style="font-family: arial;">x<sub>3</sub>, y<sub>3</sub>)</span><span style="font-family: arial;"> = R</span><span style="font-family: arial;">(</span><span style="font-family: arial;">1, 13/2</span><span style="font-family: arial;">).</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 10. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4), </b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>and (– 2, – 1) taken in </b></span><b>order. [Hint: Area of a rhombus = (1/2) (product of its diagonals)].</b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3></div><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDCI2mYiAvdv4Qt2ISmfw_6Qv4Xep-Rw6Ln6OfR9VmtggGR2yw05qaeAP5xK-OHzxPYxGCbPUqoeBAJjhLG0VDYVjR9r6zl5LaHLE4kZjhpQU6C5N6OZrpjhsiNWhNGtC4T_pDchbMvILm-LjClF0yTx7aI0bj5Ns9gBKDiIOA9Qdbh9n9iidpo0n4/s375/10.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="231" data-original-width="375" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDCI2mYiAvdv4Qt2ISmfw_6Qv4Xep-Rw6Ln6OfR9VmtggGR2yw05qaeAP5xK-OHzxPYxGCbPUqoeBAJjhLG0VDYVjR9r6zl5LaHLE4kZjhpQU6C5N6OZrpjhsiNWhNGtC4T_pDchbMvILm-LjClF0yTx7aI0bj5Ns9gBKDiIOA9Qdbh9n9iidpo0n4/s320/10.png" width="320" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">1) We know that the area of the rhombus = (1/2) (product of diagonals).</span></span></div><div><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">2) So first we will find AC and BD using the distance formula.</span></span></div><div><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;">3) First we will find AC with <span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">A(3, 0</span><span style="white-space: normal;">), </span></span>C(- 1, 4)</div></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">a) </span>x<sub>1</sub> = 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">b) </span>y<sub>1</sub> = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">c) </span>x<sub>2</sub> = - 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">d) </span>y<sub>2</sub> = 4 </span></blockquote><div><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">4) We know that:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AC) = </span><span style="font-family: arial;"><b>√</b>[(</span><span><span style="line-height: 17.12px;"><span style="font-family: arial;">x<sub>1</sub> – x<sub>2</sub></span></span><span style="font-family: arial;">)</span></span><sup>2</sup><span style="font-family: arial;"> +</span><span style="font-family: arial;"> </span><span style="font-family: arial;">(</span><span style="font-family: arial;">y<sub>1</sub> – y<sub>2</sub>)</span><sup>2</sup><span style="font-family: arial;">]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(</span><span style="font-family: arial;">AC</span><span style="font-family: arial;">) = </span><span style="font-family: arial;"><b>√</b>[(</span><span style="font-family: arial;">3 – (– 1)</span><span style="font-family: arial;">)</span><sup>2</sup><span style="font-family: arial;"> +</span><span style="font-family: arial;"> </span><span style="font-family: arial;">(0</span><span style="font-family: arial;"> – 4)</span><sup>2</sup><span style="font-family: arial;">]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">(</span><span style="white-space: normal;">AC</span><span style="font-family: arial; white-space: normal;">) = </span><span style="font-family: arial; white-space: normal;"><b>√</b>[(</span><span style="white-space: normal;">4</span><span style="white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> +</span><span style="white-space: normal;"> </span><span style="font-family: arial; white-space: normal;">(</span><span style="white-space: normal;">– 4)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;">]</span> </span></blockquote><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(</span><span style="font-family: arial;">AC</span><span style="font-family: arial;">) = </span><span style="font-family: arial;"><b>√</b>[16</span><span style="font-family: arial;"> +</span><span style="font-family: arial;"> 16</span><span style="font-family: arial;">]<br /></span><span style="font-family: arial;">(</span><span style="font-family: arial;">AC</span><span style="font-family: arial;">) = 4</span><span style="font-family: arial;"><b>√</b>2 ------------- equation 1</span></blockquote><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;">5) First we will find BD with <span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">B(4, 5</span><span style="white-space: normal;">), </span></span>C(- 2, - 1)</div></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">a) </span>x<sub>1</sub> = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">b) </span>y<sub>1</sub> = 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">c) </span>x<sub>2</sub> = - 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial;">d) </span>y<sub>2</sub> = - 1 </span></blockquote><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"></span><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">6) We know that:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BD) = </span><span style="font-family: arial;"><b>√</b>[(</span><span><span style="line-height: 17.12px;"><span style="font-family: arial;">x<sub>1</sub> – x<sub>2</sub></span></span><span style="font-family: arial;">)</span></span><sup>2</sup><span style="font-family: arial;"> +</span><span style="font-family: arial;"> </span><span style="font-family: arial;">(</span><span style="font-family: arial;">y<sub>1</sub> – y<sub>2</sub>)</span><sup>2</sup><span style="font-family: arial;">]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(</span><span style="font-family: arial;">BD</span><span style="font-family: arial;">) = </span><span style="font-family: arial;"><b>√</b>[(</span><span style="font-family: arial;">4 – (– 2)</span><span style="font-family: arial;">)</span><sup>2</sup><span style="font-family: arial;"> +</span><span style="font-family: arial;"> </span><span style="font-family: arial;">(5</span><span style="font-family: arial;"> – (</span><span style="font-family: arial;">– 1)</span><span style="font-family: arial;">)</span><sup>2</sup><span style="font-family: arial;">]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">(</span><span style="white-space: normal;">BD</span><span style="font-family: arial; white-space: normal;">) = </span><span style="font-family: arial; white-space: normal;"><b>√</b>[(</span><span style="white-space: normal;">6</span><span style="white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> +</span><span style="white-space: normal;"> </span><span style="font-family: arial; white-space: normal;">(</span><span style="white-space: normal;">6)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;">]</span> </span></blockquote><div style="white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(</span><span style="font-family: arial;">BD</span><span style="font-family: arial;">) = </span><span style="font-family: arial;"><b>√</b>[36</span><span style="font-family: arial;"> +</span><span style="font-family: arial;"> 36</span><span style="font-family: arial;">]<br /></span><span style="font-family: arial;">(</span><span style="font-family: arial;">BD</span><span style="font-family: arial;">) = 6</span><span style="font-family: arial;"><b>√</b>2 ------------- equation 2</span></blockquote><div style="white-space: normal;"><span style="font-family: arial;">7) From equations 1 and 2, we can find the area of the rhombus as follows.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">Area of the rhombus = (1/2) (product of diagonals)</span></blockquote></span></div></span></div></span></div></span></div></span></div></span></div></div></span></div></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><span style="white-space: normal;">Area of the rhombus = (1/2) (</span><span style="font-family: arial; font-size: medium; white-space: normal;">4</span><span style="font-family: arial; font-size: medium; white-space: normal;"><b>√</b>2</span><span style="white-space: normal;">) (</span><span style="font-family: arial; font-size: medium; white-space: normal;">6</span><span style="font-family: arial; font-size: medium; white-space: normal;"><b>√</b>2)</span></div></span></div></span></div></span></div></span></div></span></div></span></div></span></div></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">Area of the rhombus = (2</span><span style="font-family: arial;"><b>√</b>2</span><span style="font-family: arial;">) (</span><span style="font-family: arial;">6</span><span style="font-family: arial;"><b>√</b>2)<br /></span><span style="font-family: arial;">Area of the rhombus = (2</span><span style="font-family: arial;">) (</span><span style="font-family: arial;">6) (</span><span style="font-family: arial;"><b>√</b>2) </span><span style="font-family: arial;">(</span><span style="font-family: arial;"><b>√</b>2)<br /></span><span style="font-family: arial;">Area of the rhombus = (12</span><span style="font-family: arial;">) (</span><span style="font-family: arial;">2)<br /></span><span style="font-family: arial;">Area of the rhombus = 24 square units.</span></span></blockquote><div style="text-align: left;"> </div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #161719; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span>Coordinate geometry</span><span style="background-color: white;"><span style="color: #161719;"><span style="white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #</span></span>Coordinate geometry<span style="color: #161719;"><span style="white-space-collapse: break-spaces;"> #education #learning #students #teachers #math</span></span></span> </span></div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-7-Coordinate-geometry - Ex- 7.3</span></h2><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span>ANIL SATPUTE</span></a></div></span></div></div></span></h3>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-45847652612959580542024-01-25T15:23:00.005+05:302024-03-16T12:37:56.829+05:30170-NCERT-10-7-Coordinate geometry - Ex- 7.1<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 7.1</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 7 Coordinate geometry</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/169-ncert-10-6-triangles-ex-66.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.6</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 7.1</span></h3></div><div><b><span style="font-family: arial; font-size: medium;">Q </span><span style="font-family: arial; font-size: medium;">1. Find the distance between the following pairs of points :</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) (2, 3), (4, 1) (ii) (– 5, 7), (– 1, 3) (iii) (a, b), (– a, – b)</b></span></div></blockquote><div style="font-family: arial; font-size: large;"><h3 style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibDcF4fISoZ6LMWFcP24x3pgG-Q6nTzGa1P954iM7zsoBH0LU1XAolKhEfbSk6ptOg1c4LS89qbFc_tY4oQipNDtrSNgxAc73E9Rj7mfnkFFoLBxn_pDLHMd5qcm36XTT7BWxRXIJ7EoCvnWHVbk8IaGae3Nlu-qkPsOTpXZGywQxWRUDwQsOe_VQeJRQ/s946/1a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="946" data-original-width="913" height="632" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibDcF4fISoZ6LMWFcP24x3pgG-Q6nTzGa1P954iM7zsoBH0LU1XAolKhEfbSk6ptOg1c4LS89qbFc_tY4oQipNDtrSNgxAc73E9Rj7mfnkFFoLBxn_pDLHMd5qcm36XTT7BWxRXIJ7EoCvnWHVbk8IaGae3Nlu-qkPsOTpXZGywQxWRUDwQsOe_VQeJRQ/w610-h632/1a.png" width="610" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhzNU4wRGoVmKMyCcXv5SO1q9aH_72dqA6k6W2dD8igl0QKZZonOiXaLI_0_k5knvV1rceA6i962Di3sVvXNUPkbW3RL6AEiayyKaiXwsoI4Gopq4wi3r4v3fQ7m1NHxci2u_wh9F3tLGXlXQ_DJDC20dJzmRwFy8MPFFhxEyQRmbIeAKE7AeegwgCDt0/s904/1b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="454" data-original-width="904" height="309" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhzNU4wRGoVmKMyCcXv5SO1q9aH_72dqA6k6W2dD8igl0QKZZonOiXaLI_0_k5knvV1rceA6i962Di3sVvXNUPkbW3RL6AEiayyKaiXwsoI4Gopq4wi3r4v3fQ7m1NHxci2u_wh9F3tLGXlXQ_DJDC20dJzmRwFy8MPFFhxEyQRmbIeAKE7AeegwgCDt0/w615-h309/1b.png" width="615" /></a></h3></div><div><span style="font-family: arial; font-size: medium;"><b>Q2. Find the distance between the points (0, 0) and (36, 15). Can you now find</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>the distance </b></span><b style="font-family: arial; font-size: large;">between the two towns A and B discussed in Section 7.2.</b></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2hinZcNZEf605PogF_XvQbZIpAv_KVc8FABrzv6Cwt0kgAa0VTpUQRTCk4qDDK_nUekUVTeAg7pxa4kvnRTqY8PukCWNvVcmwrzTweUof3XWIybx_UPBum56PlT21ysCWs_q1vtLOOZ5YjJMvtZe_OPoHCdmcjr7TOiiq9cZAB2_srUbGl-rhQKqd7Bg/s919/2a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="552" data-original-width="919" height="373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2hinZcNZEf605PogF_XvQbZIpAv_KVc8FABrzv6Cwt0kgAa0VTpUQRTCk4qDDK_nUekUVTeAg7pxa4kvnRTqY8PukCWNvVcmwrzTweUof3XWIybx_UPBum56PlT21ysCWs_q1vtLOOZ5YjJMvtZe_OPoHCdmcjr7TOiiq9cZAB2_srUbGl-rhQKqd7Bg/w622-h373/2a.png" width="622" /></a></div><div class="separator" style="clear: both; text-align: left;"><b style="font-family: arial; font-size: large;">Q 3. Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.</b></div></div><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyLkwQzI2QnunLHFTTig2L30to2796XFGliGhhp3tdm_w2naBOHCvRoH8rHMgV9SMn3nvvZxNXYY4UvOQu37q716r_dpx1h5QqXvnjTu_knWQ0uNOzMoUgwf8MeYh3CEvJTWXIF-E-9b_fJeAh78Ge1lvBWG_sNbvI1TAQBmQV5x6td_CARnB31wQgqyY/s910/3a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="869" data-original-width="910" height="596" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyLkwQzI2QnunLHFTTig2L30to2796XFGliGhhp3tdm_w2naBOHCvRoH8rHMgV9SMn3nvvZxNXYY4UvOQu37q716r_dpx1h5QqXvnjTu_knWQ0uNOzMoUgwf8MeYh3CEvJTWXIF-E-9b_fJeAh78Ge1lvBWG_sNbvI1TAQBmQV5x6td_CARnB31wQgqyY/w623-h596/3a.png" width="623" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 4. </b></span><b><span style="line-height: 17.12px;"><span style="font-family: arial; font-size: medium;">Check whether (5, – 2), (6, 4), and (7, – 2) are the vertices of an isosceles</span></span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><b><span style="line-height: 17.12px;"><span style="font-family: arial; font-size: medium;">triangle.</span></span></b></div></div></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHuf_97SQVi7VZcOASgDoNAUXtg_ttp2vZtJ0C0HkRAhahhjTx4wespWKtt4tBVorOuAmekBPhpTqr7zj84DUP8qphBFQ6Ae5nfXCEYwsKADcGl-8qc2LQ3krghJ9BmG_XijMWIeDVUTWIgNlhDqX96BeqKiZWe5mk7EFxfBZuyYQ-rCsdP-NHJSjsEHs/s892/4a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="860" data-original-width="892" height="599" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHuf_97SQVi7VZcOASgDoNAUXtg_ttp2vZtJ0C0HkRAhahhjTx4wespWKtt4tBVorOuAmekBPhpTqr7zj84DUP8qphBFQ6Ae5nfXCEYwsKADcGl-8qc2LQ3krghJ9BmG_XijMWIeDVUTWIgNlhDqX96BeqKiZWe5mk7EFxfBZuyYQ-rCsdP-NHJSjsEHs/w621-h599/4a.png" width="621" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 5. In a classroom, 4 friends are </b></span><b style="font-family: arial; font-size: large;">seated at points A, B, C, and </b><b style="font-family: arial; font-size: large;">D as shown</b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b style="font-family: arial; font-size: large;">in following figure. Champa </b><b style="font-family: arial; font-size: large;">and Chameli walk into the class </b><b style="font-family: arial; font-size: large;">and after observing for a few </b><b style="font-family: arial; font-size: large;">minutes Champa asks Chameli, </b><b style="font-family: arial; font-size: large;">“Don’t you think ABCD is a </b><b style="font-family: arial; font-size: large;">square?” Chameli disagrees. </b><b style="font-family: arial; font-size: large;">Using the distance formula, find </b><b style="font-family: arial; font-size: large;">which of them is correct.</b></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrRrcqB1uxYJA-3aqBJo83-a-i8KQy90L_ycStLxXJ76cgYrC9E1qann7M8J4B0RpxiL28deAzsky_ogpR301FSuxQ6WIuo3YkIZAZaWj1M7D67LkomZ1QL1tl0ohkFnTm3Nm47VfQ2SVwxQtW_f24-wKISW2Xv_ujQhYwq0GRLQNv8n8KjQOIJ_-bP2E/s685/5a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="594" data-original-width="685" height="368" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrRrcqB1uxYJA-3aqBJo83-a-i8KQy90L_ycStLxXJ76cgYrC9E1qann7M8J4B0RpxiL28deAzsky_ogpR301FSuxQ6WIuo3YkIZAZaWj1M7D67LkomZ1QL1tl0ohkFnTm3Nm47VfQ2SVwxQtW_f24-wKISW2Xv_ujQhYwq0GRLQNv8n8KjQOIJ_-bP2E/w426-h368/5a.png" width="426" /></a></div><div class="separator" style="clear: both; text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">1) First we will find the coordinates of points A, B, C, and D, See the following </span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">figure.</span></div></div></div></blockquote></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1cDIacjy-tg1gVs3Xm6C8CqpjFGlSp17pXEJ6rc9gJRZFfK7FpP9-TWDjDCJC74prP9RSdrceKwdy_R3QAyyGkAjhoaH40AEVQg9OCqfxTi1kaUz_qcCRfTfkxoOhHHEmbxLAFO22aye9RDWtGpHa-95dg0WLfTzurlFKnzi7ucFxlHRo-q3iCfSGqb8/s685/5b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="594" data-original-width="685" height="382" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1cDIacjy-tg1gVs3Xm6C8CqpjFGlSp17pXEJ6rc9gJRZFfK7FpP9-TWDjDCJC74prP9RSdrceKwdy_R3QAyyGkAjhoaH40AEVQg9OCqfxTi1kaUz_qcCRfTfkxoOhHHEmbxLAFO22aye9RDWtGpHa-95dg0WLfTzurlFKnzi7ucFxlHRo-q3iCfSGqb8/w441-h382/5b.png" width="441" /></a></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipJontSyPE-EQBnd3-B_srLac90C9hsCL8orIPBa4C0JgCZ3Mf9eEbRxmlN9qBa_ANsAF95zvr9GvXZ99fOSyVNeLr-D6Rn-xngp8t-TdWlq-4SUFaR9jyJCgdoipIDU_GTNrDE1E-6XDX7T_-ejH9fih4ZGyq13TJygaREI6hsdF8JHRJO-l3Jvfa7LA/s912/5c.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="912" data-original-width="520" height="1087" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipJontSyPE-EQBnd3-B_srLac90C9hsCL8orIPBa4C0JgCZ3Mf9eEbRxmlN9qBa_ANsAF95zvr9GvXZ99fOSyVNeLr-D6Rn-xngp8t-TdWlq-4SUFaR9jyJCgdoipIDU_GTNrDE1E-6XDX7T_-ejH9fih4ZGyq13TJygaREI6hsdF8JHRJO-l3Jvfa7LA/w618-h1087/5c.png" width="618" /></a></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 6. Name the type of quadrilateral </b></span><b style="font-family: arial; font-size: large;">formed, if any, by the following </b><b style="font-family: arial; font-size: large;">points, and</b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b style="font-family: arial; font-size: large;">give reasons for </b><b style="font-family: arial; font-size: large;">your answer:</b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) (– 1, – 2), (1, 0), (– 1, 2), (– 3, 0)</b></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(ii) (–3, 5), (3, 1), (0, 3), (–1, – 4)</b></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(iii) (4, 5), (7, 6), (4, 3), (1, 2)</b></span></div></div></blockquote></blockquote><div><div class="separator" style="clear: both;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div class="separator" style="clear: both;"></div></blockquote><div style="text-align: left;"><b style="font-family: arial; font-size: large;">(i) (– 1, – 2), (1, 0), (– 1, 2), (– 3, 0)</b></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijkQvC0sYY6r7-Z2_l4-3N9R3wtmnfEdoFOGmoDpKK9xDiOUQxeQN2LJy774FHpSFdFgFCwrJ-CtOqYclHpei4SDk7XAmWPjmQLGm6NAo4ZAtl9S7-Zq5myPDktxtWYNZM-oX6FzpThjh2MpwTeRFsDrnid6fqPQ2QzNqGpDHvzsznRxJzBjkFxtZbzIg/s807/6a1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="741" data-original-width="807" height="552" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijkQvC0sYY6r7-Z2_l4-3N9R3wtmnfEdoFOGmoDpKK9xDiOUQxeQN2LJy774FHpSFdFgFCwrJ-CtOqYclHpei4SDk7XAmWPjmQLGm6NAo4ZAtl9S7-Zq5myPDktxtWYNZM-oX6FzpThjh2MpwTeRFsDrnid6fqPQ2QzNqGpDHvzsznRxJzBjkFxtZbzIg/w601-h552/6a1.png" width="601" /></a></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLbuCq6uLMNZiRIGDXdjgcO8Yu_LunJ6H3kBYMhVLZ4Lphp7OGvA3Mj8XMXCfTs5DY5MPrC9zLyGf2NRzPuM9580tcgJV3aPeePS5D4ad1tm5_jRCduMmVLjxfAUZZd73zQwyzNpKfNmpZrbtXaTEknpQLEKwVqZDfARXWHtctQi6huS6EDLXCNcDLQe0/s892/6a2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="765" data-original-width="892" height="532" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLbuCq6uLMNZiRIGDXdjgcO8Yu_LunJ6H3kBYMhVLZ4Lphp7OGvA3Mj8XMXCfTs5DY5MPrC9zLyGf2NRzPuM9580tcgJV3aPeePS5D4ad1tm5_jRCduMmVLjxfAUZZd73zQwyzNpKfNmpZrbtXaTEknpQLEKwVqZDfARXWHtctQi6huS6EDLXCNcDLQe0/w622-h532/6a2.png" width="622" /></a></div><div class="separator" style="clear: both; text-align: left;"><b style="font-family: arial; font-size: large;">(ii) (–3, 5), (3, 1), (0, 3), (–1, – 4)</b></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjGw0xBVrGNI9URTBxXjmjXdK8XvIcQvTwzt17Wcrvi3sTiHDuTiX_OG3EE5ZF14oKNG5XoRH-aBehQGKDEzm8mBpl1qeWRByfgZqRD2RkzU_j_wBEaNNKZGBKa8Ep4kgDhNfrmJlZTrj6NaKQ_u8NDFKqGLgWV3-Sd1up3t1nguQIRZpIzL06lg1LYYU/s895/6a3-1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="753" data-original-width="895" height="522" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjGw0xBVrGNI9URTBxXjmjXdK8XvIcQvTwzt17Wcrvi3sTiHDuTiX_OG3EE5ZF14oKNG5XoRH-aBehQGKDEzm8mBpl1qeWRByfgZqRD2RkzU_j_wBEaNNKZGBKa8Ep4kgDhNfrmJlZTrj6NaKQ_u8NDFKqGLgWV3-Sd1up3t1nguQIRZpIzL06lg1LYYU/w620-h522/6a3-1.png" width="620" /></a> </div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM8Z7Guqn_1P76NJsOj8Fybfk46LARoEFBaL65FZTGp3L8C9CAqBddNCBx-HoJJu-StHWqf9xeUpULF83IHONCQ4CvdK2sQVXkOqu9q07YA_1doS5NgiL3vFRC_ZbN0yJIBnj6-13cDHxpJn5xZuOjES7O9AYMDxuPjRbegWOjqL0SWTxhdbWerq8shss/s891/6a3-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="397" data-original-width="891" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM8Z7Guqn_1P76NJsOj8Fybfk46LARoEFBaL65FZTGp3L8C9CAqBddNCBx-HoJJu-StHWqf9xeUpULF83IHONCQ4CvdK2sQVXkOqu9q07YA_1doS5NgiL3vFRC_ZbN0yJIBnj6-13cDHxpJn5xZuOjES7O9AYMDxuPjRbegWOjqL0SWTxhdbWerq8shss/w626-h280/6a3-2.png" width="626" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXQV9mxxxsinaPYlCegtpSKAPLed44_aenAy694b-EyX0uzCdhbJ7Miqc8t-bYfxmWLADWac08jKhIEXuLQgvg46fdllJ1wB1oJOg0EeOJnMZKgpgxWVJVN8tJL5TZlgraATpizf-Syz4F8dauE2W0kO4UDgrEx1s8ZvvcnGtPpome9XPyTJgwqPVD1e8/s1117/6a4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="879" data-original-width="1117" height="434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXQV9mxxxsinaPYlCegtpSKAPLed44_aenAy694b-EyX0uzCdhbJ7Miqc8t-bYfxmWLADWac08jKhIEXuLQgvg46fdllJ1wB1oJOg0EeOJnMZKgpgxWVJVN8tJL5TZlgraATpizf-Syz4F8dauE2W0kO4UDgrEx1s8ZvvcnGtPpome9XPyTJgwqPVD1e8/w551-h434/6a4.png" width="551" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><b style="font-family: arial; font-size: large;">(iii) (4, 5), (7, 6), (4, 3), (1, 2)</b><div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjezD90DNqNm92-RlfzoXbdsjUq0-qt-Zy3Faezb8a6FRA19MgZhSzNZqBGX8YFw8VQWUh5OdPy_sUJ_XemIeYxMjGPryieEI0Ac2I_NRKmdBGMsVKkXw9QoGtipKRXzUTnvmRtk7Ffhrk09Sgs9q8Fo0-8_UAWJMW-8pyN-uFG4DW6rXUTl1KZVaeknMA/s903/6a4-1.png" style="margin-left: 1em; margin-right: 1em;"><br /><img border="0" data-original-height="750" data-original-width="903" height="525" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjezD90DNqNm92-RlfzoXbdsjUq0-qt-Zy3Faezb8a6FRA19MgZhSzNZqBGX8YFw8VQWUh5OdPy_sUJ_XemIeYxMjGPryieEI0Ac2I_NRKmdBGMsVKkXw9QoGtipKRXzUTnvmRtk7Ffhrk09Sgs9q8Fo0-8_UAWJMW-8pyN-uFG4DW6rXUTl1KZVaeknMA/w631-h525/6a4-1.png" width="631" /></a></div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZB-oe7PFjMseabGssjod07jw7TAgArr_jwsHjEFUxq6SuN_I0B_giLknorCIKwU05PB3oGBM6u-YMmi0sIrjGomOo7EZ0ufycP_oyuzzNa7emKVbd8hhAqG5VB-siYy9SaBXWbC7gzZAdHk0sD0ZPRQWv9So0UFeW9fCNl2D9-9A7mH9Uz-LwQxTzMQ/s871/6a4-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="747" data-original-width="871" height="535" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZB-oe7PFjMseabGssjod07jw7TAgArr_jwsHjEFUxq6SuN_I0B_giLknorCIKwU05PB3oGBM6u-YMmi0sIrjGomOo7EZ0ufycP_oyuzzNa7emKVbd8hhAqG5VB-siYy9SaBXWbC7gzZAdHk0sD0ZPRQWv9So0UFeW9fCNl2D9-9A7mH9Uz-LwQxTzMQ/w624-h535/6a4-2.png" width="624" /></a></div><div class="separator" style="clear: both; text-align: left;"><b style="font-family: arial; font-size: large;">Q 7. Find the point on the x-axis which is equidistant from (2, –5) </b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div class="separator" style="clear: both; text-align: left;"><b style="font-family: arial; font-size: large;">and (–2, 9).</b></div></blockquote><div class="separator" style="clear: both; text-align: left;"><div><div class="separator" style="clear: both;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div class="separator" style="clear: both;"></div></blockquote></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqKd2s5nJ7k33KK93NG-faFzXksTt4F4hpsa_PUjaa7LVF2bHwL0B8z8WjnCFOd3uHFmy4WCBT-SAkxk6Khkp5WY9YxxBdxV1gjT6qqDE9Q9QLFIC1Ghlc_gF2lpB05l6OUTXlJjK6lNP0WAUXY9BaUsoEidOKmlHYybdly83Pxm4RO724edzUGCOBJTY/s886/7a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="886" data-original-width="868" height="621" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqKd2s5nJ7k33KK93NG-faFzXksTt4F4hpsa_PUjaa7LVF2bHwL0B8z8WjnCFOd3uHFmy4WCBT-SAkxk6Khkp5WY9YxxBdxV1gjT6qqDE9Q9QLFIC1Ghlc_gF2lpB05l6OUTXlJjK6lNP0WAUXY9BaUsoEidOKmlHYybdly83Pxm4RO724edzUGCOBJTY/w607-h621/7a.png" width="607" /></a></div><div><b style="font-family: arial; font-size: large;">Q 8. Find the values of y for which the distance between the points </b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b style="font-family: arial; font-size: large;">P(2, – 3) </b><span style="font-family: arial; font-size: medium;"><b>and Q(10, y) is </b></span><b style="font-family: arial; font-size: large;">10 units.</b></div></blockquote><div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div><div class="separator" style="clear: both;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div class="separator" style="clear: both;"></div></blockquote></div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE2SQ2YDJ77CpaGvgAA4pgwcigNVU2sT-gDi_eoV6yErBaw4CdAu3pvWROWJdDU4zf8A3Xqog3kygdTSjnCeleUEjp9BTbwrgaOxf_vlFQOKYEVoLNRq5fcs5hCXLic8WsAdE74TQ8olVISk7ivjIgwX-u9ai4ggq5jbo5p76fyyzn6uyClIQW-_OiLZk/s834/8a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="673" data-original-width="834" height="485" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE2SQ2YDJ77CpaGvgAA4pgwcigNVU2sT-gDi_eoV6yErBaw4CdAu3pvWROWJdDU4zf8A3Xqog3kygdTSjnCeleUEjp9BTbwrgaOxf_vlFQOKYEVoLNRq5fcs5hCXLic8WsAdE74TQ8olVISk7ivjIgwX-u9ai4ggq5jbo5p76fyyzn6uyClIQW-_OiLZk/w601-h485/8a.png" width="601" /></a></div><div><b style="font-family: arial; font-size: large;">Q 9. If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), find the values of x. </b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><b style="font-family: arial; font-size: large;">Also </b><span style="font-family: arial; font-size: medium;"><b>find the </b></span><b style="font-family: arial; font-size: large;">distances QR and PR.</b></div></div></blockquote><div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div class="separator" style="clear: both;"><div><div class="separator" style="clear: both;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3><div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB0BGBDFQiJivGA9VMSLrp0p5tSNtEC5vk2NJ9iIJEjHPrO9aCCLtw_Ti0n8uueK14RHvfsepHeoITx_tDyCmhjO7fpFPd-E3bX03cnRj5Y3Ao6YwSFws08oMvqkC70MAQJWBEvDLa362QjSg29Tdgl2uZLJ6rymixe43srezqFqopwZ3YjGjXsjvelHc/s844/9a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="784" data-original-width="844" height="575" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB0BGBDFQiJivGA9VMSLrp0p5tSNtEC5vk2NJ9iIJEjHPrO9aCCLtw_Ti0n8uueK14RHvfsepHeoITx_tDyCmhjO7fpFPd-E3bX03cnRj5Y3Ao6YwSFws08oMvqkC70MAQJWBEvDLa362QjSg29Tdgl2uZLJ6rymixe43srezqFqopwZ3YjGjXsjvelHc/w620-h575/9a.png" width="620" /></a></div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_SVN7NdjOMpdqoHlE1jCYW9FvEwvNxGDhFlHLzHqLS0fen8TU38zxZEMwUu-o65g1RMgMRC0mcUJ-B-NOLhoBJjvw70tL_Ztr6VDtTVospEwtUoMX_CGBTYpXo5pNfHv6DAaDvqCYoFh4CBgP8xBBJSCrHD5qT5dwMjpoqw2Aezi1VwXGq8qu10csTcE/s838/9b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="561" data-original-width="838" height="415" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_SVN7NdjOMpdqoHlE1jCYW9FvEwvNxGDhFlHLzHqLS0fen8TU38zxZEMwUu-o65g1RMgMRC0mcUJ-B-NOLhoBJjvw70tL_Ztr6VDtTVospEwtUoMX_CGBTYpXo5pNfHv6DAaDvqCYoFh4CBgP8xBBJSCrHD5qT5dwMjpoqw2Aezi1VwXGq8qu10csTcE/w620-h415/9b.png" width="620" /></a></div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjo4rXy5tPOceshNDF-4HEl8dWzcTtRPFpbDF9e9Nqptfv21Sv9FasoGVKH2FZqG-cdUr3D6pqKlcPOm7x3XUMXb6Z28bBY2zPhKUWyO_QUrRxiLKxHBpa04Fp0dmQgdOBttvJh0jqBD7XwtJ6k_YxMI4Md3VGotyxDTxTkrN1Xz1zAJH4toYjBM8tEFAk/s838/9c.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="460" data-original-width="838" height="342" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjo4rXy5tPOceshNDF-4HEl8dWzcTtRPFpbDF9e9Nqptfv21Sv9FasoGVKH2FZqG-cdUr3D6pqKlcPOm7x3XUMXb6Z28bBY2zPhKUWyO_QUrRxiLKxHBpa04Fp0dmQgdOBttvJh0jqBD7XwtJ6k_YxMI4Md3VGotyxDTxTkrN1Xz1zAJH4toYjBM8tEFAk/w622-h342/9c.png" width="622" /></a></div><span style="font-family: arial; font-size: medium;"><b>Q 10. Find a relation between x and y such that the point (x, y) is equidistant</b></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div class="separator" style="clear: both;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>from the point </b></span><b style="font-family: arial; font-size: large;">(3, 6) and (– 3, 4).</b></div></div></div></div></div></blockquote><div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div class="separator" style="clear: both;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div class="separator" style="clear: both;"></div></blockquote></div></div></div><div class="separator" style="clear: both; text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;">Solution:</span></h3><div></div></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_fjdaIGtem8890siQ2hZC3xkfgntTTZQhHEkn0mN9UH4y-Xv3SHall_igRYQ4AY6jKdt6j7cPtQzyKk278U_3u1GjzvpEDfo1fViw7YG2YR-kRdUJ6tEe59MT6ET9sbf1rwPSN7BmmlHPBl6Vzm1zbxHiG3LdC0ki_139Cr_Jalw3bYidfNDeatJbsE/s945/10a.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="945" data-original-width="877" height="646" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_fjdaIGtem8890siQ2hZC3xkfgntTTZQhHEkn0mN9UH4y-Xv3SHall_igRYQ4AY6jKdt6j7cPtQzyKk278U_3u1GjzvpEDfo1fViw7YG2YR-kRdUJ6tEe59MT6ET9sbf1rwPSN7BmmlHPBl6Vzm1zbxHiG3LdC0ki_139Cr_Jalw3bYidfNDeatJbsE/w600-h646/10a.png" width="600" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div><span style="font-size: medium;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Coordinate geometry</span><span style="background-color: white; font-family: arial;"><span style="color: #161719;"><span style="white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #</span></span>Coordinate geometry<span style="color: #161719;"><span style="white-space-collapse: break-spaces;"> #education #learning #students #teachers #math</span></span></span></span></div><div><span style="font-family: arial;"><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2024/01/NcertMathsSolution.10.CoordinateGeometry.2.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-7-Coordinate-geometry - Ex- 7.2</a></span></h2><h3><span style="font-family: arial; font-size: medium; white-space-collapse: preserve;"><div style="font-weight: 400; white-space-collapse: collapse;"><span style="font-family: arial; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></span></div></span></h3></span></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-85611902390826543222023-12-21T12:24:00.003+05:302024-01-25T15:30:06.400+05:30169-NCERT-10-6-Triangles - Ex- 6.6<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.6</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/168-ncert-10-6-triangles-ex-65.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.5</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.6</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 1. In the following fig., PS is the bisector of </b><span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span><b> QPR of <span style="text-indent: -47.2667px;">∆</span> PQR. </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Prove that QS/SR = PQ/PR</b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOUDqleQmwfupZIP83swIcZExfKXX86uhKpF7Ne0fpk-iCg8sFDu4yx36vCDPFIxqPux5xua-oLI36uYJLWod1J_VXvG7Wv5WfIurQLJNMaiX8UzL-ZnBk1g1Ih_USMYEb7mmS5cKt83UqSOJFoDhhJWstmOAIrtREPy0DaDmsrizgqO8zZMevrr16/s544/59.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="544" data-original-width="449" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOUDqleQmwfupZIP83swIcZExfKXX86uhKpF7Ne0fpk-iCg8sFDu4yx36vCDPFIxqPux5xua-oLI36uYJLWod1J_VXvG7Wv5WfIurQLJNMaiX8UzL-ZnBk1g1Ih_USMYEb7mmS5cKt83UqSOJFoDhhJWstmOAIrtREPy0DaDmsrizgqO8zZMevrr16/s320/59.png" width="264" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; white-space: pre-wrap;">1) Let us draw a line parallel to PS, through R, which intersects QP at T </span></div><div style="text-align: left;"><span style="font-weight: normal; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">2) So, PS || RT.</span></span></div><div style="text-align: left;"><span style="font-weight: normal; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">3) As PS is the bisector of <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> QPR,</span></span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;">∠ QPS = <span style="background-color: white; color: #404040; text-align: center;">∠</span> SPR ------- equation 1</span></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) As <span style="white-space: pre-wrap;">PS || RT, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="background-color: white; color: #404040; font-family: arial; font-weight: 700; text-align: center;">∠</span><span style="font-family: arial;"> SPR = </span><span style="background-color: white; color: #404040; font-family: arial; font-weight: 700; text-align: center;">∠</span><span style="font-family: arial;"> PRT</span><span style="font-family: arial; white-space: pre-wrap;"> (Alertnate angles) ------- equation 2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) From equations 1 and 2 we have<span style="white-space: pre-wrap;">,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> QPS = <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> PRT<span style="white-space: pre-wrap;"> ------- equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) As <span style="white-space: pre-wrap;">PS || RT, </span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> QPS = <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> PTR<span style="white-space: pre-wrap;"> (Corresponding angles) ------- equation 4</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">7) From equations 3 and 4 we have<span style="white-space: pre-wrap;">,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> PRT = <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> PTR<span style="white-space: pre-wrap;"> ------- equation 5</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">8) From equation 5 we have<span style="white-space: pre-wrap;">,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(PR) = (PT)<span style="white-space: pre-wrap;"> ------- equation 6</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial; text-indent: -47.2667px;">In ∆ QTR,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left; text-indent: -47.2667px;"><span style="font-family: arial; font-size: medium;">(QS)/(SR) = (QP)/(PT) (By BPT)<span style="white-space: pre-wrap;"> ------- equation 7</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">10) From equations 5 and 6 we have<span style="white-space: pre-wrap;">,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">(QS)/(SR) = (QP)/(PR)</span><span style="text-indent: -47.2667px; white-space: pre-wrap;">, hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 2. In the following fig., D is a point on hypotenuse AC of </b><b><span style="text-indent: -47.2667px;">∆</span></b><b> ABC, such that</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>BD <span style="white-space: pre-wrap;">⟂</span> AC, DM <span style="white-space: pre-wrap;">⟂</span> BC, </b><b>and DN <span style="white-space: pre-wrap;">⟂</span> AB. Prove that :</b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b>(i) DM</b></span><b><span style="font-family: arial;"><sup>2</sup><span> =</span></span></b><span style="font-family: arial;"><b> DN . MC (ii) DN</b></span><b><span style="font-family: arial;"><sup>2</sup><span> =</span></span></b><span style="font-family: arial;"><b> DM . AN</b></span></span></div></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIOtfxrWnjf1W5j_GckeDz8cdNomUlZlabQItKvBtR7013-oSJVAwDHVshUrflimEhTcIlmEPz5u3n2bdQhu9vIn8wSJH_OKowemF4L5i8M7TV8Lz4MpNS1N5Ta-EHybDxS5_giiullh6u1hQw3qEzhFU55Q0bSW3t67aLkmlARHCFOAvgHeHX5uLe/s370/60.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="302" data-original-width="370" height="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIOtfxrWnjf1W5j_GckeDz8cdNomUlZlabQItKvBtR7013-oSJVAwDHVshUrflimEhTcIlmEPz5u3n2bdQhu9vIn8wSJH_OKowemF4L5i8M7TV8Lz4MpNS1N5Ta-EHybDxS5_giiullh6u1hQw3qEzhFU55Q0bSW3t67aLkmlARHCFOAvgHeHX5uLe/w222-h181/60.png" width="222" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ CMD,</span></span></span></div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">a) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> DCM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> CDM = 90</span><sup style="font-weight: 400; white-space: normal;">0</sup><span style="font-weight: 400; white-space: normal;"> ------ equation 1</span></div></span></h3></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">(sum of the remaining angles of right-angled </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; white-space: normal;">)</span></div></span></h3></div></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span face="Arial, sans-serif" style="font-weight: 400; line-height: 19.26px;">b) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> CDM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> BDM </span><span style="font-weight: 400;">= 90</span><sup style="font-weight: 400;">0</sup><span style="font-weight: 400;"> (BD </span><span style="font-weight: normal;"><span style="white-space: pre-wrap;">⟂</span> AC</span><span style="font-weight: 400;">)</span><span style="font-weight: 400; white-space: normal;"> ------ equation 2</span></div></div></span></h3></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) From equations 1 and 2 we have, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: normal;">c) </span><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400; white-space: normal;"> DCM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> CDM = </span><span style="font-weight: 400;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> CDM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> BDM</span></div></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; text-align: left; white-space: normal;">d) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> DCM = <span style="white-space: pre-wrap;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> BDM</span> ------ equation 3</div></div></span></h3></div></blockquote></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">3) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ DMB,</span></span></span></div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: 400; white-space: normal;">a) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> DBM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> BDM = 90</span><sup style="font-weight: 400; white-space: normal;">0</sup><span style="font-weight: 400; white-space: normal;"> ------ equation 4</span></div></span></h3></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: 400; white-space: normal;">(sum of the remaining angles of right-angled </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; white-space: normal;">)</span></div></span></h3></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif" style="font-weight: 400; line-height: 19.26px;">b) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> BDM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> CDM </span><span style="font-weight: 400;">= 90</span><sup style="font-weight: 400;">0</sup><span style="font-weight: 400;"> (BD </span><span style="font-weight: normal;">⟂ AC, as given in equation 2</span><span style="font-weight: 400;">)</span><span style="font-weight: 400; white-space: normal;"> equation 5</span></div></div></span></h3></blockquote><div><span style="font-family: arial; font-size: medium;">4) From equations 4 and 5 we have, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: normal;">c) </span><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400; white-space: normal;"> DBM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> BDM</span><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> </span><span style="font-weight: 400;">BDM + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> CDM</span></div></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> DBM = <span style="white-space: pre-wrap;"><span style="background-color: white; color: #404040; font-weight: 700; text-align: center; white-space: normal;">∠</span> </span><span style="white-space: pre-wrap;">CDM</span> ------ equation 6 </span></div></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;">5) From equations 3 and 6 and by AA similarity test,</div></div></span></h3><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></div></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="text-indent: -47.2667px;">a) ∆ DBM </span>~<span style="text-indent: -47.2667px;"> ∆ CDM</span> --------- equation 7</div></div></span></h3></div></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">6) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;">a) (DM)/(MC) = (BM)/(DM)</span></div></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">b) (DM) x (DM) = (BM) x (MC)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial;"><span>c) (DM</span><span>)</span><sup>2</sup><span> = </span></span><span style="white-space: pre-wrap;">(BM) x (MC)</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">7) As </span>□ <span style="font-weight: normal;">DNBM is a rectangle, BM = DN, so, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;"><span>a) </span></span><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;"><span>(DM</span><span>)</span><sup>2</sup><span> = </span></span>(DN) x (MC), </span><span style="font-family: arial;">hence proved.</span></blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">8) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ AND,</span></span></span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">a) <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> DAN + <span style="background-color: white; color: #404040; font-weight: 700; text-align: center;">∠</span> ADN = 90</span><sup style="font-weight: 400; white-space: normal;">0</sup><span style="font-weight: 400; white-space: normal;"> ------ equation 8</span></div></span></h3></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div></div></blockquote></blockquote></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">(sum of the remaining angles of right-angled </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; white-space: normal;">)</span></div></span></h3></div></blockquote></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div></div></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div></div></blockquote></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span face="Arial, sans-serif" style="line-height: 19.26px;"><span style="font-weight: 400;">b) </span><span style="background-color: white; color: #404040; font-weight: normal; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400;"> ADN + </span><span style="background-color: white; color: #404040; font-weight: 400; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400;"> BDN </span></span><span style="font-weight: 400;">= 90</span><sup style="font-weight: 400;">0</sup><span style="font-weight: 400;"> (BD </span><span style="font-weight: normal;"><span style="white-space: pre-wrap;">⟂</span> AC</span><span style="font-weight: 400;">)</span><span style="font-weight: 400; white-space: normal;"> ------ equation 9</span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div></div></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium;">9) From equations 8 and 9 we have,</span> </div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: normal;">c) </span><span style="background-color: white; color: #404040; font-weight: 400; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400; white-space: normal;"> DAN + <span style="background-color: white; color: #404040; text-align: center;">∠</span> </span><span style="font-weight: 400; white-space: normal;">ADN</span><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400;"><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> </span><span style="font-weight: 400;">ADN + <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> BDN</span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div></div></blockquote></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; text-align: left; white-space: normal;">d) <span style="background-color: white; color: #404040; text-align: center;">∠</span> DAN = <span style="white-space: pre-wrap;"><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> BDN</span> ------ equation 10</div></div></span></h3></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div></div></blockquote></blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">10) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ DNB,</span></span></span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">a) <span style="background-color: white; color: #404040; text-align: center;">∠</span> DBN + <span style="background-color: white; color: #404040; text-align: center;">∠</span> BDN = 90</span><sup style="font-weight: 400; white-space: normal;">0</sup><span style="font-weight: 400; white-space: normal;"> ------ equation 11</span></div></span></h3></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">(sum of the remaining angles of right-angled </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; white-space: normal;">)</span></div></span></h3></div></blockquote></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif" style="font-weight: 400; line-height: 19.26px;">b) <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> BDN + <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> ADN </span><span style="font-weight: 400;">= 90</span><sup style="font-weight: 400;">0</sup><span style="font-weight: 400;"> (BD </span><span style="font-weight: normal;">⟂ AC, as given in equation 9</span><span style="font-weight: 400;">)</span><span style="font-weight: 400; white-space: normal;"> equation 12</span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium;">11) From equations 11 and 12 we have,</span> </div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: normal;">c) </span><span style="background-color: white; color: #404040; font-weight: 400; text-align: center; white-space: normal;">∠</span><span style="font-weight: 400; white-space: normal;"> DBN + <span style="background-color: white; color: #404040; text-align: center;">∠</span> BDN</span><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400;"><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> </span><span style="font-weight: 400;">BDN + <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> ADN</span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; text-align: left; white-space: normal;">d) <span style="background-color: white; color: #404040; text-align: center;">∠</span> DBN = <span style="white-space: pre-wrap;"><span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> </span><span style="white-space: pre-wrap;">ADN</span> ------ equation 13 </blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;">12) From equations 10 and 13 and by AA similarity test,</div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="text-indent: -47.2667px;">a) ∆ DBN </span>~<span style="text-indent: -47.2667px;"> ∆ ADN</span> --------- equation 14</div></div></span></h3></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div></div></blockquote></blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">13) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;">a) (DN)/(AN) = (BN)/(DN)</span></div></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">b) (DN) x (DN) = (BN) x (AN)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial;"><span>c) (DN</span><span>)</span><sup>2</sup><span> = </span></span><span style="white-space: pre-wrap;">(BN) x (AN)</span></blockquote><div><span style="font-weight: normal;">14) As </span>□ <span style="font-weight: normal;">DNBM is a rectangle, BN = DM, so, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;"><span>a) </span></span><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;"><span>(DN</span><span>)</span><sup>2</sup><span> = </span></span>(DM) x (AN), </span><span style="font-family: arial;">hence proved.</span></blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;">Q 3. In the following fig., ABC is a triangle in which <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> ABC > 90° and AD <b style="white-space: normal;"><span style="white-space: pre-wrap;">⟂</span></b> CB</div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;">produced. Prove that AC<sup style="white-space: normal;">2</sup> = AB<sup style="white-space: normal;">2</sup> + BC<sup style="white-space: normal;">2</sup> + 2 BC . BD.</div></span></h3></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb3ywcPlo9Ws8B67jJAi_fCXOSjDT-mETcw--JyH3oTujJT23p_vH-YfkdHJvcXjOy-8ms1DMfd0pVXkJ61DRkdkClzGwWxcizwdjtw6XmT6DjRmyQ7dUSGBPKALOqCpUHuL136CTJeLQPKYA9LKUW9kG0Gdog3ZqgeqLEBi34iX5zyzhTlxKPhX3H/s480/61.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="280" data-original-width="480" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb3ywcPlo9Ws8B67jJAi_fCXOSjDT-mETcw--JyH3oTujJT23p_vH-YfkdHJvcXjOy-8ms1DMfd0pVXkJ61DRkdkClzGwWxcizwdjtw6XmT6DjRmyQ7dUSGBPKALOqCpUHuL136CTJeLQPKYA9LKUW9kG0Gdog3ZqgeqLEBi34iX5zyzhTlxKPhX3H/w275-h160/61.png" width="275" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) In </span><span style="text-indent: -47.2667px;">∆ ADB, b</span>y the theorem of Pythagoras, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(DB)</span><sup>2</sup><span style="font-family: arial;"> ------- equation 1.</span></blockquote><div><span style="font-weight: normal;">2) </span><span style="font-family: arial; font-weight: 400; white-space: normal;">In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ADC, b</span><span style="font-weight: 400; white-space: normal;">y the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(DC)</span><sup>2</sup></blockquote></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">(AC)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> = </span><span style="font-weight: 400;">(AD)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(DB + BC)</span><sup style="font-weight: 400;">2</sup></span></div></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>(AC)</span><sup>2</sup><span> = </span><span>(AD)</span><sup>2</sup><span> + </span><span>(DB</span><span>)</span><sup>2</sup><span> + (BC)</span><span><sup>2 </sup>+ 2 . DB . BC</span></span><span style="font-family: arial;"> ------- equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) From equations 1 and 2, we have,</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = [</span><span>(AD)</span><sup>2</sup><span> + </span><span>(DB</span><span>)</span><sup>2</sup><span>] + (BC)</span><sup>2 </sup><span>+</span><span> 2 . DB . BC<br /></span><span>(AC)</span><sup>2</sup><span> = </span><span>(AB</span><span>)</span><sup>2 </sup><span>+ (BC)</span><sup>2 </sup><span>+</span><span> 2 . BC . BD, hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 4. In the following fig., ABC is a triangle in which <span style="background: white; color: #404040; line-height: 107%;">∠</span>ABC < 90° and AD <span style="white-space: pre-wrap;">⟂</span> BC.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b>Prove that </b></span><span style="font-family: arial;"><b>AC<sup>2</sup> = AB<sup>2</sup> + BC<sup>2</sup> – 2 BC . BD.</b></span></span></div></blockquote><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:</span> </span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkHX98fxJSscxzFB0ml085gEaSaw6o63rcXEXle930PQdqOdIbyCcZNB-2y5COeTG92chxheWcVHvMD7OFsuszOEnM456G2Nu-7ENyll0n86nzmlV82gFvLEUByzVVWISaW9M9hv3qdLwZqJsQRxPyMSdSzBPjhmSOO1C4pX0YxwAV2cr9I9fqeKON/s574/62.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="338" data-original-width="574" height="140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkHX98fxJSscxzFB0ml085gEaSaw6o63rcXEXle930PQdqOdIbyCcZNB-2y5COeTG92chxheWcVHvMD7OFsuszOEnM456G2Nu-7ENyll0n86nzmlV82gFvLEUByzVVWISaW9M9hv3qdLwZqJsQRxPyMSdSzBPjhmSOO1C4pX0YxwAV2cr9I9fqeKON/w239-h140/62.png" width="239" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>1) In </span><span style="text-indent: -47.2667px;">∆ ADB, b</span>y the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(AD)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote><div style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;">2) </span><span style="font-weight: 400; white-space: normal;">In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ADC, b</span><span style="font-weight: 400; white-space: normal;">y the theorem of Pythagoras, we have,</span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(AD)</span><sup>2</sup><span> + </span><span>(DC)</span><sup>2</sup></span></blockquote></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div class="separator" style="clear: both;"><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(AD)</span><sup>2</sup><span> + </span><span>(BC - BD)</span><sup>2</sup></span></span></div></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div class="separator" style="clear: both;"><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">(AC)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> = </span><span style="font-weight: 400;">(AD)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(BC</span><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + (BD)</span><span style="font-weight: 400;"><sup>2 </sup>- 2 . BC . BD</span><span style="font-weight: 400;"> ------- equation 2.</span></span></div></div></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) From equations 1 and 2, we have,</span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + </span><span style="font-weight: 400; white-space: normal;">(BD</span><span style="font-weight: 400; white-space: normal;">)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + (BC)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>- 2 . BC . BD</span></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = [</span><span style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + </span><span style="font-weight: 400; white-space: normal;">(BD</span><span style="font-weight: 400; white-space: normal;">)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">]+ (BC)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>- 2 . BC . BD</span></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = [</span><span style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">]+ (BC)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>- 2 . BC . BD</span></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">+ (BC)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>- 2 . BC . BD, hence proved.</span></span></h3></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 5. In the following fig., AD is a median of a triangle ABC and </b></span><b>AM <span style="white-space: pre-wrap;">⟂</span> BC.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Prove that :</span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) AC<sup>2</sup> = AD<sup>2</sup> + BC . DM + (BC/2<span>)</span><sup>2</sup></b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><sup><span style="font-family: arial; font-size: medium;"><div><b>(ii) AB<sup>2</sup> = AD<sup>2</sup> - BC . DM + (BC/2<span>)</span><sup>2</sup></b></div></span></sup></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><sup><span style="font-family: arial; font-size: medium;"><div><h3><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">(iii) (AC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="white-space: normal;">(AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = 2 </span><span style="white-space: normal;">(AD</span><span style="white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + (1/2)(BC)</span><span style="white-space: normal;"><sup>2</sup></span></span> </h3></div></span></sup></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXc4KvT0KW_VcZ033poznZWEwU0ETbL-vXlKhtKG_91yLNLT1jCmFFIOo63VujPyr9Mi7SUNifgASh5_USNI59Atk3ZVsloNMH3hTsCfUl6hX3GdrGlAsSDFNKw_cjg4LoUbV7tQsfugn0KOsTkU_pKvCy8Lms1hkIQWgum9-_WI0bMb3w2mH9QH7d/s574/63.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="338" data-original-width="574" height="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXc4KvT0KW_VcZ033poznZWEwU0ETbL-vXlKhtKG_91yLNLT1jCmFFIOo63VujPyr9Mi7SUNifgASh5_USNI59Atk3ZVsloNMH3hTsCfUl6hX3GdrGlAsSDFNKw_cjg4LoUbV7tQsfugn0KOsTkU_pKvCy8Lms1hkIQWgum9-_WI0bMb3w2mH9QH7d/w285-h168/63.png" width="285" /></a></div><div class="separator" style="clear: both; text-align: left;"><b style="white-space: normal;">(i) AC<sup>2</sup> = AD<sup>2</sup> + BC . DM + (BC/2<span>)</span><sup>2</sup></b></div><div class="separator" style="clear: both; text-align: left;"><b style="white-space: normal;"><sup><br /></sup></b></div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) In </span><span style="text-indent: -47.2667px;">∆ AMD, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AM)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(MD)</span><sup>2</sup><span style="font-family: arial;"> ------- equation 1.</span></blockquote><div><span style="font-weight: normal;">2) </span><span style="font-family: arial; font-weight: 400; white-space: normal;">In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ AMC, b</span><span style="font-weight: 400; white-space: normal;">y the theorem of Pythagoras, we get,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AM)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(MC)</span><sup>2</sup></blockquote></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> = </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(AM)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> + </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(MD + DC)</span><sup style="font-weight: 400; white-space: normal;">2</sup></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3 style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AM)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + </span><span style="font-weight: 400; white-space: normal;">(MD</span></span><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + (DC)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: normal;"><span><sup> </sup>+ 2 (MD) (DC) </span><span>------- equation 2.</span></span></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>3) From equations 1 and 2, we have,</span></span></div><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AM)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + </span><span style="font-weight: 400; white-space: normal;">(MD</span></span><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + (DC)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: normal;"><span><sup> </sup>+ 2 (MD) (DC)</span></span></span></h3><h3><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = [</span><span style="font-weight: 400; white-space: normal;">(A</span><span style="font-weight: 400; white-space: normal;">D</span><span style="font-weight: 400; white-space: normal;">)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">] + (DC)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>+ 2 (MD) (DC)</span></span><span style="font-family: arial; font-weight: normal;"> </span><span style="font-family: arial; font-weight: normal;">------- equation 3.</span></span></h3></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) As (DC) = (1/2) (BC), so from equation 3 we have</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = [</span><span style="font-weight: 400; white-space: normal;">(A</span><span style="font-weight: 400; white-space: normal;">D</span><span style="font-weight: 400; white-space: normal;">)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">] + (BC/2)</span><span style="font-weight: 400; white-space: normal;"><sup>2 </sup>+ 2 (MD) (DC)</span></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = [</span><span style="font-weight: 400; white-space: normal;">(A</span><span style="font-weight: 400; white-space: normal;">D</span><span style="font-weight: 400; white-space: normal;">)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">] + (BC/2)</span><span style="white-space: normal;"><sup><span style="font-weight: 400;">2</span><span style="font-weight: normal;"><span> </span></span></sup><span style="font-weight: normal;">+ [2 </span></span></span><span style="font-weight: normal;"><span style="font-family: arial;">(DC)] (MD) [since 2 (DC) = (BC)],</span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">(AC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = [</span><span style="white-space: normal;">(A</span><span style="white-space: normal;">D</span><span style="white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;">] + (BC/2)</span><span style="white-space: normal;"><sup>2<span> </span></sup>+ </span></span><span style="font-family: arial;">(BC) (MD)</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = (AD)</span><sup>2</sup><span> + (BC) (DM) + (BC/2</span><span>)</span><sup>2 </sup><span>------- equation 4.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) (AC)</span><sup>2</sup><span> = (AD)</span><sup>2</sup><span> + (BC) (DM) + (BC/2</span><span>)</span><sup>2</sup><span>, hence proved.</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <b>(ii) AB<sup>2</sup> = AD<sup>2</sup> - BC . DM + (BC/2<span>)</span><sup>2</sup></b><span> </span></span></div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">6) In </span><span style="text-indent: -47.2667px;">∆ AMB, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AM)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BM)</span><sup>2</sup></blockquote></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> = </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(AM)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> + </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(BD - DM)</span><sup style="font-weight: 400; white-space: normal;">2</sup></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AM)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> + </span><span style="font-weight: 400; white-space: normal;">(BD</span></span><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + (DM)</span><sup style="font-weight: 400;"><span>2</span><span> </span></sup><span>- 2 (BD) (DM)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = [</span><span style="white-space: normal;">(AM)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="white-space: normal;">(DM</span></span><span>)</span><sup>2</sup><span>] + (BD)</span><sup><span>2</span><span> </span></sup><span>- 2 (BD) (DM)<br /></span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = [</span><span style="white-space: normal;">(AD</span></span><span>)</span><sup>2</sup><span>] + (BC/2)</span><sup><span>2</span><span> </span></sup><span>- 2 (BC/2) (DM)</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = </span><span style="white-space: normal;">(AD</span></span><span>)</span><sup>2</sup><span> + (BC/2)</span><sup><span>2</span><span> </span></sup><span>- (BC) (DM) </span><span>------- equation 5.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">7) (AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = </span><span style="white-space: normal;">(AD</span></span><span>)</span><sup>2</sup><span> + (BC/2)</span><sup><span>2</span><span> </span></sup><span>- (BC) (DM), </span><span>hence proved</span></span></div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="white-space: normal;">(iii) (AC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="white-space: normal;">(AB)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = 2 </span><span style="white-space: normal;">(AD</span><span style="white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + (1/2)(BC)</span><span style="white-space: normal;"><sup>2</sup></span></span></h3><div><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">8) Adding equations 4 and 5</span>, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(AC)<sup>2</sup> + <span style="font-family: arial;">(AB)</span><sup>2 </sup>= (AD)<sup>2</sup> + (BC) (DM) + (BC/2<span>)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">(AD</span></span><span style="font-family: arial;">)</span><sup>2</sup><span style="font-family: arial;"> + (BC/2)</span><sup><span style="font-family: arial;">2</span><span style="font-family: arial;"> </span></sup><span style="font-family: arial;">- (BC) (DM)</span></blockquote></span></h3></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;"><h3><span style="font-family: arial; white-space: pre-wrap;"><span style="font-size: medium; font-weight: normal;">(AC)<sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="font-family: arial; white-space: normal;">(AB)</span><sup style="white-space: normal;">2 </sup><span style="white-space: normal;">= 2 </span><span style="font-family: arial;"><span style="font-family: arial; white-space: normal;">(AD</span></span><span style="font-family: arial; white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> + 2 (BC/2)</span><sup style="white-space: normal;"><span style="font-family: arial;">2</span></sup></span></span></h3></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-size: medium; white-space: normal;"><div style="text-align: left;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-weight: normal;">(AC)<sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="font-family: arial; white-space: normal;">(AB)</span><sup style="white-space: normal;">2 </sup><span style="white-space: normal;">= 2 </span><span style="font-family: arial;"><span style="font-family: arial; white-space: normal;">(AD</span></span><span style="font-family: arial; white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> + 2 (BC)</span><sup style="white-space: normal;"><span style="font-family: arial;">2</span></sup></span></span><span style="font-family: arial;">/4</span></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;"><h3><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; font-size: medium; font-weight: 400;">(AC)<sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="font-family: arial; white-space: normal;">(AB)</span><sup style="white-space: normal;">2 </sup><span style="white-space: normal;">= 2 </span><span style="font-family: arial;"><span style="font-family: arial; white-space: normal;">(AD</span></span><span style="font-family: arial; white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> + 1/2 (BC)</span><sup style="white-space: normal;"><span style="font-family: arial;">2</span></sup></span></span></h3></span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; font-weight: 400;">9) (AC)<sup style="white-space: normal;">2</sup><span style="white-space: normal;"> + </span><span style="font-family: arial; white-space: normal;">(AB)</span><sup style="white-space: normal;">2 </sup><span style="white-space: normal;">= 2 </span><span style="font-family: arial;"><span style="font-family: arial; white-space: normal;">(AD</span></span><span style="font-family: arial; white-space: normal;">)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> + 1/2 (BC)</span><sup style="white-space: normal;"><span style="font-family: arial;">2</span></sup></span></span><span style="font-family: arial;">, </span><span style="font-family: arial;">hence proved.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 6. Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum </b></span><b>of the squares of its sides.</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglYGkAY_IHBn0fF1OlKLOdSLtoHJWhYB16594xpRY9mWPXCubjIRkbmMQzhQoQMvEznHG6S9npbrJ2CnZOo14PrfSkob1LnG7DuTfSo1O4Pq2tse4w_9r3YbFwXscfcCK1UEEvkXqNh0as2TjxS1uzeXab62541EV9-Pp0ltbIKgQPwyDQAyt2Ta16/s562/64.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="322" data-original-width="562" height="152" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglYGkAY_IHBn0fF1OlKLOdSLtoHJWhYB16594xpRY9mWPXCubjIRkbmMQzhQoQMvEznHG6S9npbrJ2CnZOo14PrfSkob1LnG7DuTfSo1O4Pq2tse4w_9r3YbFwXscfcCK1UEEvkXqNh0as2TjxS1uzeXab62541EV9-Pp0ltbIKgQPwyDQAyt2Ta16/w266-h152/64.png" width="266" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) In </span><span style="text-indent: -47.2667px;">∆ ACF, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AF)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(CF)</span><sup>2 </sup>------- equation 1.</blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">2) In </span><span style="text-indent: -47.2667px;">∆ AFD, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AF)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> - </span><span style="font-family: arial;">(DF)</span><sup>2 </sup>------- equation 2.</blockquote><div style="text-align: left;"><span style="font-weight: normal;">3) According to the diagram, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">(CF) = (DC - DF)</span><sup> </sup><span>------- equation 3.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) From equations 1, 2, and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(AF)</span><sup>2</sup><span> + </span><span>(CF)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = [</span><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> - (DF</span></span><span>)</span><sup>2</sup><span>] + </span><span>(DC - DF)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> - </span><span style="color: red; font-weight: 400; white-space: normal;">(DF</span></span><span style="color: red;"><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup></span><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(DC)</span><sup style="font-weight: 400;">2 </sup><span style="font-weight: normal;"><span>+ </span></span><span style="color: red; font-weight: 400;">(DF)</span><sup style="font-weight: 400;"><span style="color: red;">2</span> </sup><span style="font-weight: normal;"><span>- 2 (DC) (DF)</span></span></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> </span></span><span style="font-weight: 400;">+ </span><span style="font-weight: 400;">(DC)</span><sup style="font-weight: 400;">2 </sup><span style="font-weight: normal;"><span>- 2 (DC) (DF)</span></span><span> ------- equation 4.</span></span></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">5) In </span><span style="text-indent: -47.2667px;">∆ BDE, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BD)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(ED)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BE)</span><sup>2 </sup>------- equation 5.</blockquote></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">6) In </span><span style="text-indent: -47.2667px;">∆ DEA, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(ED)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> - </span><span style="font-family: arial;">(AE)</span><sup>2 </sup>------- equation 6.</blockquote><div><span style="font-weight: normal;">7) According to the diagram, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(BE) = (BA + AE)</span><sup> </sup><span>------- equation 7.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">8) From equations 5, 6, and 7, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(BD)</span><sup>2</sup><span> = </span><span>(ED)</span><sup>2</sup><span> + </span><span>(BE)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">(BD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = [</span><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> - (AE</span></span><span>)</span><sup>2</sup><span>] + </span><span>(BA + AE)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-weight: 400; white-space: normal;">(BD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> = </span><span style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;"> - </span><span style="color: red; font-weight: 400; white-space: normal;">(AE</span></span><span style="color: red;"><span style="font-weight: 400;">)</span><sup style="font-weight: 400;">2</sup></span><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(BA)</span><sup style="font-weight: 400;">2 </sup><span style="font-weight: normal;"><span>+ </span></span><span style="color: red; font-weight: 400;">(AE)</span><sup style="font-weight: 400;"><span style="color: red;">2</span> </sup><span style="font-weight: normal;"><span>+ 2 (BA) (AE)</span></span></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><span style="white-space: normal;">(BD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> = </span><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(BA)</span><sup>2 </sup><span>+ 2 (BA) (AE)</span><span> ------- equation 8.</span></span></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">9) Adding equations 4 and 8</span>, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(AC)<sup>2</sup> + <span style="font-family: arial;">(BD)</span><sup>2 </sup>= [<span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> </span></span><span style="font-family: arial;">+ </span><span style="font-family: arial;">(DC)</span><sup>2 </sup><span><span style="font-family: arial;">- 2 (DC) (DF)]</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">+ [</span><span style="font-family: arial; white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="font-family: arial; white-space: normal;"> </span></span><span style="font-family: arial;">+ </span><span style="font-family: arial;">(BA)</span><sup>2 </sup><span style="font-family: arial;">+ 2 (BA) (AE)] </span></blockquote></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;"><span>------- equation 9</span></span>.</span></div></h3></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">10) According to the diagram, we know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) (AD) = (BC)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) (</span><span style="font-family: arial;">BA</span><span style="font-family: arial;">) = (DC)<br /></span><span style="font-family: arial;">a) (</span><span style="font-family: arial;">AE</span><span style="font-family: arial;">) = (DF)<br /></span></span></blockquote><span style="font-family: arial; font-size: medium;">11) So, we will change the red-colored terms of equation 9 as follows:</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2 </sup><span>= [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(DC)</span><sup>2 </sup><span>- 2 (DC) (DF)]</span><span> </span><span>+ [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(<span style="color: red;">AD</span>)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(BA)</span><sup>2 </sup><span>+ 2 (<span style="color: red;">BA</span>) (<span style="color: red;">AE</span>)]</span></span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2 </sup><span>= [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(DC)</span><sup>2 </sup><span>- <span style="color: red;">2 (DC) (DF)</span>]</span><span> </span><span>+ [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(BC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(BA)</span><sup>2 </sup><span>+ <span style="color: red;">2 (DC) (DF)</span>]<br /></span><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2 </sup><span>= [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(AD)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(DC)</span><sup>2</sup><span>]</span><span> </span><span>+ [</span><span style="white-space: pre-wrap;"><span style="white-space: normal;">(BC)</span><sup style="white-space: normal;">2</sup><span style="white-space: normal;"> </span></span><span>+ </span><span>(BA)</span><sup>2</sup><span>] </span><span><span>------- equation 10.</span></span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">12) From equation 10, we can say that,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>The sum of the squares of the diagonals of a parallelogram is equal to the sum </b></span><b>of the squares of its sides.</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 7. In the following fig., two chords AB and CD intersect each other </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>at point P. </b></span><b>Prove that :</b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(i) <span style="text-indent: -47.2667px;">∆</span> APC ~ <span style="text-indent: -47.2667px;">∆</span> DPB <span> </span>(ii) AP . PB = CP . DP</b></span> </span></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtBBdNQxpQol8s93NdSntXdjvqloGNruFUeZAduIfMnLItBLKzEjq139AU_EpqgZ9vSUbSgCz4LffnXFIGdlpEXVQQkO9aKUF18LZ5MAk-Qutsks0OOfom9VwQbokMY23whER9y33VGegJNCpoQpQuUT7JjbmtL-qNinU2NmhBoXmv68byaCWB6As_/s350/65.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="293" data-original-width="350" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtBBdNQxpQol8s93NdSntXdjvqloGNruFUeZAduIfMnLItBLKzEjq139AU_EpqgZ9vSUbSgCz4LffnXFIGdlpEXVQQkO9aKUF18LZ5MAk-Qutsks0OOfom9VwQbokMY23whER9y33VGegJNCpoQpQuUT7JjbmtL-qNinU2NmhBoXmv68byaCWB6As_/w212-h178/65.png" width="212" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><b>(i) <span style="text-indent: -47.2667px;">∆</span> APC ~ <span style="text-indent: -47.2667px;">∆</span> DPB</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;">1) </span><span style="font-family: arial; font-size: medium;">In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ APC and </span><span style="text-indent: -47.2667px;">∆ DPB</span><span style="font-family: arial; text-indent: -47.2667px;">,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) < APC = < DPB (vertically opposite angles are equal)</blockquote></div><div style="white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: 400;">b) < CAP = < BDP (angles inscribed in the same arc are equal)</span></div><div style="font-weight: 400;">2) So, by AA similarity test, <span style="font-family: arial; text-indent: -47.2667px;">∆ APC <span style="background-color: white; color: #333333; text-indent: 0px;">~</span> </span><span style="text-indent: -47.2667px;">∆ DPB, hence proved.</span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><br /></span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><b style="text-indent: 0px;">(ii) AP . PB = CP . DP</b></span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><b style="text-indent: 0px;"><br /></b></span></div><div><div style="font-weight: 400;">1) We proved that,</div></div></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="white-space: normal;"><div><div style="font-weight: 400; text-align: left;"><span style="font-family: arial; text-indent: -47.2667px;">∆ APC <span style="background-color: white; color: #333333; text-indent: 0px;">~</span> </span><span style="text-indent: -47.2667px;">∆ DPB</span></div></div></div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="white-space: normal;"><div><div style="font-weight: 400;">2) So, as corresponding sides are proportional, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">(AP)/(DP) = (CP)/(PB)</blockquote></div></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="white-space: normal;"><div><div style="text-align: left;"><span style="font-weight: 400;">(AP) . (PB) = (CP) . (DP), </span><span style="font-weight: 400;">hence proved.</span></div></div></div></div></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q 8. In the following fig., two chords AB and CD of a circle intersect each</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>other at the point P </b></span><b>(when produced) outside the circle. Prove that</b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(i) </b></span><b><span style="text-indent: -47.2667px;">∆</span></b><span><b> PAC ~ </b></span><b><span style="text-indent: -47.2667px;">∆</span></b><span><b> PDB <span> </span>(ii) PA . PB = PC . PD</b></span></span></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipNYOKtUqAQy-eZs-PHX0pxNRgCdVLjdcfKaEMI67vMBHAqBMf2GeBzz2ejULFaxvNxndJxTQdg7CAjYe2Ac8rH8H9mnnMIwcrX7p-lBYPeqGNCs5vBFBUyuF7EM7jYAMdEFpgcMii-OPkg7-OGI6H6qZWMhNsMpmsfvH4OGOHNSozwcnhO-a6vkbN/s529/66.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="351" data-original-width="529" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipNYOKtUqAQy-eZs-PHX0pxNRgCdVLjdcfKaEMI67vMBHAqBMf2GeBzz2ejULFaxvNxndJxTQdg7CAjYe2Ac8rH8H9mnnMIwcrX7p-lBYPeqGNCs5vBFBUyuF7EM7jYAMdEFpgcMii-OPkg7-OGI6H6qZWMhNsMpmsfvH4OGOHNSozwcnhO-a6vkbN/w247-h164/66.png" width="247" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><b>(i) <span style="text-indent: -47.2667px;">∆</span> PAC ~ <span style="text-indent: -47.2667px;">∆</span> PDB</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;">1) </span><span style="font-family: arial; font-size: medium;">In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PAC and </span><span style="text-indent: -47.2667px;">∆ PDB</span><span style="font-family: arial; text-indent: -47.2667px;">,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) < APC = < DPB (common angles)</blockquote></div><div style="white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: 400;">b) < CAP = < BDP (Exterior angle of a cyclic quadrilateral is equal to the</span></div></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: 400;">opposite interior)</span></div></div></div></span></h3></blockquote></blockquote></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div style="font-weight: 400;">2) So, by AA similarity test, <span style="font-family: arial; text-indent: -47.2667px;">∆ PAC <span style="background-color: white; color: #333333; text-indent: 0px;">~</span> </span><span style="text-indent: -47.2667px;">∆ PDB, hence proved.</span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><br /></span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><b style="text-indent: 0px;">(ii) </b></span><b>PA . PB = PC . PD</b></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;"><b style="text-indent: 0px;"><br /></b></span></div><div><div style="font-weight: 400;">1) We proved that,</div></div></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div><div style="font-weight: 400; text-align: left;"><span style="font-family: arial; text-indent: -47.2667px;">∆ PAC <span style="background-color: white; color: #333333; text-indent: 0px;">~</span> </span><span style="text-indent: -47.2667px;">∆ PDB</span></div></div></div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"></blockquote></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"></span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div><div style="font-weight: 400;">2) So, as corresponding sides are proportional, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">(PA)/(PD) = (PC)/(PB)</blockquote></div></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div style="text-align: left;"><span style="font-weight: 400;">(PA) . (PB) = (PC) . (PD), </span><span style="font-weight: 400;">hence proved.</span></div></div></div></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><span style="white-space: pre-wrap;">Q 9. In the following fig., D is a point on side BC of </span></b><b><span style="text-indent: -47.2667px;">∆</span></b><b><span style="white-space: pre-wrap;"> ABC </span></b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">such that </span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">BD/CD = AB/AC<br /></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Prove that AD is the </span><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">bisector of </span></b><b><span style="background: white; color: #404040; line-height: 107%;"><span style="font-family: arial; font-size: medium;">∠ </span></span></b><b><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">BAC.</span></b></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJgiOvV0HzvjsEt-56uWHi-viJhBOB4nmEZ43e067MSBvVKSerc3LKDUeHuHvKIPGcaSTkKShWvQBRR2tW2m-bCeWoNEERqe-wGr6PB74yHmZ5ZvsBmOSTrcljTJXPTaGo3oYyuBK78ZyqfD37adHhw9oPs_F-8kHQC-CmMXTUJzJjDIcnG0U_f2ZX/s537/68.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="537" data-original-width="457" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJgiOvV0HzvjsEt-56uWHi-viJhBOB4nmEZ43e067MSBvVKSerc3LKDUeHuHvKIPGcaSTkKShWvQBRR2tW2m-bCeWoNEERqe-wGr6PB74yHmZ5ZvsBmOSTrcljTJXPTaGo3oYyuBK78ZyqfD37adHhw9oPs_F-8kHQC-CmMXTUJzJjDIcnG0U_f2ZX/w232-h273/68.png" width="232" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) It is given that,</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; font-weight: 400; text-align: left;">a) (BD)/(CD) = (BA)/(AC)</div></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) So, using the converse of the basic proportionality theorem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) AD || EC</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) <span style="background: white; color: #404040; line-height: 107%;">∠ BAD = </span></span><span style="background-color: white; color: #404040; font-family: arial;">∠ AEC (corresponding angles) ------ equation 1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="background-color: white; color: #404040; font-family: arial;">∠ CAD = </span><span style="background-color: white; color: #404040; font-family: arial;">∠ ACE (alternet interior angles)</span><span style="background-color: white; color: #404040; font-family: arial;"> ------ equation 2</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) As (AE) = (AC),</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="background-color: white; color: #404040; font-family: arial;">∠ AEC = </span><span style="background-color: white; color: #404040; font-family: arial;">∠ ACE</span><span style="background-color: white; color: #404040; font-family: arial;"> ------ equation 3</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) From equations 1 and 3, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a)</span><span style="font-family: arial;"> </span><span style="background: white; color: #404040; font-family: arial; line-height: 19.26px;">∠ BAD = </span><span style="background-color: white; color: #404040; font-family: arial;">∠ ACE</span><span style="background-color: white; color: #404040; font-family: arial;"> ------ equation 4</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">5) From equations 2 and 4, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a)</span><span style="font-family: arial;"> </span><span style="background: white; color: #404040; font-family: arial; line-height: 19.26px;">∠ BAD = </span><span style="background-color: white; color: #404040; font-family: arial;">∠ CAD</span><span style="background-color: white; color: #404040; font-family: arial;"> ------ equation 5</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">6) From equation 5, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a)</span><span style="font-family: arial;"> <span style="color: #404040;"><span style="background-color: white;">AD is the bisector of </span></span></span><span style="background-color: white; color: #404040; font-family: arial;">∠ BAC, hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 10. Nazima is fly fishing in a stream. The tip of </b></span><b>her fishing rod is 1.8 m above the surface </b><b>of the water and the fly at the end of the </b><b>string rests on the water 3.6 m away and </b><b>2.4 m from a point directly under the tip of </b><b>the rod. Assuming that her string </b><b>(from the tip of her rod to the fly) is taut, </b><b>how much string does she have out </b><b>(see the following fig.)? If she pulls in the string at</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>the rate of 5 cm per second, what will be </b></span><b>the horizontal distance of the fly from her </b><b>after 12 seconds?</b></span></div><div><b><span style="font-family: arial; font-size: medium;">(Note: See the original figure in the text-book)</span></b></div><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiWDJeddzD00wFn7UoKeV9tfoyusoOpPlAUaOOuWRssrTdEjpORz3LmLkT1PT_ETJLq-bMOUQIp77HhypQCzC1Sa152xcxhjsZSUTnKegj9454-I4NSjRMQCP2yUCKpBYXY2SoeOdHSKfIqlQ1KV59-_nyDzCO9t2PyxoCSxk_h9ALEC1ZakLy893W/s512/70.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="325" data-original-width="512" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiWDJeddzD00wFn7UoKeV9tfoyusoOpPlAUaOOuWRssrTdEjpORz3LmLkT1PT_ETJLq-bMOUQIp77HhypQCzC1Sa152xcxhjsZSUTnKegj9454-I4NSjRMQCP2yUCKpBYXY2SoeOdHSKfIqlQ1KV59-_nyDzCO9t2PyxoCSxk_h9ALEC1ZakLy893W/w279-h177/70.png" width="279" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) Let AB be the distance between the tip of the fishing rod and the water surface.</span></div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">2) Let BC be the distance </span>between the fly and the rod.</div><div style="font-weight: 400; white-space: normal;">3)<span style="font-family: arial;"> In </span><span style="text-indent: -47.2667px;">∆ ABC, b</span>y the theorem of Pythagoras, we get,</div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup></blockquote></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><span style="font-family: arial; font-weight: 400; white-space: normal;">(AC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> = </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(1.8)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> + </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(2.4)</span><sup style="font-weight: 400; white-space: normal;">2</sup></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(3 x 0.6)</span><sup>2</sup><span> + </span><span>(3 x 0.8)</span><sup>2<br /></sup><span>(AC)</span><sup>2</sup><span> = [</span><span>(3</span><span>)</span><sup>2</sup><span> x (0.6)</span><sup>2</sup><span>] + [</span><span>(3</span><span>)</span><sup>2</sup><span> x (0.8)</span><sup>2</sup><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(3</span><span>)</span><sup>2</sup><span> x [(0.6)</span><sup>2</sup><span> + </span><span>(0.8)</span><sup>2</sup><span>]<br /></span><span>(AC)</span><sup>2</sup><span> = </span><span>(3</span><span>)</span><sup>2</sup><span> x [(0.36)</span><span> + </span><span>(0.64)</span><span>]<br /></span><span>(AC)</span><sup>2</sup><span> = 9</span><span> x 1<br /></span><span>(AC)</span><span> = 3 ----------- equation 1</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) So, the length of the string is 3 m.</span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) According to the problem, <span>Nazima pulls in the string at </span></span><span style="font-family: arial;">the rate of 5 cm per</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">second, so, for 12 seconds, the fly comes to point D, so the string pulled by Nazima can be calculated using the formula, distance = speed x time, so we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>distance of the string pulled = speed x time</span><br /><span>distance of the string pulled</span><span> = 5 x 12<br /></span><span>distance of the string pulled</span><span> = 60 cm<br /></span><span>distance of the string pulled</span><span> = 0.6 m</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) According to the figure, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">AD = AC - </span><span style="font-family: arial;">distance of the string pulled</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">AD = 3.0 - 0.6<br /></span><span style="font-family: arial;">AD = 2.4 m</span></span></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">7) In </span><span style="text-indent: -47.2667px;">∆ ABD, b</span>y the theorem of Pythagoras, we get,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BD)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AD)</span><sup>2</sup><span style="font-family: arial;"> - </span><span style="font-family: arial;">(AB)</span><sup>2</sup></blockquote></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; white-space: normal;">(BD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> = </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(2.4)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-family: arial; font-weight: 400; white-space: normal;"> - </span><span style="font-family: arial; font-weight: 400; white-space: normal;">(1.8)</span><sup style="font-weight: 400; white-space: normal;">2</sup></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(BD)</span><sup>2</sup><span> = </span><span>(3 x 0.8)</span><sup>2</sup><span> - </span><span>(3 x 0.6)</span><sup>2<br /></sup><span>(BD)</span><sup>2</sup><span> = </span><span>(3</span><span>)</span><sup>2</sup><span> x [(0.8)</span><sup>2</sup><span> - </span><span>(0.6)</span><sup>2</sup><span>]<br /></span><span>(BD)</span><sup>2</sup><span> = </span><span>(3</span><span>)</span><sup>2</sup><span> x [(0.64)</span><span> - </span><span>(0.36)</span><span>]<br /></span><span>(BD)</span><sup>2</sup><span> = 9</span><span> x (0.28)<br /></span><span>(BD)</span><span> = 3 x </span><span style="line-height: 107%;"><span>√(0.28)<br /></span></span><span>(BD)</span><span> = 3 x </span><span style="line-height: 17.12px;"><span>0.529<br /></span></span><span>(BD)</span><span> = 1.587</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) The horizontal distance of a fly from Nazima is DE, so</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(DE) = (EB) + (BD)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(DE) = (1.2) + (1.587)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(DE) = 2.787<br /></span><span style="font-family: arial;">(DE) = 2.79 m</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">9) So, the horizontal distance of the fly from Nazima </span><span style="font-family: arial;">after 12 seconds is 2.79 m.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></span></div><div style="text-align: left;"><span style="font-family: arial;"><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2024/01/NcertMathsSolution.10.CoordinateGeometry.1.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-7-Coordinate-geometry - Ex- 7.1</a></span></h2><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></span></div></div></span></h3></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-41329640429215541912023-12-20T12:05:00.001+05:302023-12-21T12:25:22.828+05:30168-NCERT-10-6-Triangles - Ex- 6.5<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.5</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/167-ncert-10-6-triangles-ex-64.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.4</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.5</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q1. The sides of the triangles are given below. Determine which of them are right triangles. </b></span><b>In the case of a right triangle, write the length of its hypotenuse.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 7 cm, 24 cm, 25 cm</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 3 cm, 8 cm, 6 cm</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 50 cm, 80 cm, 100 cm</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iv) 13 cm, 12 cm, 5 cm</b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwAeYitd8N_fVryCcWJijklzRIX4bUAlpF0ge6jD-i5NkCJVZb8kLomTIC__CgEd5z8xEeVojxoihctJBNkblJbsXLbzOUuEqO3ADSRJLkf7s_QD9xKfj3pGZNoigSNBqTknrXgxiKbMw4Q5BtBRv8WCGwkPTf9uny8Dz8KVWha2gyhox8AKG5faax/s370/44.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="370" data-original-width="271" height="202" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwAeYitd8N_fVryCcWJijklzRIX4bUAlpF0ge6jD-i5NkCJVZb8kLomTIC__CgEd5z8xEeVojxoihctJBNkblJbsXLbzOUuEqO3ADSRJLkf7s_QD9xKfj3pGZNoigSNBqTknrXgxiKbMw4Q5BtBRv8WCGwkPTf9uny8Dz8KVWha2gyhox8AKG5faax/w148-h202/44.png" width="148" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) In a right-angled triangle ABC,</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">a) <span style="background-color: white; color: #404040; text-align: center; white-space: normal;">∠</span> ABC = 90</span><sup style="font-weight: normal;">0</sup></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><span style="font-weight: normal;">b) (AC) is </span><span style="font-weight: normal; white-space: normal;">hypotenuse</span><span style="font-weight: normal;">.</span></div></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote><h3 style="text-align: left;"><span style="color: red; font-family: arial; font-size: medium;">Note: Here AC is the largest side, so always take the greatest side as the value of (AC)</span></h3><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) 7 cm, 24 cm, 25 cm</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) Let (AB) = 7, (BC) = 24, and (AC) = 25.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) Let us check these values in Equation 1.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(7)</span><sup>2</sup><span> + </span><span>(24)</span><sup>2<br /></sup><span>LHS = 49</span><span> + 576<br /></span><span>LHS = 625</span><span> ------- equation 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(AC)</span><sup>2</sup></span></div><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(25)</span><sup>2<br /></sup><span>RHS = 625</span><span> ------- equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) From equations 2 and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">LHS = RHS </span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) So, here, 7 cm, 24 cm, and 25 cm are the sides of the </span><span style="font-family: arial; white-space: pre-wrap;">right-angled triangle.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">6) Here the length of the hypotenuse is 25 cm.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 3 cm, 8 cm, 6 cm</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote></div><div><span style="font-family: arial; font-size: medium;">2) Let (AB) = 3, (BC) = 6, and (AC) = 8.</span></div><div><span style="font-family: arial; font-size: medium;">3) Let us check these values in Equation 1.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(3)</span><sup>2</sup><span> + </span><span>(6)</span><sup>2<br /></sup><span>LHS = 9</span><span> + 36<br /></span><span>LHS = 45</span><span> ------- equation 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(AC)</span><sup>2</sup></span></div><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(8)</span><sup>2<br /></sup><span>RHS = 64</span><span> ------- equation 3</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) From equations 2 and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">LHS </span><span style="background-color: white; color: #404040; text-align: center;"><span style="font-family: arial; font-size: medium;">≠</span></span><span style="font-family: arial; font-size: medium;"> RHS </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) So, 3 cm, 8 cm, and 6 cm are not the sides of the </span><span style="font-family: arial; white-space: pre-wrap;">right-angled triangle.</span></span></div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 50 cm, 80 cm, 100 cm</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> ------- equation 1.</span></blockquote></div><div><span style="font-family: arial;">2) Let (AB) = 50, (BC) = 80, and (AC) = 100.</span></div><div><span style="font-family: arial;">3) Let us check these values in Equation 1.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AC)</span><sup>2</sup></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(50)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(80)</span><sup>2<br /></sup><span style="font-family: arial;">LHS = 2500</span><span style="font-family: arial;"> + 6400<br /></span><span style="font-family: arial;">LHS = 8900</span><span style="font-family: arial;"> ------- equation 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(AC)</span><sup>2</sup></div><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(100)</span><sup>2<br /></sup><span style="font-family: arial;">RHS = 10000</span><span style="font-family: arial;"> ------- equation 3</span></blockquote><div><span style="font-family: arial;">4) From equations 2 and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">LHS </span><span style="background-color: white; color: #404040; text-align: center;"><span style="font-family: arial;">≠</span></span><span style="font-family: arial;"> RHS </span></blockquote><div><span style="font-family: arial;">5) So, 50 cm, 80 cm, and 100 cm are not the sides of the </span><span style="font-family: arial; white-space: pre-wrap;">right-angled triangle.</span></div><div><span style="font-family: arial; white-space: pre-wrap;"><br /></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iv) 13 cm, 12 cm, 5 cm</b></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote></div><div><span style="font-family: arial; font-size: medium;">2) Let (AB) = 5, (BC) = 12, and (AC) = 13.</span></div><div><span style="font-family: arial; font-size: medium;">3) Let us check these values in equation 1.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(5)</span><sup>2</sup><span> + </span><span>(12)</span><sup>2<br /></sup><span>LHS = 25</span><span> + 144<br /></span><span>LHS = 169</span><span> ------- equation 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(AC)</span><sup>2</sup></span></div><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(13)</span><sup>2<br /></sup><span>RHS = 169</span><span> ------- equation 3</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) From equations 2 and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">LHS = RHS </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) So, here, 5 cm, 12 cm, and 13 cm are the sides of the </span><span style="font-family: arial; white-space: pre-wrap;">right-angled triangle.</span></span></div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">6) Here the length of the hypotenuse is 13 cm.</span></div><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b><span style="white-space: pre-wrap;">Q</span><span style="white-space: pre-wrap;">2. PQR is a triangle right angled at P and M is a point on QR such that </span></b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b><span style="white-space: pre-wrap;">PM </span></b></span><span style="white-space: pre-wrap;">⟂</span><span><b><span style="white-space: pre-wrap;"> QR. Show that PM</span></b></span><sup><b>2</b></sup><b><span style="white-space: pre-wrap;"> = QM x MR.</span></b></span></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHo_3OQxzDvoOoqFUm_CWXTEqWETWuv5JMmpwFyoxam3Sf62XQ9OMUROY7HlBfhHXw9BdeRjOy_J4F7OIQXTUa2PV6BVkJ0exQWZBaBY4cvBMkM72KZaXYjN3ubaHOu9n50NPXXK2EkIKg9p1DOX3PbyegA-iCMdA6JMw44g71QyVCYa8VNQotlPpH/s398/45.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="208" data-original-width="398" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHo_3OQxzDvoOoqFUm_CWXTEqWETWuv5JMmpwFyoxam3Sf62XQ9OMUROY7HlBfhHXw9BdeRjOy_J4F7OIQXTUa2PV6BVkJ0exQWZBaBY4cvBMkM72KZaXYjN3ubaHOu9n50NPXXK2EkIKg9p1DOX3PbyegA-iCMdA6JMw44g71QyVCYa8VNQotlPpH/w274-h143/45.png" width="274" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ PQR, and ∆ MQP,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ PQR and </span><span style="text-indent: -47.2667px;">∆ <span style="text-indent: -47.2667px;">MQP</span>, points P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ M, </span></span><span style="text-indent: -47.2667px;">Q</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ Q, and </span></span><span style="text-indent: -47.2667px;">R</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">i) point P is associated with poin M,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">ii) point Q is associated with poin Q,<br /></span></blockquote><span style="font-family: arial; font-weight: 400; white-space: normal;"><span> </span><span> </span>iii) point R is associated with point P,</span><br style="white-space: normal;" /><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) <span style="background-color: white; color: #404040; text-align: center;">∠</span> QPR = <span style="background-color: white; color: #404040; text-align: center;">∠</span> QMP = 90<sup>0</sup> -------- (given)</div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif" style="line-height: 19.26px;">b) <span style="background-color: white; color: #404040; text-align: center;">∠</span> PQR = <span style="background-color: white; color: #404040; text-align: center;">∠</span> MQP (same angles)<br /></span></div></blockquote><div style="font-weight: 400; white-space: normal;">2) By AA similarity test, </div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="text-indent: -47.2667px;">a) ∆ PQR </span>~<span style="text-indent: -47.2667px;"> ∆ MQP</span> --------- equation 1</div></div></span></h3></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><span><span style="font-family: arial;">3) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ PQR, and ∆ MPR,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ PQR and </span><span style="text-indent: -47.2667px;">∆ <span style="text-indent: -47.2667px;">MPR</span>, points P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ M, </span></span><span style="text-indent: -47.2667px;">Q</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P, and </span></span><span style="text-indent: -47.2667px;">R</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ R. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">i) point P is associated with poin M,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">ii) point Q is associated with poin P,<br /></span></blockquote><span style="font-family: arial; font-weight: 400; white-space: normal;"><span> </span><span> </span>iii) point R is associated with point R,</span><br style="white-space: normal;" /><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) <span style="background-color: white; color: #404040; text-align: center;">∠</span> QPR = <span style="background-color: white; color: #404040; text-align: center;">∠</span> PMR = 90<sup>0</sup> -------- (given)</div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif" style="line-height: 19.26px;">b) <span style="background-color: white; color: #404040; text-align: center;">∠</span> PRQ = <span style="background-color: white; color: #404040; text-align: center;">∠</span> MRP (same angles)<br /></span></div></blockquote><div style="font-weight: 400; white-space: normal;">4) By AA similarity test,</div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="text-indent: -47.2667px;">a) ∆ PQR </span>~<span style="text-indent: -47.2667px;"> ∆ MPR</span> --------- equation 2</div></div></span></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) From equations 1 and 2 we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆ MQP </span><span style="font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> ∆ MPR</span><span style="font-family: arial;"> --------- equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) (PM)/(RM) = (QM)/(PM)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) (PM) x (PM) = (QM) x (RM)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>c) (PM</span><span>)</span><sup>2</sup><span> = (QM) x (RM) hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q3. In the following fig., ABD is a triangle right-angled at A </b></span><b>and AC </b><span style="white-space: pre-wrap;"><b>⟂</b></span><b> BD.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Show that</span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><b><span style="font-family: arial; font-size: medium;"><span>(i) AB</span><sup>2</sup><span> = BC . BD</span></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><b><span style="font-family: arial; font-size: medium;"><span>(ii) AC</span><sup>2</sup><span> = BC . DC</span></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(iii) AD</span><sup>2</sup><span> = BD . CD</span></span></b></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEOUnun7lKfOhY5vo5pXw7DfdRuNhl5f8GclwHcKVD37mJqzFU3_ord871HvQH7hxSgCA3ZyaWiUGa96ndBqZDy7U3OMNGD0knQefvwlChVkoO2kqlUVkY37IltZ75G5xLviccAH_zakOgT7JKCXRqrJsR_VvvdW2htAJ2TmL7Cjy-d97IB76PPEpP/s387/46.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="306" data-original-width="387" height="194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEOUnun7lKfOhY5vo5pXw7DfdRuNhl5f8GclwHcKVD37mJqzFU3_ord871HvQH7hxSgCA3ZyaWiUGa96ndBqZDy7U3OMNGD0knQefvwlChVkoO2kqlUVkY37IltZ75G5xLviccAH_zakOgT7JKCXRqrJsR_VvvdW2htAJ2TmL7Cjy-d97IB76PPEpP/w245-h194/46.png" width="245" /></a></div></span><div style="text-align: left;"><div style="font-weight: 400;"><span><span style="font-family: arial; font-size: medium;"><b><span style="font-family: arial;">(i) AB</span><sup>2</sup><span style="font-family: arial;"> = BC . BD</span></b></span></span></div><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></span></span></div><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ ADB and ∆ CAB,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) <span style="background-color: white; color: #404040; text-align: center;">∠</span> DAB = <span style="background-color: white; color: #404040; text-align: center;">∠</span> ACB = 90<sup>0</sup> -------- (given)</span></div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; line-height: 19.26px;">b) <span style="background-color: white; color: #404040; text-align: center;">∠</span> ABD = <span style="background-color: white; color: #404040; text-align: center;">∠</span> CBA (same angles)<br /></span></div></blockquote><div style="font-weight: 400;"><span style="font-family: arial; font-size: medium;">2) By AA similarity test,</span></div></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><div style="font-weight: 400; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">a) ∆ ADB </span>~<span style="text-indent: -47.2667px;"> ∆ CAB</span> --------- equation 1</span></div></div></h3></blockquote><div><span style="font-family: arial; font-size: medium;">3) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AB)/(CB) = (DB)/(AB)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) (AB) x (AB) = (CB) x (DB)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>c) (AB</span><span>)</span><sup>2</sup><span> = </span><span>(CB) x (DB)</span><span> hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(ii) AC</span><sup>2</sup><span> = BC . DC</span></span></b></div><div style="text-align: left;"><h3><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;">1) As per the following theorem,</span></span></div></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><div style="font-weight: 400; text-align: left;"><div><span style="font-family: arial; font-size: medium;">If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other. So we have,</span></div></div></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">a) ∆ ABC </span><span style="font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> ∆ DAC</span><span style="font-family: arial;"> --------- equation 1</span></span></div></blockquote></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">2) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AC)/(DC) = (BC)/(AC)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) (AC) x (AC) = (BC) x (DC)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>c) (AC</span><span>)</span><sup>2</sup><span> = </span><span>(BC) x (DC)</span><span> hence proved.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(iii) AD</span><sup>2</sup><span> = BD . CD</span></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><h3><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ ACD and ∆ BAD,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) <span style="background-color: white; color: #404040; text-align: center;">∠</span> ACD = <span style="background-color: white; color: #404040; text-align: center;">∠</span> BAD = 90<sup>0</sup> -------- (given)</span></div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; line-height: 19.26px;">b) <span style="background-color: white; color: #404040; text-align: center;">∠</span> ADC = <span style="background-color: white; color: #404040; text-align: center;">∠</span> BDA (same angles)<br /></span></div></blockquote><div style="font-weight: 400;"><span style="font-family: arial; font-size: medium;">2) By AA similarity test,</span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">a) ∆ ACD </span>~<span style="text-indent: -47.2667px;"> ∆ BAD</span> --------- equation 1</span></div></h3></blockquote><div><span style="font-family: arial; font-size: medium;">3) As corresponding sides of similar triangles are proportional, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AD)/(BD) = (CD)/(AD)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) (AD) x (AD) = (BD) x (CD)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>c) (AD</span><span>)</span><sup>2</sup><span> = </span><span>(BD) x (CD)</span><span> hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q4. ABC is an isosceles triangle right angled at C. Prove that AB</b></span><b><sup>2</sup><span> = 2AC</span><sup>2</sup><span>. </span></b></span></div><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6jPNtw9ON8uYcCDJH6vURD9f76Lw_mzBlzArbRGDpozKLCwmKdllD9VhPZmVJs8Wh69JH8nbcPF5vWImKD0kKrjbsp--7zykW9gPxTYCVbr5deQUe7pv33yLlH3bKRm5sCTh4IVCJZG3BfgR-Xbnltw91JKN08aVWZrnkLJEWhdD5nsiRXQWEdUxQ/s424/47.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="424" data-original-width="376" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6jPNtw9ON8uYcCDJH6vURD9f76Lw_mzBlzArbRGDpozKLCwmKdllD9VhPZmVJs8Wh69JH8nbcPF5vWImKD0kKrjbsp--7zykW9gPxTYCVbr5deQUe7pv33yLlH3bKRm5sCTh4IVCJZG3BfgR-Xbnltw91JKN08aVWZrnkLJEWhdD5nsiRXQWEdUxQ/w173-h195/47.png" width="173" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AC)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> ------- equation 1.</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) It is given that </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ABC is </span><span style="font-weight: normal;"><span style="white-space: normal;">an isosceles triangle, so,</span></span></div></div></span></h3></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; text-align: left;">(BC) = (AC) ------- equation 2.</div></div></span></h3></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) From equations 1 and 2, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> + </span><span>(AC)</span><sup>2<br /></sup><span>(AB)</span><sup>2</sup><span> = 2</span><span>(AC)</span><sup>2</sup><span>, hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q5. ABC is an isosceles triangle with AC = BC. If </b></span><span><b>AB</b></span><b><sup>2</sup><span> = 2AC</span><sup>2</sup></b><span><b>, </b></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>prove that ABC is a right </b></span><b>triangle. </b></span></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTvFrDSs99Xbtq-RbEycaG9w2tzmLUkwWOyRduPhETCrB9TbqWVX-vhlsgczS-8qJ9NuNE0fRDq0S5SPBThZnw3KJvnb_Ro_70DKeDAMuwMS7SEqaycTPyNJEjySp2x0tdmpacnSBzB08Nw8tNSsxfrr-OY6afS_FbD_pZnqFJ4_4avlbhlJksezJk/s424/48.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="424" data-original-width="376" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTvFrDSs99Xbtq-RbEycaG9w2tzmLUkwWOyRduPhETCrB9TbqWVX-vhlsgczS-8qJ9NuNE0fRDq0S5SPBThZnw3KJvnb_Ro_70DKeDAMuwMS7SEqaycTPyNJEjySp2x0tdmpacnSBzB08Nw8tNSsxfrr-OY6afS_FbD_pZnqFJ4_4avlbhlJksezJk/w155-h174/48.png" width="155" /></a></div><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium;">1) </span><span style="font-family: arial; white-space: pre-wrap;">It is given that </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC is </span><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">an isosceles triangle, so,</span></span></div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both;"><div style="font-weight: 400;">(BC) = (AC) ------- equation 1.</div></div></span></h3></blockquote><div><span style="font-family: arial; font-size: medium;">2) It is given that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = 2</span><span>(AC)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> + </span><span>(AC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> </span><span>------- equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) From equation 2, by Pythagoras theorem, we can say that AB is the </span><span style="font-family: arial; white-space: pre-wrap;">hypotenuse,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">and AC and BC are </span><span style="font-family: arial; white-space: pre-wrap;">other </span><span style="font-family: arial; white-space: pre-wrap;">two sides of a right-angled triangle, hence proved.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q6. ABC is an equilateral triangle of side 2a. Find each of its altitudes.</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8WLEDRaUavW8Ld2rfXX_QRqdeJtTS4YTmyXgTEpWjA3f1RZVdZP-fU8eGvatY40pVKpbPzfIx2IgaYpo9NvkZ66PFWyEdFmlFtZxCUaZMZWYXYec9RkT1zdKbjtam5fyIwKe_GbPIZKkcLdCF28rrFQaYGGgfH033Gz5TUhfztQ4sPww1oPIa_b5f/s386/49.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="310" data-original-width="386" height="162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8WLEDRaUavW8Ld2rfXX_QRqdeJtTS4YTmyXgTEpWjA3f1RZVdZP-fU8eGvatY40pVKpbPzfIx2IgaYpo9NvkZ66PFWyEdFmlFtZxCUaZMZWYXYec9RkT1zdKbjtam5fyIwKe_GbPIZKkcLdCF28rrFQaYGGgfH033Gz5TUhfztQ4sPww1oPIa_b5f/w202-h162/49.png" width="202" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) In ∆ ABC </span><span style="font-weight: normal; white-space: normal;">an equilateral triangle of side 2a.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">2) AD </span><span style="font-weight: normal;">⟂ </span><span style="font-weight: normal;">BC and AB = 2a, BD = a.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">3) In ∆ ABD, by Pythagoras theorem, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> = </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AD)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> + </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(BD)</span><sup style="font-weight: 400; white-space: normal;">2</sup></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(2a)</span><sup>2</sup><span> = </span><span>(AD)</span><sup>2</sup><span> + </span><span>(a)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = </span><span>(2a)</span><sup>2</sup><span> - </span><span>(a)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = </span><span>4a</span><sup>2</sup><span> - </span><span>a</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = </span><span>3a</span><sup>2<br /></sup><span>AD</span><span> </span><span><span face="Arial, sans-serif">= </span>√3 a</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>4) So, the altitude of a triangle ABC is </span><span>√3 a.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q7. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the </b></span><b>squares of its diagonals.</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPagt_YKxP2WvKxA_1Epf1-6InmycEsqBbgtXdZ-7c3W32kOgeUglxzO2TjjuZ-QMm6XtKyT1WT8ouajie5ZOP-fcobQeSlyYXH6oeUmisNwpHGcGUl1vmXZeuJuKMjQdE01JMn7NdFCGIsYCo6tAVIjwTVuScHvRrxykIfCu3SeI8HCqocC25WXFu/s264/50.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="221" data-original-width="264" height="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPagt_YKxP2WvKxA_1Epf1-6InmycEsqBbgtXdZ-7c3W32kOgeUglxzO2TjjuZ-QMm6XtKyT1WT8ouajie5ZOP-fcobQeSlyYXH6oeUmisNwpHGcGUl1vmXZeuJuKMjQdE01JMn7NdFCGIsYCo6tAVIjwTVuScHvRrxykIfCu3SeI8HCqocC25WXFu/w201-h168/50.png" width="201" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) 囗 ABCD is a rhombus, so AC ⟂ BD.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">2) In ∆ AOB, by Pythagoras theorem, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;"> </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> = </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AO)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> + </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(BO)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">------- equation 1.</span></div></span></h3></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">3) In ∆ BOC, by Pythagoras' theorem, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><h3><span style="font-family: arial; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-size: medium; font-weight: normal;"><span face="Arial, sans-serif">(BC)</span><sup>2</sup><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(BO)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(CO)</span><sup>2</sup>------- equation 2.</span></div></span></h3></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">4) In ∆ COD, by Pythagoras theorem, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = </span><span style="white-space: pre-wrap;">(CO)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DO)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;">------- equation 3.</span></span></div></blockquote><div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><span style="font-weight: normal;">5) In ∆ DOA, by Pythagoras' theorem, we have,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = </span><span style="white-space: pre-wrap;">(DO)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(AO)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;">------- equation 4.</span></span></p></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) Adding equations 1, 2, 3, and 4, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = 2[</span><span>(AO)</span><sup>2</sup><span> + </span><span>(BO)</span><sup>2 </sup><span style="white-space: pre-wrap;">+ </span><span>(CO)</span><sup>2</sup><span> + </span><span>(DO)</span><sup>2</sup><span style="white-space: pre-wrap;">]</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AB)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> + </span><span face="Arial, sans-serif" style="font-weight: 400;">(BC)</span><sup style="font-weight: 400;">2</sup><span face="Arial, sans-serif" style="font-weight: 400;"> + </span><span face="Arial, sans-serif" style="font-weight: 400;">(CD)</span><sup style="font-weight: 400;">2</sup><span face="Arial, sans-serif" style="font-weight: 400;"> + </span><span face="Arial, sans-serif" style="font-weight: 400;">(DA)</span><sup style="font-weight: 400;">2</sup><span face="Arial, sans-serif" style="font-weight: 400;"> = 2[</span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AC/2)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> + </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(BD/2)</span><sup style="font-weight: 400; white-space: normal;">2 </sup><span face="Arial, sans-serif" style="font-weight: 400;">+ </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(AC/2)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> + </span><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(BD/2)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400;">]</span></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = 2[</span><span>(AC)</span><sup>2</sup><span>/4 + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">/4 + </span><span>(AC)</span><sup>2</sup><span>/4 + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">/4]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = 2/4[</span><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;"> + </span><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">]<br /></span><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = 2/4[2</span><span>(AC)</span><sup>2</sup><span> + 2</span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">]<br /></span><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = 4/4[</span><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">]<br /></span><span>(AB)</span><sup>2</sup><span> + </span><span style="white-space: pre-wrap;">(BC)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(CD)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> + </span><span style="white-space: pre-wrap;">(DA)</span><sup style="white-space: pre-wrap;">2</sup><span style="white-space: pre-wrap;"> = [</span><span>(AC)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span style="white-space: pre-wrap;">], hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q8. In the following fig., O is a point in the interior of a triangle </b></span><b>ABC, </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>OD </b><span style="white-space: pre-wrap;"><b>⟂</b></span><b> BC, OE </b><b style="white-space: pre-wrap;">⟂</b><b> AC, and OF </b><b style="white-space: pre-wrap;">⟂</b><b> AB. Show that </b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) OA</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + OB</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + OC</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> – OD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> – OE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> – OF</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> = AF</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + BD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + CE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b>, </b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) </b></span><b>AF</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + BD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + CE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><span><b> = </b></span><b>AE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + CD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + BF</b><sup style="white-space: pre-wrap;"><b>2</b></sup></span></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7y3uWBfEahKeduWBqFewfDe9O7_JjPuZ30q17Vkjk9unu5ZIyeo0oIhW0eWEd_jGlC923VqJx5U0woX0P-zBXGPaf5Eo8b75OAIqdbjuPDMQMXsx0vAkdaDVNxI6UoLrLZtyDLljMAxGzWpOSyYE3auuMgnyUkGjRshySiQYu4yWO8yDjcvkgoxeo/s340/51.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="340" data-original-width="318" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7y3uWBfEahKeduWBqFewfDe9O7_JjPuZ30q17Vkjk9unu5ZIyeo0oIhW0eWEd_jGlC923VqJx5U0woX0P-zBXGPaf5Eo8b75OAIqdbjuPDMQMXsx0vAkdaDVNxI6UoLrLZtyDLljMAxGzWpOSyYE3auuMgnyUkGjRshySiQYu4yWO8yDjcvkgoxeo/w171-h183/51.png" width="171" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><b style="white-space: normal;">(i) OA</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> + OB</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> + OC</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> – OD</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> – OE</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> – OF</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> = AF</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> + BD</b><sup style="font-weight: 400;"><b>2</b></sup><b style="white-space: normal;"> + CE</b><sup style="font-weight: 400;"><b>2</b></sup></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><sup style="font-weight: 400;"><b><br /></b></sup></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-weight: normal;">1) In ∆ AOF, </span></span><span style="font-family: arial; font-weight: normal;">∆ BOD, and ∆ COE </span><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">by Pythagoras theorem, we have,</span></span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">(OA)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> = </span><span style="font-weight: 400;">(AF)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(OF)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;">------- equation 1.</span></span></div></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(OB)</span><sup>2</sup><span> = </span><span>(BD)</span><sup>2</sup><span> + </span><span>(OD)</span><sup>2</sup><span>------- equation 2.<br /></span><span>(OC)</span><sup>2</sup><span> = </span><span>(CE)</span><sup>2</sup><span> + </span><span>(OE)</span><sup>2</sup><span>------- equation 3.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) Adding equations 1, 2, and 3, we get, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">(OA)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OB)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OC)</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">= </span><span face="Arial, sans-serif">(AF)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OF)</span><sup>2 </sup>+ <span face="Arial, sans-serif">(BD)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OD)</span><sup>2 </sup><span face="Arial, sans-serif">+ (CE)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OE)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span face="Arial, sans-serif">(OA)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OB)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OC)</span><sup>2</sup><span face="Arial, sans-serif"> - </span></span><span>(OE)</span><sup>2 </sup><span>- </span><span>(OD)</span><sup>2 </sup><span>- </span><span>(OF)</span><sup>2 </sup><span>= </span><span>(AF)</span><sup>2</sup><span> </span><span>+ </span><span>(BD)</span><sup>2</sup><span> </span><span>+ (CE)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>Hence proved.</span> </span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) </b></span><b>AF</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + BD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + CE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><span><b> = </b></span><b>AE</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + CD</b><sup style="white-space: pre-wrap;"><b>2</b></sup><b> + BF</b><sup style="white-space: pre-wrap;"><b>2</b></sup> </span></div><div style="text-align: left;"><h3><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;"><span style="font-weight: normal;">1) In ∆ AOF, </span></span><span style="font-family: arial; font-weight: normal;">∆ BOD, and ∆ COE </span><span style="font-family: arial; font-weight: normal; white-space: pre-wrap;">by Pythagoras theorem, we have,</span></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">(OA)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> = </span><span style="font-weight: 400;">(AF)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;"> + </span><span style="font-weight: 400;">(OF)</span><sup style="font-weight: 400;">2</sup><span style="font-weight: 400;">------- equation 1.</span></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(OB)</span><sup>2</sup><span> = </span><span>(BD)</span><sup>2</sup><span> + </span><span>(OD)</span><sup>2</sup><span>------- equation 2.<br /></span><span>(OC)</span><sup>2</sup><span> = </span><span>(CE)</span><sup>2</sup><span> + </span><span>(OE)</span><sup>2</sup><span>------- equation 3.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) Adding equations 1, 2, and 3, we get, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span face="Arial, sans-serif">(AF)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OF)</span><sup>2 </sup>+ <span face="Arial, sans-serif">(BD)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OD)</span><sup>2 </sup><span face="Arial, sans-serif">+ (CE)</span><sup>2</sup><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">(OE)</span><sup>2 </sup></span><span>= (OA)</span><sup>2</sup><span> + </span><span>(OB)</span><sup>2</sup><span> + </span><span>(OC)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span face="Arial, sans-serif">(AF)</span><sup>2 </sup><span face="Arial, sans-serif">+ </span><span face="Arial, sans-serif">(BD)</span><sup>2 </sup>+ <span face="Arial, sans-serif">(CE)</span><sup>2</sup><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(OA)</span><sup>2 </sup><span face="Arial, sans-serif">+ (OB)</span><sup>2 </sup><span face="Arial, sans-serif">+ </span><span face="Arial, sans-serif">(OC)</span><sup>2 </sup></span><span style="color: red;"><span>- (OF)</span><sup>2 </sup><span>- </span><span>(OD)</span><sup>2 </sup><span>- </span><span>(OE)</span><sup>2</sup></span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span face="Arial, sans-serif">(AF)</span><sup>2 </sup><span face="Arial, sans-serif">+ </span><span face="Arial, sans-serif">(BD)</span><sup>2 </sup>+ <span face="Arial, sans-serif">(CE)</span><sup>2</sup><span face="Arial, sans-serif"> = [</span><span face="Arial, sans-serif">(OA)</span><sup>2 </sup></span><span style="color: red;">- </span><span style="color: red;">(OE)</span><sup style="color: red;">2</sup><span><span face="Arial, sans-serif">] + [(OC)</span><sup>2 </sup></span><span style="color: red;">- </span><span style="color: red;">(OD)</span><sup style="color: red;">2</sup><span><span face="Arial, sans-serif">] + [</span><span face="Arial, sans-serif">(OB)</span><sup>2 </sup></span><span style="color: red;"><span>-</span></span> <span style="color: red;">(OF)</span><sup style="color: red;">2</sup><span><span face="Arial, sans-serif">]</span></span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span face="Arial, sans-serif">(AF)</span><sup>2 </sup><span face="Arial, sans-serif">+ </span><span face="Arial, sans-serif">(BD)</span><sup>2 </sup>+ <span face="Arial, sans-serif">(CE)</span><sup>2</sup><span face="Arial, sans-serif"> = (AE)</span></span><sup>2</sup><span><span face="Arial, sans-serif"> + (CD)</span><sup>2 </sup></span><span><span face="Arial, sans-serif">+ </span><span face="Arial, sans-serif">(BF)</span><sup>2</sup></span></span></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Hence proved.</span> </span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 9. A ladder 10 m long reaches a window 8 m above the </b></span><b>ground. </b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Find the distance of the foot of the ladder </b><b>from the base of the wall.</b></span></div></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuZQnB-O4GkLsW3cCVti9WUqmh110D2SMCOfVArnoYIGHbSEgI7RQ3OiIsWBocqnDCOQGYSr0Ob0h_xcKTqZZkNnuk5OxKsexdpA5ajvQU0heuQk1dMdNeA7ZtZn35aSQCG_oWCbfyDwd-RzYk37n8pM3gOb_lhcJ0ji0iJCqiWJ4iZD4U2cOj3vNb/s370/52.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="370" data-original-width="271" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuZQnB-O4GkLsW3cCVti9WUqmh110D2SMCOfVArnoYIGHbSEgI7RQ3OiIsWBocqnDCOQGYSr0Ob0h_xcKTqZZkNnuk5OxKsexdpA5ajvQU0heuQk1dMdNeA7ZtZn35aSQCG_oWCbfyDwd-RzYk37n8pM3gOb_lhcJ0ji0iJCqiWJ4iZD4U2cOj3vNb/w156-h213/52.png" width="156" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AC)</span><sup>2</sup></blockquote></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div style="text-align: left;"><span style="font-family: arial;">(8)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(10)</span><sup>2</sup></div></div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(10)</span><sup>2 </sup><span style="font-family: arial;">- </span><span style="font-family: arial;">(8)</span><sup>2</sup></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = 100</span><sup> </sup><span style="font-family: arial;">- 64</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = 36</span></div><sup><span>(BC) = 6</span></sup></blockquote><div style="white-space: normal;"><span style="font-family: arial; font-weight: 400;">2) </span><span style="font-weight: normal;"><span style="font-family: arial;">The distance between the foot of the ladder </span><span style="font-family: arial;">from the base of the wall is 6 m.</span></span></div><div style="white-space: normal;"><span style="font-weight: normal;"><span style="font-family: arial;"><br /></span></span></div><div style="white-space: normal;">Q 10. A guy wire attached to a vertical pole of height 18 m is 24 m long and</div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left; white-space: normal;">has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?</div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkEIT29h1dGxY7kKUQP0lNy_RzTa8dH2NChVJZl-O4BjrDJazBCalAfCSWoFWzHGMjW86AFOuj_Qy82Mn_b0LnpR1yq75nwIhtzTl33BeawU0momNgBmeyDy3-eOEeqDy1_PDlSnp650s2uGe5745fZq9EiXIka8atos3kMW9DgZUcWoVKRY_iPlNL/s370/53.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="370" data-original-width="271" height="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkEIT29h1dGxY7kKUQP0lNy_RzTa8dH2NChVJZl-O4BjrDJazBCalAfCSWoFWzHGMjW86AFOuj_Qy82Mn_b0LnpR1yq75nwIhtzTl33BeawU0momNgBmeyDy3-eOEeqDy1_PDlSnp650s2uGe5745fZq9EiXIka8atos3kMW9DgZUcWoVKRY_iPlNL/w160-h218/53.png" width="160" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial;">1) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(AB)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(AC)</span><sup>2</sup></blockquote></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="font-family: arial;">(18)</span><sup>2</sup><span style="font-family: arial;"> + </span><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(24)</span><sup>2</sup></div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"></blockquote></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = </span><span style="font-family: arial;">(24)</span><sup>2 </sup><span style="font-family: arial;">- </span><span style="font-family: arial;">(18)</span><sup>2</sup></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = (24</span><sup> </sup><span style="font-family: arial;">- 18) x (24 + 18)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;">(BC)</span><sup>2</sup><span style="font-family: arial;"> = (6) x (42)</span></div></blockquote></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="text-align: left;"><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;">(BC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span face="Arial, sans-serif" style="font-weight: 400; white-space: normal;"> = (6) x (6 x 7)</span></div></div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(BC) = </span><span style="font-family: arial; font-size: medium;">6√7</span></div></blockquote><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both;"><p class="MsoNormal" style="white-space: pre-wrap;"><o:p></o:p></p>
<div><span style="font-family: arial; font-size: medium; font-weight: 400; white-space: normal;">2) </span><span style="font-weight: 400;">The distance between the pole and the stake </span><span style="font-weight: normal;">is </span><span style="font-family: arial; font-size: medium; font-weight: 400; white-space: normal;"> </span><span style="font-family: arial; font-size: medium; font-weight: 400; white-space: normal;">6√7</span><span style="font-weight: normal; white-space: normal;"> m.</span></div><div><span style="font-weight: normal; white-space: normal;"><br /></span></div><div><span style="white-space: normal;">Q</span><span style="font-weight: normal; white-space: normal;"> </span>11. An aeroplane leaves an airport and flies due north at a speed of 1000 km</div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both;"><div style="text-align: left;">per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 1½ hours?</div></div></span></h3></blockquote><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both;"><div></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwiIucjjs9MISbGRtSVFvvRQHQcaODEO5vi8WO1EvPhgEz14vGg3IkRXvygaV8QkVnhh5NulbONkS5zIdjWPmZNg5tOr4H_MqCFp98hP--lnL4onyw5LwA5LHqYgmrTDDUZ4fZ1VPvKLmOrmO8OL-pL6Q70mQXkqItT442rxAlO2KYrR4NeNLsVoPl/s373/54.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="292" data-original-width="373" height="161" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwiIucjjs9MISbGRtSVFvvRQHQcaODEO5vi8WO1EvPhgEz14vGg3IkRXvygaV8QkVnhh5NulbONkS5zIdjWPmZNg5tOr4H_MqCFp98hP--lnL4onyw5LwA5LHqYgmrTDDUZ4fZ1VPvKLmOrmO8OL-pL6Q70mQXkqItT442rxAlO2KYrR4NeNLsVoPl/w205-h161/54.png" width="205" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) It is given that,</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; font-weight: 400; text-align: left;">a) The speed of the aeroplane travelling to north is 1000 km per hour.</div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">b) The speed of the aeroplane travelling to west is 1200 km per hour.</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) We know that the Speed = (Distance)/(Time), so Distance = Speed x Time.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) So after </span><span>1½ hours</span> <span>i.e. after 3/2 hours, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">the distances traveled by the plans </span><span style="font-family: arial;">towards <span style="color: red;">north</span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Distance of a plane = Speed x Time</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Distance of a plane = 1000 x 3/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Distance of a plane = 500 x 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Distance of a plane = 1500 km. -------- equation 1</span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) So after </span><span>1½ hours</span> <span>i.e. after 3/2 hours, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">the distances traveled by the plans </span><span style="font-family: arial;">towards <span style="color: red;">west</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Distance of a plane = Speed x Time</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Distance of a plane = 1200 x 3/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Distance of a plane = 600 x 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Distance of a plane = 1800 km. -------- equation 2</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) Here, (OP) = 1500 km, and (OQ) = 1800 km. and <span style="background-color: white; color: #404040; text-align: center;">∠</span> POQ = </span><span>90</span><sup>0</sup><span>.</span> </span></div><div><span style="font-family: arial; font-size: medium;">6) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(PQ)</span><sup>2</sup><span> = </span><span>(OP)</span><sup>2</sup><span> + </span><span>(OQ)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(PQ)</span><sup>2</sup><span> = </span><span>(1500)</span><sup>2</sup><span> + </span><span>(1800)</span><sup>2<br /></sup><span>(PQ)</span><sup>2</sup><span> = </span><span>(3 x 5 x 100)</span><sup>2</sup><span> + </span><span>(3 x 6 x 100)</span><sup>2<br /></sup><span>(PQ)</span><sup>2</sup><span> = [</span><span>(3 x 100)</span><sup>2</sup><span>] x [(5</span><span>)</span><sup>2 </sup><span>+</span> <span>(6</span><span>)</span><sup>2</sup><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(PQ)</span><sup>2</sup><span> = [</span><span>(3 x 100)</span><sup>2</sup><span>] x </span><span><span face="Arial, sans-serif">[25</span><sup> </sup>+ 36]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(PQ)</span><sup>2</sup><span> = [</span><span>(3 x 100)</span><sup>2</sup><span>] x </span><span><span face="Arial, sans-serif">[61</span>]<br /></span><span>(PQ)</span><span> = </span><span>(3 x 100)</span><span> x </span><span>√</span><span>61</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(PQ)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">300</span><span style="font-family: arial;"> x </span><span style="font-family: arial;">√</span><span style="font-family: arial;">61</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7) The</span><span style="font-family: arial;"> distance between two planes after </span><span style="font-family: arial;">1½ hours will be </span><span style="font-family: arial;">300</span><span style="font-family: arial;">√</span><span style="font-family: arial;">61 km.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b><span face="Arial, sans-serif">Q </span>12. Two poles of heights 6 m and 11 m stand on a </b></span><b>plane ground. If the</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>distance between the feet </b><b>of the poles is 12 m, find the distance between</b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">their tops.</span></b></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU3zORpopCp06K196lE0twP41-JddnpSHQqzrWef1_llfhQUAAetnksZ7c65fpPcMYlbNWtaPDLOvtrw52o6BPcA44swgNxqm_uQPfF98k7y6hBP4wD2emNwXLa0lJsmFVyk4fi2fFqRpkdfm_8myrZcZ94vf5k0FHaFKuJi7OBTTSW5iT16gfX_1J/s564/55.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="367" data-original-width="564" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU3zORpopCp06K196lE0twP41-JddnpSHQqzrWef1_llfhQUAAetnksZ7c65fpPcMYlbNWtaPDLOvtrw52o6BPcA44swgNxqm_uQPfF98k7y6hBP4wD2emNwXLa0lJsmFVyk4fi2fFqRpkdfm_8myrZcZ94vf5k0FHaFKuJi7OBTTSW5iT16gfX_1J/w253-h164/55.png" width="253" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) The distance between the poles RS and PQ is 12 m.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">2) So, according to the figure, in </span><span style="font-weight: normal;">∆ SVQ,</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; font-weight: 400; text-align: left;">(VQ) = 11 - 6 = 5 m and (SV) = 12 m.</div></span></h3></blockquote><div><span style="font-family: arial; font-size: medium;">3) By the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(QS)</span><sup>2</sup><span> = </span><span>(SV)</span><sup>2</sup><span> + </span><span>(VQ)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(QS)</span><sup>2</sup><span> = </span><span>(12)</span><sup>2</sup><span> + </span><span>(5)</span><sup>2<br /></sup><span>(QS)</span><sup>2</sup><span> = 144</span><span> + 25<br /></span><span>(QS)</span><sup>2</sup><span> = 169<br /></span><span>(QS)</span><span> = 13</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) So, the distance between the tips of the poles is 13 m.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 13. D and E are points on the sides CA and CB </b></span><b>respectively of a triangle</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>ABC right angled at C. </b><b><span>Prove that AE</span><sup>2</sup><span> + BD</span><sup>2</sup><span> = AB</span><sup>2</sup><span> + DE</span><sup>2</sup><span>. </span></b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgju4SC1cLbIsGVzO0MVWjZ5dq_0JBgVB7o_TYlImwdb8xnfiRiVoYNlwUsZuQtlZWVMpgdQEBvqeT_CuorFE_boD6sxh87xBoU-UptmLkr32xqThctd4zuuuA0vGc5f3jj01NmI7VyDAKNLsHlT_CZHQdeVWLrQwivYWFJEb8HaEgZBBuhaqJZgyc-/s246/56.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="213" data-original-width="246" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgju4SC1cLbIsGVzO0MVWjZ5dq_0JBgVB7o_TYlImwdb8xnfiRiVoYNlwUsZuQtlZWVMpgdQEBvqeT_CuorFE_boD6sxh87xBoU-UptmLkr32xqThctd4zuuuA0vGc5f3jj01NmI7VyDAKNLsHlT_CZHQdeVWLrQwivYWFJEb8HaEgZBBuhaqJZgyc-/w211-h183/56.png" width="211" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial;">1) In </span><span style="text-indent: -47.2667px;">∆ ACE,</span> by the theorem of Pythagoras, we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>(AE)</span><sup>2</sup><span> = </span><span>(EC)</span><sup>2</sup><span> + </span><span>(AC)</span><sup>2</sup><span> ------- equation 1.</span></span></blockquote></div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">2) </span><span style="font-family: arial;">In </span><span style="text-indent: -47.2667px;">∆ BCD,</span> by the theorem of Pythagoras, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;"><span>(BD)</span><sup>2</sup><span> = </span><span>(BC)</span><sup>2</sup><span> + </span><span>(DC)</span><sup>2</sup><span> ------- equation 2.</span></span></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">3) Adding equation 1 and equation 2, we get,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;"><span>(AE)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span> = [</span><span>(EC)</span><sup>2 </sup></span><span>+ (BC)</span><sup>2</sup>] + <span style="font-family: arial;"><span>[</span><span>(AC)</span><sup>2 </sup></span><span>+ (DC)</span><sup>2</sup>] ------- equation 3.</div></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">4) Re-arranging the terms of equation 3, we get,</span></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; white-space: normal;"><span>(AE)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span> = [</span><span>(EC)</span><sup>2 </sup></span><span style="font-weight: 400; white-space: normal;">+ (DC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">] + </span><span style="font-family: arial; font-weight: 400; white-space: normal;"><span>[</span><span>(AC)</span><sup>2 </sup></span><span style="font-weight: 400; white-space: normal;">+ (BC)</span><sup style="font-weight: 400; white-space: normal;">2</sup><span style="font-weight: 400; white-space: normal;">] </span><span style="font-weight: 400; white-space: normal;">------- equation 4.</span></div></div></span></h3></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5)</span><span style="font-family: arial;"> In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC,</span><span style="font-family: arial;"> by the theorem of Pythagoras, we have,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(BC)</span><sup>2</sup><span> + </span><span>(AC)</span><sup>2</sup><span> ------- equation 5.</span></span></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>6) </span><span>In </span><span style="text-indent: -47.2667px;">∆ DCE,</span> by the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(DE)</span><sup>2</sup><span> = </span><span>(EC)</span><sup>2</sup><span> + </span><span>(DC)</span><sup>2</sup><span> ------- equation 6.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) From equations 4, 5, and 6, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>(AE)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span> = [</span><span>(EC)</span><sup>2 </sup></span><span>+ (DC)</span><sup>2</sup><span>] + </span><span><span>[</span><span>(AC)</span><sup>2 </sup></span><span>+ (BC)</span><sup>2</sup><span>]</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>(AE)</span><sup>2</sup><span> + </span><span>(BD)</span><sup>2</sup><span> = [</span><span>(DE</span></span><span>)</span><sup>2</sup><span>] + </span><span><span>[</span><span>(AB</span></span><span>)</span><sup>2</sup><span>]</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>8) So, we have, AE</span><sup>2</sup><span> + </span><span>BD</span><sup>2</sup><span> = </span><span>DE</span></span><sup>2</sup><span> + </span><span><span>AB</span></span><sup>2</sup><span>],</span><span> hence proved.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 14. The perpendicular from A on side BC of an </b></span><span style="text-indent: -47.2667px;"><b>∆</b></span><b> ABC intersects BC at D such that DB = 3 CD </b><span><b>(see the following fig). Prove that 2 AB</b></span><b><sup>2</sup><span> = 2 AC</span><sup>2</sup><span> + BC</span><sup>2</sup><span>.</span></b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZFvod_F2oqiYEcrhokNKU0MLezH96aiDCmXWeYptYed_mdHTpymNOIfy6vGO10ievv2jhohguQbZZDcKt4mHP-59VKb1RMhBDBvSLIsaqlwpHydDu-lE-aKTtMrbNiG8oXQBJbtVLgJ30hCPf5a4IXyYGfb5DM_wiit8O363_GZlmjEtSzYhS9nWr/s574/57.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="338" data-original-width="574" height="155" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZFvod_F2oqiYEcrhokNKU0MLezH96aiDCmXWeYptYed_mdHTpymNOIfy6vGO10ievv2jhohguQbZZDcKt4mHP-59VKb1RMhBDBvSLIsaqlwpHydDu-lE-aKTtMrbNiG8oXQBJbtVLgJ30hCPf5a4IXyYGfb5DM_wiit8O363_GZlmjEtSzYhS9nWr/w263-h155/57.png" width="263" /></a></div></span></h3></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>1) </span><span>In </span><span style="text-indent: -47.2667px;">∆ ADB,</span> by the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>(AD)</span><sup>2</sup><span> = </span><span>(AB)</span><sup>2</sup><span> - </span><span>(DB)</span><sup>2</sup><span> ------- equation</span></span> 1.</span></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>In </span><span style="text-indent: -47.2667px;">∆ ADC,</span> by the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>(AD)</span><sup>2</sup><span> = </span><span>(AC)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup><span> ------- equation</span></span> 2.</span></blockquote><div><span style="font-family: arial; font-size: medium;">3) From equations 1 and 2, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span><span>(AB)</span><sup>2</sup><span> - </span><span>(DB)</span><sup>2</sup> = </span></span><span>(AC)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>(AB)</span><sup>2</sup><span> - </span><span>(AC)</span><sup>2</sup> = </span><span>(DB)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup> <span>------- equation</span><span> 3.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) It is given that DB = 3 CD</span> <span>------- equation</span><span> 4.</span></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">5) From equations 3 and 4, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>(AB)</span><sup>2</sup><span> - </span><span>(AC)</span><sup>2</sup> = </span><span>(DB)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup> </span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>(AB)</span><sup>2</sup><span> - </span><span>(AC)</span><sup>2</sup> = </span><span>(3 CD)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>(AB)</span><sup>2</sup><span> - </span><span>(AC)</span><sup>2</sup> = </span><span>9 (CD)</span><sup>2</sup><span> - </span><span>(CD)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>(AB)</span><sup>2</sup><span> - </span><span>(AC)</span><sup>2</sup> = 8</span><span> </span><span>(CD)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>2 (AB)</span><sup>2</sup><span> - 2 </span><span>(AC)</span><sup>2</sup> = 16</span><span> </span><span>(CD)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>2 (AB)</span><sup>2</sup><span> - 2 </span><span>(AC)</span><sup>2</sup> = </span><span>(4 CD)</span><sup>2<br /></sup><span><span>2 (AB)</span><sup>2</sup><span> - 2 </span><span>(AC)</span><sup>2</sup> = </span><span>(CD + 3 CD)</span><sup>2<br /></sup><span><span>2 (AB)</span><sup>2</sup><span> - 2 </span><span>(AC)</span><sup>2</sup> = </span><span>(CD + DB)</span><sup>2<br /></sup><span><span>2 (AB)</span><sup>2</sup><span> - 2 </span><span>(AC)</span><sup>2</sup> = </span><span>(BC)</span><sup>2<br /></sup><span><span>2 (AB)</span><sup>2</sup><span> = 2 </span><span>(AC)</span><sup>2</sup> + </span><span>(BC)</span><sup>2</sup><span>,</span><span> hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Q 15. In an equilateral triangle ABC, D is a point on side BC such that </span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>BD = 1/3 BC. Prove that </span><span>9 AD</span><sup>2</sup><span> = 7 AB</span><sup>2</sup><span>.</span></span></b></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmIK_Rs0wdWN-KZhjCLr7LifIf2WaT7gMk86SH2dWK0EEARMHi6ZiHxULqDhTfWvbceaui7LGWpfDxn_D9vGYfCVEyBY6tPnrCWOXI357ToqfULyBPVVBMtjuDMoZLDT_wWz--NLdT-K8nMq0xE44twdKMiPZQ7Yo3aQ8vrrF7mDi8EBLSxQoa4tzi/s637/58.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="310" data-original-width="637" height="156" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmIK_Rs0wdWN-KZhjCLr7LifIf2WaT7gMk86SH2dWK0EEARMHi6ZiHxULqDhTfWvbceaui7LGWpfDxn_D9vGYfCVEyBY6tPnrCWOXI357ToqfULyBPVVBMtjuDMoZLDT_wWz--NLdT-K8nMq0xE44twdKMiPZQ7Yo3aQ8vrrF7mDi8EBLSxQoa4tzi/s320/58.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">1) </span><span style="font-weight: 400; white-space: normal;">In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ABC, </span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">a) AB = BC = AC</span></div></span></h3></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">b) AE = (</span><span style="font-weight: normal;">√3/2) AB --------- equation 1</span></div></span></h3></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">c) BE = (</span><span style="font-weight: normal;">1/2) AB --------- equation 2</span></div></span></h3></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">d) BD = (</span><span style="font-weight: normal;">1/3) AB (given) --------- equation 3</span></div></span></h3></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>In </span><span style="text-indent: -47.2667px;">∆ AED,</span> by the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-size: medium;"><span>(AD)</span><sup>2</sup><span> = </span><span>(AE)</span><sup>2</sup><span> + </span><span>(DE)</span><sup>2</sup></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AD)</span><sup>2</sup><span> = </span><span>(AE)</span><sup>2</sup><span> + </span><span>(BE - BD)</span><sup>2</sup><span style="white-space: pre-wrap;"> --------- equation 4</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) From equations 1, 2, 3, and 4, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AD)</span><sup>2</sup><span> = </span><span>(AE)</span><sup>2</sup><span> + </span><span>(BE - BD)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">√3/2) AB]</span><sup>2</sup><span> + </span><span>[(<span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">1/2) AB)</span> - (<span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">1/3) AB)</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">3/4) (AB)</span><sup>2</sup><span>] + </span><span>[(<span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">1/6) AB)</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">3/4) (AB)</span><sup>2</sup><span>] + </span><span>[(1/36)<span style="white-space: pre-wrap;"> (AB)</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">3/4) + (1/36)] [</span><span><span style="white-space: pre-wrap;">AB</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">27/36) + (1/36)] [</span><span><span style="white-space: pre-wrap;">AB</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">27 + 1)/36)] [</span><span><span style="white-space: pre-wrap;">AB</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">28)/36)] [</span><span><span style="white-space: pre-wrap;">AB</span>]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [</span><span style="text-indent: -47.2667px;">(</span><span style="white-space: pre-wrap;">7)/9)] [</span><span><span style="white-space: pre-wrap;">AB</span>]</span><sup>2<br /></sup><span>9 (AD)</span><sup>2</sup><span> = </span><span style="white-space: pre-wrap;">7 (</span><span><span style="white-space: pre-wrap;">AB)</span></span><sup>2</sup><span>,</span><span> hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 16. In an equilateral triangle, prove that three times the square of one side is equal to four </b></span><b>times the square of one of its altitudes. </b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjd3sNVKBTGV8sr44JKrPX46-xXbVPB_YWzrP26nmVGR0yuJM0OjUQICDonCARBmtCO01rtKTaaGzA7_Gs-BHuxX0rMmpNgHecjf4a8m0KKwwjK83uFAPJJfIlfjNz7WuOJ0i5uG5jT301qIowbZJUAGymIKxJFPsLfZ4aMN2ZGRndNry5Khizon-Ud/s386/49.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="310" data-original-width="386" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjd3sNVKBTGV8sr44JKrPX46-xXbVPB_YWzrP26nmVGR0yuJM0OjUQICDonCARBmtCO01rtKTaaGzA7_Gs-BHuxX0rMmpNgHecjf4a8m0KKwwjK83uFAPJJfIlfjNz7WuOJ0i5uG5jT301qIowbZJUAGymIKxJFPsLfZ4aMN2ZGRndNry5Khizon-Ud/w210-h169/49.png" width="210" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; white-space: normal;">1) </span><span style="font-weight: 400; white-space: normal;">In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ABC,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">a) AD = (</span><span style="font-weight: normal;">√3/2) AB --------- equation 1</span></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"></div><div style="font-weight: 400; white-space: normal;"></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">b) BD = (</span><span style="font-weight: normal;">1/2) AB --------- equation 2</span></div></span></h3></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>In </span><span style="text-indent: -47.2667px;">∆ ADB,</span> by the theorem of Pythagoras, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span><span>(AD)</span><sup>2</sup><span> = </span><span>(AB)</span><sup>2</sup><span> - </span><span>(BD)</span><sup>2</sup></span></span><span style="font-family: arial; white-space: pre-wrap;"> --------- equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) from equations 1, 2, and 3, we have, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AD)</span><sup>2</sup><span> = </span><span>(AB)</span><sup>2</sup><span> - </span><span>[(1/2) (AB)]</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = </span><span>(AB)</span><sup>2</sup><span> - (</span><span>1/4) (AB)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [1 - </span><span>(</span><span>1/4)] (AB)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [(4/4) - </span><span>(</span><span>1/4)] (AB)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = [(4 - </span><span>1)/4)] (AB)</span><sup>2<br /></sup><span>(AD)</span><sup>2</sup><span> = (3</span><span>/4) (AB)</span><sup>2<br /></sup><span>4 (AD)</span><sup>2</sup><span> = 3</span><span> (AB)</span><sup>2</sup><span>,</span><span> </span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #333333;">4) 4 × (Square of altitude) = 3 × (Square of one side), </span>hence proved.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q 17. Tick the correct answer and justify: In </b></span><b style="text-indent: -47.2667px;">∆</b><span><b> ABC, AB = 6</b></span><span style="white-space: pre-wrap;"><b>√</b></span><b>3 cm, AC = 12 cm,</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>and BC = 6 cm. </b></span><b>The angle B is : </b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(A) 120° (B) 60°</span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(C) 90° (D) 45°</b></span></div></blockquote></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium;"><span>1) </span><span>In </span><span style="text-indent: -47.2667px;">∆ ABC,</span> it is given that,</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left; white-space: normal;">a) AB = <span style="font-family: arial; font-size: medium;">6</span><span style="white-space: pre-wrap;">√</span>3</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) AC = 12</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">c) BC = 6</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) Now we will find </span><span>(AB)</span><sup>2</sup></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) (AB)</span><sup>2</sup><span> = </span><span>(<span>6</span><span style="white-space: pre-wrap;">√</span>3)</span><sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) (AB)</span><sup>2</sup><span> = </span><span>(36 x 3)<br /></span><span>c) (AB)</span><sup>2</sup><span> = 108 ------- equation 1</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) Now we will find </span><span>(AC)</span><sup>2</sup></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (AC)</span><sup>2</sup><span> = </span><span>(12)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) (AC)</span><sup>2</sup><span> = 144</span><span> ------- equation 2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) Now we will find </span><span>(BC)</span><sup>2</sup></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (BC)</span><sup>2</sup><span> = </span><span>(6)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) (BC)</span><sup>2</sup><span> = 36</span><span> ------- equation 3</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) From equations 1, 2, and 3, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">144 = 108 + 36</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AC)</span><sup>2</sup><span> = </span><span>(AB)</span><sup>2</sup><span> + </span><span>(BC)</span><sup>2</sup><span> ------- equation 4</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>6) From equation 4, we have,</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">This is a right angled triangle at point B, so angle B is 90° hence answer is C.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></span></div><div style="text-align: left;"><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/169-ncert-10-6-triangles-ex-66.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.6</a></span></h2><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></span></div></div></span></h3></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-29572310812771427352023-12-19T12:47:00.004+05:302023-12-20T12:09:02.409+05:30167-NCERT-10-6-Triangles - Ex- 6.4<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.4</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/166-ncert-10-6-triangles-ex-63.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.3</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.4</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><span>Q</span>1. Let </b><b style="text-indent: -47.2667px;">∆</b><b> ABC ~ </b><b style="text-indent: -47.2667px;">∆</b><b> DEF and their areas be 64 cm2 and 121 cm2 respectively</b><b>. </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>If EF = </b><b>15.4 cm, find BC.</b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">1) As </span><span style="text-indent: -47.2667px;">∆</span> ABC ~ <span style="text-indent: -47.2667px;">∆</span> DEF, we know that ratios of the areas of similar triangles are equal</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">to the ratios of the square of corresponding sides of the tringles, so, we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">A(<span style="text-indent: -47.2667px;">∆</span> ABC)/A(<span style="text-indent: -47.2667px;">∆</span> DEF) = [(BC)<sup>2</sup>/(EF)<sup>2</sup>]</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">64/121 = [(BC)/(EF)]<sup>2</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">[(BC)/(EF)]<sup>2</sup> = 64/121<br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">[(BC)</span><span style="font-family: arial; font-size: medium;">/(</span><span style="font-family: arial; font-size: medium;">15.4)]</span><span style="font-family: arial; font-size: medium;"> = 8/11<br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BC) = </span><span style="font-family: arial; font-size: medium;">(</span><span style="font-family: arial; font-size: medium;">15.4 x </span><span style="font-family: arial; font-size: medium;">8)/11<br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BC) = </span><span style="font-family: arial; font-size: medium;">(</span><span style="font-family: arial; font-size: medium;">1.4 x </span><span style="font-family: arial; font-size: medium;">8)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BC) = </span><span style="font-family: arial; font-size: medium;">1</span><span style="font-family: arial; font-size: medium;">1.2 cm.</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) So, BC = 11.2 cm.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q2. Diagonals of a trapezium ABCD with AB || DC intersect each other at the</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>point O. </b><b>If AB = 2 CD, find the ratio of the areas of triangles AOB and COD. </b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNF0cs-4ZiDR5xio9cJNqOSn9M0JOLdVAjLokOxV0FUVJWdjZXpJDM61J-JUwi9HcraHEspKWjaMy6Kwonv_gWW-19BVOFEPmL1PkKuA3CvghN4ONkW1QZ4zp7uUpsXAz7PvGX1W0x51hw7_cf8tTdXA18y34FynERtSlHo8PEmM4S-nUa3lilc5Ll/s793/38.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="342" data-original-width="793" height="138" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNF0cs-4ZiDR5xio9cJNqOSn9M0JOLdVAjLokOxV0FUVJWdjZXpJDM61J-JUwi9HcraHEspKWjaMy6Kwonv_gWW-19BVOFEPmL1PkKuA3CvghN4ONkW1QZ4zp7uUpsXAz7PvGX1W0x51hw7_cf8tTdXA18y34FynERtSlHo8PEmM4S-nUa3lilc5Ll/s320/38.png" width="320" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; font-weight: 400; white-space: pre-wrap;">1) In trapezium </span><span style="font-family: arial; font-size: medium;"><span style="font-weight: 400;">ABCD, </span><span style="font-weight: normal;"> </span></span><span style="font-weight: normal;">AB || DC, and in </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium; font-weight: 400;"> OAB and </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium; font-weight: 400;"> OCD, </span><span style="font-weight: 400;">we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; text-align: left; white-space: normal;">a) < OAB = < OCD --------- (alternate angles)<br /><span style="font-family: arial; font-size: medium;">b) < OBA = < ODC</span> --------- (alternate angles)<br /><span style="font-family: arial; font-size: medium;">c) < AOB = < COD</span> --------- (vertically opposite angles)</blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) So by AAA similarity test, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium;"> OAB </span>~<span style="font-family: arial; font-size: medium;"> </span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium;"> OCD<br /></span></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">3) W</span><span style="font-family: arial; font-size: medium;">e know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial; font-size: medium;">to the ratios of</span></div></span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; text-align: left; white-space: normal;"><span style="font-family: arial; font-size: medium;">the </span><span style="font-family: arial; font-size: medium;">square of </span>corresponding sides of the triangles, so, we have,</div></span></div></h3></blockquote><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OAB)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OCD) = [(AB)</span><sup>2</sup><span style="font-family: arial;">/(</span><span style="font-family: arial;">CD)</span><sup>2</sup><span style="font-family: arial;">]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OAB)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OCD) = [(AB)</span><span style="font-family: arial;">/(</span><span style="font-family: arial;">CD)]</span><sup>2</sup></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OAB)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OCD) = [2(CD)</span><span style="font-family: arial;">/(</span><span style="font-family: arial;">CD)]</span><sup>2</sup></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OAB)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OCD) = [2</span><span style="font-family: arial;">/1</span><span style="font-family: arial;">]</span><sup>2</sup></blockquote></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OAB)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> OCD) = 4/1</span></blockquote></blockquote><div style="white-space: normal;"><span style="font-family: arial; font-weight: 400;">4) So, </span><span style="font-weight: normal;">the ratio of the areas of triangles AOB and COD is 4:1</span><span style="font-family: arial; font-weight: 400;">.</span></div><div style="white-space: normal;"><br /></div><div style="white-space: normal;">Q3. In the following fig., ABC and DBC are two triangles on the same base BC.</div></span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left; white-space: normal;">If AD intersects BC at O, show that </div></span></div></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left; white-space: normal;">[area (<b style="text-indent: -47.2667px;">∆ </b>ABC)]/[area (<b style="text-indent: -47.2667px;">∆ </b>DBC)] = AO/DO.</div></span></div></h3></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN4LWNeBGXrz-J3aRVPRBtjGOX4TQMgv3nRhXcZqwRSlsrbxuxWEga-bAJAIMRx-j5y6tVeSxZdZ_xeprb4C_M_Rf5mu0ZtY6A--cXpCfUG-93lzlRzFfGImSYDWaADoA4RQdZaQp65_JLyNPBDEuAPL-DrsakubPUCvMhM1Y17I3RZw92dPmHRpzm/s486/39.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="290" data-original-width="486" height="191" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN4LWNeBGXrz-J3aRVPRBtjGOX4TQMgv3nRhXcZqwRSlsrbxuxWEga-bAJAIMRx-j5y6tVeSxZdZ_xeprb4C_M_Rf5mu0ZtY6A--cXpCfUG-93lzlRzFfGImSYDWaADoA4RQdZaQp65_JLyNPBDEuAPL-DrsakubPUCvMhM1Y17I3RZw92dPmHRpzm/s320/39.png" width="320" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-family: arial; font-size: medium; font-weight: 400; white-space: pre-wrap;">1) We know that the area of a triangle = 1/2 x base x height.</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">a) <span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> ABC)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> DCB) = [1/2 x (BC) x (AM)]/</span>[1/2 x (BC) x (DN)]<br /><span style="font-family: arial; font-size: medium;">b) </span><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> ABC)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> DCB) = (AM)/</span>(DN) -------- equation 1</blockquote><div><span style="font-weight: normal;">2) In </span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-family: arial; font-weight: 400; white-space: normal;"> AMO and </span>∆<span style="font-family: arial; font-weight: 400; white-space: normal;"> DNO,</span></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; text-align: left;">a) < AMO = < DNO ------ (AM ⊥ BC and AM ⊥ BC)</div></span></h3></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) < AOM = < DON ------- (vertically opposite angles)</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) <span style="white-space: pre-wrap;">So by AA similarity test, we have,</span></span></div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium;"> AMO </span>~<span style="font-family: arial; font-size: medium;"> </span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-size: medium;"> DNO<br /></span></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">4) Here we have</span>, </div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"></blockquote></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(AM)/(DN) = (AO)/(DO) -------- equation 2</blockquote><div style="text-align: left;"><span style="font-weight: normal;">5) From equations 1 and 2 we have,</span> </div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> ABC)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> DCB) = (AO)/</span>(DO), hence proved.</blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;">Q4. If the areas of two similar triangles are equal, prove they are congruent. </div><div style="text-align: left;"></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5Ccb5mVibivZbASOoiqpR8vpLQ4EpQLYV2f_C23H3Wimo-oBYN9pYoZ5JWlT-xPTQAe9S4M7TsrDnJQLFaZ3LUDtG2aYEJbo8hiv-rj4jfK4jEPArXmcUK3T562h-ae4GM7tW6gz3iiSt_a3rt49QiGwjlEVQSdV4DatyucINMku0p-a8ts6h5Wg3/s735/40.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="316" data-original-width="735" height="138" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5Ccb5mVibivZbASOoiqpR8vpLQ4EpQLYV2f_C23H3Wimo-oBYN9pYoZ5JWlT-xPTQAe9S4M7TsrDnJQLFaZ3LUDtG2aYEJbo8hiv-rj4jfK4jEPArXmcUK3T562h-ae4GM7tW6gz3iiSt_a3rt49QiGwjlEVQSdV4DatyucINMku0p-a8ts6h5Wg3/s320/40.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) </span><span style="font-weight: 400;">As </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> ABC ~ </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> PQR, we know that ratios of the areas of similar triangles are equal</span></div></span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">to the ratios of the square of corresponding sides of the tringles, so, we have,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">A(<span style="text-indent: -47.2667px;">∆</span> ABC)/A(<span style="text-indent: -47.2667px;">∆</span> PQR) = [(AB)<sup>2</sup>/(PQ)<sup>2</sup>]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>1 = </span><span>[(AB)</span><sup>2</sup><span>/(PQ)</span><sup>2</sup><span>]</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>[</span><span>(AB)</span><sup>2</sup><span>/</span><span>(PQ)</span><sup>2</sup><span>] = 1<br /></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AB)</span><sup>2</sup><span> = </span><span>(PQ)</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AB) = </span><span style="font-family: arial; font-size: medium;">(PQ)</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;">2) Similarly we can prove that,</span> </span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AC) = </span><span style="font-family: arial; font-size: medium;">(PR)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BC) = </span><span style="font-family: arial; font-size: medium;">(QR</span><span style="font-family: arial; font-size: medium;">)</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;">3) So by the SSS </span></span><span style="font-family: arial; font-size: medium; font-weight: normal;">test of congruence, we have,</span></div></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; font-weight: 400; white-space: pre-wrap;"> ABC </span><span style="background-color: white; color: #404040; font-weight: 400; text-align: center;"><span style="font-family: arial;">≅</span></span><span style="font-family: arial; font-weight: 400; white-space: pre-wrap;"> </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; font-weight: 400; white-space: pre-wrap;"> PQR, hence proved.</span></span></div></h3></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q5. D, E, and F are respectively the mid-points of sides AB, BC, and CA of </b></span><b style="text-indent: -47.2667px;">∆</b><span><b> ABC. Find the </b></span><b>ratio of the areas of </b><b style="text-indent: -47.2667px;">∆</b><b> DEF and </b><b style="text-indent: -47.2667px;">∆</b><b> ABC.</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuGpMtArOxU0v3DGJR_Q6MqEahM8aq6jmbujDjfktdtErpH0O2Kk7y5ZxlXbmfUqkTFAQbelXPwp8ABAN3CJIZVKVlEhCsAzU_6js9INGFevYLpQRtUwwQKBvrDfIyekcoQkCIP4NXphdNLQWa-AMdVW0kfdo2XBiWviGIgb-i9MQROlWVCzV-AUKi/s480/41.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="416" data-original-width="480" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuGpMtArOxU0v3DGJR_Q6MqEahM8aq6jmbujDjfktdtErpH0O2Kk7y5ZxlXbmfUqkTFAQbelXPwp8ABAN3CJIZVKVlEhCsAzU_6js9INGFevYLpQRtUwwQKBvrDfIyekcoQkCIP4NXphdNLQWa-AMdVW0kfdo2XBiWviGIgb-i9MQROlWVCzV-AUKi/w217-h188/41.png" width="217" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white; color: #333333; font-weight: 400; white-space: normal;">1) As </span><span style="background-color: white; color: #333333; font-weight: 400; white-space: normal;">D, E, and F are the mid-points of </span><span class="mjx-chtml" id="MathJax-Element-1-Frame" style="background-color: white; box-sizing: border-box; color: #333333; font-weight: 400; white-space: normal;"><span class="mjx-math" id="MJXc-Node-6465" style="box-sizing: border-box;"><b style="color: black; text-indent: -47.2667px;">∆ </b>ABC, </span></span><span class="mjx-chtml" style="background-color: white; box-sizing: border-box; color: #333333; font-weight: 400; white-space: normal;"><span class="mjx-math" style="box-sizing: border-box;">we have </span></span><span style="background-color: white; color: #333333; font-weight: 400; white-space: normal;">DE || AC, and by mid-point</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white; color: #333333; font-weight: 400; white-space: normal;">theorem, we have,</span></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="background-color: white; color: #333333; font-family: arial; font-size: medium;">DF = (1/2) CB ----------- equation 1</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="background-color: white; color: #333333; font-family: arial; font-size: medium;">DF = BE (As E is mid-point of BC) ----------- equation 2</span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) So, <b>▱</b> </span><span style="font-family: arial;">DFEB is a </span><span style="font-family: arial;">parallelogram,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) < DFE = < DBE --- (</span><span style="font-family: arial;">opposite angles of a parallelogram) </span><span style="background-color: white; color: #333333; font-family: arial;">---- equation 3</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) So, <b>▱</b> </span><span style="font-family: arial;">FDEC is a </span><span style="font-family: arial;">parallelogram,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) < FDE = < FCE --- (</span><span style="font-family: arial;">opposite angles of a parallelogram)</span><span style="font-family: arial;"> </span><span style="background-color: white; color: #333333; font-family: arial;">---- equation 4</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) In </span><b style="background-color: white; text-indent: -47.2667px;">∆ </b><span style="background-color: white; color: #333333;">DEF and </span><b style="background-color: white; text-indent: -47.2667px;">∆ </b><span style="background-color: white; color: #333333;">ABC, and from equations 3 and 4, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) < DFE = < CBA --- from equation 3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) < FDE = < BCA --- from equation 4</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) By AA similarity test, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) </span><b style="background-color: white; text-indent: -47.2667px;">∆ </b><span style="background-color: white; color: #333333;">DEF </span><span style="white-space: pre-wrap;">~</span><span style="background-color: white; color: #333333;"> </span><b style="background-color: white; text-indent: -47.2667px;">∆ </b><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span></span></div></blockquote><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial; white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial;">to the ratios of</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">the square of corresponding sides of the triangles, so, we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">DEF</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span><span>) = [(DF)</span><sup>2</sup><span>/(CB)</span><sup>2</sup><span>]</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">DEF</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span><span>) = [(DF)</span><span>/(CB)]</span><sup>2<br /></sup><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">DEF</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span><span>) = [(1/2)(CB)</span><span>/(CB)]</span><sup>2</sup><span> ---- from equation 1<br /></span><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">DEF</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span><span>) = [(1/2)</span><span>]</span><sup>2<br /></sup><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">DEF</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; text-indent: -47.2667px;">C</span><span style="background-color: white; color: #333333;">AB</span><span>) = 1/4</span></span></blockquote></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7) So, </span><span style="font-family: arial;">the </span><span style="font-family: arial;">ratio of the areas of </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> DEF and </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> ABC = 1/4.</span></span></div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q6. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio </b></span><span><b>of their corresponding medians.</b></span> </span></div></div><div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZgvppNWDi0Tw3romt2JYKOJ18I3jl1DRSX9rjTVddRwRCtUiw_5dvgJxaQVA1Ei6C_yU8DtwjxDnNfRT1NMwNP9JYlKwteEDpJ37uDAnNfTt9cf7WLtrTuHyat74p0O7qRyX7wKB7YlA4FqjQZRbtxG_I2-XGkLNdbZ6owTBKkHjZKyMskxCZVCzK/s716/37.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="299" data-original-width="716" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZgvppNWDi0Tw3romt2JYKOJ18I3jl1DRSX9rjTVddRwRCtUiw_5dvgJxaQVA1Ei6C_yU8DtwjxDnNfRT1NMwNP9JYlKwteEDpJ37uDAnNfTt9cf7WLtrTuHyat74p0O7qRyX7wKB7YlA4FqjQZRbtxG_I2-XGkLNdbZ6owTBKkHjZKyMskxCZVCzK/s320/37.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400;">1) It is given that </span>∆<span style="font-weight: 400;"> ABC ~ </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> PQR,</span></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) < ABC = < PQR --------- </span><span style="background-color: white; color: #333333; font-family: arial;">equation 1.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) < BCA = < QRP ---------</span><span style="font-family: arial;"> </span><span style="background-color: white; color: #333333; font-family: arial;">equation 2.</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">a) < CAB = < RPQ ---------</span><span style="font-family: arial;"> </span><span style="background-color: white; color: #333333; font-family: arial;">equation 3.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) Similarity, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">a) (AB)/(PQ) = (BC)/(QR) = (CA)/(RP)</span><span style="font-family: arial;"> --------- </span><span style="background-color: white; color: #333333; font-family: arial;">equation 4.</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial; white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial;">to the ratios of</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the square of corresponding sides of the triangles, so, we have,</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left; text-indent: -47.2667px;"><span style="font-family: arial; font-size: medium;"><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">ABC</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> <span style="background-color: white;">PQR</span></span><span>) = (AB)</span><sup>2</sup><span>/(PQ)</span><sup>2</sup><span> = </span><span style="text-indent: -47.2667px;">(BC)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(QR)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;"> = </span><span style="text-indent: -47.2667px;">(AC)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(PR)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333;">--- equation 5.</span></span></blockquote></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) In </span><span style="font-family: arial; font-weight: 700; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> ABD and </span><span style="font-family: arial; font-weight: 700; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> PQM, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) < ABD = < PQM ---- from equation 1.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) (AB)/(PQ) = (BD)/(QM) ----- (D and M are mid-points </span><span style="font-family: arial;">of BC and QR).</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) So, by the SAS similarity test,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> ABD </span><span style="font-family: arial; white-space: pre-wrap;">~</span><span style="font-family: arial; white-space: pre-wrap;"> </span><span style="font-family: arial; font-weight: 700; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> PQM</span><span style="font-family: arial;"> --------- </span><span style="background-color: white; color: #333333; font-family: arial;">equation 6.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>6)</span> <span style="white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span>to the ratios of</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the square of corresponding sides of the triangles, so, we have,</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: -47.2667px;"><span style="font-family: arial; font-size: medium;"><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="background-color: white; color: #333333;">ABD</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> <span style="background-color: white;">PQM</span></span><span>) = (AB)</span><sup>2</sup><span>/(PQ)</span><sup>2</sup><span> = </span><span style="text-indent: -47.2667px;">(BD)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(QM)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;"> = </span><span style="text-indent: -47.2667px;">(AD)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(PM)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333;">--- equation 7.</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) So from equation 7, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333; text-indent: -47.2667px;">ABD</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> <span style="background-color: white;">PQM</span></span><span style="text-indent: -47.2667px;">) = </span><span style="text-indent: -47.2667px;">(AD)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(PM)</span><sup style="text-indent: -47.2667px;">2</sup></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) Hence, it is proved that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">the ratio of the areas of two similar triangles is equal to the square of the ratio </span><span style="font-family: arial; font-size: medium;">of their corresponding medians.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q7. Prove that the area of an equilateral triangle described on one side of a square is equal </b></span><b>to half the area of the equilateral triangle described on one of its diagonals. </b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKsylCAkqtMo1y2FqtnXKRFYQWiXD1ynJtW8X1BNyqD0ioLuLeODjgYQyPZsAFWFwkH3yVmIQEKgzTfetRKG7bygiVeGFh02kuyFqmufzhdFpQWbDWc7WZ7yCTcc6A3hRzkKSqiAovFoOksYN7n0viiH02vv0lZgIT5G1sUyWQlL_1wB2gxPgWplXm/s463/42.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="463" data-original-width="411" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKsylCAkqtMo1y2FqtnXKRFYQWiXD1ynJtW8X1BNyqD0ioLuLeODjgYQyPZsAFWFwkH3yVmIQEKgzTfetRKG7bygiVeGFh02kuyFqmufzhdFpQWbDWc7WZ7yCTcc6A3hRzkKSqiAovFoOksYN7n0viiH02vv0lZgIT5G1sUyWQlL_1wB2gxPgWplXm/s320/42.png" width="284" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400;">1) </span>∆<span style="font-weight: 400;"> PAB and </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> QAC are equilateral triangles, so we have,</span></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-weight: 700; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> PAB </span><span style="font-family: arial; white-space: pre-wrap;">~</span><span style="font-family: arial; white-space: pre-wrap;"> </span><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> QAC</span></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">2) </span><span style="font-family: arial; white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial;">to the ratios of</span></div><div style="font-weight: 400; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">the square of corresponding sides of the triangles, so, we have,</span></blockquote></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: -47.2667px;"><span style="font-family: arial;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial; text-indent: 0px; white-space: pre-wrap;">PAB</span><span style="font-family: arial;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial; text-indent: 0px; white-space: pre-wrap;">QAC</span><span style="font-family: arial;">) = (AB)</span><sup>2</sup><span style="font-family: arial;">/(AC)</span><sup>2</sup><span style="font-family: arial;"> </span><span style="background-color: white; color: #333333; font-family: arial;">--- equation 1</span></blockquote></blockquote><div style="text-align: left;"><span style="font-weight: normal;">3) In </span><span style="font-family: arial; font-weight: 400; white-space: normal;"><b>▱</b> </span><span style="font-weight: 400; white-space: normal;">ABCD is a </span><span style="font-weight: 400; white-space: normal;">square, so we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">(AC) = </span><span style="white-space: pre-wrap;">√2 (AB)</span><span style="text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333; text-indent: -47.2667px;">--- equation 2</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">4) From equations 1 and 2,</span></div></div></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"> <span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">A(</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"> </span><span style="font-weight: 400;">PAB</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">)/A(</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"> </span><span style="font-weight: 400;">QAC</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">) = (AB)</span><sup style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">2</sup><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">/(AC)</span><sup style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">2</sup></div></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">A(</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"> </span><span style="font-weight: 400;">PAB</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">)/A(</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"> </span><span style="font-weight: 400;">QAC</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">) = (AB)</span><sup style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">2</sup><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">/(</span><span style="font-weight: 400;">√2 (AB)</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">)</span><sup style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">2</sup></div></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">PAB</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">QAC</span><span style="text-indent: -47.2667px;">) = 1</span><span style="text-indent: -47.2667px;">/(</span><span style="white-space: pre-wrap;">√2</span><span style="text-indent: -47.2667px;">)</span><sup style="text-indent: -47.2667px;">2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">PAB</span><span style="font-family: arial; text-indent: -47.2667px;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">QAC</span><span style="font-family: arial; text-indent: -47.2667px;">) = 1</span><span style="font-family: arial; text-indent: -47.2667px;">/2<br /></span><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">PAB</span><span style="font-family: arial; text-indent: -47.2667px;">)</span><span style="font-family: arial; text-indent: -47.2667px;"> = (1/2) x [</span><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">QAC</span><span style="font-family: arial; text-indent: -47.2667px;">)] hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q8. ABC and BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of </b></span><b>the areas of triangles ABC and BDE is </b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(A) 2: 1 (B) 1: 2 (C) 4: 1 (D) 1: 4 </span></b></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><h3 style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMidUagKQiNkOzYHRpTjLYuXw8AP7OpmDx1z7Qtmr5q6Hwg5aGYDy3kuFTmxDo7mCFM5ZtnnrT01ExMYfcLux6-hYB_xmVivVrqxvAM8-u06F6cLlejzBssQ10ZhNWQf_cVXAU3uhDEQLasrk85ICAREULKBoNYTan8Xp_ynQMGM5qdrekaFRDg-a1/s535/43.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="354" data-original-width="535" height="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMidUagKQiNkOzYHRpTjLYuXw8AP7OpmDx1z7Qtmr5q6Hwg5aGYDy3kuFTmxDo7mCFM5ZtnnrT01ExMYfcLux6-hYB_xmVivVrqxvAM8-u06F6cLlejzBssQ10ZhNWQf_cVXAU3uhDEQLasrk85ICAREULKBoNYTan8Xp_ynQMGM5qdrekaFRDg-a1/s320/43.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) As D is the mid-point of BC,</span></div></span></h3></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="text-align: left;">(BC) = 2 (BD) -------- equation 1</div></span></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) </span><span style="font-family: arial; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> ABC and </span><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> EBD are equilateral triangles, so we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; white-space: pre-wrap;">∆ </span><span style="font-family: arial; white-space: pre-wrap;">ABC </span><span style="font-family: arial; white-space: pre-wrap;">~</span><span style="font-family: arial; white-space: pre-wrap;"> </span><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> EBD.</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial; white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial;">to the ratios of</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">the square of corresponding sides of the triangles, so, we have,</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: -47.2667px;"><span style="font-family: arial; font-size: medium;"><span>A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="text-indent: 0px; white-space: pre-wrap;">ABC</span><span>)/A(</span><span style="text-indent: -47.2667px;">∆</span><span> </span><span style="text-indent: 0px; white-space: pre-wrap;">EBD</span><span>) = (BC)</span><sup>2</sup><span>/(BD)</span><sup>2</sup><span> </span><span style="background-color: white; color: #333333;">--- equation 2.</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) From equations 1 and 2 we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">ABC</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">EBD</span><span style="text-indent: -47.2667px;">) = (BC)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(BD)</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;"> <br /></span><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">ABC</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">EBD</span><span style="text-indent: -47.2667px;">)</span><span style="text-indent: -47.2667px;"> = [2(BD)]</span><sup style="text-indent: -47.2667px;">2</sup><span style="text-indent: -47.2667px;">/(BD)</span><sup style="text-indent: -47.2667px;">2<br /></sup><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">ABC</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">EBD</span><span style="text-indent: -47.2667px;">)</span><span style="text-indent: -47.2667px;"> = [2]</span><sup style="text-indent: -47.2667px;">2<br /></sup><span style="text-indent: -47.2667px;">A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">ABC</span><span style="text-indent: -47.2667px;">)/A(</span><span style="text-indent: -47.2667px;">∆</span><span style="text-indent: -47.2667px;"> </span><span style="white-space: pre-wrap;">EBD</span><span style="text-indent: -47.2667px;">)</span><span style="text-indent: -47.2667px;"> = 4/1<br /></span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">The ratio of </span><span style="font-family: arial;">the areas of triangles ABC and EBD is 4:1</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) Answer: (C) 4:1.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q9. Sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio </b></span><b>(A) 2:3 (B) 4:9 (C) 81:16 (D) 16:81</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbCEl6yvWbory3Nt3hzwleSR5TxOHCgGo-yfjLypNmqYNzn0sew6ScYm_kRcitheQIZWMVE-eksxd_2tEEJIzlPuQrt7ZHj29t4QjmSHqKvuuRknzetCzexZEn-NOvp4p9Wm2Mo9aUTbUNejL_2fsKnn7cAVX3iNyfmNPXGP4Q_ReFV3eLw0F7B6wF/s658/33.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="264" data-original-width="658" height="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbCEl6yvWbory3Nt3hzwleSR5TxOHCgGo-yfjLypNmqYNzn0sew6ScYm_kRcitheQIZWMVE-eksxd_2tEEJIzlPuQrt7ZHj29t4QjmSHqKvuuRknzetCzexZEn-NOvp4p9Wm2Mo9aUTbUNejL_2fsKnn7cAVX3iNyfmNPXGP4Q_ReFV3eLw0F7B6wF/s320/33.png" width="320" /></a></div></span></h3><h3 style="white-space: normal;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) </span><span style="font-weight: 400;">∆</span><span style="font-weight: 400;"> ABC and </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> PQR are similar triangles, so we have,</span></div></span></h3><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; white-space: pre-wrap;">∆ </span><span style="font-family: arial; white-space: pre-wrap;">ABC </span><span style="font-family: arial; white-space: pre-wrap;">~</span><span style="font-family: arial; white-space: pre-wrap;"> </span><span style="font-family: arial; text-indent: -47.2667px; white-space: pre-wrap;">∆</span><span style="font-family: arial; white-space: pre-wrap;"> PQR.</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) It is given that </span><span style="font-weight: 400; white-space: normal;">(AB)/(PQ)</span><span style="font-weight: 400; white-space: normal;"> </span><span face="Arial, sans-serif" style="font-weight: 400; line-height: 25.68px; white-space: normal;"><span>=</span> </span><span style="font-weight: 400; white-space: normal;">(BC)/(QR)</span><span style="font-weight: 400; white-space: normal;"> = (AC)/(PR) = 4/9</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"> </span><span style="background-color: white; color: #333333; font-weight: 400; text-indent: -47.2667px; white-space: normal;">--- equation 1.</span> </div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">3) </span><span style="font-family: arial; white-space: pre-wrap;">We know that ratios of the areas of similar triangles are equal </span><span style="font-family: arial;">to the ratios of</span></div><div style="font-weight: 400; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">the square of corresponding sides of the triangles, so, we have,</span></blockquote></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: -47.2667px;"><span style="font-family: arial;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial; text-indent: 0px; white-space: pre-wrap;">ABC</span><span style="font-family: arial;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial; text-indent: 0px; white-space: pre-wrap;">PQR</span><span style="font-family: arial;">) = (BC)</span><sup>2</sup><span style="font-family: arial;">/(QR)</span><sup>2</sup><span style="font-family: arial;"> </span><span style="background-color: white; color: #333333; font-family: arial;">--- equation 2.</span></blockquote></blockquote><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">4) From equations 1 and 2 we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">ABC</span><span style="font-family: arial; text-indent: -47.2667px;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">PQR</span><span style="font-family: arial; text-indent: -47.2667px;">) = (BC)</span><sup style="text-indent: -47.2667px;">2</sup><span style="font-family: arial; text-indent: -47.2667px;">/(QR)</span><sup style="text-indent: -47.2667px;">2</sup><span style="font-family: arial; text-indent: -47.2667px;"> <br /></span><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">ABC</span><span style="font-family: arial; text-indent: -47.2667px;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">PQR</span><span style="font-family: arial; text-indent: -47.2667px;">)</span><span style="font-family: arial; text-indent: -47.2667px;"> = [4/9]</span><sup style="text-indent: -47.2667px;">2</sup><sup style="text-indent: -47.2667px;"><br /></sup><span style="font-family: arial; text-indent: -47.2667px;">A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">ABC</span><span style="font-family: arial; text-indent: -47.2667px;">)/A(</span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">PQR</span><span style="font-family: arial; text-indent: -47.2667px;">)</span><span style="font-family: arial; text-indent: -47.2667px;"> = 16/81</span></blockquote><span style="font-weight: normal;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">The ratio of </span><span style="font-family: arial;">the areas of triangles ABC and PQR is 16:81</span></span><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;">6) Answer: (D) 16:81.</span></div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;"><br /></span></div><div style="font-weight: 400; white-space: normal;"><span style="font-family: arial;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></span></div></span></div></h3><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/168-ncert-10-6-triangles-ex-65.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.5</a></span></h2><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><div style="font-family: "Times New Roman";"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></span></div></div></span></div></h3></div><div><h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"></div></span></div></h3></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-91841097767983832812023-12-18T13:03:00.002+05:302023-12-19T21:00:53.399+05:30166-NCERT-10-6-Triangles - Ex- 6.3<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.3</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/165-ncert-10-6-triangles-ex-62.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.2</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.3</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q1. State which pairs of triangles in the following Fig, are similar. Write the similarity criterion used by </b></span><b>you for answering the question and also write the pairs of similar triangles in the symbolic </b><b>form : </b></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJFz8dErMokeAO7o-TBqgHrSR7UAN2TWlXvsFzzIZWxhdvWIo0BI7BAp4P2vJgVEYjT1JkCiDa84PmEmVdfgoOCeNady4kWoM0YmMX6FNop0-uxRy9ZLmatEmfNPOcf3dQF3xPCuN87XPV-sxTmjXSZYC6hstfe0UxXb1Wl90ftUFnFKgSZBegNRtZ/s805/21.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="663" data-original-width="805" height="434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJFz8dErMokeAO7o-TBqgHrSR7UAN2TWlXvsFzzIZWxhdvWIo0BI7BAp4P2vJgVEYjT1JkCiDa84PmEmVdfgoOCeNady4kWoM0YmMX6FNop0-uxRy9ZLmatEmfNPOcf3dQF3xPCuN87XPV-sxTmjXSZYC6hstfe0UxXb1Wl90ftUFnFKgSZBegNRtZ/w527-h434/21.png" width="527" /></span></a></div><span style="font-size: medium;"><span style="font-family: arial;"><div><h3 style="white-space: pre-wrap;"><span style="font-family: arial;">Solution:</span></h3></div><div><div class="separator" style="clear: both; text-align: left;">(i)</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgD4Sz47VO9Cfbzy3UBGINuH59Z4ujOz4SL87Us9B_UmXVz6L1q6MH8ihvCwDeItxCf7w-tklGMc0u9CHaA3oey8Ey7NskbeQUISIJ4-rlvzaulMBkCiyNf6anzAlFLatxQC4w_o4wIKGqi1T52VLj9U0jiL0lJIszm61S-ybfTTEsaRPk1ZIAmMVt/s485/15.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="253" data-original-width="485" height="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgD4Sz47VO9Cfbzy3UBGINuH59Z4ujOz4SL87Us9B_UmXVz6L1q6MH8ihvCwDeItxCf7w-tklGMc0u9CHaA3oey8Ey7NskbeQUISIJ4-rlvzaulMBkCiyNf6anzAlFLatxQC4w_o4wIKGqi1T52VLj9U0jiL0lJIszm61S-ybfTTEsaRPk1ZIAmMVt/w302-h158/15.png" width="302" /></a></div></div>1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQR,</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) < A = < P = 60<sup>0</sup></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) < B = < Q = 80</span><sup>0<br /></sup><span>c) < C = < R = 40</span><sup>0</sup></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) So by AAA similarity rule, </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC </span><span style="background-color: white; color: #333333; font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQR.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(ii)</span></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuUFrNjWhBzeLen3SDeYSEUFsvIjis8qZwsPPMf2mrVs_fBLrcsAZzUOQ0rbkjkhVLMdxuiDydIpGkKEITT9cEL4wh3IX8-4tlgngBZ9deYVh82Q9-DaYuvGfexwfgNroqcr5O9CYLEYvIXeEA1YI62SE6bk3OL0lee4IMpkU02KCjnymhtFmLDGrK/s485/16.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="324" data-original-width="485" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuUFrNjWhBzeLen3SDeYSEUFsvIjis8qZwsPPMf2mrVs_fBLrcsAZzUOQ0rbkjkhVLMdxuiDydIpGkKEITT9cEL4wh3IX8-4tlgngBZ9deYVh82Q9-DaYuvGfexwfgNroqcr5O9CYLEYvIXeEA1YI62SE6bk3OL0lee4IMpkU02KCjnymhtFmLDGrK/w264-h176/16.png" width="264" /></span></a></div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ QRP,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ABC and </span><span style="text-indent: -47.2667px;">∆ QRP, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ Q, </span></span><span style="text-indent: -47.2667px;">B</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ R, </span></span><span style="text-indent: -47.2667px;">C</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P. </span></span><span>That is</span> </span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">i) point A is associated with poin Q, </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point B is associated with poin R,<br />iii) point B is associated with poin R,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) (AB)/(QR) = 2/4 = 1/2 -------- equation 1</span></div></blockquote></div></div><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) (BC)/(RP) = 2.5/5 = 1/2</span><span style="font-family: arial;"> -------- equation 2</span></span></blockquote></div></div><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) (AC)/(QP) = 3/6 = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">2) So from equations 1, 2, and 3, the corresponding ratios are proportional, so by SSS</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">test, </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC </span><span style="background-color: white; color: #333333; font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ QRP.</span></span></div></div></div></blockquote><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(iii)<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMroUtRUjk6VlzYDAjwewXYeGfDvuWg7qbZxpk-7QjYlDRguvNEGiIO-pP2P2CppulTX6rv1g2xBvkm1WYP0v1p6lK0e6HD4Oj-iE_4iQ4XelWe-VUARQRmPkpHss_6_w8-tdDiA-oWXEPjOOFcH2igPrDnnLRgV7PrKP8dij6JByb0kzAxelS8qRH/s381/17.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="229" data-original-width="381" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMroUtRUjk6VlzYDAjwewXYeGfDvuWg7qbZxpk-7QjYlDRguvNEGiIO-pP2P2CppulTX6rv1g2xBvkm1WYP0v1p6lK0e6HD4Oj-iE_4iQ4XelWe-VUARQRmPkpHss_6_w8-tdDiA-oWXEPjOOFcH2igPrDnnLRgV7PrKP8dij6JByb0kzAxelS8qRH/w284-h170/17.png" width="284" /></a></div></span><div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ LMP and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ FED,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ LMP and </span><span style="text-indent: -47.2667px;">∆ FED, points L</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ F, </span></span><span style="text-indent: -47.2667px;">M</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ E, and </span></span><span style="text-indent: -47.2667px;">P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ D. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) point L is associated with poin F, </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point M is associated with poin E,<br />iii) point P is associated with poin D,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (LM)/(FE) = 2.7/5 = 27/50 -------- equation 1</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) (MP)/(ED) = 2/4 = 1/2</span><span style="font-family: arial;"> -------- equation 2</span></span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) (LP)/(FD) = 3/6 = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) So from equations 1, 2, 3, (LM)/(FE) </span><span face="Arial, sans-serif" style="line-height: 107%;">≠ </span><span>(MP)/(ED) = (LP)/FD)</span><span>, so </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC </span><span id="docs-internal-guid-0b486645-7fff-4f80-4bed-10a243c31edb"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">≁</span></span></span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ QRP.</span></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(iv)<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoSuRIm_mkm2itNAUlnJvjWwc9Vlck0XnipXOkVdYqLlWTnO-58sC4pGQ3O_RV3vGb1IEEgXLjik5yn3a35dJMqhSuWXnSm_HT64eQNMocxfYrENZocJjIBNgjr175kThztMHvpUfW8rxo-mSImGhRpWzZ1to4X7JAPp-QcRKknxU8zwYXzrHwfGo6/s650/18.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="281" data-original-width="650" height="138" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoSuRIm_mkm2itNAUlnJvjWwc9Vlck0XnipXOkVdYqLlWTnO-58sC4pGQ3O_RV3vGb1IEEgXLjik5yn3a35dJMqhSuWXnSm_HT64eQNMocxfYrENZocJjIBNgjr175kThztMHvpUfW8rxo-mSImGhRpWzZ1to4X7JAPp-QcRKknxU8zwYXzrHwfGo6/s320/18.png" width="320" /></a></div></span><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ LMN and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ RQP,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ LMN and </span><span style="text-indent: -47.2667px;">∆ RQP, points L</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ R, </span></span><span style="text-indent: -47.2667px;">M</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ Q, and </span></span><span style="text-indent: -47.2667px;">N</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) point L is associated with poin R, </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point M is associated with poin Q,<br />iii) point N is associated with poin P,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (LM)/(RQ) = 5/10 = 1/2 -------- equation 1</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) < NML = < PQR</span><span> = </span><span>70</span><sup>0</sup><span>-------- equation 2</span></span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="font-family: arial;">(MN)/(QP) = 2.5/5</span><span style="font-family: arial;"> = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) So from equations 1, 2, 3, </span><span>(LM)/(RQ)</span><span> </span><span face="Arial, sans-serif" style="line-height: 25.68px;"><span>=</span> </span><span>(MN)/(QP)</span><span> = 1/2</span><span>, so by SAS property,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; text-indent: -47.2667px;">LMN</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333; font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; text-indent: -47.2667px;">RQP</span><span style="font-family: arial; text-indent: -47.2667px;">.</span></span></blockquote><span style="font-family: arial; font-size: medium;">(v)<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOq_z6qt50OfF7kcw2vNRHAm36taFRZD8HV8BjR9x1KMBVzFY7Da50m4EFhY21ivK70QSmvm4Z2ds2ddH3VAA4Zo6GkcAyCxQ3RPPsu6hQqFIpb5wzsHorHDx1GAjQEipjvILg20lNkEHWlzUKmCvjkfNn5ir33hsP2i-p7m6Y4nspWioau1UzznHb/s531/19.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="287" data-original-width="531" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOq_z6qt50OfF7kcw2vNRHAm36taFRZD8HV8BjR9x1KMBVzFY7Da50m4EFhY21ivK70QSmvm4Z2ds2ddH3VAA4Zo6GkcAyCxQ3RPPsu6hQqFIpb5wzsHorHDx1GAjQEipjvILg20lNkEHWlzUKmCvjkfNn5ir33hsP2i-p7m6Y4nspWioau1UzznHb/s320/19.png" width="320" /></a></div></span><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ DFE,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ABC and </span><span style="text-indent: -47.2667px;">∆ DFE, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ D, </span></span><span style="text-indent: -47.2667px;">B</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ F, and </span></span><span style="text-indent: -47.2667px;">C</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ E. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) point A is associated with poin D, </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point B is associated with poin F,<br />iii) point C is associated with poin E,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AB)/(DF) = 2.5/5 = 1/2 -------- equation 1</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) < ABC </span><span face="Arial, sans-serif">≠</span><span> < DFE</span><span> </span><span>-------- equation 2</span></span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="font-family: arial;">(BC)/(FE) = 3/6</span><span style="font-family: arial;"> = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) So from equations 1, 2, 3, </span><span>(AB)/(DF)</span><span> </span><span face="Arial, sans-serif" style="line-height: 25.68px;"><span>=</span> </span><span>(BC)/(FE)</span><span> = 1/2</span><span>, but </span><span>< ABC </span><span face="Arial, sans-serif">≠</span><span> < DFE so,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; text-indent: -47.2667px;">ABC</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; white-space: pre-wrap;">≁</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial;">DFE</span><span style="font-family: arial; text-indent: -47.2667px;">.</span></span></div></div></blockquote><span style="font-family: arial; font-size: medium;">(vi)<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSQ3wrs6QF9ch3n_L0JECUhB12dlV9W4tcpFrtsk0qgKcMAlufuPjqbGUJFIwgyC4vTrod85ZlOm8oY8bhnPU1j_Da2ferI6HR0c52QWWJ3LKUgbC0Jqk3XAoH6NIMansV2-WFDbxTes2ReWHoJkZR5sLn2beFX2GRDwkeZB6SLvlv0Zu11ca5IiST/s450/20.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="173" data-original-width="450" height="123" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSQ3wrs6QF9ch3n_L0JECUhB12dlV9W4tcpFrtsk0qgKcMAlufuPjqbGUJFIwgyC4vTrod85ZlOm8oY8bhnPU1j_Da2ferI6HR0c52QWWJ3LKUgbC0Jqk3XAoH6NIMansV2-WFDbxTes2ReWHoJkZR5sLn2beFX2GRDwkeZB6SLvlv0Zu11ca5IiST/s320/20.png" width="320" /></a></div></span><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ DEF and </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQR,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ DEF and </span><span style="text-indent: -47.2667px;">∆ PQR, points D</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P, </span></span><span style="text-indent: -47.2667px;">E</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ Q, and </span></span><span style="text-indent: -47.2667px;">F</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ R. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) point D is associated with poin P, </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point E is associated with poin Q,<br />iii) point F is associated with poin R,</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">in </span><span style="font-family: arial; text-indent: -47.2667px;">∆ DEF,</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">< D + < E + < F = 180</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">70 + 80 + < F = 180</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">150 + < F = 180</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">< F = 180 - 150</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">< F = 30</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-indent: -47.2667px;">< DFE = 30</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">< DFE = < PRQ = 30</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 1</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">in </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQR,</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-indent: -47.2667px;">< P + < Q + < R = 180<br /></span><span style="text-indent: -47.2667px;">< P + 80 + 30 = 180<br /></span><span style="text-indent: -47.2667px;">< P + 110 = 180<br /></span><span style="text-indent: -47.2667px;">< P = 180 - 110<br /></span><span style="text-indent: -47.2667px;">< P = 70<br /></span><span style="text-indent: -47.2667px;">< QPR = 70<br /></span><span style="text-indent: -47.2667px;">< QPR = < EDF = 70</span><span> </span><span>-------- equation 2</span> </span></blockquote></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) < DEF </span><span style="font-family: arial;">=</span><span style="font-family: arial;"> < PQR</span><span style="font-family: arial;"> = 80 </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) From equations 1, 2, and 3, </span><span style="font-family: arial;">by AAA similarity rule, </span><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial;">DEF</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="background-color: white; color: #333333; font-family: arial;">~</span><span style="font-family: arial; text-indent: -47.2667px;"> </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQR.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. In the following Fig., </b></span><span style="text-indent: -47.2667px;"><b>∆</b></span><span><b> ODC ~ </b></span><span style="text-indent: -47.2667px;">∆</span><span><b> OBA, < BOC = 125° </b></span><b>and < CDO = 70°. </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Find < DOC, </b><b>< DCO and </b><b>< OAB.</b></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYfcU3zBum1Sn5m2uffzIg2odQ_GjgVzgYOknBhnb0Pyr5YOefyrEn25K6ye47vY4H3ptWfIgM8LsBlIhoQsFK7oOalluThOGFbWFJxlZ43jBWi80Ses4NLGtYL5WhUxbAWMdAhgjvh-0KJIa_YsJex1Nfvb7doW9yc97pAUHvd-GbsGDwkMCSP9xu/s440/22.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="211" data-original-width="440" height="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYfcU3zBum1Sn5m2uffzIg2odQ_GjgVzgYOknBhnb0Pyr5YOefyrEn25K6ye47vY4H3ptWfIgM8LsBlIhoQsFK7oOalluThOGFbWFJxlZ43jBWi80Ses4NLGtYL5WhUxbAWMdAhgjvh-0KJIa_YsJex1Nfvb7doW9yc97pAUHvd-GbsGDwkMCSP9xu/s320/22.png" width="320" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In the above figure</span><span style="font-family: arial; text-indent: -47.2667px;">,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) < DOC and < COB are angles in linear pairs, so</blockquote></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">< DOC + < COB = 180</blockquote></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="text-indent: -47.2667px;">< </span>DOC<span style="text-indent: -47.2667px;"> + 125</span><span style="text-indent: -47.2667px;"> = 180</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="text-indent: -47.2667px;"><span style="font-weight: normal; white-space: pre-wrap;">< DOC = </span><span style="font-weight: 400;">180 - 125</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="font-weight: 400; text-indent: -47.2667px;">< DOC = 55</span><span style="font-family: arial; font-weight: 400;"> </span><span style="font-family: arial; font-weight: 400;">-------- equation 1</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">b) So < DOC = 55</span></blockquote><div style="text-align: left;"><div style="font-weight: 400;"><div><span style="font-family: arial; font-size: medium;">2) In </span><span style="font-family: arial; text-indent: -47.2667px;"><span style="text-indent: -47.2667px;">∆ PQR,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">< DOC + < DCO + < CDO = 180</blockquote></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="text-indent: -47.2667px;">< </span>55<span style="text-indent: -47.2667px;"> + </span>< DCO + 70<span style="text-indent: -47.2667px;"> = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="text-indent: -47.2667px;"><span style="font-weight: normal; white-space: pre-wrap;">< DCO + 125 = </span><span style="font-weight: 400;">180</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px;">< DCO = 180 - 125</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px;">< DCO = 55</span><span style="font-family: arial; font-weight: 400;"> </span><span style="font-family: arial; font-weight: 400;">-------- equation 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">So < DCO = 55</span></blockquote><div><div><span style="font-family: arial; font-size: medium; font-weight: 400;">3) It's given that, </span><span style="font-weight: normal;"><span style="text-indent: -47.2667px;">∆</span><span> ODC ~ </span><span style="text-indent: -47.2667px;">∆</span><span> OBA,</span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">< OAB = < OCD<span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">4) From equations 2 and 3, we have,</span> </div></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="text-indent: -47.2667px;">< OAB</span><span style="text-indent: -47.2667px;"> = 55</span></blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;">Q3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at point O. Using a similarity criterion for two triangles, show that </div><div style="text-align: left;">(OA)/(OB) = (OC)/(OD).</div><div style="text-align: left;"></div></div></span></div></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb0kj8C5p7WLwqY-fC1ri7gUOK1Zu3kNY5Nq-SfTFt1QK-XnPus9nX40t-FZ_IhRn0AWJApUqfx1h1wlmMT416qsMp-x2v3c2KOcM50YisPMzgPIslhASEx3edon3bHCtR9CmwKQO7RqwtmnvVJHpZsI5UV9i1I3jJHkjPXfwVphKgJQJScpbw7i77/s517/24.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="286" data-original-width="517" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb0kj8C5p7WLwqY-fC1ri7gUOK1Zu3kNY5Nq-SfTFt1QK-XnPus9nX40t-FZ_IhRn0AWJApUqfx1h1wlmMT416qsMp-x2v3c2KOcM50YisPMzgPIslhASEx3edon3bHCtR9CmwKQO7RqwtmnvVJHpZsI5UV9i1I3jJHkjPXfwVphKgJQJScpbw7i77/s320/24.png" width="320" /></a></div><div style="white-space: normal;"><div style="font-weight: 400;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">trapezium ABCD, </span><span style="font-family: arial; text-indent: -47.2667px;">AB ‖ CD, and diagonals AC and BD intersect at O.</span></div><div><span style="font-family: arial; text-indent: -47.2667px;">2) </span><span style="font-family: arial; font-size: medium;">In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ AOB and </span><span style="text-indent: -47.2667px;">∆ COD</span><span style="font-family: arial; text-indent: -47.2667px;">,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a) < AOB = < COD (vertically opposite angles are equal)</blockquote></div><div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: 400;">b) < ABO = < CDO (alternate interior angles are equal)</span></div><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-weight: 400;">c) < BAO = < DCO (</span><span style="font-weight: 400;">alternate </span><span style="font-weight: 400;">interior</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">angles are equal</span><span style="font-weight: 400;">)</span></div><div style="font-weight: 400;">3) So, by AAA similarity test, <span style="font-family: arial; text-indent: -47.2667px;">∆ AOB <span style="background-color: white; color: #333333; text-indent: 0px;">~</span> </span><span style="text-indent: -47.2667px;">∆ COD</span></div><div style="font-weight: 700;"><div style="font-weight: 400;">3) So, as corresponding sides are proportional, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">(OA)/(OB) = (OC)/(OD)<span style="text-indent: -47.2667px;">. </span>Hence proved.</blockquote></div><div style="font-weight: 400;"><br /></div><div>Q4. In the following Fig., (QR)/(QS) = (QT)/(PR) and < 1 = < 2. Show that</div><div><span style="text-indent: -47.2667px;">∆</span> PQS ~ <span style="text-indent: -47.2667px;">∆</span> TQR.</div></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7pKlpX_-LXiieJxzS3aeYZqK9IYse3BnFLxE1CML_rHCCOyZYA7kaSohmOeYtPPNlXVnLH8fxuv0EQMUFGcHJiWR_qJ2s9WrxbXJfqGralW3BAHHrxRTu7FIhWAG-I8aaeY3wJEB3JbLnGso_5HWvQLaSl2Ckbcv2uvZoSYmy5C0phE90KlsXLNPt/s463/23.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="244" data-original-width="463" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7pKlpX_-LXiieJxzS3aeYZqK9IYse3BnFLxE1CML_rHCCOyZYA7kaSohmOeYtPPNlXVnLH8fxuv0EQMUFGcHJiWR_qJ2s9WrxbXJfqGralW3BAHHrxRTu7FIhWAG-I8aaeY3wJEB3JbLnGso_5HWvQLaSl2Ckbcv2uvZoSYmy5C0phE90KlsXLNPt/s320/23.png" width="320" /></a></div><div><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; text-indent: -47.2667px;">PQR,</span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">< 1 = < 2, so</blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(QP) = (PR)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 1</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) It is given that,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(QR)/(QS) = (QT)/(PR)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 2</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">3) From equations 1 and 2, we have,</span></div></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(QR)/(QS) = (QT)/(QP)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">4)</span> <span style="font-family: arial; font-weight: 400;">In </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">PQS and</span><span style="font-family: arial; font-weight: 400;"> </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">TQR,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;">< PQS = < TQR <span style="text-indent: -47.2667px;">(same angle)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 4</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal; text-indent: -47.2667px;">5) From equations 3 and 4 and by SAS property of similarity we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">PQS </span><span style="font-family: arial; font-weight: 400;">~ </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">∆ </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">TQR. Hence proved.</span></blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;">Q5. S and T are points on sides PR and QR of <span style="text-indent: -47.2667px;">∆</span> PQR such that < P = < RTS. Show that <span style="text-indent: -47.2667px;">∆</span> RPQ ~ <span style="text-indent: -47.2667px;">∆</span> RTS.</div></div></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYXDpITYH0F8aqqpg09YAo4MFzZQ8altESKKsDimO_gpzTnmpKddnXJnPYrANUTxGTz51UinwMChM638iurI3aEUbkD5x5Rv2AWfUeOgfJ8GdcSPkSbHctTgOyfgALESyReQVNSRFmhMpVzC3EWK7Yu-cwY_savD6a2RxfE5hjndSe_9u6gdmLT57e/s370/25.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="353" data-original-width="370" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYXDpITYH0F8aqqpg09YAo4MFzZQ8altESKKsDimO_gpzTnmpKddnXJnPYrANUTxGTz51UinwMChM638iurI3aEUbkD5x5Rv2AWfUeOgfJ8GdcSPkSbHctTgOyfgALESyReQVNSRFmhMpVzC3EWK7Yu-cwY_savD6a2RxfE5hjndSe_9u6gdmLT57e/w234-h223/25.png" width="234" /></a></div><div style="white-space: normal;"><div><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><div><div style="font-weight: 400;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ PQR,
and ∆ TSR,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ PQR and </span><span style="text-indent: -47.2667px;">∆ TSR, points P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ T, </span></span><span style="text-indent: -47.2667px;">Q</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ S, and </span></span><span style="text-indent: -47.2667px;">R</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ R. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) point P is associated with poin T,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) point Q is associated with poin S,<br /></span></blockquote><span style="font-family: arial; font-weight: 400;"><span> </span><span> </span>iii) point R is associated with point R,</span><br /><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;">a) < QPR = < STR
(given)<br /><span face="Arial, sans-serif" style="line-height: 107%;">b) < PRQ = <
TRS (same angles)<br /></span></div></blockquote><div style="font-weight: 400; text-align: left;">2) By AA similarity test, <span style="text-indent: -47.2667px;">∆ PQR </span>~<span style="text-indent: -47.2667px;"> ∆ TSR. Hence proved.</span></div><div style="font-weight: 400; text-align: left;"><span style="text-indent: -47.2667px;"><br /></span></div><div style="text-align: left;"><span style="text-indent: -47.2667px;">Q</span>6. In the following fig., if <span style="text-indent: -47.2667px;">∆</span> ABE <span style="background-color: white;">≅</span> <span style="text-indent: -47.2667px;">∆</span> ACD, show that <span style="text-indent: -47.2667px;">∆</span> ADE ~ <span style="text-indent: -47.2667px;">∆</span> ABC.</div></div><div style="text-align: left;"></div></div></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMkyT7JRzhPoirVkLJJbvFJajC_i53pSMFOnYDxM6RcApd8GbBVxDKVYaKZQyeB8xZMRRbgN_VOGOA9i59IrDunK7DP2Be3OrtGSrUHfYJqsB0Rnt6JiuP9jwxFvgzBD2hyxCwoijoS5FUtPZUmz09htmeta3v_MdwnLF5rLufrJ96lZsvWNTZM6yJ/s391/26.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="364" data-original-width="391" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMkyT7JRzhPoirVkLJJbvFJajC_i53pSMFOnYDxM6RcApd8GbBVxDKVYaKZQyeB8xZMRRbgN_VOGOA9i59IrDunK7DP2Be3OrtGSrUHfYJqsB0Rnt6JiuP9jwxFvgzBD2hyxCwoijoS5FUtPZUmz09htmeta3v_MdwnLF5rLufrJ96lZsvWNTZM6yJ/w223-h208/26.png" width="223" /></a></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><div style="line-height: normal; margin-bottom: 0cm;"><div style="font-weight: 400; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">It is given that </span></span></span><span style="text-indent: -47.2667px;">∆</span> ABE <span style="background-color: white;">≅</span> <span style="text-indent: -47.2667px;">∆</span> ACD, corresponding sides of congruent triangles are</div><div style="font-weight: 400; text-align: left;"><span> </span>congruent, so,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) AD = AE ------- equation 1</span></span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) AB = AC ------- equation 2</span></blockquote><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;">2) In </span></span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> DAE <span style="background-color: white;">and </span></span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> BAC, and f</span><span style="font-weight: normal;">rom equations 1 and 2, we have,</span></div><div style="line-height: normal; margin-bottom: 0cm;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AD)/(AB) = (AE)/(AC) </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) < DAE = <BAC (same angle)</span></blockquote><span style="font-family: arial; font-size: medium; font-weight: normal;">3) </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">DAE </span><span style="background-color: white; color: #333333; font-weight: 400;">~</span><span style="font-weight: 400;"><span style="background-color: white;"> </span></span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> BAC.</span><span style="font-weight: 400; text-indent: -47.2667px;"> Hence proved.</span></div><div style="line-height: normal; margin-bottom: 0cm;"><span style="font-weight: 400; text-indent: -47.2667px;"><br /></span></div><div style="line-height: normal; margin-bottom: 0cm;"><span style="text-indent: -47.2667px;">Q</span>7. In the following fig., altitudes AD and CE of <span style="text-indent: -47.2667px;">∆</span> ABC intersect each other at point P. Show that:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRBCoGthwZnhuANwZ-Jv2CERABDc76aDxTQmCneorwslpy4m82OpEWbLnwkOzrauU_6L0QgfgLqLeuWwuDe500zDvI15CNudBHTm4F2tVN6_3NXo0iY3j23Ucdalvkh1Ao71dLCww89aA1kdc3s0Ac4nNePkGjJeESLj8R-Za-4cU0cpBr9xGl3lZ6/s798/28.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="171" data-original-width="798" height="116" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRBCoGthwZnhuANwZ-Jv2CERABDc76aDxTQmCneorwslpy4m82OpEWbLnwkOzrauU_6L0QgfgLqLeuWwuDe500zDvI15CNudBHTm4F2tVN6_3NXo0iY3j23Ucdalvkh1Ao71dLCww89aA1kdc3s0Ac4nNePkGjJeESLj8R-Za-4cU0cpBr9xGl3lZ6/w542-h116/28.png" width="542" /></a></div></div></div></span></h3><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:</span></h3><div><b><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">(i) </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP </span><span style="background-color: white; color: #333333;">~</span><span><span style="background-color: white;"> </span></span><span style="text-indent: -47.2667px;">∆</span><span> CDP</span></span></b></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><span style="font-family: arial; font-size: medium;"><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP and </span><span style="text-indent: -47.2667px;">∆</span><span> CDP</span></span></div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ AEP and </span><span style="text-indent: -47.2667px;">∆ CDP, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ C, </span></span><span style="text-indent: -47.2667px;">E</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ D, and </span></span><span style="text-indent: -47.2667px;">P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>i) point A is associated with poin C,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>ii) point E is associated with poin D,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>iii) point P is associated with poin P, </span></blockquote><div style="text-align: left;">2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP and </span><span style="text-indent: -47.2667px;">∆</span><span> CDP</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < AEP = < CDP = 90 (given) -------- equation 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) < APE = < CPD (Vertically oposite angles) -------- equation 2</span></blockquote><div style="text-align: left;">3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> CDP</span>, hence proved.</span></blockquote><div style="text-align: left;"><br /></div><div><b><span style="white-space: pre-wrap;">(ii) </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABD </span><span style="background-color: white; color: #333333;">~</span><span><span style="background-color: white;"> </span></span><span style="text-indent: -47.2667px;">∆</span><span> CBE</span></b></div><div><b><span><br /></span></b></div><div><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABD and </span><span style="text-indent: -47.2667px;">∆</span><span> CBE</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ABD and </span><span style="text-indent: -47.2667px;">∆ CBE, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ C, </span></span><span style="text-indent: -47.2667px;">B</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ B, and </span></span><span style="text-indent: -47.2667px;">D</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ E. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>i) point A is associated with poin C,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>ii) point B is associated with poin B,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>iii) point D is associated with poin E, </span></blockquote><div>2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABD and </span><span style="text-indent: -47.2667px;">∆</span><span> CBE</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < ADB = < CEB = 90 (given) -------- equation 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) < ABD = < CBE (common angles) -------- equation 2</span></blockquote><div>3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABD <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> CBE</span>, hence proved.</span></blockquote><div style="text-align: left;"><br /></div><div><b><span style="white-space: pre-wrap;">(iii) </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP </span><span style="background-color: white; color: #333333;">~</span><span><span style="background-color: white;"> </span></span><span style="text-indent: -47.2667px;">∆</span><span> ADB</span></b></div><div><b><span><br /></span></b></div><div><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP and </span><span style="text-indent: -47.2667px;">∆</span><span> ADB</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ AEP and </span><span style="text-indent: -47.2667px;">∆ ADB, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ A, </span></span><span style="text-indent: -47.2667px;">E</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ D, and </span></span><span style="text-indent: -47.2667px;">P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ B. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>i) point A is associated with poin A,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>ii) point E is associated with poin D,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>iii) point P is associated with poin B, </span></blockquote><div>2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP and </span><span style="text-indent: -47.2667px;">∆</span><span> ADB</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < AEP = < ADB = 90 (given) -------- equation 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) < PAE = < BAD (common angles) -------- equation 2</span></blockquote><div>3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>AEP <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> </span>ADB, hence proved.</span></blockquote></span></div></span></div><div style="text-align: left;"><br /></div><div><b><span style="white-space: pre-wrap;">(iv) </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>PDC </span><span style="background-color: white; color: #333333;">~</span><span><span style="background-color: white;"> </span></span><span style="text-indent: -47.2667px;">∆</span><span> BEC</span></b></div><div><b><span><br /></span></b></div><div><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>PDC and </span><span style="text-indent: -47.2667px;">∆</span><span> BEC</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ PDC and </span><span style="text-indent: -47.2667px;">∆ BEC, points P</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ B, </span></span><span style="text-indent: -47.2667px;">D</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ E, and </span></span><span style="text-indent: -47.2667px;">C</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ C. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>i) point P is associated with poin B,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>ii) point D is associated with poin E,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>iii) point C is associated with poin C, </span></blockquote><div>2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>PDC and </span><span style="text-indent: -47.2667px;">∆</span><span> BEC</span></div><div><span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < PDC = < BEC = 90 (given) -------- equation 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) < PCD = < BCE (common angles) -------- equation 2</span></blockquote><div>3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>PDC <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> </span>BCE, hence proved.</span></blockquote></span></div></span></div></span></div></span></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Q8. E is a point on the side AD produced of a parallelogram ABCD and BE</b></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><b>intersect </b><b>CD at F. Show that </b><span style="text-indent: -47.2667px;">∆</span><b> ABE ~ </b><span style="text-indent: -47.2667px;">∆</span><b> CFB.</b></div></span></div></span></div></blockquote><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQKD0UFPfaLAzcSt6sTyoPnzc2eBYuRryedN2LyqYlRxSpzs1DvPiX4pGBFow1hH9uFGoYIhgF6TDD4FXB4SJBgX3y9R62QmXANk9wuqJJ0KLGg9p6z0LAjOTE9FqB6-Q23QijumblgU4MP1_7b87Y8DJfDJnb659q4kzFOxOu06rp7skuNvAtSyE7/s531/29.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="238" data-original-width="531" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQKD0UFPfaLAzcSt6sTyoPnzc2eBYuRryedN2LyqYlRxSpzs1DvPiX4pGBFow1hH9uFGoYIhgF6TDD4FXB4SJBgX3y9R62QmXANk9wuqJJ0KLGg9p6z0LAjOTE9FqB6-Q23QijumblgU4MP1_7b87Y8DJfDJnb659q4kzFOxOu06rp7skuNvAtSyE7/s320/29.png" width="320" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="font-weight: 400; white-space: normal;"><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABE and </span><span style="text-indent: -47.2667px;">∆</span><span> CFB</span></div><div style="white-space: normal;"><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ABE and </span><span style="text-indent: -47.2667px;">∆ CFB, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ C, </span></span><span style="text-indent: -47.2667px;">B</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ F, and </span></span><span style="text-indent: -47.2667px;">E</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ B. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>i) point A is associated with poin C,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>ii) point B is associated with poin F,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>iii) point E is associated with poin B, </span></blockquote><div style="font-weight: 400;">2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABE and </span><span style="text-indent: -47.2667px;">∆</span><span> </span>CFB</div><div><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>a) < BAE = < FCB (<span style="background-color: white; color: #333333;">Opposite angles of a parallelogram</span>) -------- equation 1</span></span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span><span>b) < AEB = < CBF (</span><span style="background-color: white; color: #333333;"><span>Alternate interior angles as AE || BC</span></span><span>) -------- equation 2</span></span></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABE <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> </span>CFB, hence proved.</span></blockquote><div style="font-weight: 400; text-align: left;"><br /></div><div style="text-align: left;"><span><span>Q9. In the following fig., ABC and AMP are two right </span><span>triangles, right angled at B and M </span></span><span><span>respectively. Prove that: </span></span></div><div style="text-align: left;"><span><span>(i) </span></span><span style="text-indent: -47.2667px;"><span>∆</span></span><span><span> ABC ~ </span></span><span style="text-indent: -47.2667px;">∆</span><span><span> AMP </span><span>(ii) (</span></span><span>CA)/(PA) = (BC)/(</span><span>MP) </span></div></span></div></span></div></span></div></h3><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:</span><div class="separator" style="clear: both; text-align: center; white-space: pre-wrap;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_dIqt9ywNNa2YRkuJMhXtoyN7UuTq80xfzqETmA9acTOQEWn8LYeywe6vaxgvvdEWMR-v3diC3j8jXEv6K73FpQOFDzwfgSWHIfzMoPrcknAiZQhmyeHxZaGWNxcuvh7nDfcIncw2s4NTbcnp7IIKxtPHOt9TDzAv2YhEfgi91ecpn_6bYXKQFvj5/s409/30.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="219" data-original-width="409" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_dIqt9ywNNa2YRkuJMhXtoyN7UuTq80xfzqETmA9acTOQEWn8LYeywe6vaxgvvdEWMR-v3diC3j8jXEv6K73FpQOFDzwfgSWHIfzMoPrcknAiZQhmyeHxZaGWNxcuvh7nDfcIncw2s4NTbcnp7IIKxtPHOt9TDzAv2YhEfgi91ecpn_6bYXKQFvj5/s320/30.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left; white-space: pre-wrap;"><span style="font-size: medium; white-space: normal;">(i) </span><span style="text-indent: -47.2667px; white-space: normal;"><span style="font-size: medium;">∆</span></span><span style="font-size: medium; white-space: normal;"> ABC ~ </span><span style="text-indent: -47.2667px; white-space: normal;">∆</span><span style="font-size: medium; white-space: normal;"> AMP</span></div><div class="separator" style="clear: both; text-align: left; white-space: pre-wrap;"><span style="font-size: medium; white-space: normal;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABC and </span><span style="text-indent: -47.2667px;">∆</span><span> AMP</span></div><div><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ABC and </span><span style="text-indent: -47.2667px;">∆ <span style="text-indent: 0px;">AMP</span>, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ A, </span></span><span style="text-indent: -47.2667px;">B</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ M, and </span></span><span style="text-indent: -47.2667px;">C</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ P. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span>i) point A is associated with poin A,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span>ii) point B is associated with poin M,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span>iii) point C is associated with poin P, </span></blockquote><div style="font-weight: 400; white-space: normal;">2) <span>In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABC and </span><span style="text-indent: -47.2667px;">∆</span><span> </span>AMP</div><div><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span><span style="font-size: medium;">a) < ABC = < AMP = 90 -------- equation 1</span></span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span><span style="font-size: medium;">b) < BAC = < MAP (</span><span style="background-color: white; color: #333333;"><span style="font-size: medium;">Common angle</span></span><span style="font-size: medium;">) -------- equation 2</span></span></blockquote><div style="font-weight: 400; white-space: normal;">3) From equations 1 and 2, by AA similarity test, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABC <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> </span>AMP, hence proved.</span></blockquote><p style="white-space: normal;"> <span style="font-size: medium;"><span>(ii) (</span></span><span>CA)/(PA) = (BC)/(</span><span>MP)</span></p><div style="text-align: left; white-space: normal;"><span style="font-weight: 400;">1) In </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">ABC and </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> AMP</span></div><div style="text-align: left;"><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span><span style="font-size: medium;">a) </span></span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABC <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>AMP (we proved this)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span><span style="font-size: medium;">b) </span></span><span>(CA)/(PA) = (BC)/(MP) (By basic propotinality theorem)</span></blockquote><div style="text-align: left;"><span style="font-weight: 400;"><br /></span></div><div style="text-align: left;">Q10. CD and GH are respectively the bisectors of < ACB and < EGF such that D and H lie on sides AB and FE of <span style="text-indent: -47.2667px;">∆</span> ABC and <span style="text-indent: -47.2667px;">∆</span> EFG respectively. </div><div style="text-align: left;">If <span style="text-indent: -47.2667px;">∆</span> ABC ~ <span style="text-indent: -47.2667px;">∆</span> FEG, show that: </div><div style="text-align: left;">(i) (CD)/(GH) = (AC)/(FG) </div><div style="text-align: left;">(ii) <span style="text-indent: -47.2667px;">∆</span> DCB ~ <span style="text-indent: -47.2667px;">∆</span> HGE </div><div style="text-align: left;">(iii) <span style="text-indent: -47.2667px;">∆</span> DCA ~ <span style="text-indent: -47.2667px;">∆</span> HGF</div></span></div></span></div></span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:</span></span></h3><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: left;"><div><span><div><span><div style="text-align: left;"><span><div style="text-align: left;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMOXPTmbn4vQ6CjE0AQPnl23MqxZEwAZMdXkidngQBHsnHbU9j7Rx3MFPxWh7CZPxrZyzKLYR7jlHt4rGmQLsu2LxOiGN2Anvf_KXixBK9Xi6dzFFWPlBcFp6HH9s1m8eKytt4cSy0CykD6Xwbt4PF_IYUn1a9j6O4LEpDd2rtJZJ4uG5f1OQ1_1L_/s871/31.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="370" data-original-width="871" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMOXPTmbn4vQ6CjE0AQPnl23MqxZEwAZMdXkidngQBHsnHbU9j7Rx3MFPxWh7CZPxrZyzKLYR7jlHt4rGmQLsu2LxOiGN2Anvf_KXixBK9Xi6dzFFWPlBcFp6HH9s1m8eKytt4cSy0CykD6Xwbt4PF_IYUn1a9j6O4LEpDd2rtJZJ4uG5f1OQ1_1L_/w353-h150/31.png" width="353" /></a></div></span></div></span></div></span></div></div></span></h3><h3><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; white-space: pre-wrap;"><span style="font-size: medium; white-space: normal;">(i) </span>(CD)/(GH) = (AC)/(FG)</div><div class="separator" style="clear: both; white-space: pre-wrap;"><span style="font-size: medium; white-space: normal;"><br /></span></div><div class="separator" style="clear: both;"><div style="font-weight: 400;"><span>1) </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABC <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> FEG (given), so we have,</span></div><div><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < ACB = < FGE --------- equation 1</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) CD is bisector of < ACB and GH is the bisector of < FGE</span> </blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>c) (< ACB)/2 = (< FGE)/2 </span>--------- equation 2</blockquote><div style="text-align: left;"><span style="font-weight: normal;">2)</span> <span style="font-weight: normal;">From equations 1 and 2, and in </span><span style="font-weight: normal; text-indent: -47.2667px;">∆</span><span style="font-weight: normal;"> </span><span style="font-weight: normal;">DCA and </span><span style="font-weight: normal; text-indent: -47.2667px;">∆</span><span style="font-weight: normal;"> HGF</span><span style="font-weight: normal;"> we have</span></div></span></div></div></span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< DCA = < HGF </span><span style="font-family: arial;">--------- equation 3</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< CAD = < GFH</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">From equations 3 and 4, and the AA similarity test, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial;">DCA <span style="background-color: white; color: #333333;">~</span> </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> HGF</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 5</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) So, (CD)/(GH) = (AC)/(FG) hence proved.</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700;">(ii) </span><span style="font-family: arial; font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-weight: 700;"> DCB ~ </span><span style="font-family: arial; font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-family: arial; font-weight: 700;"> HGE</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) In </span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>DCB <span style="background-color: white; color: #333333;">and</span> </span><span style="text-indent: -47.2667px;">∆</span><span> HGE (</span>CD is bisector of < ACB and GH is the bisector of < FGE)</span></div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < DCB = < HGE --------- equation 1</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>b) < DBC = < HEG (Common angle)</span> --------- equation 2</blockquote><div style="text-align: left;">2) From equations 1 and 2, we have,</div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="text-indent: -47.2667px;">∆</span><span> </span><span>DCB <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> HGE,</span> </span>hence proved.</blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;"><span style="font-weight: 700;">(iii) </span><span style="font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-weight: 700;"> DCA ~ </span><span style="font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-weight: 700;"> HGF</span></div><div style="text-align: left;"><br /></div><div style="text-align: left;">It is already proved in example: (i) above. (see equation 5 of example (i)).</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><span><b>Q11. In the following fig., E is a point on side CB </b></span><b>produced of an isosceles triangle ABC </b><b>with AB = AC. If AD </b><span id="docs-internal-guid-5a38655d-7fff-58dc-d833-b403dc6b4b2a"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>⟂</span></span></span><b> BC and EF </b><span style="white-space: pre-wrap;">⟂</span><b> AC, </b></div><div style="text-align: left;"><b>prove that </b><span style="font-weight: 700; text-indent: -47.2667px;">∆</span><b> ABD ~ </b><span style="font-weight: 700; text-indent: -47.2667px;">∆</span><b> ECF.</b></div></span></div></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZNdUEowZqzZEf-EdAdMveCrgiA8ki12YMxNL0nP6SU8GnkG-QshJBGJ_6F_ftkKb8p5jxqc0nm7wGGaD8G9GsbHFlpNfC6UvHr9a0cKMLcMpRq8KyxD3-eikaCjwL-l37VRpQrfU2A4lmRFcrbM-hNuyGi6VRGTqOVAxPprOV22H3d9WiOK-pAmbh/s350/32.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="301" data-original-width="350" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZNdUEowZqzZEf-EdAdMveCrgiA8ki12YMxNL0nP6SU8GnkG-QshJBGJ_6F_ftkKb8p5jxqc0nm7wGGaD8G9GsbHFlpNfC6UvHr9a0cKMLcMpRq8KyxD3-eikaCjwL-l37VRpQrfU2A4lmRFcrbM-hNuyGi6VRGTqOVAxPprOV22H3d9WiOK-pAmbh/w259-h223/32.png" width="259" /></a></div></span></span><div style="text-align: left;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><span style="font-weight: 400;">1) </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">ABC <span style="color: #333333;"><span style="background-color: white;">is an </span></span></span><span style="font-weight: normal;">isosceles triangle with AB = AC, so we have,</span></div><div style="white-space: normal;"><span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) < ABC = < ACB --------- equation 1</span></blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) In </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">ABD and </span><span style="font-weight: 400; text-indent: -47.2667px;">∆</span><span style="font-weight: 400;"> </span><span style="font-weight: 400;">ECF and from equation 1</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) </span>< ABD = < ECF (same angle) --------- equation 2 </blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>c) </span>< ADB = < EFC (each angle is 90<sup>0</sup>) --------- equation 3</blockquote><div><span style="font-weight: normal;">3)</span> <span style="font-weight: normal;">From equations 2 and 3, and the </span><span style="font-weight: 400;">AA similarity test, we have,</span></div></span><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span>a) <span style="text-indent: -47.2667px;">∆</span><span> </span><span>ABD <span style="background-color: white; color: #333333;">~</span> </span><span style="text-indent: -47.2667px;">∆</span><span> E</span>CF, hence proved.</span></blockquote><div style="text-align: left;"><br /></div><div style="text-align: left;">Q12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of <span style="text-indent: -47.2667px;">∆</span> PQR (see the following fig.). Show that <span style="text-indent: -47.2667px;">∆</span> ABC ~ <span style="text-indent: -47.2667px;">∆</span> PQR.</div></div></span></span></div></h3><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinBNegOsL_YD-b31KQQezLWDC6Tqtwosuj6V9nXcf-JzB4XzwQi4yEzooyCqlNjPbGRDKgRjR-2QCHgr3mb5Ey1lcioXahD7GslXvcofwcaQ9JxXrME2mTCQzMuPLNVQNksp0ozCEgQYwhLpiWG_4Ia4TsSsovAG3cx6UBV0p8iLeHYy9kIw4c_qK2/s658/33.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="264" data-original-width="658" height="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinBNegOsL_YD-b31KQQezLWDC6Tqtwosuj6V9nXcf-JzB4XzwQi4yEzooyCqlNjPbGRDKgRjR-2QCHgr3mb5Ey1lcioXahD7GslXvcofwcaQ9JxXrME2mTCQzMuPLNVQNksp0ozCEgQYwhLpiWG_4Ia4TsSsovAG3cx6UBV0p8iLeHYy9kIw4c_qK2/s320/33.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-weight: normal;">1) It is given that:</span></div></span></span></h3></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><div class="separator" style="clear: both; font-weight: 400; text-align: left;">a) (AB)/(PQ) = (BC)/(QR) = (AD)/(PM) ---------- equation 1.</div></span></span></h3></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial; white-space: pre-wrap;">(AB)/(PQ) = [(BC)/2]/[(QR)/2] = (AD)/(PM)</span></span></div><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="font-family: arial; white-space: pre-wrap;">(AB)/(PQ) = (BD)/(QM) = (AD)/(PM)</span><span style="font-family: arial; white-space: pre-wrap;"> ---------- equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) From equation 2 and the basic proportionality theorem, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial;">ABD <span style="background-color: white; color: #333333;">~</span> </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> PQM</span><span style="font-family: arial; white-space: pre-wrap;"> ---------- equation 3.</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial;">ABC <span style="background-color: white; color: #333333;">and</span> </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> PQR,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial; white-space: pre-wrap;">(AB)/(PQ) = (BC)/(QR) ---------- equation 4.</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) As </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial;">ABD <span style="background-color: white; color: #333333;">~</span> </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> PQM, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial; white-space: pre-wrap;">< ABC = < PQR ---------- equation 5.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) From equations 4 and 5 and by SAS test, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> </span><span style="font-family: arial;">ABC <span style="background-color: white; color: #333333;">~</span> </span><span style="font-family: arial; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"> PQR, hence proved.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q13. D is a point on the side BC of a triangle </b></span><b>ABC such that < ADC = < BAC. Show </b><b>that CA</b><sup><b><span>2</span></b></sup><b> = CB.CD.</b></span></div><span style="font-family: arial; font-size: medium;"></span><h3><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBVganmODjl7WjxZ3TAmdmGu_RDJpZ2lDcTEg7MyFd5IbtmDRn-nN7ejPLgByDk_AQh9mbl8WbQfgePVqF7STRfFe0NHT47LEBcGKZK_Q_9vxUsVNp3UKOZCprLzNpF8kWHm3XizocQqlUcZcHr_TXE_ouGv7mzA6zyMHDISGCTBe4eGBac1dp2P_A/s398/34.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="208" data-original-width="398" height="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBVganmODjl7WjxZ3TAmdmGu_RDJpZ2lDcTEg7MyFd5IbtmDRn-nN7ejPLgByDk_AQh9mbl8WbQfgePVqF7STRfFe0NHT47LEBcGKZK_Q_9vxUsVNp3UKOZCprLzNpF8kWHm3XizocQqlUcZcHr_TXE_ouGv7mzA6zyMHDISGCTBe4eGBac1dp2P_A/s320/34.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">In ∆ ADC and ∆ BAC,</span><br /></span></span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;"><span>Note: For </span><span style="text-indent: -47.2667px;">∆ ADC and </span><span style="text-indent: -47.2667px;">∆ BAC, points A</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ B, </span></span><span style="text-indent: -47.2667px;">D</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ A, and </span></span><span style="text-indent: -47.2667px;">C</span><span><span style="text-indent: -47.2667px;"> </span><span style="text-align: center;">↔ C. </span></span><span>That is</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">i) point A is associated with poin B,</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">ii) point D is associated with poin A,<br /></span></blockquote><span style="font-family: arial; font-weight: 400; white-space: normal;"><span> </span><span> </span>iii) point C is associated with point C,</span><br style="white-space: normal;" /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-weight: 400;">a) </span><span style="font-weight: normal;">< ADC = < BAC</span><span style="font-weight: 400;"> (given)</span><br /><span face="Arial, sans-serif" style="font-weight: 400; line-height: 19.26px;">b) < ACD = < BCA (common angles)<br /></span></div></blockquote><div style="font-weight: 400; white-space: normal;">2) By AA similarity test, <span style="text-indent: -47.2667px;">∆ ADC </span>~<span style="text-indent: -47.2667px;"> ∆ BAC.</span></div><div style="font-weight: 400; white-space: normal;"><span style="text-indent: -47.2667px;">3) By the basic proportionality theorem, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">(CA)/(CB) = (CD)/(CA)</span> </span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="white-space: pre-wrap;">(CA) x (CA) = (CD) x (CB)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">(CA)</span><sup><span style="font-family: arial; font-size: medium;">2</span></sup><span> = (CB) x (CD), hence proved.</span></blockquote></div></span></span></h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q14. Sides AB and AC and median AD of a </b></span><b>triangle ABC are respectively</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>proportional to sides PQ and PR and </b></span><b>median PM of another triangle PQR.</b></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b>Show that </b></span><span style="font-family: arial; font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"><b> ABC ~ </b></span><span style="font-family: arial; font-weight: 700; text-indent: -47.2667px;">∆</span><span style="font-family: arial;"><b> PQR.</b></span></span></div><div style="text-align: left;"><span style="font-size: medium;"><h3><span style="font-size: medium;"><span style="white-space: pre-wrap;"><span style="font-family: arial;">Solution:</span><div class="separator" style="clear: both; font-family: "Times New Roman"; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi09fsqKprmKw1vf1rCs-e0W6mWp6WmUAMp6fVqFSWqMzMxwkEDu6v-Eh0j8MX3JcZH4er_cpeQqhu5s70zUFAjzN2otoum8pdjZz3lxK3n8lhtN0dTH-Yca0LCiaBIJ4_xTYVOoBUNfo_DMzzcmq8EF17YSFENOWdrKnicmBw286OyhlY6_fEBN7mX/s801/35.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="453" data-original-width="801" height="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi09fsqKprmKw1vf1rCs-e0W6mWp6WmUAMp6fVqFSWqMzMxwkEDu6v-Eh0j8MX3JcZH4er_cpeQqhu5s70zUFAjzN2otoum8pdjZz3lxK3n8lhtN0dTH-Yca0LCiaBIJ4_xTYVOoBUNfo_DMzzcmq8EF17YSFENOWdrKnicmBw286OyhlY6_fEBN7mX/s320/35.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-family: "Times New Roman"; font-weight: 400; white-space: normal;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">Extend AD to E such that AD = ED</span></span></span><span style="font-family: arial; text-indent: -47.2667px;">.</span></div><div style="font-family: "Times New Roman"; font-weight: 400; white-space: normal;"><span style="font-size: medium;"><span style="font-family: arial; text-indent: -47.2667px;"><span face="Arial, sans-serif" style="text-indent: -47.2667px;">2) </span></span></span><span style="font-family: arial; text-indent: -47.2667px;">Extend PM to N such that PM = NM.</span></div><div style="font-family: "Times New Roman"; font-weight: 400; white-space: normal;"><span style="font-family: arial; text-indent: -47.2667px;">3) In ∆ ABC and ∆ PQR,</span></div><blockquote style="border: none; font-family: "Times New Roman"; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">(AB/PQ) = (AC/PR) = (AD/PM) ---------- equation 1.</span></blockquote><div style="font-family: "Times New Roman"; text-align: left;"><span style="font-weight: normal;">4)</span> <span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">In ∆ ABD and ∆ ECD,</span></div><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) (AD) = (ED) (construction)</span></blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) < ADB = < EDC (vertically opposite angles)</span></blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">c) (BD) = (CD) (AD is median of </span><span style="text-indent: -47.2667px;">∆ ABC)</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-weight: normal;">5) So by SAS test </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ABD </span><span style="background-color: white; color: #404040; font-family: arial; font-weight: 400; text-align: center; white-space: normal;">≅</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"><span style="font-family: arial;"> ∆ ECD</span>.</span></div><div style="text-align: left;"><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"><span style="font-family: arial;">6) So here, (AB) = (EC)</span></span><span style="font-family: arial; font-weight: 400; white-space: normal;"> </span><span style="font-family: arial; font-weight: 400; white-space: normal;">---------- equation 2</span></div><div style="text-align: left;"><div><span style="font-weight: normal;">7)</span> <span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">In ∆ PQM and ∆ NRM,</span></div><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) (PM) = (NM) (construction)</span></blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) < PMQ = < NMR (vertically opposite angles)</span></blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">c) (QM) = (RM) (PM is median of </span><span style="text-indent: -47.2667px;">∆ PQR)</span></blockquote><div><span style="font-family: arial; font-weight: normal;">8) So by SAS test </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ PQM </span><span style="background-color: white; color: #404040; font-family: arial; font-weight: 400; text-align: center; white-space: normal;">≅</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"><span style="font-family: arial;"> ∆ NRM</span>.</span></div><div><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;"><span style="font-family: arial;">9) So here, (PQ) = (NR)</span></span><span style="font-family: arial; font-weight: 400; white-space: normal;"> </span><span style="font-family: arial; font-weight: 400; white-space: normal;">---------- equation 3</span></div><div><span style="font-family: arial; font-weight: 400; white-space: normal;">10) From equations 1, 2, and 3, we have,</span></div><div><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) </span>(AB/PQ) = (AC/PR) = (AD/PM)</blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) </span>(EC/NR) = (AC/PR) = 2(AD)/2(PM)</blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">c) </span>(EC/NR) = (AC/PR) = (AE)/(PN) ------- (As 2AD = AE, and 2PM = PN)</blockquote><div style="text-align: left;"><span style="font-family: arial;"><span style="font-weight: normal;">11) So, b</span></span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">y SSS similarity test</span><span style="font-family: arial;"><span style="font-weight: normal;"> </span></span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">∆ ACE </span><span style="font-family: arial; font-weight: 400; white-space: normal;">~</span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;"> ∆ PRN,</span></div><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">12) So we have,</span></div><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="font-family: arial; font-size: medium;">a) </span>< CAD = < RPM,</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="font-family: arial; font-size: medium;">b) </span>[2(< CAD)] = [2(< RPM)] (as AD and PM are medians of <span style="text-indent: -47.2667px;">∆ ABC and ∆ PQR)</span> </blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-indent: 0px;"><span style="font-family: arial; font-size: medium;">c) </span>< CAB = < RPQ ---------- equation 4</blockquote></span></div><div style="text-align: left;"><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px; white-space: normal;">13) From equations 1 and 4, we have,</span></div><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) </span>(AB/PQ) = (AC/PR) and < CAB = < RPQ by SAS similarity test, we have,</blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) </span><span style="text-indent: -47.2667px;">∆ ABC </span>~<span style="text-indent: -47.2667px;"> ∆ PQR hence proved.</span></blockquote><div style="text-align: left;"><br /></div></div><div style="text-align: left;"><span style="font-family: arial;">Q15. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time, a tower casts a shadow 28 m long. Find the height of the tower.</span></div><div style="text-align: left;"></div></div></div></span></span></h3><h3><span style="font-size: medium;"><span><span style="font-family: arial;"><span style="white-space: pre-wrap;">Solution:</span><div class="separator" style="clear: both; text-align: center; white-space: pre-wrap;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh51fyXHUep69JNIXyOW06AG2iJBUJ9AKhGqWT6RGjBNVCnBtX7qR5O_SlzUt3NfWsNbaHwkINvsi5VSdGcNjcVitpE160A5kCS_Luqp0IqkkCezTFhDRkbSFvTrPQELgCozE91Lilt5FN9FPanhVdgDnoGWzn3YHltA75k-gveM_839Pv19PTHhhW1/s586/36.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="499" data-original-width="586" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh51fyXHUep69JNIXyOW06AG2iJBUJ9AKhGqWT6RGjBNVCnBtX7qR5O_SlzUt3NfWsNbaHwkINvsi5VSdGcNjcVitpE160A5kCS_Luqp0IqkkCezTFhDRkbSFvTrPQELgCozE91Lilt5FN9FPanhVdgDnoGWzn3YHltA75k-gveM_839Pv19PTHhhW1/s320/36.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="white-space: pre-wrap;">1)</span><span style="white-space: pre-wrap;"> </span><span style="text-indent: -47.2667px;">In ∆ ABC and ∆ PQR,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) < BCA = < QRP (angle of elevation)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) < ABC = < PQR = 90 (angle between pole or tower and their shadows)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">c) </span><span style="text-indent: -47.2667px;">∆ ABC </span>~<span style="text-indent: -47.2667px;"> ∆ PQR by AA similarity test,</span></blockquote><div style="font-family: "Times New Roman"; white-space: pre-wrap;"><span style="font-family: arial; font-weight: normal;">2) So by basic proportionality theorem</span><span style="font-weight: 400; text-indent: -47.2667px; white-space: normal;">.</span></div><div><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">(AB)/(PQ) = (BC)/(QR)</span></blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(6)/(PQ) = (4)/(28)</blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(6)/(PQ) = (1)/(7)</blockquote><div style="font-family: "Times New Roman"; white-space: pre-wrap;"><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">6 x 7 = (PQ)</blockquote><blockquote style="border: none; font-family: arial; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;">(PQ) = 42 m</blockquote><div><span style="font-family: arial;"><span style="font-weight: normal;">3) So, the height of the tower is 42 m.</span></span></div></div><div style="font-family: "Times New Roman"; white-space: pre-wrap;"><span style="font-family: arial;"><span style="font-weight: normal;"><br /></span></span></div><div><span style="font-family: arial; white-space: pre-wrap;">Q</span><span style="white-space: pre-wrap;">16. If AD and PM are medians of triangles ABC and PQR, respectively where</span></div><span style="text-indent: -47.2667px;">∆</span><span style="white-space: pre-wrap;"> ABC ~ </span><span style="text-indent: -47.2667px;">∆</span><span style="white-space: pre-wrap;"> PQR, prove that AB/PQ = AD/PM.</span></div><div></div></div></span></span></span></h3><h3><span style="font-size: medium;"><span><span style="font-family: arial;"><span style="white-space: pre-wrap;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2Ujf9uxEbLWt4a6BtEAUc_Ot-JEnuOrLIhsAKkrwrNlaPmJEv2ivXibx1_Yp_PPsDEgab-ARjLmeTdDHIxnRC5HzfAKXpDjmtRbbvHPMniKd8pTWrxskTV_u180278GdHFsb6cuh5qRWk8ASO9-9wE5WHgyBEhtMkDVJCuZxGrRaXqSMFnB6Fzq3n/s716/37.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="299" data-original-width="716" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2Ujf9uxEbLWt4a6BtEAUc_Ot-JEnuOrLIhsAKkrwrNlaPmJEv2ivXibx1_Yp_PPsDEgab-ARjLmeTdDHIxnRC5HzfAKXpDjmtRbbvHPMniKd8pTWrxskTV_u180278GdHFsb6cuh5qRWk8ASO9-9wE5WHgyBEhtMkDVJCuZxGrRaXqSMFnB6Fzq3n/s320/37.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><span style="white-space: pre-wrap;">1)</span><span style="white-space: pre-wrap;"> </span><span style="text-indent: -47.2667px;">As ∆ ABC </span>~<span style="text-indent: -47.2667px;"> ∆ PQR,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) (AB)/(PQ) = (AC)/(PR) = (BC)/(QR)</span></blockquote><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">b) </span>(AB)/(PQ) = (AC)/(PR) = [(BC)/2]/[(QR)/2] </blockquote><div style="text-align: left;"><span style="font-weight: normal;">2) </span><span style="font-weight: 400; white-space: normal;">As D and M are midpoints of BC and QR respectively, we have,</span></div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><span style="font-family: arial; font-size: medium;">a) </span>(AB)/(PQ) = (AC)/(PR) = (BD)/(QM) ---------- equation 1</blockquote><span style="font-weight: normal;"><span style="white-space: pre-wrap;">3) As AD and PM are the medians of </span><span style="text-indent: -47.2667px;">∆ ABC </span>and<span style="text-indent: -47.2667px;"> ∆ PQR,</span></span></div></span></span></span></span><blockquote style="border: none; font-family: arial; font-size: large; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) < BAD = < QPM</span> ---------- equation 2</blockquote><div style="text-align: left;"><span style="font-weight: normal;"><span style="font-family: arial; font-size: medium;">4) In </span></span><span style="font-family: arial; font-size: medium; font-weight: normal;">∆ ABD and<span style="text-indent: -47.2667px;"> ∆ PQM, and from equations 1 and 2, we have,</span></span></div><blockquote style="border: none; font-family: arial; font-size: large; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">a) (AB)/(PQ) = (AD)/(PM) = (BD)/(QM)</blockquote><blockquote style="border: none; font-family: arial; font-size: large; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;">b) < BAD = < QPM</blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal;">5) By the SAS similarity test, we have </span></span><span style="font-family: arial; font-size: medium; font-weight: normal;">∆ ABD ~<span style="text-indent: -47.2667px;"> ∆ PQM</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: normal;"><span style="text-indent: -47.2667px;">6) So by basic </span></span><span style="font-family: arial; font-size: medium;"><span style="font-weight: normal; white-space: pre-wrap;">proportionality theorem</span><span style="font-weight: 400; text-indent: -47.2667px;">.</span></span></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) (AB)/(PQ) = (AD)/(PM), hence proved.</span></blockquote></div></h3></span></div><h3><span style="font-family: arial; font-size: medium; white-space: pre-wrap;"><div style="white-space: normal;"><div><div style="font-weight: 400; line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></div></div></div></span></h3><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/167-ncert-10-6-triangles-ex-64.html" rel="nofollow" target="_blank"><span style="color: #0400ff; font-family: arial;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.4</a></span></h2><div style="font-family: "Times New Roman"; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></span></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-66339537232857589452023-12-14T15:18:00.001+05:302023-12-19T21:01:44.114+05:30165-NCERT-10-6-Triangles - Ex- 6.2<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.2</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/11/164-ncert-10-6-triangles-ex-61.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-6-Triangles - Ex- 6.1</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.2</span></h3></div><div><span style="font-family: arial; font-size: medium;">Q1. In the following Fig, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).</span></div><div><span style="font-family: arial; font-size: medium;"><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgkaT3gP5ZyJQIREoQlR4URJZPALvfyckmSW7j3cIgpHXq2gU25QWzxxfcbxrhkYDOTEf7SKBYDsXDKzdKDqGPpkQ7bWnRTULErw6zwlUsM2D8gQZ45XsLwm3Er4WhElOLMeeKRA8WqXgGpHI9OHS7AHZDNP4-hCPDPeL9G34HcshIo0rjnupW_2nH/s722/6a.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="274" data-original-width="722" height="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgkaT3gP5ZyJQIREoQlR4URJZPALvfyckmSW7j3cIgpHXq2gU25QWzxxfcbxrhkYDOTEf7SKBYDsXDKzdKDqGPpkQ7bWnRTULErw6zwlUsM2D8gQZ45XsLwm3Er4WhElOLMeeKRA8WqXgGpHI9OHS7AHZDNP4-hCPDPeL9G34HcshIo0rjnupW_2nH/w394-h149/6a.png" width="394" /></span></a></div></div><div><span style="font-size: medium;"><span style="font-family: arial;">(i) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">DE ‖ BC, find EC in fig (i)</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">In ∆ ABC and ∆ ADE,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< ABC = < ADE<span> (</span></span><span style="font-family: arial;">corresponding angles</span><span style="font-family: arial;">)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< ACB = < AED</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">corresponding</span><span style="font-family: arial;"> angles)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">< BAC = < DAE</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">common</span><span style="font-family: arial;"> angle)<br /></span><span style="font-family: arial;">So, ∆ ABC </span><span style="background-color: white; color: #333333;"><span style="font-family: arial;">~</span></span><span style="font-family: arial;"> ∆ ADE</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(AD)/(DB) = (AE)/(EC)<br /></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(1.5)/3 = 1/(EC)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1/2 = 1/(EC)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(EC) = 2</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">c) So, here EC = 2 cm.</span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">(ii) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">DE ‖ BC, find AD in fig (ii)</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">In ∆ ABC and ∆ ADE,</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< ABC = < ADE (</span><span style="font-family: arial;">corresponding angles</span><span style="font-family: arial;">)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< ACB = < AED</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">corresponding</span><span style="font-family: arial;"> angles)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">< BAC = < DAE</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">common</span><span style="font-family: arial;"> angle)<br /></span><span style="font-family: arial;">So, ∆ ABC </span><span style="background-color: white; color: #333333;"><span style="font-family: arial;">~</span></span><span style="font-family: arial;"> ∆ ADE</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AD)/(DB) = (AE)/(EC)<br /></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AD)/(7.2) = (1.8)/(5.4)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AD) = [(1.8)(7.2)]/(5.4)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AD) = (1.8)[(72)/(54)]</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(AD) = (1.8)[(8)/(6)]<br /></span><span>(AD) = (18/10)[(8)/(6)]<br /></span><span>(AD) = (3/10)[(8)]<br /></span><span>(AD) = [(3)(8)]/10<br /></span><span>(AD) = 24/10<br /></span><span>(AD) = 2.4</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>c) So, here AD = 2.4 cm.</span> </span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q2. E and F are points on the sides PQ and PR respectively of a </b></span><span style="text-indent: -47.2667px;"><span style="font-family: arial; font-size: medium;">∆</span></span><span style="font-family: arial; font-size: medium;"><b> PQR. For each of the following cases, state whether EF || QR:</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm</b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm</b></span></div></blockquote><div style="text-align: left;"><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBAOwNOOzWDvoYFD4RKTyh8l8I6BpADq9-F1eLKntxL7CzEbZiQFUIAFP23zAQdt29n2-_zX0aeqWAnQLOZs0Tlu-V98I8JvbE9Rhd6gcJDlPVc0eVkkdlBQIlDFgLIVTeANwO-x2YP2tCcfeFxefsJZpistG3tdfrqkCmlcgfwiqT39AUDLfvxvoi/s329/7.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="279" data-original-width="329" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBAOwNOOzWDvoYFD4RKTyh8l8I6BpADq9-F1eLKntxL7CzEbZiQFUIAFP23zAQdt29n2-_zX0aeqWAnQLOZs0Tlu-V98I8JvbE9Rhd6gcJDlPVc0eVkkdlBQIlDFgLIVTeANwO-x2YP2tCcfeFxefsJZpistG3tdfrqkCmlcgfwiqT39AUDLfvxvoi/w232-h196/7.png" width="232" /></span></a></div></div><div><span style="font-family: arial; font-size: medium;">(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) If </span><span style="font-family: arial; text-indent: -47.2667px;">EF ‖ QR, then we must have,</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(PE)/(EQ) = (PF)/(FR) ----------- equation 1</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) LHS = </span><span style="font-family: arial;">(PE)/(EQ)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(3.9)/(3)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">1.3</span><span style="font-family: arial;"> ----------- equation 2</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">c) RHS = </span><span style="font-family: arial;">(PF)/(FR)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(3.6)/(2.4)<br /></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(36)/(24)<br /></span><span>RHS = </span><span>3/2</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>1.5</span><span> ----------- equation 3</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2) So, from equations 2 and 3, we can say that (PE)/(EQ) <span style="line-height: 107%;">≠ </span>(PF)/(FR).</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) So, EF is not paraller to QR.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div><span style="font-size: medium;"><span style="font-family: arial;">(ii) </span><span style="font-family: arial;">PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) If </span><span style="font-family: arial; text-indent: -47.2667px;">EF ‖ QR, then we must have,</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(PE)/(EQ) = (PF)/(FR) ----------- equation 1</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) LHS = </span><span style="font-family: arial;">(PE)/(EQ)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(4)/(4.5)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(40)/(45)</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">8/9</span><span style="font-family: arial;"> ----------- equation 2</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) RHS = </span><span style="font-family: arial;">(PF)/(FR)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(8)/(9)<br /></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>8/9</span><span> ----------- equation 3</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2) So, from equations 2 and 3, we can say that (PE)/(EQ) </span><span style="font-family: arial;">=</span><span style="font-family: arial;"><span style="line-height: 19.26px;"> </span>(PF)/(FR).</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3) So, EF is paraller to QR.</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;">(iii) </span><span style="font-family: arial;">PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) First we will find EQ and FR</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">EQ = PQ - PE</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">EQ = 1.28 - 0.18</span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">EQ = 1.10</span><span style="font-family: arial;"> ----------- equation 1</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">FR = PR - PF</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">FR = 2.56 - 0.36</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>FR = 2.20</span><span> ----------- equation 2</span> </span></blockquote></blockquote></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) If </span><span style="font-family: arial; text-indent: -47.2667px;">EF ‖ QR, then we must have,</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(PE)/(EQ) = (PF)/(FR) ----------- equation 3</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) LHS = </span><span style="font-family: arial;">(PE)/(EQ)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(0.18)/(1.10)</span></span></blockquote></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">(18)/(110)</span></span></blockquote></blockquote></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span>(9)/(55)</span> </span></blockquote></blockquote></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span style="font-family: arial;">9/55</span><span style="font-family: arial;"> ----------- equation 4</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) RHS = </span><span style="font-family: arial;">(PF)/(FR)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(0.36)/(2.20)<br /></span></span></blockquote></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial;">(36)/(220)</span></span></div></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>(18)/(110)<br /></span><span>RHS = </span><span>(9)/(55)</span> </span></blockquote></blockquote></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>RHS = </span><span>9/55</span><span> ----------- equation 5</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2) So, from equations 4 and 5, we can say that (PE)/(EQ) </span><span style="font-family: arial;">=</span><span style="font-family: arial;"><span style="line-height: 19.26px;"> </span>(PF)/(FR).</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3) So, EF is paraller to QR.</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q3. In the following fig. , if LM || CB and LN || CD, prove that </b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(AM)/(AB) = (AN)/(AD).</b></span></div></div></blockquote><div style="text-align: left;"><div><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc9BPv4q09UnF8QhzULZmJKXcyt2a85SYgj6wJLUUIvFGryfnAAZTnJ_HtzsMAyvbT9bJBRj9lpJWsOiP1feD8rAU7AADrfY_gZ1m38BgssqiyrzdDmm48d1rhcjXW9J1H5szGCdd4D2pT7x9k8h3LCpTJCYSuOG2ILSwMZxOTk5Ua1SggZSSkdCTe/s533/8.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="448" data-original-width="533" height="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc9BPv4q09UnF8QhzULZmJKXcyt2a85SYgj6wJLUUIvFGryfnAAZTnJ_HtzsMAyvbT9bJBRj9lpJWsOiP1feD8rAU7AADrfY_gZ1m38BgssqiyrzdDmm48d1rhcjXW9J1H5szGCdd4D2pT7x9k8h3LCpTJCYSuOG2ILSwMZxOTk5Ua1SggZSSkdCTe/w188-h158/8.png" width="188" /></span></a></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">LM ‖ CB,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">In ∆ AML and ∆ ABC,</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< AML = </span><span style="font-family: arial;">< ABC (</span><span style="font-family: arial;">corresponding angles</span><span style="font-family: arial;">)</span></span></blockquote></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< ALM = </span><span style="font-family: arial;">< ACB</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">corresponding</span><span style="font-family: arial;"> angles)</span></span></blockquote></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< MAL = </span><span style="font-family: arial;">< BAC</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">common</span><span style="font-family: arial;"> angle)</span></span></blockquote></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">So, ∆ </span><span style="font-family: arial;">AML </span><span style="background-color: white; color: #333333;"><span style="font-family: arial;">~</span></span><span style="font-family: arial;"> ∆ ABC</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(AM)/(AB) = (AL)/(AC)</span><span style="font-family: arial;"> ----------- equation 1</span></span></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ADC, </span><span style="font-family: arial; text-indent: -47.2667px;">LN ‖ CD,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">In ∆ ANL and ∆ ADC,</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< ANL = </span><span style="font-family: arial;">< ADC (</span><span style="font-family: arial;">corresponding angles</span><span style="font-family: arial;">)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< ALN = </span><span style="font-family: arial;">< ACD</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">corresponding</span><span style="font-family: arial;"> angles)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< NAL = </span><span style="font-family: arial;">< DAC</span><span style="font-family: arial;"> (</span><span style="font-family: arial;">common</span><span style="font-family: arial;"> angle)</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">So, ∆ </span><span style="font-family: arial;">ANL </span><span style="background-color: white; color: #333333;"><span style="font-family: arial;">~</span></span><span style="font-family: arial;"> ∆ ADC</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Using basic proportionality theorem,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(AN)/(AD) = (AL)/(AC)</span><span style="font-family: arial;"> ----------- equation 2</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) From equations 1 and 2, we have </span><span style="font-family: arial;">(AM)/(AB)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">=</span><span style="font-family: arial;"><span style="line-height: 19.26px;"> </span></span><span style="font-family: arial;">(AN)/(AD)</span><span style="font-family: arial;">.</span><span style="font-family: arial;"> </span><span style="font-family: arial;">Hence proved.</span></span></div></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q4. In the following Fig., DE || AC and DF || AE. Prove that BF/FE = BE/EC</b></span></div><div style="text-align: left;"><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUAAT8oONlUxbwRs8ynut6EuNo1pxoO-3uOJ053C5efEXMW15zBW3aEeY7dcpfJ2UMfPDS0uTeG7xV7RszTYbIUyzjGZVfJnGCzEzoaO0DgGfOkQdAIQGjDq8GjVwMSV2DY_XeP03Obmfrrd-koYjaezj1PeYi-WUnkFEGGMTDdGDuYKo-Qg0T8uUv/s404/9.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="218" data-original-width="404" height="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUAAT8oONlUxbwRs8ynut6EuNo1pxoO-3uOJ053C5efEXMW15zBW3aEeY7dcpfJ2UMfPDS0uTeG7xV7RszTYbIUyzjGZVfJnGCzEzoaO0DgGfOkQdAIQGjDq8GjVwMSV2DY_XeP03Obmfrrd-koYjaezj1PeYi-WUnkFEGGMTDdGDuYKo-Qg0T8uUv/w293-h158/9.png" width="293" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">DE ‖ AC,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(BD)/(DA) = (BE)/(EC)</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400; text-align: left;">2) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ BAE, </span><span style="text-indent: -47.2667px;">DF ‖ AE,</span> </div><div style="text-align: left;"><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BD)/(DA) = (BF)/(FE)</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400; text-align: left;">3) From equations 1 and 2, we have (BE)/(EC) = (BF)/(FE). Hence proved.</div><div style="font-weight: 400; text-align: left;"><br /></div><div style="text-align: left;">Q5. In the following Fig., DE || OQ and DF || OR. Show that EF || QR.</div><div style="text-align: left;"></div></div></div></div></span></div></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS9yj4JveUtRPQgozQ0C-wM9PdOK_DPTJO7VlPfEXIz6XzwmMIqYHzGWusODxozTveJc5taVYrKTOXde0KEpBcVpXAUaotAiRgn_42D5gBgLvjjovOi34OlzpWl3SoOqM1LN_o24EmBKSPXCN2hTcHxeRyuBKcaaIwGc6z0JEaNL4htFmNcX165SxG/s387/10.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="261" data-original-width="387" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS9yj4JveUtRPQgozQ0C-wM9PdOK_DPTJO7VlPfEXIz6XzwmMIqYHzGWusODxozTveJc5taVYrKTOXde0KEpBcVpXAUaotAiRgn_42D5gBgLvjjovOi34OlzpWl3SoOqM1LN_o24EmBKSPXCN2hTcHxeRyuBKcaaIwGc6z0JEaNL4htFmNcX165SxG/w246-h166/10.png" width="246" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ PQO, </span><span style="font-family: arial; text-indent: -47.2667px;">DE ‖ OQ,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(PD)/(DO) = (PE)/(EQ)</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400;">2) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ POR, </span><span style="text-indent: -47.2667px;">DF ‖ OR,</span> </div><div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(PD)/(DO) = (PF)/(FR)</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, we have (PE)/(EQ) = (PF)/(FR).</div><div style="font-weight: 400;">4) So, <span style="text-indent: -47.2667px;">EF ‖ QR. </span>Hence proved.</div></div></div><div style="font-weight: 400;"><br /></div><div>Q6. In the following Fig., A, B, and C are points on OP, OQ, and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.</div><div></div></div></span></div></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwpLAlh2WV4-XX6IXFKTC9lKquhyIsu-Ihk8z-sW7xyG_vaqDn-odCc9ktW-1rRaaxGsbfutNmjqNQ__9xxTNOIOXmfafUHK_UdbPOgGl7gSUPEJ5Sc_qfEBlNSRxNZY9w_psMK_7NmRNVBM-FgNVzOQ_m_v2BGjBlr8IT_tJ1CIoRPCifK8bxWUVB/s397/11.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="365" data-original-width="397" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwpLAlh2WV4-XX6IXFKTC9lKquhyIsu-Ihk8z-sW7xyG_vaqDn-odCc9ktW-1rRaaxGsbfutNmjqNQ__9xxTNOIOXmfafUHK_UdbPOgGl7gSUPEJ5Sc_qfEBlNSRxNZY9w_psMK_7NmRNVBM-FgNVzOQ_m_v2BGjBlr8IT_tJ1CIoRPCifK8bxWUVB/w198-h182/11.png" width="198" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ OPQ, </span><span style="font-family: arial; text-indent: -47.2667px;">AB ‖ PQ,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(OA)/(AP) = (OB)/(BQ)</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400;">2) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ OPR, </span><span style="text-indent: -47.2667px;">AC ‖ PR,</span> </div><div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(OA)/(AP) = (OC)/(CR)</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, we have (OB)/(BQ) = (OC)/(CR).</div><div style="font-weight: 400;">4) So, <span style="text-indent: -47.2667px;">BC ‖ QR. </span>Hence proved.</div></div></div><div style="font-weight: 400;"><br /></div><div>Q7. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).</div><div></div></div></div></span></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoWwDKQNFYxgH4Z_AlS9_kjEbu_-ZHnjADc42PSaKO0DEcX4ExjBjOdnXxYWf4Fby2bLsimekuA1-tJmjIdYNYPZ89Gv5fCMSxYQcRGK49ykf1INlCyxDT9q6ilPCskJUe6MZdHni1df1atMYB01x83u9jQr0Kfi00gn91XwJGscZ6Uct52gNbkidT/s429/12.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="347" data-original-width="429" height="184" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoWwDKQNFYxgH4Z_AlS9_kjEbu_-ZHnjADc42PSaKO0DEcX4ExjBjOdnXxYWf4Fby2bLsimekuA1-tJmjIdYNYPZ89Gv5fCMSxYQcRGK49ykf1INlCyxDT9q6ilPCskJUe6MZdHni1df1atMYB01x83u9jQr0Kfi00gn91XwJGscZ6Uct52gNbkidT/w227-h184/12.png" width="227" /></a></div></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">point P is the midpoint of AB,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) So we have,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AP)/(PB) = 1</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400;">2) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ ABC, </span><span style="text-indent: -47.2667px;">PQ ‖ BC,</span> </div><div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) Using basic proportionality theorem,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AP)/(PB) = (AQ)/(QC)</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, we have (AQ)/(QC) = 1.</div><div style="font-weight: 400;">4) Here, <span style="text-indent: -47.2667px;">AQ = QC, so Q is the midpoint of AC.</span></div><div style="font-weight: 400;"><span style="text-indent: -47.2667px;">5) So, line PQ bisects the third side AC of </span><span style="text-indent: -47.2667px;">∆ ABC.</span><span style="text-indent: -47.2667px;"> </span>Hence proved.</div></div></div><div style="font-weight: 400;"><br /></div><div>Q8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in</div><div>Class IX).</div><div></div></div></span></div></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoWwDKQNFYxgH4Z_AlS9_kjEbu_-ZHnjADc42PSaKO0DEcX4ExjBjOdnXxYWf4Fby2bLsimekuA1-tJmjIdYNYPZ89Gv5fCMSxYQcRGK49ykf1INlCyxDT9q6ilPCskJUe6MZdHni1df1atMYB01x83u9jQr0Kfi00gn91XwJGscZ6Uct52gNbkidT/s429/12.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="347" data-original-width="429" height="184" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoWwDKQNFYxgH4Z_AlS9_kjEbu_-ZHnjADc42PSaKO0DEcX4ExjBjOdnXxYWf4Fby2bLsimekuA1-tJmjIdYNYPZ89Gv5fCMSxYQcRGK49ykf1INlCyxDT9q6ilPCskJUe6MZdHni1df1atMYB01x83u9jQr0Kfi00gn91XwJGscZ6Uct52gNbkidT/w227-h184/12.png" width="227" /></a></div></span><div><span style="font-family: arial; font-size: medium;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">point P is the midpoint of AB,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) So we have,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AP)/(PB) = 1</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400;">2) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ ABC, </span><span style="text-indent: -47.2667px;">point Q is the midpoint of AC,</span> </div><div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) So we have,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AQ)/(QC) = 1</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, we have (AP)/(PB) = (AQ)/(QC) = 1.</div><div style="font-weight: 400;">4) So, <span style="text-indent: -47.2667px;">PQ ‖ BC. </span>Hence proved.</div></div></div><div style="font-weight: 400;"><br /></div><div>Q9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at point O. Show that AO/BO = CO/DO.</div><div></div></div></span></div></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigipHJ7rddCnGdcjL0km3x9V1Ft5CyO2nXdjb8Jr5GTncuZ0pl6hMHr82DV_h2XnuFfYh2gti7qn4nvvIYzAHFv_8OTCPM_5urmp7y4lNrDRxFzKCmOip9hGQtk1sjIiBVpNAZe0cdCPVTR3B4gIBtRyLfsfxksQlOj9ZkgYnyifFLwEKPQEiQqgdK/s517/13.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="286" data-original-width="517" height="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigipHJ7rddCnGdcjL0km3x9V1Ft5CyO2nXdjb8Jr5GTncuZ0pl6hMHr82DV_h2XnuFfYh2gti7qn4nvvIYzAHFv_8OTCPM_5urmp7y4lNrDRxFzKCmOip9hGQtk1sjIiBVpNAZe0cdCPVTR3B4gIBtRyLfsfxksQlOj9ZkgYnyifFLwEKPQEiQqgdK/w245-h135/13.png" width="245" /></a></div><div class="separator" style="clear: both; text-align: left;"><div style="font-weight: 400; white-space: normal;"><div><span style="font-family: arial; font-size: medium;">1) In </span><span style="font-family: arial; text-indent: -47.2667px;">trapezium ABCD, </span><span style="font-family: arial; text-indent: -47.2667px;">AB ‖ CD, and diagonals AC and BD intersect at O.</span></div><div><span style="font-family: arial; text-indent: -47.2667px;">2) Construct line PQ such that </span><span style="font-family: arial; text-indent: -47.2667px;">PQ ‖ AB and PQ</span><span style="font-family: arial; text-indent: -47.2667px;"> ‖ DC.</span></div><div><span style="font-family: arial; text-indent: -47.2667px;">3) </span><span style="font-family: arial; font-size: medium;">In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">OQ ‖ AB,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">Using basic proportionality theorem,</span></blockquote></div><div style="white-space: normal;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BQ)/(CQ) = (AO)/(CO)</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div style="font-weight: 400;">4) <span style="font-family: arial; font-size: medium;">In </span><span style="text-indent: -47.2667px;">∆ BDC, </span><span style="text-indent: -47.2667px;">OQ ‖ DC,</span> </div><div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">Using basic proportionality theorem,</span></blockquote></div><div><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BQ)/(CQ) = (BO)/(DO)</span><span style="font-family: arial;"> ----------- equation 2</span></div></blockquote><div style="font-weight: 400;">3) From equations 1 and 2, we have (AO)/(CO) = (BO)/(DO).</div><div style="font-weight: 400;">4) So, <span style="text-indent: -47.2667px;">we have </span>(AO)/(BO) = (CO)/(DO)<span style="text-indent: -47.2667px;">. </span>Hence proved.</div></div></div><div style="font-weight: 400;"><br /></div><div>Q10. The diagonals of a quadrilateral ABCD intersect each other at the point O</div><div>such that (AO)/(BO) = (CO)/(DO). Show that ABCD is a trapezium.</div></div></div></span></h3><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVQPf7FrZTWa9vEpG5UQuQ8Z-9JanIWJYZ4KWBwMs1poBBLA9G8_pt6ijSs-K8B3WG-qTn4HFqWcDcDJXZZQF_5-9iH5a_5NKju39TNnu4AfUtRQBcLNEAsjPybVDXldEW4lh93riTFNmA_oP2thBiW1XhGHfOPAslAjzwTnt0WVjCNnUXW8rjFVLK/s548/14.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="294" data-original-width="548" height="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVQPf7FrZTWa9vEpG5UQuQ8Z-9JanIWJYZ4KWBwMs1poBBLA9G8_pt6ijSs-K8B3WG-qTn4HFqWcDcDJXZZQF_5-9iH5a_5NKju39TNnu4AfUtRQBcLNEAsjPybVDXldEW4lh93riTFNmA_oP2thBiW1XhGHfOPAslAjzwTnt0WVjCNnUXW8rjFVLK/w277-h149/14.png" width="277" /></a></div></span></h3><h3 style="white-space: pre-wrap;"><div class="separator" style="clear: both;"><div style="white-space: normal;"><div><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both;"><div style="font-weight: 400;"><div><span style="font-family: arial; font-size: medium;">1) In quadrilateral </span><span style="font-family: arial; text-indent: -47.2667px;">ABCD, </span><span style="font-family: arial; text-indent: -47.2667px;">diagonals AC and BD intersect at O.</span></div><div><span style="font-family: arial; text-indent: -47.2667px;">2) Construct line PQ such that </span><span style="font-family: arial; text-indent: -47.2667px;">PQ ‖ AB</span><span style="font-family: arial; text-indent: -47.2667px;">.</span></div><div><span style="font-family: arial; text-indent: -47.2667px;">3) </span><span style="font-family: arial; font-size: medium;">In </span><span style="font-family: arial; text-indent: -47.2667px;">∆ ABC, </span><span style="font-family: arial; text-indent: -47.2667px;">OQ ‖ AB,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a) </span><span style="font-family: arial;">Using basic proportionality theorem,</span></blockquote></div><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BQ)/(CQ) = (AO)/(CO)</span><span style="font-family: arial;"> ----------- equation 1</span></div></blockquote><div>4) As per the problem<span style="text-indent: -47.2667px;">,</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(AO)/(BO) = (CO)/(DO)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(AO)/(CO) = (BO)/(DO) </span><span style="font-family: arial;">----------- equation 2</span></div></blockquote></blockquote><div style="text-align: left;">5) From equations 1 and 2, we have,</div></div></div><div style="text-align: left;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(BO)/(DO) = (BQ)/(CQ)</span><span style="font-family: arial;"> ----------- equation 3</span></div></blockquote><div style="font-weight: 400; text-align: left;">6) So, from equation 3 and the converse of the basic proportionality theorem,</div><div style="text-align: left;"><blockquote style="border: none; font-weight: 400; margin: 0px 0px 0px 40px; padding: 0px;"><div style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="text-indent: -47.2667px;">PQ ‖ AB, so </span><span style="text-indent: -47.2667px;">AB ‖ CD.</span></div></blockquote><div style="text-align: left;"><span style="font-weight: 400;">7) Therefore </span><span style="font-family: arial; font-size: medium; font-weight: 400;">quadrilateral </span><span style="font-family: arial; font-weight: 400; text-indent: -47.2667px;">ABCD is the </span><span style="font-weight: normal;">trapezium.</span></div></div></div></div></span></div></div></div></h3></div><div style="text-align: left;"><h3 style="white-space: pre-wrap;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="white-space: normal;"><div style="text-align: left;"><div style="font-weight: 400; text-align: left;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></div></div></div></span></div></h3></div><div><span style="font-family: arial; font-size: medium;"><h2 style="clear: both; color: #0400ff; text-align: left;"><span style="font-size: medium;"><span style="color: #0400ff; font-family: arial;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.3</span></h2><div style="font-family: "Times New Roman"; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a> </div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-12075037071695428422023-11-23T12:40:00.003+05:302023-12-19T21:01:18.237+05:30164-NCERT-10-6-Triangles - Ex- 6.1<div style="text-align: left;"><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 6.1</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 6 Triangles</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/11/163-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.4</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 6.1</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><div>Q1. Fill in the blanks using the correct word given in brackets :</div></b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(i) All circles are _________. (congruent, similar)</div></b></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">Ans: similar, "All circles are <b>similar</b>".</div></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(ii) All squares are_________ . (similar, congruent)</div></b></span></div></blockquote><p><span style="font-family: arial; font-size: medium;"> <span>Ans: similar, "All squares are <b>similar</b>".</span></span></p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(iii) All _________ triangles are similar. (isosceles, equilateral)</div></b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">Ans: similar, "All <b>equilateral</b> tringles are similar".</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(iv) Two polygons of the same number of sides are similar, if (a) their</div></b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">corresponding angles are _________ and (b) their corresponding sides are_________ . (equal, proportional)</div></b></span></div></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">Ans: a) </span><span style="font-family: arial;">equal, b) proportional</span><span style="font-family: arial;">, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">"</span><span style="font-family: arial;">Two polygons of the same number of sides are </span><span style="font-family: arial;">similar, if </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(a) their </span><span style="font-family: arial;">corresponding angles are </span><span style="font-family: arial; font-weight: 700;">equal</span><span style="font-family: arial;"> and </span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(b) their corresponding sides are </span><span style="font-family: arial; font-weight: 700;">proportional.</span></span></div></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div>2. Give two different examples of a pair of</div></b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(i) similar figures. (ii) non-similar figures.</div></b></span></div></blockquote><h3 style="white-space: pre-wrap;"><span style="font-family: arial; font-size: medium;">Solution:</span></h3><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(i) similar figures:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) All equilateral tringles are similar.</span></div></blockquote></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6aAtIivrI9mZN2hZM4EXi12o5DAWdOUnSLWkxqQ1a-l0BmEgyahW-xm3xNtnX9u2RiZ1ff_aM0c_664iKzPTs7QivdEuVT_Mm-g-oqy2TCXpqn6X-pTdkLtyZt04VTaK_TyI9gaz2j6RDezdljfb2sk3kIzbSW9baLO3iqUh-6YHGGThwKR_Mx0XV/s415/1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="199" data-original-width="415" height="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6aAtIivrI9mZN2hZM4EXi12o5DAWdOUnSLWkxqQ1a-l0BmEgyahW-xm3xNtnX9u2RiZ1ff_aM0c_664iKzPTs7QivdEuVT_Mm-g-oqy2TCXpqn6X-pTdkLtyZt04VTaK_TyI9gaz2j6RDezdljfb2sk3kIzbSW9baLO3iqUh-6YHGGThwKR_Mx0XV/s320/1.png" width="320" /></span></a></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">All squares are similar.</span></span></div></blockquote></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj83mz_o4qTV7TJO3R-zuL90SOcR-U0N12-VyevXudsEaW26lHa7FxpwJRf_uDPdqDDkULVpyCXWGqDH8t2XEBDZqsfM3hg5oCnQZnrjhtkgH1R3xTctFLG8x5cjiLxaUgUvCRODnOUxbw5IPFDYPNDS_D7nbzMR8B0AQZRhQz8xKeSCRwgdixEsEv9/s430/2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="197" data-original-width="430" height="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj83mz_o4qTV7TJO3R-zuL90SOcR-U0N12-VyevXudsEaW26lHa7FxpwJRf_uDPdqDDkULVpyCXWGqDH8t2XEBDZqsfM3hg5oCnQZnrjhtkgH1R3xTctFLG8x5cjiLxaUgUvCRODnOUxbw5IPFDYPNDS_D7nbzMR8B0AQZRhQz8xKeSCRwgdixEsEv9/s320/2.png" width="320" /></span></a></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700;">(ii) </span><span style="font-family: arial; font-weight: 700;">non-</span><span style="font-family: arial; font-weight: 700;">similar figures:</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a) Equilateral tringle and </span><span color="var(--r1Clr)" style="font-family: arial;">isosceles </span><span style="font-family: arial;">traingle are non-similar.</span></span></div></blockquote></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilAIl4kiBEVYeQtK4439RzZezFv6uT6_NeeeAXnudLpKnawDADPVnWVgE9mSVvM3IBAgei7qBQbBAMks6g3Hw0v8WFHmpItpbIDPDhS-wltxaP-e11YvxkdMo54RPkoUxOqKv3D3LQIWa_ipqtvBR3UATyDqSvtxaAncngvTUPg8fHn9gjcNgSeGSz/s433/3.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="149" data-original-width="433" height="110" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilAIl4kiBEVYeQtK4439RzZezFv6uT6_NeeeAXnudLpKnawDADPVnWVgE9mSVvM3IBAgei7qBQbBAMks6g3Hw0v8WFHmpItpbIDPDhS-wltxaP-e11YvxkdMo54RPkoUxOqKv3D3LQIWa_ipqtvBR3UATyDqSvtxaAncngvTUPg8fHn9gjcNgSeGSz/s320/3.png" width="320" /></span></a></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b) Square and rectangle</span><span> are non-similar.</span> </span></div></blockquote></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7nE2fw2xTNWlCxeoirDyhLAgZfFSTEYeF93KtM6EljnPdedEtawFv0rAQT8LQiRWx3GwvmkmZvdySPk_eomKGFvo2GxrXdkMEIupIP3V7pnpv2YJ5_JrycDo82hl_bCLBi6F4K6wpYJpdsVkjpEA25zOMcx_LOC-RJsXr9nJzl--RQlwkezwkCIib/s396/4.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="120" data-original-width="396" height="97" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7nE2fw2xTNWlCxeoirDyhLAgZfFSTEYeF93KtM6EljnPdedEtawFv0rAQT8LQiRWx3GwvmkmZvdySPk_eomKGFvo2GxrXdkMEIupIP3V7pnpv2YJ5_JrycDo82hl_bCLBi6F4K6wpYJpdsVkjpEA25zOMcx_LOC-RJsXr9nJzl--RQlwkezwkCIib/s320/4.png" width="320" /></span></a></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div>3. State whether the following quadrilaterals are similar or not:</div></b></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZdE9FceoxsyW58__sH6ogh0FAnbJ3G404lpwwBMVUjW_DA3M1RvwBRTav60L9HypZMy2TFCyJ6XrTi3iZ2MRnmjOeTjhZ0mgVnuzH_SprxgBr_k91hEy-STFGiZ1Wvqi1aO2tTPe1dU9XzwbD_dbG5KD4V4S1zFRPI_ObxtsalIu45ZVjheV6pEz_/s677/5.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="354" data-original-width="677" height="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZdE9FceoxsyW58__sH6ogh0FAnbJ3G404lpwwBMVUjW_DA3M1RvwBRTav60L9HypZMy2TFCyJ6XrTi3iZ2MRnmjOeTjhZ0mgVnuzH_SprxgBr_k91hEy-STFGiZ1Wvqi1aO2tTPe1dU9XzwbD_dbG5KD4V4S1zFRPI_ObxtsalIu45ZVjheV6pEz_/s320/5.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: medium;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on </span><span style="font-family: arial;">Triangles</span><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #<span style="color: black; white-space-collapse: collapse;">Triangles</span> #education #learning #students #teachers #math</span></span></div><div><span style="font-family: arial; font-size: medium;"><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/12/165-ncert-10-6-triangles-ex-62.html" rel="nofollow" target="_blank"><span style="color: #0400ff; font-family: arial;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.2</a></span></h2><div style="font-family: "Times New Roman"; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-27381086069395460472023-11-05T12:54:00.002+05:302023-11-23T12:42:00.044+05:30163-NCERT-10-5-Arithmetic Progressions - Ex-5.4<div style="text-align: left;"><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 5.4</span></span></div><div style="color: black; font-family: "Times New Roman"; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 5 Arithmetic Progressions</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/10/162-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.3</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 5.4</span></h3></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>1. Which term of the AP: 121, 117, 113, . . ., is its first negative term?</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">[Hint: Find n for <span face="Arial, sans-serif">a</span><sub>n</sub> < 0]</span></b></div></div></blockquote><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><span style="font-family: arial;"><div><h3><span style="font-family: arial;">Solution:</span></h3></div><div><span style="font-family: arial;">1) </span><span style="font-family: arial;">According to the problem, </span><span>a</span><sub>1</sub><sub> </sub><span>= 121</span><sub>, </sub><span>a</span><sub>2</sub><sub> </sub><span>= 117</span><sub>, </sub><span>a</span><sub>3</sub><sub> </sub><span><span>= 113</span><span>. . . </span></span><span>so d = - 4, </span></div></span></div><div><span style="font-family: arial;"><span></span><span>2) </span></span><span>We will have to find n using the above information.</span><span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;">a</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = a + (n – 1) d</span></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;">a</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = 121 + (n – 1) (- 4)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span><span style="font-family: arial;"><span style="font-family: arial; white-space: normal;">a</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = 121 + (- 4n + 4)</span></span></span></blockquote></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 121 - 4n + 4<br /></span><span>a</span><sub>n</sub><span> = 125 - 4n</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">3) As we have to find which first term is a negative term. </span><br /></span><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;">a</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> < 0</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">125 - 4n < 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">125 < 4n</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">4n > 125</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">n > 125/4</span></blockquote></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">n > 31.25</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) So the first negative term of this AP is the 32nd term.</span> </span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>2. The sum of the third and the seventh terms of an AP is 6 and their product</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>is 8. Find the sum of first sixteen terms of the AP.</b></span></div></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Let the first term be "a" and the common difference be "d".</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span>a</span><sub>3</sub><span> +</span><span> </span><span><span>a</span><sub>7</sub><span> = 6 ---------- equation 1</span></span></span></div></div><div><span style="font-family: arial; font-size: medium;"><span>3) Here </span><span>a</span><sub>3</sub><span> x</span><span> </span><span><span>a</span><sub>7</sub><span> = 8 ---------- equation 2</span></span></span></div><div><span style="font-family: arial; font-size: medium;"><span><span>4) Now we will find </span></span><span>a</span><sub>3</sub><span> and</span><span> </span><span><span>a</span><sub>7</sub></span><span>,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a)</span><span> </span><span>First we will find </span><span>a</span><sub>3</sub> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> = a + (3 – 1) d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> = a + 2d</span><span> ---------- equation 3</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b)</span><span> </span><span>Now we will find </span><span>a</span><sub>7</sub> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>7</sub><span> = a + (7 – 1) d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>7</sub><span> = a + 6d</span><span> ---------- equation 4</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) From equations 1, 3, and 4, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> +</span><span> </span><span><span>a</span><sub>7</sub><span> = 6</span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a + 2d +</span><span style="font-family: arial;"> </span><span style="font-family: arial;"><span>a + 6d = 6<br /></span></span><span style="font-family: arial;">2a + 8d</span><span style="font-family: arial;"> = 6<br /></span><span style="font-family: arial;">2(a + 4d)</span><span style="font-family: arial;"> = 6<br /></span><span style="font-family: arial;">(a + 4d)</span><span style="font-family: arial;"> = 6/2<br /></span><span style="font-family: arial;">(a + 4d)</span><span style="font-family: arial;"> = 3<br /></span><span style="font-family: arial;">a = 3 - 4d</span><span style="font-family: arial;"> ---------- equation 5</span></span></blockquote><span style="font-family: arial; font-size: medium;">6) From equations 2, 3, 4, and 5, we have,</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> x</span><span> </span><span><span>a</span><sub>7</sub><span> = 8</span></span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(a + 2d) x</span><span> (</span><span>a + 6d) = 8</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(3 - 4d + 2d) x</span><span> (</span><span>3 - 4d + 6d) = 8</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(3 - 2d) x</span><span> (</span><span>3 + 2d) = 8</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(3</span><sup>2</sup><span> - 4d</span><sup>2</sup><span>)</span><span> = 8</span></span></div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>(9</span><span> - 4d</span><sup>2</sup><span>)</span><span> = 8</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>4d</span><sup>2</sup><span> = 9 - 8<br /></span><span>4d</span><sup>2</sup><span> = 1<br /></span><span>d</span><sup>2</sup><span> = 1/4<br /></span><span>d</span><sup>2</sup><span> = 1/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = </span><span style="text-align: center;"><span style="font-family: arial;">± 1/2<br /></span></span><span style="font-family: arial;">d = </span><span style="text-align: center;"><span style="font-family: arial;">1/2 or - 1/2</span></span><span style="font-family: arial;"> ---------- equation 6</span></span></div></blockquote><span style="font-family: arial; font-size: medium;">7) Put d = 1/2 and d = - 1/2 from equation 6 in equation 5,</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) First we take d = 1/2, we get</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - 4d</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - 4(1/2)</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - 2</span></blockquote></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a = 1</span><span style="font-family: arial;"> ---------- equation 7</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) First we take d = - 1/2, we get</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - 4d</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - 4(- 1/2)</span></blockquote></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 - (- 2)</span></blockquote></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 3 + 2</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a = 5</span><span style="font-family: arial;"> ---------- equation 8</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">8) We know that the sum of the first n terms of an AP is given by:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub> = (n/2)[2a + (n - 1) d]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) First we will find sum of first 16 terms with a = 1 and d = 1/2:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub> = (n/2)[2a + (n - 1) d]</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>16</sub><span> = (16/2)[2(1) + (16 - 1) (1/2)]</span><br /><span>S</span><sub>16</sub><span> = 8[2 + (15) (1/2)]<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>16</sub><span> = 8[2 + (15/2)]</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span></span><span>S</span><sub>16</sub><span> = 8[(4 + 15)]/2<br /></span><span>S</span><sub>16</sub><span> = 4(19)<br /></span><span>S</span><sub>16</sub><span> = 76</span><span> ---------- equation 9</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">b) First we will find sum of first 16 terms with a = 5 and d = (- 1/2):</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub> = (n/2)[2a + (n - 1) d]</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>16</sub><span> = (16/2)[2(5) + (16 - 1) (- 1/2)]</span><br /><span>S</span><sub>16</sub><span> = 8[10 + (15) (- 1/2)]<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>16</sub><span> = 8[10 - (15/2)]</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span></span><span>S</span><sub>16</sub><span> = 8[(20 - 15)]/2<br /></span><span>S</span><sub>16</sub><span> = 4(5)<br /></span><span>S</span><sub>16</sub><span> = 20</span><span> ---------- equation 10</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>9) From equations 9 and 10, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) </span><span>S</span><sub>16</sub><span> = 76 when a = 1 and d = 1/2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) </span><span>S</span><sub>16</sub><span> = 20 when a = 5 and d = - 1/2.</span></span></blockquote><div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>3. A ladder has rungs 25 cm apart. (see the following fig.). The rungs decrease</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are 2½ m apart, what is the length of the wood required for the rungs?</b></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">[Hint : Number of rungs = (250/25) + 1]</span></b></p></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQKnBzo93rDRyubWP__Z1E8-6xEBJ-EgCKOLeZ8NpQOFFC9we4CDHvrj11_mLqwj05d9ysIMDIQf5UoaNHWmyzuCjNrk_AmqpTViQo5spqrLNYKri1AFAFZgJkgc7RARZdg7uKt3gFZe_vGe6RjnWAdO1GtucphONu7L7c3BTtA-U-Ld1j5LugDx4g/s640/5.4-3.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="640" data-original-width="492" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQKnBzo93rDRyubWP__Z1E8-6xEBJ-EgCKOLeZ8NpQOFFC9we4CDHvrj11_mLqwj05d9ysIMDIQf5UoaNHWmyzuCjNrk_AmqpTViQo5spqrLNYKri1AFAFZgJkgc7RARZdg7uKt3gFZe_vGe6RjnWAdO1GtucphONu7L7c3BTtA-U-Ld1j5LugDx4g/w246-h320/5.4-3.png" width="246" /></span></a></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;">1) The distance between the rungs is 25 cm.</span></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span>2) </span></span><span style="font-family: arial;">Distance between the top rung and the bottom</span></span></div></div></div></div></div></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> rung is </span><span style="font-family: arial;">2½</span><span style="font-family: arial;"> m. i.e. 5/2 m = 2.5 m = 250 cm.</span></span></div></blockquote></blockquote></blockquote><div><div><div><div><span style="font-family: arial; font-size: medium;">3) So total number of rungs = [(250/25) + 1] = 11.</span></div><div><span style="font-family: arial; font-size: medium;">4) The length of rungs is decreasing from bottom to<span> </span>top uniformly, so they are in AP.</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>5)</span><span> </span><span>Here, </span><span>a</span><sub>1</sub> <span>= 45, l = 25 and n = 11, so,</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span> </span><span> </span>S</span><sub>n</sub><span> = (n/2)[a + l]</span></span></div></blockquote></blockquote></blockquote></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span> S</span><sub>11</sub><span> = (11/2)[45 + 25]</span></span></div><div><span style="font-family: arial; font-size: medium;"><span> S</span><sub>11</sub><span> = (11/2)[70]</span></span></div><div><span style="font-family: arial; font-size: medium;"><span> S</span><sub>11</sub><span> = 11(35)</span></span></div><div><span style="font-family: arial; font-size: medium;"><span> S</span><sub>11</sub><span> = 385</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial;">The length of the wood required for the rungs is <span> </span><span> </span>385 cm.</span></span></div></div></blockquote></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>4. The houses of a row are numbered consecutively from 1 to 49. Show that</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : <span face="Arial, sans-serif">s</span><sub>x - 1</sub> = S<sub>49</sub> – S<sub>x</sub>]</b></span></div></div></blockquote><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><span style="font-family: arial;"><div><h3><span style="font-family: arial;">Solution:</span></h3></div><div><span style="font-family: arial;">1) </span><span style="font-family: arial;">Row houses are numbered 1, 2, 3, . . 49.</span></div><div><span style="font-family: arial;">2) So, </span><span>a</span><sub>1</sub><sub> </sub><span>= 1</span><sub>, </sub><span>a</span><sub>2</sub><sub> </sub><span>= 2</span><sub>, </sub><span>a</span><sub>3</sub><sub> </sub><span><span>= 3</span><span>. . . </span></span><span>so d = 1, </span></div></span></div><div><span style="font-family: arial;"><span></span><span>3) The sum</span></span><span> of the number of houses preceding xth house will be </span><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">(x-1)</sub><span style="font-family: arial; white-space: normal;">.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = (n/2)[2a + (n – 1) d]</span></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>(x-1)</sub><span style="font-family: arial;"> = ((x-1)/2)[2(1) + (x - 1 - 1) (1)]</span></span></blockquote></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div style="text-align: left;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">(x-1)</sub><span style="font-family: arial; white-space: normal;"> = ((x-1)/2)[2 + (x - 2)]</span> </div></div></span></span></div></blockquote><div><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">(x-1)</sub><span style="font-family: arial; white-space: normal;"> = ((x-1)/2)[x]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">(x-1)</sub><span style="font-family: arial; white-space: normal;"> = x(x-1)/2</span><span style="white-space: normal;"> ---------- equation 1<br /></span></blockquote></div></span></span></div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">4) Now we will find </span><span>S</span><sub>49</sub></span></div><div><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="white-space: normal;"><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = (n/2)[2a + (n – 1) d]</span><div><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>49</sub><span style="font-family: arial;"> = (49/2)[2(1) + (49 - 1) (1)]</span></span></div></span></span></div></div><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><div><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">49</sub><span style="font-family: arial; white-space: normal;"> = (49/2)[2 + 48]</span> </div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="white-space: normal;"><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><div><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">49</sub><span style="font-family: arial; white-space: normal;"> = (49/2)(2)[1 + 24]</span></div></span></span></div></div></blockquote></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">49</sub><span style="font-family: arial; white-space: normal;"> = (49)[25]</span></div></div></span></span></div></blockquote><div><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="white-space: normal;"><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><div><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">49</sub><span style="font-family: arial; white-space: normal;"> = 1225</span><span style="white-space: normal;"> ---------- equation 2</span></div></span></span></div></div></blockquote><div style="text-align: left;">5) Now we will find <span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">x</sub></div></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><div style="text-align: left;"><div style="white-space: normal;"><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><span style="font-family: arial; white-space: normal;">S</span><sub style="white-space: normal;">n</sub><span style="font-family: arial; white-space: normal;"> = (n/2)[2a + (n – 1) d]</span><div><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>x</sub><span style="font-family: arial;"> = (x/2)[2(1) + (x - 1) (1)]</span></span></div></span></span></div></div><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><div><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>x</sub> = (x/2)[2 + (x - 1)]</span> </div></span></span></div><div style="white-space: normal;"><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>x</sub> = (x/2)[x + 1]</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; white-space: normal;"><span style="font-family: arial;">S</span><sub>x</sub> = x(x + 1)/2</span><span style="white-space: normal;"> ---------- equation 3</span></div></span></span></div></div></div></div></span></span></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) Now we will find </span><span style="font-family: arial;">S<sub>49</sub> – S<sub>x </sub></span><span style="font-family: arial;">using equations 2 and 3.</span></span></div><div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>S</span><sub>49</sub><span> - </span></span><span>S</span><sub>x</sub><span> = 1225 - </span><span>[x(x + 1)/2]</span><span> ---------- equation 4</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>7) According to the problem, </span><span>S</span><sub>(x-1)</sub><span> = </span> <span><span>S</span><sub>49</sub><span> - </span></span><span>S</span><sub>x </sub><span> ---------- equation 5</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) From equations 1, 4, and 5, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">S</span><sub style="font-family: arial;">(x-1)</sub><span style="font-family: arial;"> = </span><span style="font-family: arial;"> </span><span style="font-family: arial;"><span>S</span><sub>49</sub><span> - </span></span><span style="font-family: arial;">S</span><sub style="font-family: arial;">x</sub></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x(x-1)/2 = </span><span style="font-family: arial;"> </span><span style="font-family: arial;">1225 - </span><span style="font-family: arial;">[x(x + 1)/2]</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x(x-1)/2 = </span><span style="font-family: arial;"> [</span><span style="font-family: arial;">2450 - (</span><span style="font-family: arial;">x(x + 1))]/2</span></span></div><span style="font-size: medium;"><span style="font-family: arial;">x(x-1) = </span><span style="font-family: arial;"> [</span><span style="font-family: arial;">2450 - (</span><span style="font-family: arial;">x(x + 1))]<br /></span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2 </sup><span style="font-family: arial;">- x = 2450 - </span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2 </sup><span style="font-family: arial;">- x<br /></span><span style="font-family: arial;">2x</span><sup style="font-family: arial;">2 </sup><span style="font-family: arial;">= 2450<br /></span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2 </sup><span style="font-family: arial;">= 1225<br /></span><span style="font-family: arial;"><span>x</span><sup> </sup><span>= </span><span style="text-align: center;">± </span>35</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><sup style="font-family: arial;"> </sup><span style="font-family: arial;">= </span><span style="font-family: arial;">35 or - 35</span></span></blockquote><span style="font-family: arial; font-size: medium;">9) As the number of houses can't be negative, the value of x will be 35.</span><div><span style="font-size: medium;"> </span><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>5. A small terrace at a football ground comprises of 15 steps each of which is</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>50 m long and built of solid concrete. </b></span><b>Each step has a rise of 1/4 m and a tread of 1/2 m. (see the following fig.). Calculate the total volume of concrete required to build the terrace. </b><b>[Hint: Volume of concrete required to build the first step = (1/4) x (1/2) x 50 <span>m</span><sup>3</sup>]</b></span></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"> </span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFMMKJ4Kp9Zh0wWaw9xnQuI6507W5yCaL7fL_lfMkmc-Y5DqHYLvF1BeQPOqfLhnsmFl4VQF-l1xOFKqjs2cwCX7WznEZi9c425GcNwtP5PKuxLL-xqoa6_deYy9a3-g6jtHatQmZSG1NtyYAqvI_RTvQcDgt6M-YmRqiteqLk6G8F0jroqFrSq86W/s442/5.4-5.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: medium;"><img border="0" data-original-height="210" data-original-width="442" height="190" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFMMKJ4Kp9Zh0wWaw9xnQuI6507W5yCaL7fL_lfMkmc-Y5DqHYLvF1BeQPOqfLhnsmFl4VQF-l1xOFKqjs2cwCX7WznEZi9c425GcNwtP5PKuxLL-xqoa6_deYy9a3-g6jtHatQmZSG1NtyYAqvI_RTvQcDgt6M-YmRqiteqLk6G8F0jroqFrSq86W/w400-h190/5.4-5.png" width="400" /></span></a></div><span style="font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div>1) According to the problem and the figure, we have,</div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">i) The heights of the steps are given bellow:</div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a) The height of the first step is (1/4) m</div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) The height of the second step is (1/4) + (1/4) = (1/2) m</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) The height of the 3rd step is (1/2) + (1/4) = (3/4) m</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) The height of the 4th step is (3/4) + (1/4) = (1) m</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) If we consider the height in the changing form, the width will be the same for all</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">the </span><span style="font-family: arial; font-size: medium;">steps. i.e. the width will be 1/2 m for all the steps.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) Here the length of the terrace is 50 m.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Here we can find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">The volume of the steps = volume of the cuboid</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">The volume of the steps = Length x Breadth x Height</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Now we will find the volumes of the steps:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) The volume of the first step </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= (1/4) x (1/2) x (50)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (1/8) x (50)<br /></span><span style="font-family: arial;">= (50/8)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (25/4)</span><span style="font-family: arial;"> ---------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) The volume of the second step </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= (1/2) x (1/2) x (50)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>= (1/4) x (50)<br /></span><span>= (50/4)</span><span> ---------- equation 2</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">c) The volume of the 3rd step </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= (3/4) x (1/2) x (50)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>= (3/4) x (25)<br /></span><span>= (75/4)</span><span> ---------- equation 3</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>d) The volume of the 4th step is (3/4) + (1/4) = (1) m</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (1) x (1/2) x (50)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (1) x (25)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (25)</span><span style="font-family: arial;"> ---------- equation 4</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) From equations 1, 2, 3, and 4, volumes of the steps are in AP.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) So, here, a = 25/4, d = 50/4 - 25/4 = 25/4, and n = 15,</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) so the sum of these 15 steps will be,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> = (n/2)[2a + (n – 1) d]<br /></span><span>S</span><sub>15</sub><span> = (15/2)[2(25/4) + (15 – 1) (25/4)]<br /></span><span>S</span><sub>15</sub><span> = (15/2)[2(25/4) + (14) (25/4)]<br /></span><span>S</span><sub>15</sub><span> = (15/2) x (2(25/4)) x [1 + (7)]<br /></span><span>S</span><sub>15</sub><span> = (15/2) x (2(25/4)) x [8]<br /></span><span>S</span><sub>15</sub><span> = [(15 x 2 x 25)/8] x [8]<br /></span><span>S</span><sub>15</sub><span> = (15 x 2 x 25)<br /></span><span>S</span><sub>15</sub><span> = (30 x 25)<br /></span><span>S</span><sub>15</sub><span> = (750)</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>9) So, the volume of concrete required to build the terrace is 750 </span><span><span>m</span><sup>3</sup></span>.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on Arithmetic Progressions, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #ArithmeticProgressions #education #learning #students #teachers #math</span><span style="font-family: arial;"> </span></span></div><div><div><a href="https://anil7pute.blogspot.com/2023/11/164-ncert-10-6-triangles-ex-61.html" rel="nofollow" target="_blank"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><span style="color: #0400ff; font-family: arial;">Click here for</span><span style="color: #0400ff;"> </span>⇨ NCERT-10-6-Triangles - Ex- 6.1</span></h2></div></span></a></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span>ANIL SATPUTE</span></a> </span><br /></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-50035015013470347952023-10-12T16:39:00.004+05:302023-11-05T15:04:34.709+05:30162-NCERT-10-5-Arithmetic Progressions - Ex-5.3<h2 style="clear: both; color: #0400ff;"><span style="color: #0400ff; font-family: arial; font-size: medium;"></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 5.3</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 5 Arithmetic Progressions</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><a href="https://anil7pute.blogspot.com/2023/09/161-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff; font-family: arial; font-size: medium;">Click here for </span></a><a href="https://anil7pute.blogspot.com/2023/09/160-ncert-10-5-arithmetic-progressions.html" rel="nofollow" style="font-family: arial; font-size: large;" target="_blank">⇨ </a><a href="https://anil7pute.blogspot.com/2023/09/161-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="font-family: arial; font-size: medium;">NCERT-10-5-Arithmetic Progressions - Ex- 5.2</span></a></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 5.3</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><div>Q1. Find the sum of the following APs:</div></b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(i) 2, 7, 12, . . ., to 10 terms. <span> </span>(ii) –37, –33, –29, . . ., to 12 terms.</div></b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(iii) 0.6, 1.7, 2.8, . . ., to 100 terms. <span> </span>(iv) 1/15, 1/12, 1/10, . . ., to 11 terms.</div></b></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial;">Explanation:</span></h3><div><span style="font-family: arial;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></blockquote><div><span style="font-family: arial;"><span>2) The </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">S = (n/2)[2a + (n – 1) d]</div></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) If </span><span>a</span><sub>n</sub><span> = l is the last term of an AP, then</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">S = (n/2)[2a + (n – 1) d]</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S = (n/2)[a + (a + (n – 1) d)]<br /></span><span>S = (n/2)[a + </span><span>a</span><sub>n</sub><span>]<br /></span><span>S = (n/2)[a + l</span><span>].</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(i) 2, 7, 12, . . ., to 10 terms.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;">1) Here, a = 2, d = 7 - 2 = 5, and n = 10</span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (n/2)[2a + (n – 1) d]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">S = (10/2)[2(2) + (10 – 1) (5)]</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">S = (5)[4 + 5(9)]<br /></span><span style="font-family: arial;">S = (5)[4 + 45]<br /></span><span style="font-family: arial;">S = (5)(49)<br /></span><span style="font-family: arial;">S = 245</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>3) So, the </span><span>sum of the given AP is 245.</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"> </div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(ii) –37, –33, –29, . . ., to 12 terms.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) Here, a = - 37, d = - 33 + 37 = 4, and n = 12</span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (n/2)[2a + (n – 1) d]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (12/2)[2(- 37) + (12 – 1) (4)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (6)[- 74 + 4(11)]<br /></span><span style="font-family: arial;">S = (6)[- 74 + 44]<br /></span><span style="font-family: arial;">S = (6)(- 30)<br /></span><span style="font-family: arial;">S = - 180</span></blockquote><span style="font-family: arial; font-size: medium;">3) So, the </span>sum of the given AP is - 180.<span style="font-size: medium;"> </span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(iii) 0.6, 1.7, 2.8, . . ., to 100 terms.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) Here, a = 0.6, d = 1.7 - 0.6 = 1.1, and n = 100</span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (n/2)[2a + (n – 1) d]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (100/2)[2(0.6) + (100 – 1) (1.1)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (50)[1.2 + 1.1(99)]<br /></span><span style="font-family: arial;">S = (50)[1.2 + 108.9]<br /></span><span style="font-family: arial;">S = (50)(110.1)<br /></span><span style="font-family: arial;">S = 5505</span></blockquote><span style="font-family: arial; font-size: medium;">3) So, the </span>sum of the given AP is 5505.<span style="font-size: medium;"> </span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(iv) 1/15, 1/12, 1/10, . . ., to 11 terms.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) Here, a = 1/15, </span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = 1/12 - 1/15</span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = (15 - 12)/(12x15)</span></div><span style="font-size: medium;"><span style="font-family: arial;">d = </span><span style="font-family: arial;">(3)/(12x15)<br /></span><span style="font-family: arial;">d = </span><span style="font-family: arial;">1/(4x15)<br /></span><span style="font-family: arial;">d = </span><span style="font-family: arial;">1/60 and</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">n = 11</span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (n/2)[2a + (n – 1) d]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S = (11/2)[2(1/15) + (11 – 1) (1/60)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (11/2)[2/15 + 10/60]<br /></span><span style="font-family: arial;">S = (11/2)[2/15 + 1/6]<br /></span><span style="font-family: arial;">S = </span><span style="font-family: arial;">(11/2)/[(4/30) + (5/30)]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">S = (11/2)/(4 + 5)/30 </div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (11/2)/(9/30)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">S = (11/2)/(3/10)</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">S = 33/20 </div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3) So, the </span>sum of the given AP is 33/20.</span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-size: medium;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><b><span>Q</span>2. Find the sums given below :</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 7 + 10<span style="text-align: center;">½ + </span>14 + . . . + 84</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 34 + 32 + 30 + . . . + 10</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iii) –5 + (–8) + (–11) + . . . + (–230)</b></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><div><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></div></div></div></div></div><div><div><span style="font-family: arial;"><b>(i) 7 + 10<span style="text-align: center;">½ + </span>14 + . . . + 84</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><div><span style="font-family: arial;">1) Here, a = 7, d = 10<span style="text-align: center;">½</span> - 7 = 3<span style="text-align: center;">½ = 7/2</span>, nth term is </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = l = 84</span></div><div><span style="font-family: arial;">2) We know that,</span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">84 = 7 + (n – 1) (7/2)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(n – 1) (7/2) = 84 - 7<br /></span><span>(n – 1) (7/2) = 77<br /></span><span>(n – 1) = 77(2/7)<br /></span><span>(n – 1) = 11(2)<br /></span><span>(n – 1) = 22<br /></span><span>n = 22 + 1<br /></span><span>n = 23</span> </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[a + l]</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (23/2)[7 + 84]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (23/2)[91]<br /></span><span style="font-family: arial;">S = 2093/2<br /></span><span style="font-family: arial;">S = 1046.5</span></blockquote><span style="font-family: arial;">4) So, the </span><span style="font-family: arial;">sum of the given AP is </span><span style="font-family: arial;">1046.5</span><span style="font-family: arial;">.</span></div><div><span style="font-family: arial;"><br /></span></div><div><b>(ii) 34 + 32 + 30 + . . . + 10</b></div><div><b><br /></b></div><div><span style="font-family: arial;">1) Here, a = 34, d = 32 - 34 = - 2, nth term is </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = 10</span></div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">10 = 34 + (n – 1) (- 2)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n – 1) (- 2) = 10 - 34<br /></span><span style="font-family: arial;">(n – 1) (- 2) = - 24<br /></span><span style="font-family: arial;">(n – 1) = (- 24)/(- 2)<br /></span><span style="font-family: arial;">(n – 1) = 12</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n = 12 + 1<br /></span><span style="font-family: arial;">n = 13</span> </blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[a + l]</span></blockquote></span></div></span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">S = (13/2)[34 + 10]</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">S = (13/2)[44]</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S = (13)[22]<br /></span><span>S = 286</span> </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;">4) So, the sum of the given AP is 286.</span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><b>(iii) –5 + (–8) + (–11) + . . . + (–230)</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = - 5, d = (- 8) - (- 5) = - 3, nth term is </span>a<sub>n</sub> = - 230</span></div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- 230 = - 5 + (n – 1) (- 3)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n – 1) (- 3) = 5 - 230<br /></span><span style="font-family: arial;">(n – 1) (- 3) = - 225<br /></span><span style="font-family: arial;">(n – 1) = (- 225)/(- 3)<br /></span><span style="font-family: arial;">(n – 1) = 75</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n = 75 + 1<br /></span><span style="font-family: arial;">n = 76</span></blockquote></div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[a + l]</span></blockquote></span></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (76/2)[- 5 + (- 230)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (38)[- 235]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = - 8930</span> </blockquote></div><div><span style="font-family: arial;">4) So, the sum of the given AP is - 8930.</span></div><div><span style="font-family: arial;"><br /></span></div><div><b><span style="font-family: arial;">Q</span>3. In an AP:</b></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><b>(i) given a = 5, d = 3, a<sub>n</sub> = 50, find n and S<sub>n</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(ii) given a = 7, a<sub>13</sub> = 35, find d and S<sub>13</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(iii) given a<sub>12</sub> = 37, d = 3, find a and S<sub>12</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(iv) given a<sub>3</sub> = 15, S<sub>10</sub> = 125, find d and a<sub>10</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(v) given d = 5, S<sub>n</sub> = 75, find a and a<sub>9</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(vi) given a = 2, d = 8, S<sub>n</sub> = 90, find n and a<sub>n</sub>.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(vii) given a = 8, a<sub>n</sub> = 62, S<sub>n</sub> = 210, find n and d.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(viii) given a<sub>n</sub> = 4, d = 2, S<sub>n</sub> = – 14, find n and a.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(ix) given a = 3, n = 8, S = 192, find d.</b></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><b>(x) given l = 28, S = 144, and there are total 9 terms. Find a.</b></div></div></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><h3><span style="font-family: arial;">Solution:</span></h3></div><div><div><span style="font-family: arial;"><div><b>(i) given a = 5, d = 3, a<sub>n</sub> = 50, find n and S<sub>n</sub>.</b></div><div><b><br /></b></div><div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = 5, d = 3, nth term is </span>a<sub>n</sub> = 50.</span></div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">50 = 5 + (n – 1) (3)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n – 1) (3) = 50 - 5<br /></span><span style="font-family: arial;">(n – 1) (3) = 45<br /></span><span style="font-family: arial;">(n – 1) = 45/3<br /></span><span style="font-family: arial;">(n – 1) = 15</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n = 15 + 1<br /></span><span style="font-family: arial;">n = 16</span></blockquote></div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[a + l]</span></blockquote></span></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (16/2)[5 + 50]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (8)[55]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = 440</span> </blockquote></div><div><span style="font-family: arial;">4) So, </span><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = 440, n</span><span style="font-family: arial;"> = 16</span><span style="font-family: arial;">.</span></div></div><div><b><br /></b></div></span></div><div><span style="font-family: arial;"><div><b>(ii) given a = 7, a<sub>13</sub> = 35, find d and S<sub>13</sub>.</b></div><div><b><br /></b></div><div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = 7, d = ?, 13th term is </span>a<sub>13</sub> = 35. Find d and </span><span style="font-family: arial;">S</span><sub>13</sub></div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>13</sub><span style="font-family: arial;"> = 7 + (13 – 1) (d)</span></blockquote></div></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>13</sub><span style="font-family: arial;"> = 7 + 12d</span></div></div></div></span></div></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">35 = 7 + 12d</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">12d = 35 - 7</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">12d = 28</span></blockquote></div></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">d = 28/12</span> </div></div></div></span></div></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 7/3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 1</span></blockquote></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S<sub>n</sub> = (n/2)[a + l]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>13</sub><span style="font-family: arial;"> </span><span style="font-family: arial;">= (13/2)[7 + 35]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>13</sub><span style="font-family: arial;"> </span><span style="font-family: arial;">= (13/2)[42]</span></blockquote></div></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>13</sub><span> </span><span>= (13)[21]<br /></span><span>S</span><sub>13</sub><span> </span><span>= 273</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">4) So, </span><span style="font-family: arial;">S</span><sub>13</sub><span style="font-family: arial;"> = 273, d</span><span style="font-family: arial;"> = 7/3</span><span style="font-family: arial;">.</span></div></div><div><b><br /></b></div></span></div><div><span style="font-family: arial;"><div><b>(iii) given a<sub>12</sub> = 37, d = 3, find a and S<sub>12</sub>.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = ?, d = 3, 12th term is </span>a<sub>12</sub> = 37. Find a and </span><span style="font-family: arial;">S</span><sub>12</sub></div><div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>12</sub><span style="font-family: arial;"> = a + (12 – 1) (3)</span></blockquote></div></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">37 = a + 3(11)</span></div></div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">37 = a + 33<br />a = 37 - 33<br />a = 4 --------- equation 1</div></span></div></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) We know that the </span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> = (n/2)[a + l]</span><span><br /></span><span>S</span><sub>12</sub><span> = (12/2)[4 + 37]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> = (6)[41]<br /></span><span>S</span><sub>12</sub><span> = 246</span> </span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) So, </span><span>S</span><sub>12</sub><span> = 246, a</span><span> = 4</span><span>.</span><span> </span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><b>(iv) given a<sub>3</sub> = 15, S<sub>10</sub> = 125, find d and a<sub>10</sub>.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a<sub>3</sub> = 15, </span>S<sub>10</sub> = 125. Find d and </span><span style="font-family: arial;">a<sub>10</sub>.</span></div><div><span style="font-family: arial;">2) We know that,</span></div></div></span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = a + (3 – 1) (d)</span></div></div></span></div></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">15 = a + 2d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div>a + 2d = 15 --------- equation 1 <br /></div></span></div></div></span></blockquote></div></span></div></div></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">3) We know that the </span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = (n/2)[2a + (n - 1) d]</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">S</span><sub>10</sub><span style="font-family: arial;"> = (10/2)[2(a) + (10 - 1) d]</span></blockquote></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">125 = (5)[2(a) + 9d] </div></div></span></div></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">[2(a) + 9d]</span><span style="font-family: arial;"> = 125/5</span></blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">[2(a) + 9d] = 25 </div></div></span></div></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2a + 9d = 25</span><span style="font-family: arial;"> --------- equation 2</span><br /></blockquote><div><span style="font-family: arial;">4) Subtract double of equation 1 from equation 2</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2a + 9d = 25</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2a + 4d = 30</span></blockquote><div><span style="font-family: arial; font-size: medium;"> ( - ) ( - ) ( - ) </span></div><div><span style="font-family: arial; font-size: medium;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"> 5d = - 5</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">d = - 5/5</span></div></blockquote></blockquote></div></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">d = - 1</span><span style="font-family: arial;"> --------- equation 3</span></div></div></span></div></div></span></blockquote></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = - 1 from</span><span style="font-family: arial;"> equation 3 in equation 1, and we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a + 2d = 15</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>a + 2(- 1) = 15<br /></span><span>a - 2 = 15<br /></span><span>a = 15 + 2<br /></span><span>a = 17</span></span><span> --------- equation 4</span> </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">6) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span></blockquote></div></span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">a<sub>10</sub> = 17 + (10 – 1) (- 1) </div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>10</sub><span style="font-family: arial;"> = 17 + 9 (- 1)</span></div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>10</sub><span> = 17 - 9<br /></span><span>a</span><sub>10</sub><span> = 8</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>7) So, </span><span>a</span><sub>10</sub><span> = 8, d</span><span> = - 1</span><span>.</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"></span></div></div></span></div><div><span style="font-family: arial;"><div><b>(v) given d = 5, S<sub>9</sub> = 75, find a and a<sub>9</sub>.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, d = 5, </span>S<sub>9</sub> = 75. Find a and </span><span style="font-family: arial;">a</span><sub>9.</sub></div><div><div><span style="font-family: arial;">2) We know that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S<sub>n</sub> = (n/2)[2a + (n - 1) d]</span></blockquote></div></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">S</span><sub>9</sub><span style="font-family: arial;"> = (9/2)[2a + (9 - 1) (5)]</span></div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">75 = (9/2)[2a + 8 (5)]<br /></span><span style="font-family: arial;">75 = (9/2)[2a + 40]<br /></span><span style="font-family: arial;">75 = 9[a + 20]<br /></span><span style="font-family: arial;">[a + 20] = 75/9<br /></span><span style="font-family: arial;">[a + 20] = 25/3<br /></span><span style="font-family: arial;">a = (25/3) - 20<br /></span><span style="font-family: arial;">a = (25 - 60)/3<br /></span><span style="font-family: arial;">a = - 35/3</span></span></blockquote><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"></span><span style="font-family: arial;">3) We know that the </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>9</sub><span style="font-family: arial;"> = - 35/3 + (9 – 1) (5)</span></div></div></span></div></div></span></blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">a<sub>9</sub> = - 35/3 + (8)(5)</div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>9</sub><span> = - 35/3 + 40<br /></span><span>a</span><sub>9</sub><span> = (- 35 + 120)/3<br /></span><span>a</span><sub>9</sub><span> = 85/3</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">4) So, </span><span style="font-family: arial;">a</span><sub>9</sub><span style="font-family: arial;"> = 85/3, a</span><span style="font-family: arial;"> = - 35/3</span><span style="font-family: arial;">.</span></div></div><div><b><br /></b></div></span></div><div><span style="font-family: arial;"><div><b>(vi) given a = 2, d = 8, S<sub>n</sub> = 90, find n and a<sub>n</sub>.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = 2, d = 8, </span>S<sub>n</sub> = 90. Find n and </span><span style="font-family: arial;">a</span><sub>n.</sub></div>2) We know that,<br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial;">S<sub>n</sub> = (n/2)[2a + (n - 1) d]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = (n/2)[2(2) + (n - 1)(8)]<br /></span><span style="font-family: arial;">90 = (n/2)[4 + 8(n - 1)]<br /></span><span style="font-family: arial;">90 = n[2 + 4(n - 1)]</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">90 = n(2 + 4n - 4)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">90 = n(4n - 2)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">90 = 4</span><span style="font-family: arial;">n</span><sup>2 </sup><span style="font-family: arial;">- 2n</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">45 = 2</span><span style="font-family: arial;">n</span><sup>2 </sup><span style="font-family: arial;">- n</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">2</span><span style="font-family: arial;">n</span><sup>2 </sup><span style="font-family: arial;">- n - 45 = 0</span></blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">2n<sup>2 </sup>- 10n + 9n - 45 = 0</div></div></span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2n(n</span><sup> </sup><span>- 5) + 9(n - 5) = 0<br /></span><span>(n</span><sup> </sup><span>- 5)(2n + 9) = 0<br /></span><span>(n</span><sup> </sup><span>- 5) = 0 or (2n + 9) = 0<br /></span><span>n</span><sup> </sup><span>= 5 or 2n = - 9<br /></span><span>n</span><sup> </sup><span>= 5 or n = - 9/2</span> </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">3) As n is always a positive integer, ignore n = - 9/2. So we have n = 5.</span></div>4) We know that,<br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span></div></span></div></div></span>a<sub>5</sub> = 2 + (5 – 1) (8)</div></span></div></div></span></div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div>a<sub>5</sub> = 2 + (8)(4)</div></span></div></div></span>a<sub>5</sub> = 2 + 32<br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote>a<sub>5</sub> = 34</span></div></div></span></blockquote><div><span style="font-family: arial;">5) So,</span><span style="font-family: arial;"> </span><span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> = 34, n</span><span style="font-family: arial;"> = 5</span><span style="font-family: arial;">.</span><span style="font-family: arial;"> </span></div></div><div><b><br /></b></div></span></div><div><span style="font-family: arial;"><div><b>(vii) given a = 8, a<sub>n</sub> = 62, S<sub>n</sub> = 210, find n and d.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, a = 8, a<sub>n</sub> = 62, </span>S<sub>n</sub> = 210. Find n and d.</span></div><div><span style="font-family: arial;">2) We know that,<br /></span></div></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = (n/2)[a + l]</span></div></span></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">210 = (n/2)[8 + 62]<br /></span><span style="font-family: arial;">210 = (n/2)[70]<br /></span><span style="font-family: arial;">210 = 35n<br /></span><span style="font-family: arial;">35n = 210<br /></span><span style="font-family: arial;">n = 210/35<br /></span><span style="font-family: arial;">n = 6</span></span></blockquote><span style="font-family: arial; font-size: medium;"> <span>3) Now we will find d using the formula:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span></span></div></span>62 = 8 + (6 – 1) (d)</span></div><span style="font-family: arial;">62 = 8 + 5d</span><br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial;">5d = 62 - 8</span></div></span></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial;">5d = 54</span></div></span></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = 54/5<br /></span><span style="font-family: arial;">d = 10.8</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">5) So,</span><span style="font-family: arial;"> d</span><span style="font-family: arial;"> = 10.8, n</span><span style="font-family: arial;"> = 6</span><span style="font-family: arial;">.</span><span style="font-family: arial;"> </span></div><div><span style="font-family: arial;"><br /></span></div></span></div><div><span style="font-family: arial;"><div><b>(viii) given a<sub>n</sub> = 4, d = 2, S<sub>n</sub> = – 14, find n and a.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, d = 2, a<sub>n</sub> = 4, </span>S<sub>n</sub> = - 14. Find n and a.</span></div><div><span style="font-family: arial;">2) We know that,<br /></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = (n/2)[a + l]</span></blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div style="text-align: left;">- 14 = (n/2)[a + 4] </div></span></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n/2)[a + 4] = - 14</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n[a + 4] = - 14(2)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n[a + 4] = - 28</span><span style="font-family: arial;"> --------- equation 1</span></blockquote><div style="text-align: left;"> 3) We know that,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">a<sub>n</sub> = a + (n – 1) d<br /></span></span></div></span>4 = a + (n – 1) (2)</span></div></span></div><span style="font-family: arial;"><div>4 = a + 2n – 2</div></span>a + 2n = 4 + 2<br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote>a + 2n = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a = 6 - 2n</span><span style="font-family: arial;"> --------- equation 2</span></blockquote><div style="text-align: left;">4) Put a = 6 - 2n from equation 2 in equation 1, we get, </div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n(a + 4) = - 28</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n((6 - 2n) + 4) = - 28</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">n(6 - 2n + 4) = - 28</span></blockquote></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div style="text-align: left;">n(10 - 2n) = - 28 </div></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2n(5 - n) = - 28</span></blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div style="text-align: left;">n(5 - n) = - 14</div></span></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">5n - </span><span style="font-family: arial;">n</span><sup>2 </sup><span style="font-family: arial;">= - 14</span></blockquote></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">n</span><sup>2 </sup>- 5n - 14 = 0</div></span></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n</span><sup>2 </sup><span style="font-family: arial;">- 7n + 2n - 14 = 0</span></blockquote></div></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">n(n</span><sup> </sup>- 7) + 2(n - 7) = 0</div></span></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(n</span><sup> </sup><span>- 7)(n + 2) = 0<br /></span><span>(n</span><sup> </sup><span>- 7) = 0 or (n + 2) = 0<br /></span><span>n</span><sup> </sup><span>= 7 or n = - 2<br /></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">5) As n is always a positive integer, ignore n = - 2. So we have n = 7.</span></div><div><span style="font-family: arial;">6) Put n = 7 in equation 2, and we get,</span><br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">a = 6 - 2n</span></span></div></span>a = 6 - 2(7)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a = 6 - 14</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a = - 8</span></blockquote><div><span style="font-family: arial;">7) So, a = - 8, n = 7.</span></div></div><div><b><br /></b></div></span></div><div><span style="font-family: arial;"><div><b>(ix) given a = 3, n = 8, S = 192, find d.</b></div><div><b><br /></b></div><div><span style="font-family: arial;">1) Here, a = 3, n = 8, sum of first n terms is </span>S<sub>n</sub> = 192, find d.</div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>2) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[2a + (n - 1) d]</span></blockquote></span></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">192 = (8/2)[2(3) + (8 - 1) d]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">192 = 4[6 + 7d]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">4[6 + 7d] = 192</blockquote></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">[6 + 7d] = 192/4</div></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">[6 + 7d] = 48</blockquote></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">7d = 48 - 6</div></span></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7d = 42<br /></span><span style="font-family: arial;">d = 42/7<br /></span><span style="font-family: arial;">d = 6</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) So, d = 6.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"> </div></span></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><b>(x) given l = 28, S = 144, and there are total 9 terms. Find a.</b></div></span></div></div><div><span style="font-family: arial;"><br /></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">1) Here, l = 28, </span>S<sub>n</sub> = 144, n = 9, Find a.</span></div><div><span style="font-family: arial;">2) We know that,<br /></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub><span style="font-family: arial;"> = (n/2)[a + l]</span></blockquote></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><span style="font-family: arial;">144 = (9/2)[a + 28] </span></div></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(9/2)[a + 28] = 144</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">[a + 28] = 144(2/9)</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">[a + 28] = 16 (2)</span></blockquote></span></div></span></div></span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a + 28 = 32</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a = 32 - 28</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a = 4</span></div></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) So, a = 4.</span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Q4. How many terms of the AP: 9, 17, 25, . . . must be taken to give a sum of</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">636?</span></b></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">a</span><sub>1</sub> = a = 9, <span face="Arial, sans-serif">a</span><sub>2</sub> = 17, <span face="Arial, sans-serif">a</span><sub>3</sub> = 25,<span style="font-family: arial;"> sum of first n terms is </span>S<sub>n</sub> = 636, find n.</div><div><div><span style="font-family: arial;">2) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>d = </span><span>a</span><sub>2</sub><span> - </span><span>a</span><sub>1</sub></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = 17</span><span style="font-family: arial;"> - 9<br /></span><span style="font-family: arial;">d = 8</span></span></blockquote></div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[2a + (n - 1) d]</span></blockquote></span></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">636 = (n/2)[2(9) + (n - 1) (8)]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">636 = (n/2)[18 + 8(n - 1)]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">636 = n[9 + 4(n - 1)]</blockquote></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>636 = n[9 + 4n - 4]</div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">636 = n[4n + 5]</blockquote></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>636 = 4<span style="font-family: arial;">n</span><sup>2 </sup>+ 5n</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>4</span><span>n</span><sup>2 </sup><span>+ 5n - 636 = 0</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4</span><span>n</span><sup>2 </sup><span>+ 53n - 48n - 636 = 0</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>n(4</span><span>n</span><sup> </sup><span>+ 53) - 12(4n + 53) = 0</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(4</span><span>n</span><sup> </sup><span>+ 53)(n - 12) = 0<br /></span><span>(4</span><span>n</span><sup> </sup><span>+ 53) = 0 or (n - 12) = 0<br /></span><span>n</span><sup> </sup><span>= - 53/4 or n = 12</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) As n is always a positive integer, ignore n = - 53/4. So we have n = 12.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms </b></span><b>and the common difference.</b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">a</span><sub>1</sub> = a = 5, <span face="Arial, sans-serif">a</span><sub>n</sub> = 45, <span face="Arial, sans-serif">S</span><sub>n</sub> = 400,<span style="font-family: arial;"> </span>find n and d.</div><div>2) <span style="font-family: arial;"><span>We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S = (n/2)[a + l]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">400 = (n/2)[(5) + 45]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">400 = (n/2)[50]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">400 = n[25]</blockquote></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>n[25] = 400</div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">n = 400/25</blockquote></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">n = 16</span></blockquote><span style="font-family: arial; font-size: medium;"><span>3) Now we will find the value of d.</span> <br /></span><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></span></blockquote></span></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">45 = 5 + d (16 – 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">15d = 45 - 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">15d = 40</blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>d = 40/15</div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>d = 8/3</span> </span></blockquote></div><div><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">So, d = 8/3, n = 16.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b><span>Q</span><span>6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference </span></b></span><b>is 9, how many terms are there, and what is their sum?</b></span></div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">a</span><sub>1</sub> = a = 17, <span face="Arial, sans-serif">a</span><sub>n</sub> = 350, d = 9,<span style="font-family: arial;"> </span>find n and <span face="Arial, sans-serif">S</span><sub>n</sub>.</div><div><div><span style="font-family: arial;"><span>2) Now we will find the value of d.</span> <br /></span><div><span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></span></blockquote></span></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">350 = 17 + 9(n – 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">9(n – 1) = 350 - 17</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">9(n – 1) = 333</blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div>(n – 1) = 333/9</div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div>(n – 1) = 37</div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n = 37 + 1</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n = 38</span> </blockquote></div><div>3) <span style="font-family: arial;"><span>We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[a + l]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (38/2)[(17) + 350]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>n</sub> = 19[17 + 350]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>n</sub> = 19(367)</blockquote></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span face="Arial, sans-serif">S</span><sub>n</sub> = 6973</div></span></blockquote></div></div></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>4) </span><span>So, n = 38, </span><span>S</span><sub>n</sub><span> = 6973</span><span>.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q7. Find the sum of the first 22 terms of an AP in which d = 7 and the 22nd term is 149.</b></span></div><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">a</span><sub>22</sub> = 149, d = 7,<span style="font-family: arial;"> </span>find and <span face="Arial, sans-serif">S</span><sub>22</sub>.</div><div><div><span style="font-family: arial;"><span>2) Now we will find the value of d.</span> <br /></span><div><span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></span></span></blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div></div></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>22</sub><span style="font-family: arial;"> = a + (22 – 1) (7)</span> </div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div></div></span></div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">149 = a + 7(22 – 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a = 149 - 7(21)</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a = 149 - 147</blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div>a = 2<span> </span></div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></div></blockquote></div></div><div>3) <span style="font-family: arial;"><span>We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[a + l]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>22</sub> = (22/2)[(2) + 149]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>22</sub> = (11)[151]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>22</sub> = 1661<br /></blockquote></div></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><span>4) </span><span>So, a = 2, </span><span>S</span><sub>n</sub><span> = 1661</span><span>.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b><span>Q</span><span>8. Find the sum of the first 51 terms of an AP whose second and third terms are 14 and 18 </span></b><b>respectively.</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">a</span><sub>2</sub> = 14, <span face="Arial, sans-serif">a</span><sub>3</sub> = 18, find and <span face="Arial, sans-serif">S</span><sub>51</sub>.</div><div><div><span style="font-family: arial;">2) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>d = </span><span>a</span><sub>3</sub><span> - </span><span>a</span><sub>2</sub></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = 18</span><span style="font-family: arial;"> - 14<br /></span><span style="font-family: arial;">d = 4</span></span></blockquote></div><div><div><span style="font-family: arial;"><span>3) Now we will find the value of d.</span> <br /></span><div><span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></span></span></blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div></div></div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = a + (2 – 1) (4)</span> </div></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div></div></span></div></span></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">14 = a + 4(1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a = 14 - 4</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a = 10</blockquote></span></span></span></div></span></span></span></div></span></span></div></span></span></span></span></span></span></div></span></span></span></div></span></div></span></span></div></div></div><div>4) <span style="font-family: arial;"><span>We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[2a + (n - 1)d]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>51</sub> = (51/2)[2(10) + (51 - 1)(4)]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>51</sub> = (51/2)[20 + (50)(4)]</blockquote></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span face="Arial, sans-serif">S</span><sub>51</sub> = (51/2)[20 + 200]</div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>51</sub><span> = (51/2)[220]<br /></span><span>S</span><sub>51</sub><span> = 51[110]<br /></span><span>S</span><sub>51</sub><span> = 5610</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>5) </span><span>So, </span><span>S</span><sub>51</sub><span> = 5610</span><span>.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q9. If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of the </b></span><b>first n terms.</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) Here, </span><span face="Arial, sans-serif">S</span><sub>7</sub> = 49, <span face="Arial, sans-serif">S</span><sub>17</sub> = 289, find and <span face="Arial, sans-serif">S</span><sub>n</sub>. Let the first term be 'a' and the common</div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">difference be 'd'. </div></span></div></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">2) </span><span style="font-family: arial;"><span>We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></div></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[2a + (n - 1)d]</span></blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>7</sub> = (7/2)[2a + (7 - 1)(d)]</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">49 = (7/2)[2a + (6)(d)]</blockquote></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">49 = 7[a + 3d]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a + 3d = 49/7</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">a + 3d = 7</span><span style="font-family: arial;"> --------- equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Now we will use </span><span>S</span><sub>17</sub><span> = 289 to get another equation</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> = (n/2)[2a + (n - 1)d]</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>17</sub><span> = (17/2)[2a + (17 - 1)d]</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>17</sub><span> = (17/2)[2a + (16)d]</span></span></div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>17</sub><span> = 17[a + 8d]<br /></span><span>289 = 17[a + 8d]<br /></span><span>17[a + 8d] = 289<br /></span><span>a + 8d = 289/17<br /></span><span>a + 8d = 17</span><span> --------- equation 2</span></span></blockquote><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) Subtract equation 1 from </span><span style="font-family: arial;">equation 2</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a + 8d = 17</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + 3d = 7</span></div></div></blockquote><div><span style="font-family: arial; font-size: medium;"> <span> </span>( - ) ( - ) ( - )</span></div><div><span style="font-family: arial; font-size: medium;">--------------------------</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> 5d = 10</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">d = 10/5</span></div></div></span></div></div></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = 2 --</span><span style="font-family: arial;">------- equation 3</span></span></div></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = 2 from</span><span style="font-family: arial;"> equation 3 in equation 1, and we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a + 3d = 7</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a + 3(2) = 7</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">a + 6 = 7</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a = 7 - 6</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 1</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) </span><span style="font-family: arial;">We know that the </span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[2a + (n - 1)d] </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">S</span><sub>n</sub> = (n/2)[2(1) + (n - 1)(2)]</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">S</span><sub>n</sub> = n[1 + (n - 1)]</blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> = n[n]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> = </span><span>n</span><sup>2</sup> </span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>5) </span><span>So, </span><span>S</span><sub>n</sub><span> = </span><span>n</span><sup>2</sup><span>.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;"><span>Q</span><span>10. Show that </span><span>a</span><sub>1,</sub><span> </span><span>a</span><sub>2,</sub><span> . . ., </span><span>a</span><sub>n</sub><span>, . . . form an AP where </span><span>a</span><sub>n</sub><span> </span><span>is defined as below :</span></span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(i) </span><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n <span> </span>(ii) </span><span>a</span><sub>n</sub><span> =</span><span> 9 – 5n. </span></span></b></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Also, find the sum of the first 15 terms in each case.</span></b></div></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(i) </span><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n.</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>1) Put n = 1, 2, 3 in equation </span><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n to get </span><span>a</span><sub>1</sub><span>, </span><span>a</span><sub>2</sub><span>, </span><span>a</span><sub>3 </sub><span>and so on.</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Put n = 1,</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 3 + 4(1)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 3 + 4</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 7</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b) Put n = 2,</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> 3 + 4(2)</span><br /><span></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> 3 + 8</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> 11</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>c) Put n = 3,</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> 3 + 4(3)</span><br /><span></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> 3 + 12</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> 15</span> </span></div></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span><span>2) </span></span><span>Here, </span><span>a</span><sub>1</sub><span> =</span><span> 7, </span><span>a</span><sub>2</sub><span> =</span><span> 11, </span><span>a</span><sub>3</sub><span> =</span><span> 15.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub></span></blockquote></blockquote></div></div><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = 11 - 7</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span face="Arial, sans-serif">d = 4</span></span><span style="font-family: arial;"> --------- equation 1</span></span></div></div></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = 15</span><span face="Arial, sans-serif"> - 11</span></span></div></div></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = </span><span face="Arial, sans-serif">4 --------- equation 2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span><span>3) Here </span><span face="Arial, sans-serif">the common difference d = </span></span><span>a</span><sub>2</sub><span> - </span><span>a</span><sub>1 </sub><span>= </span><span>a</span><sub>3</sub><span> - </span><span>a</span><sub>2 </sub><span>= 4, so the equation </span><span>a</span><sub>n</sub><span> =</span><span> 3 + 4n</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">form an AP and their terms are 7, 11, 15 . . .</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Now we will find the sum of the first 15 terms.</span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>5) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub> = (n/2)[2a + (n – 1) d]</blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15/2)[2(7) + (15 – 1) (4)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15/2)[2(7) + 4(14)]<br /></span><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15)[7 + 2(14)]<br /></span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">S</span><sub>15</sub> = (15)[7 + 28] </div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = </span><span style="font-family: arial;">(15)(35)</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub></span><span style="font-family: arial;"> = 525</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>6) </span><span>So, here </span><span>S</span><sub>n</sub><span> = 525</span><span>.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(ii) </span><span>a</span><sub>n</sub><span> =</span><span> 9 – 5n.</span></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Put n = 1, 2, 3 in equation </span><span>a</span><sub>n</sub><span> =</span><span> 9 - 5n to get </span><span>a</span><sub>1</sub><span>, </span><span>a</span><sub>2</sub><span>, </span><span>a</span><sub>3 </sub><span>and so on.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) Put n = 1,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> 9 - 5n</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 9 - 5(1)<br /></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 9 - 5</span></span><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span> =</span><span> 4</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) Put n = 2,</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> </span><span>9 - 5n</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> </span><span>9 - 5(2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> </span><span>9 - 10</span></span><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> - 1</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>c) Put n = 3,</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> =</span><span> </span><span>9 - 5n</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> </span><span>9 - 5(3)</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> </span><span>9 - 15</span><br /></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> =</span><span> - 6</span></span></div></blockquote></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>Here, </span><span>a</span><sub>1</sub><span> =</span><span> 4, </span><span>a</span><sub>2</sub><span> =</span><span> - 1, </span><span>a</span><sub>3</sub><span> =</span><span> - 6.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub></span></blockquote></blockquote></div></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d = - 1 - 4</span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">d = - 5</span><span style="font-family: arial;"> --------- equation 1</span></span></div></div></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = - </span><span face="Arial, sans-serif"> 6 - (-1)</span></span></div></div></blockquote></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>d = - </span><span> 6 + 1</span> </span></div></blockquote></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = </span><span face="Arial, sans-serif">- 5 --------- equation 2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span><span>3) Here </span><span face="Arial, sans-serif">the common difference d = </span></span><span>a</span><sub>2</sub><span> - </span><span>a</span><sub>1 </sub><span>= </span><span>a</span><sub>3</sub><span> - </span><span>a</span><sub>2 </sub><span>= - 5, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>so the equation </span><span>a</span><sub>n</sub><span> =</span><span> 9 - 5n </span><span>form an AP and their terms are 4, - 1, - 6 . . .</span></span></div></div></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">4) Now we will find the sum of the first 15 terms.</span></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>5) We know that the </span></span><span style="font-family: arial;">sum of the first n terms of an AP is given by:</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">S</span><sub>n</sub> = (n/2)[2a + (n – 1) d]</blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15/2)[2(4) + (15 – 1) (- 5)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15/2)[2(4) - 5(14)]<br /></span><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = (15)[4 - 5(7)]<br /></span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">S<sub>15</sub> = (15)[4 - 35] </span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub> = </span><span style="font-family: arial;">(15)[- 31] </span></span></div></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">S</span><sub>15</sub></span><span style="font-family: arial;"> = - 465</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"> <span>6) </span><span>So, here </span><span>S</span><sub>n</sub><span> = - 465</span><span>.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span><b>Q11. If the sum of the first n terms of an AP is 4n – </b></span><b><span>n</span><sup>2</sup></b><span><b>, what is the first term (that is S</b></span><sub><b>1</b></sub><span><b>)? What </b></span><b>is the sum of the first two terms? What is the second term? Similarly, find the 3rd, the 10th, and </b><b>the nth terms.</b></span></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Here </span><span>S</span><sub>n</sub><span> =</span><span> </span><span>4n – </span><span>n</span><sup>2</sup><span> so,</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) So, first term </span><span>S</span><sub>1</sub><span> will be,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> </span><span>4n – </span><span>n</span><sup>2</sup></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>1</sub><span> =</span><span> </span><span>4(1) – (1)</span><sup>2</sup></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>1</sub><span> =</span><span> </span><span>4 – 1</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>1</sub><span> =</span><span> 3 = </span><span>a</span><sub>1</sub><span> = a</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b) To get sum of first two terms, put n = 2, in </span><span>S</span><sub>n</sub><span> =</span><span> </span><span>4n – </span><span>n</span><sup>2</sup></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> </span><span>4n – </span><span>n</span><sup>2</sup><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>2</sub><span> =</span><span> </span><span>4(2) – (2)</span><sup>2</sup><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>2</sub><span> =</span><span> </span><span>8 – 4</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>2</sub><span> =</span><span> 4</span></span></div></blockquote></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>We know that the second term </span><span>a</span><sub>2</sub><span> =</span><span> </span><span>S</span><sub>2</sub><span> </span><span>- </span><span>S</span><sub>1</sub><span>.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> </span><span>S</span><sub>2</sub><span> </span><span>- </span><span>S</span><sub>1</sub></span></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> 4</span><span> </span><span>- 3</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> =</span><span> 1</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) The common difference will be:</span></div><div style="text-align: left;"><div><div><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub></span></blockquote></div></div><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d = 1 - 3</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">d = - 2</span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Now we will find 3rd term:</span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> = 3 + (3 – 1) (- 2)</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>3</sub><span> = 3 + (2) (- 2)<br /></span><span>a</span><sub>3</sub><span> = 3 - 4<br /></span><span>a</span><sub>3</sub><span> = - 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) Now we will find the 10th term:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>10</sub><span> = 3 + (10 – 1) (- 2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>10</sub><span> = 3 + (9) (- 2)<br /></span><span>a</span><sub>10</sub><span> = 3 - 18<br /></span><span>a</span><sub>10</sub><span> = - 15</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">6) Now we will find the nth term:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 3 + (n – 1) (- 2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 3 - 2n + 2<br /></span><span>a</span><sub>n</sub><span> = 5 - 2n</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) So here we have:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) First term <span>S</span><sub>1</sub> = a = 3.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b) </span><span>Sum of first two terms</span><span> </span><span>S</span><sub>2</sub><span> = 4.</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>c) The second term </span><span>a</span><sub>2</sub><span> =</span><span> 1</span></span></div><span style="font-family: arial; font-size: medium;"><span>d) The third term </span><span>a</span><sub>3</sub><span> =</span><span> - 1<br /></span><span>e) The 10th term </span><span>a</span><sub>10</sub><span> =</span><span> - 15<br /></span><span>f) The nth term </span><span>a</span><sub>n</sub><span> =</span><span> </span><span>5 - 2n</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span> </span><br /></span><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q12. Find the sum of the first 40 positive integers divisible by 6.</b></span></div></div></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Here </span><span>a</span><sub>1</sub><span> will be 6, and </span><span>a</span><sub>2</sub><span> will be 12, so here d = 6.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">So, now we will find sum of first 40 integers</span><span style="font-family: arial; font-size: medium;">,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>40</sub><span> =</span><span> </span><span>(</span><span>40/2)[2(6) + (</span><span><span>40 – </span><span>1</span>) (6)]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>40</sub><span> =</span><span> 20</span><span>[12 + 6(39</span><span>)]</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>40</sub><span> =</span><span> 20</span><span>[12 + 234</span><span>]</span></span></div></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>40</sub><span> =</span><span> 20</span><span>[246</span><span>]</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>40</sub><span> =</span><span> 4920</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">2) So the sum of the first 40 terms will be 4920.</span></div></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><span style="font-family: arial; font-size: medium;"><b>Q</b><span><b>13. Find the sum of the first 15 multiples of 8.</b></span></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Here </span><span>a</span><sub>1</sub><span> will be 8, and </span><span>a</span><sub>2</sub><span> will be 16, so here d = 8.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">So, </span><span style="font-family: arial; font-size: medium;">now we will find sum of first 15 multiples of 8</span><span style="font-family: arial; font-size: medium;">,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>15</sub><span> =</span><span> </span><span>(</span><span>15/2)[2(8) + (</span><span><span>15 – </span><span>1</span>) (8)]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>15</sub><span> =</span><span> 15</span><span>[8 + 4(14</span><span>)]</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>15</sub><span> =</span><span> 15</span><span>[8 + 56</span><span>]</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>15</sub><span> =</span><span> 15</span><span>[64</span><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>15</sub><span> =</span><span> 960</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) So the sum of the first 15 multiples of 8 will be 960.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q14. Find the sum of the odd numbers between 0 and 50.</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here, our odd numbers are 1, 3, 5, . . 47, 49.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) We have </span><span>a</span><sub>1</sub><span> = 1, </span><span>a</span><sub>2</sub><span> = 3, d = 2 and </span><span>a</span><sub>n</sub><span> = 49,</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">49 = 1 + 2(n – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">2(n – 1) = 49 - 1</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2(n – 1) = 48</span><div><span style="font-family: arial;">(n – 1) = 48/2</span></div></blockquote></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">(n – 1) = 24 <br /></div></span></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>n = 25</span></span></div></blockquote><div style="text-align: left;"> 3) According to the problem,</div></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">So, </span><span style="font-family: arial; font-size: medium;">now we will find sum of odd numbers from 0 to 50.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[a + l]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>25</sub><span> =</span><span> </span><span>(</span><span>25/2)[1 + 49</span><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>25</sub><span> =</span><span> (25/2)</span><span>[50</span><span>]</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>25</sub><span> =</span><span> 25</span><span>[25</span><span>]</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>25</sub><span> =</span><span> 625</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) The sum of all odd numbers between 0 and 50 is 625.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: justify;"><span style="font-family: arial; font-size: medium;"><span><b>Q15. A contract on a construction job specifies a penalty for delay of completion beyond a </b></span><b>certain date as follows: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third </b><b>day, etc., the penalty for each succeeding day being Rs 50 more than for the preceding day. </b><b>How much money does the contractor have to pay as a penalty if he has delayed the work by 30 </b><b>days?</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Here </span><span>a</span><sub>1</sub><span> will be 200, and </span><span>a</span><sub>2</sub><span> will be 250, so here d = 50.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>So, </span><span>now we will find total penalty for the delayed work by 30 days, so we will have to find </span><span>S</span><sub>n</sub><span>.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>30</sub><span> =</span><span> </span><span>(</span><span>30/2)[2(200) + (</span><span><span>30 – </span><span>1</span>) (50)]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>30</sub><span> =</span><span> 15</span><span>[400 + 50(29</span><span>)]</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>30</sub><span> =</span><span> 15</span><span>[400 + 1450</span><span>]</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>30</sub><span> =</span><span> 15</span><span>[1850</span><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>30</sub><span> =</span><span> 27750</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) So the </span><span style="font-family: arial;">total penalty for the delayed work by 30 days is 27750</span><span style="font-family: arial;">.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q16. A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their </b></span><b>overall academic performance. If each prize is Rs 20 less than its preceding prize, find the </b><b>value of each of the prizes.</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the cost of the first prize be Rs x.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the problem, the cost of the second prize is (x - 20).</span></div><div><span style="font-family: arial; font-size: medium;">3) The cost of the third prize is (x - 40). So the common difference is d = - 20.</span></div><div><span style="font-family: arial; font-size: medium;"><span>4) As the sum of Rs 700 is to be used give 7 cash prizes, </span><span>S</span><sub>7</sub><span> = 700</span><span>.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>So, </span><span>now we will find total penalty for the delayed work by 30 days, so we will have to find </span><span>S</span><sub>n</sub><span>.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>7</sub><span> =</span><span> </span><span>(</span><span>7/2)[2(x) + (</span><span><span>7 – </span><span>1</span>) (- 20)]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>7</sub><span> =</span><span> </span><span>(</span><span>7)[(x) + (6</span><span>) (- 10)]</span><br /></span><div><span style="font-size: medium;"><span style="font-family: arial;">700 =</span><span style="font-family: arial;"> </span><span style="font-family: arial;">(</span><span style="font-family: arial;">7)[(x) - 6</span><span style="font-family: arial;">0]</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">100 =</span><span style="font-family: arial;"> x - 60</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x =</span><span style="font-family: arial;"> 100 + 60</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x =</span><span> 160</span> <br /></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">5) So, the value of each of the prizes was (i) Rs 160, (ii) Rs 140, (iii) Rs 120, </span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(iv) Rs 100, (v) Rs 80, (vi) </span><span style="font-family: arial;">Rs </span><span style="font-family: arial;">60, and (vii) Rs 40.</span></span></div></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q17. In a school, students thought of planting trees in and around the school to reduce air </b></span><b>pollution. It was decided that the number of trees, that each section of each class will </b><b>plant, will be the same as the class, in which they are studying, e.g., a section of Class I </b><b>will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are </b><b>three sections of each class. How many trees will be planted by the students?</b></span></div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) Here </span><span>a</span><sub>1</sub><span> will be 1, and </span><span>a</span><sub>2</sub><span> will be 2, </span><span>a</span><sub>n</sub><span> will be 12, </span><span>so here d = 1.</span></span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>So, </span><span>now we will find total penalty for the delayed work by 30 days, so we will have to find </span><span>S</span><sub>n</sub><span>.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> =</span><span> </span><span>(</span><span>12/2)[2(1) + (</span><span><span>12 – </span><span>1</span>) (1)]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> =</span><span> 6</span><span>[2 + 1(11</span><span>)]</span><br /></span><div><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> =</span><span> 6</span><span>[2 + 11</span><span>]</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> =</span><span> 6</span><span>[13</span><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>12</sub><span> =</span><span> 78</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) So one section of a school will plant 78 trees. So 3 sections will plant </span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">78 x 3 = 234 plants.</span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div><span style="font-family: arial; font-size: medium;"><b><span>Q18. A spiral is made up of successive semicircles, with centers alternately at A and B, </span><span>starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . . as shown in </span></b><b>Fig. What is the total length of such a spiral made up of thirteen consecutive </b><b><span>semicircles? (Take p = </span><span>22/7</span><span>)</span></b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-t1W_B5CZLyX-LLLZdMsHnE_MLmPhOEvejO58bHNlrd2dXfuynY2rb-2SEX4o99GZahTJo71_sBFbHE6JuokdIfDFaZkTuxvwdWh43UDQJNPldgITX4wwWaxNE-Mm5OzHKROIubrOTt-ufVmc_Vizs4FNtQfEqrJ7-hQMImo26qpTa-kJsp4VXkE2/s707/5.3-18.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="385" data-original-width="707" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-t1W_B5CZLyX-LLLZdMsHnE_MLmPhOEvejO58bHNlrd2dXfuynY2rb-2SEX4o99GZahTJo71_sBFbHE6JuokdIfDFaZkTuxvwdWh43UDQJNPldgITX4wwWaxNE-Mm5OzHKROIubrOTt-ufVmc_Vizs4FNtQfEqrJ7-hQMImo26qpTa-kJsp4VXkE2/s320/5.3-18.png" width="320" /></a></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial; font-size: medium;">1) We know that the perimeter of the first semicircle is </span><span style="white-space: pre-wrap;">𝞹r</span>.</div><div class="separator" style="clear: both; text-align: left;"><div><span style="font-family: arial; font-size: medium;">2) According to the problem, the perimeters of the semicircle are:</span></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: left;"><div style="text-align: left;"><span style="font-family: arial;">a) r = 0.5 so </span><span> </span><span>a</span><sub>1 </sub><span>= 0.5 </span><span style="white-space: pre-wrap;">𝞹 </span><span>= </span><span style="white-space: pre-wrap;">𝞹/2</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;">b) r = 1 so </span><span> </span><span>a</span><sub>2 </sub><span>= </span><span style="white-space: pre-wrap;">𝞹 </span><span>= </span><span style="white-space: pre-wrap;">𝞹</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">c) r = 1.5 so </span><span> </span><span>a</span><sub>3 </sub><span>= 1.5 </span><span style="white-space: pre-wrap;">𝞹 </span></span><span style="font-family: arial;">= 3</span><span style="font-family: arial; white-space: pre-wrap;">𝞹/2</span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span></span><span>3) Here </span><span>a</span><sub>1, </sub><span>a</span><sub>2, </sub><span>and </span><span>a</span><sub>3</sub><sub>, </sub><span>are the semicircle's perimeter and are in AP.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>4) Here d = </span><span>a</span><sub>2 </sub><span>- </span><span>a</span><sub>1 </sub><span>= </span><span style="white-space: pre-wrap;">𝞹 - (</span><span style="white-space: pre-wrap;">𝞹/2) = (</span><span style="white-space: pre-wrap;">𝞹/2)</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">5) We will have to find </span><span>S</span><sub>13</sub><span>, so we have,</span><span style="white-space: pre-wrap;"> </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> </span><span>(</span><span>13/2)[2(</span></span><span style="font-family: arial; white-space: pre-wrap;">𝞹/2</span><span style="font-family: arial;"><span>) + (</span><span><span>13 – </span><span>1</span>) (</span></span><span style="font-family: arial; white-space: pre-wrap;">𝞹/2</span><span style="font-family: arial;"><span>)]</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> </span><span>(</span><span>13/2)</span></span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">𝞹/2</span><span style="font-family: arial;">)</span><span style="font-family: arial;"><span>[2</span></span><span style="font-family: arial;"><span> + 12</span></span><span style="font-family: arial;">]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> </span><span>(</span><span>13/2)</span></span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">𝞹/2</span><span style="font-family: arial;">)</span><span style="font-family: arial;">[14</span><span style="font-family: arial;">]</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> </span></span><span style="font-family: arial;">(</span><span style="font-family: arial;">13/2)</span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">𝞹</span><span style="font-family: arial;">)</span><span style="font-family: arial;">[7</span><span style="font-family: arial;">]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> (</span><span>13/2)</span></span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">22/7</span><span style="font-family: arial;">)</span><span style="font-family: arial;">[7</span><span style="font-family: arial;">]</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> (</span><span>13/2)</span></span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">22)</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span>S</span><sub>13</sub><span> =</span><span> (</span><span>13)</span></span><span style="font-family: arial;">(</span><span style="font-family: arial; white-space: pre-wrap;">11)</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>S</span><sub>13</sub><span> =</span><span> </span><span>143</span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">6) </span></span><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">So, the length of such a spiral of thirteen consecutive semicircles is 143 cm.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>Q19. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it, and so on. In how many rows are the 200 logs placed and how many logs are in the top row?</b></span></span></div><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><span style="font-family: arial;"><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial;">1) Here a number of logs in the rows are in an AP</span>.</div><div class="separator" style="clear: both;"><span style="font-family: arial;">2) According to the problem, </span><span>a</span><sub>1</sub><sub> </sub><span>= 20</span><sub>, </sub><span>a</span><sub>2</sub><sub> </sub><span>= 19</span><sub>, </sub><span>a</span><sub>3</sub><sub> </sub><span><span>= 18</span><span>. . . </span></span><span>and </span><span>S</span><sub>n </sub><span>= 200</span><span style="vertical-align: sub;">, </span><span>so d = - 1, </span></div></span></div><div><div><span style="font-family: arial;"><span></span><span>3) </span></span><span>We will have to find n using the above information.</span><span> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;"><span>S</span><sub>n</sub><span> =</span><span> </span><span>(</span><span>n/2)[2(</span></span><span style="font-family: arial;">20</span><span style="font-family: arial;"><span>) + (</span><span><span>n – </span><span>1</span>) (</span></span><span style="font-family: arial;">- 1</span><span style="font-family: arial;"><span>)]</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span><span style="font-family: arial;"><span>200 =</span><span> </span><span>(</span><span>n/2)[40 - (n - 1)</span></span><span style="font-family: arial;">]</span></span></blockquote></div></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div style="text-align: left;"><span style="font-family: arial;"><span>200 =</span><span> </span><span>(</span><span>n/2)[40 - n + 1)</span></span><span style="font-family: arial;">]</span> </div></div></span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;"><span>400 =</span><span> n</span></span><span style="font-family: arial;">(</span><span style="font-family: arial;">41 - n</span><span style="font-family: arial;">)</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;"><span>400 =</span><span> 41n - </span></span></span><span style="white-space: normal;"> </span><span style="white-space: normal;">n</span><sup style="white-space: normal;">2</sup></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">n</span><sup style="white-space: normal;">2</sup> - 41n + 400 = 0</blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">n</span><sup style="white-space: normal;">2</sup> - 16n - 25n + 400 = 0</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="white-space: normal;">n(n</span> - 16) - 25(n - 25) = 0</blockquote></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="text-align: left;"><span style="white-space: normal;">(n</span> - 16)(n - 25) = 0</div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(n</span><span style="font-family: arial; white-space: pre-wrap;"> - 16) = 0 or (n - 25) = 0<br /></span><span style="font-family: arial;">n</span><span style="font-family: arial; white-space: pre-wrap;"> = 16 or n = 25</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) Now we will find <span style="font-family: arial;">a</span><sub>16</sub><span style="font-family: arial;"> and </span><span style="font-family: arial;">a</span><sub>25</sub><span style="font-family: arial;"> using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span>.</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a) First we will find <span style="font-family: arial;">a</span><sub>16</sub> </div></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></span></blockquote></span></span></span></div></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><div><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">a</span><sub>16</sub> = 20 + (- 1) (16 – 1)</span></blockquote></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial;">a</span><sub>16</sub> = 20 - (15)</blockquote></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>16</sub> = 5</blockquote></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span><span style="font-family: arial; font-size: medium;">b) First we will find a<sub>25</sub> </span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></span></div><div><span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">a</span><sub>25</sub> = 20 + (- 1) (25 – 1)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>25</sub><span> = 20 - (24)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>25</sub><span> = - 4</span></span></blockquote></blockquote><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div style="text-align: left; white-space: normal;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">5) </span></span><span style="font-family: arial;"><span style="white-space: pre-wrap;">As the number of logs can't be negative, </span></span><span style="font-family: arial;">a</span><sub>25</sub> = - 4 is impossible<span style="white-space: pre-wrap;">. So 200 logs</span></div></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div style="text-align: left; white-space: normal;"><span style="white-space: pre-wrap;">can be placed in 16 rows and there are 5 logs in the 16th log.</span></div></div></span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div><span style="font-family: arial;"><span style="white-space: pre-wrap;"><br /></span></span></div></div></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>Q20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.</b></span></span></div><div><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-WS4_-p45QYGb_AN-uiOsqELdq_b909uAOSQcTPh27o4Z0dZOmP9HwMkKlEd-lursdYVLgn8HD8UMvb4KCR3rzeLkd1E2Zj8HhUvq4tlVLm6wixG9JrX-93XD85YhTXbmvY6t-aJ66zrS3ElBOwp8jdXp_cohIaZq0T7Rxypo_kdBnRIX5RchdDpx/s1087/5.3-20.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="202" data-original-width="1087" height="107" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-WS4_-p45QYGb_AN-uiOsqELdq_b909uAOSQcTPh27o4Z0dZOmP9HwMkKlEd-lursdYVLgn8HD8UMvb4KCR3rzeLkd1E2Zj8HhUvq4tlVLm6wixG9JrX-93XD85YhTXbmvY6t-aJ66zrS3ElBOwp8jdXp_cohIaZq0T7Rxypo_kdBnRIX5RchdDpx/w576-h107/5.3-20.png" width="576" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div><b>A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops </b><b>it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and </b><b>continues in the same way until all the potatoes are in the bucket. What is the total </b><b>distance the competitor has to run?</b></div><div><b>[Hint: To pick up the first potato and the second potato, the total distance (in meters) </b><b>run by a competitor is 2 × 5 + 2 × (5 + 3)]</b></div><span style="white-space: pre-wrap;"><div style="font-family: "Times New Roman"; font-size: medium; white-space: normal;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><span style="font-family: arial;"><div><h3><span style="font-family: arial;">Solution:</span></h3></div><div><span style="font-family: arial;">1) The potatoes are placed at 5, 8, 11, 14, 17 . . . distances from the bucket.</span></div><div><span style="font-family: arial;">2) These distances form an AP.</span></div><div class="separator" style="clear: both;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">According to the problem, a competitor travels distances as 10, 16, 22, 28, . . . </span></div><div class="separator" style="clear: both;"><span style="font-family: arial;">4) So </span><span>a</span><sub>1</sub><sub> </sub><span>= 10</span><sub>, </sub><span>a</span><sub>2</sub><sub> </sub><span>= 16</span><sub>, </sub><span>a</span><sub>3</sub><sub> </sub><span><span>= 22, </span></span><span>a</span><sub>4</sub><sub> </sub><span><span>= 28</span></span><span><span>. . . </span></span><span style="vertical-align: sub;">, </span><span>so d = 6, </span></div></span></div><div><span style="font-family: arial;"><span></span><span>5) Now </span></span><span>we will have to find <span>S</span><sub>10</sub> using the above information.</span><span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>S</span><sub>n</sub><span> =</span><span> (</span><span>n/2)[2a + (n – </span><span>1) d</span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;"><span>S</span><sub>10</sub><span> =</span><span> </span><span>(</span><span>10/2)[2(</span></span><span style="font-family: arial;">10</span><span style="font-family: arial;"><span>) + (</span><span><span>10 – </span><span>1</span>) (</span></span><span style="font-family: arial;">6</span><span style="font-family: arial;"><span>)]</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span><span style="font-family: arial;"><span style="font-family: arial;"><span>S</span><sub>10</sub><span> =</span><span> </span><span>(</span><span>10/2)(2)</span><span>[10</span></span><span style="font-family: arial;"><span> + (9</span><span>) (</span></span><span style="font-family: arial;">3</span><span style="font-family: arial;"><span>)]</span></span></span></span></blockquote></div></span></span></div><blockquote style="border: none; font-family: "Times New Roman"; font-size: medium; margin: 0px 0px 0px 40px; padding: 0px; white-space: normal;"><div><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><span style="font-family: arial;"><span>S</span><sub>10</sub><span> =</span><span> 10</span><span>[10</span></span><span style="font-family: arial;"><span> + 27</span></span><span style="font-family: arial;"><span>]</span></span> </div></span></span></div></blockquote><div style="font-family: "Times New Roman"; white-space: normal;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>S</span><sub>10</sub><span> =</span><span> 10</span><span>[37</span></span><span style="font-family: arial;">]</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>S</span><sub>10</sub><span> =</span><span> 370</span></span></blockquote></div></span></span></div><div style="white-space: normal;"><span style="font-family: arial;"><span style="font-size: medium; white-space: pre-wrap;"><div style="font-family: "Times New Roman"; white-space: normal;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">6) A competitor will have to run a total distance of 370 m</span></span><span style="white-space: pre-wrap;">.</span></div><div style="font-family: "Times New Roman"; white-space: normal;"><span style="white-space: pre-wrap;"><br /></span></div><div style="font-family: "Times New Roman"; white-space: normal;"><span style="white-space: pre-wrap;"><span style="background-color: white; color: #161719; font-family: arial; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on Arithmetic Progressions, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #ArithmeticProgressions #education #learning #students #teachers #math</span></span></div><h3 style="text-align: left; white-space: normal;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/11/163-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff; font-family: arial;">Click here for</span><span style="color: #0400ff;"> </span><span style="color: #0400ff;">⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.4</span></a></span></h3><h3 style="text-align: left; white-space: normal;"><span style="font-weight: normal;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; font-family: "Times New Roman"; font-size: medium; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a><span style="font-family: "Times New Roman"; font-size: medium;"> </span></span></h3></span></span></div></span></span></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-11337138788628769852023-09-21T17:42:00.002+05:302023-10-12T16:40:09.128+05:30161-NCERT-10-5-Arithmetic Progressions - Ex-5.2<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 5.2</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 5 Arithmetic Progressions</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/09/160-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.1</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 5.2</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q1. Fill in the blanks in the following table, given that a is the first term, d is the common </b></span><b>difference, and </b><span><b><span face="Arial, sans-serif">a</span><sub>n</sub></b></span><span face="Arial, sans-serif"> </span><b>the nth term of the AP: </b></span></div><div style="text-align: left;"><span id="docs-internal-guid-0d9c2fb0-7fff-7b42-7458-b01f46476dbb" style="font-family: arial; font-size: medium;"><div align="center" dir="ltr" style="margin-left: 0pt;"><br /></div></span></div><div style="text-align: left;"><div align="center">
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; mso-yfti-tbllook: 1184;">
<tbody><tr style="height: 24.65pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid black; height: 24.65pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif"> </span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-left: none; border: 1pt solid black; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><b><span face="Arial, sans-serif">a</span></b><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-left: none; border: 1pt solid black; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><b><span face="Arial, sans-serif">d</span></b><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-left: none; border: 1pt solid black; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><b><span face="Arial, sans-serif">n</span></b><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-left: none; border: 1pt solid black; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><b><span face="Arial, sans-serif">a</span></b><b><sub><span face="Arial, sans-serif">n</span></sub></b><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
<tr style="height: 24.65pt; mso-yfti-irow: 1;">
<td style="border-top: none; border: 1pt solid black; height: 24.65pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">i</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">7</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">3</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">8</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-----</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
<tr style="height: 24.65pt; mso-yfti-irow: 2;">
<td style="border-top: none; border: 1pt solid black; height: 24.65pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">ii</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-18</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-----</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">10</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">0</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
<tr style="height: 24.65pt; mso-yfti-irow: 3;">
<td style="border-top: none; border: 1pt solid black; height: 24.65pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">iii</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-----</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-3</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">18</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-5</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
<tr style="height: 24.65pt; mso-yfti-irow: 4;">
<td style="border-top: none; border: 1pt solid black; height: 24.65pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">iv</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-18.9</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">2.5</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-----</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">3.6</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
<tr style="height: 24.65pt; mso-yfti-irow: 5; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: 1pt solid black; height: 24.65pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 29.45pt;" valign="top" width="39">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">v</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">3.5</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.4pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">0</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">105</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
<td style="border-bottom: 1pt solid black; border-left: none; border-right: 1pt solid black; border-top: none; height: 24.65pt; mso-border-left-alt: solid black 1.0pt; mso-border-top-alt: solid black 1.0pt; padding: 0cm; width: 35.45pt;" valign="top" width="47">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 6.0pt; margin: 6pt 0cm 0cm; text-align: center;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">-----</span><span style="mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></span></p>
</td>
</tr>
</tbody></table>
</div></div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) In general, </span><span face="Arial, sans-serif">a</span><sub>n </sub><span face="Arial, sans-serif">is known as the nth term of an AP, and </span><span face="Arial, sans-serif">a</span><sub>r</sub> is known as the general</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">term or the rth term of an AP.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) Sometimes </span><span>a</span><sub>n </sub><span>is also called the last term of an AP and is denoted by 'l'.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><span>i) </span></b><b><span>a = 7, d = 3, n</span><span> = 8, find </span><span>a</span><sub>n.</sub></b><b><span> </span></b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a = 7, d = 3, n = 8,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 7 + 3(8 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 7 + 3(8 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 7 + 3(7)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 7 + 21</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 28</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) So, here </span><span>a</span><sub>n</sub><span> = 28.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><span>ii) </span></b><b><span>a = - 18, </span></b><b><span>a</span><sub>n</sub></b><b><span> = 0, n</span><span> = 10, find d</span><sub>.</sub></b></span></div><div><b><sub><span style="font-family: arial; font-size: medium;"><br /></span></sub></b></div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a = - 18, </span><span>a</span><sub>n</sub><span> = 0, n = 10,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">0 = - 18 + d(10 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">0 = - 18 + 9d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">9d = 18</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">d = 18/9</span></div><div><span style="font-family: arial; font-size: medium;">d = 2.</span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, here d</span><span style="font-family: arial;"> = 2.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b><span>iii) </span></b><b><span>d = - 3, </span></b><b><span>a</span><sub>n</sub></b><b><span> = - 5, n</span><span> = 18, find a</span><sub>.</sub></b></span></div><div><b><sub><span style="font-family: arial; font-size: medium;"><br /></span></sub></b></div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>d = - 3, </span><span>a</span><sub>n</sub><span> = - 5, n = 18,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">- 5 = a + (- 3)(18 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">- 5 = a + (- 3)(17)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">- 5 = a - 51</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 51 - 5</span></div><div><span style="font-family: arial; font-size: medium;">a = 46.</span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, here a</span><span style="font-family: arial;"> = 46.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b><span>iv) </span></b><b><span>a = - 18.9, d = 2.5, </span></b><b><span>a</span><sub>n</sub></b><b><span> = 3.6, find n</span><sub>.</sub></b><b><span> </span></b></span></div><div><b><sub><span style="font-family: arial; font-size: medium;"><br /></span></sub></b></div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a = - 18.9, d = 2.5, </span><span>a</span><sub>n</sub><span> = 3.6,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">3.6 = - 18.9 + (2.5)(n – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(2.5)(n – 1)</span><span style="font-family: arial;"> = 3.6 + 18.9</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">(2.5)(n – 1)</span><span style="font-family: arial;"> = 22.5</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">(n – 1)</span><span style="font-family: arial;"> = 22.5/2.5</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(n – 1)</span><span style="font-family: arial;"> = 225/25</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(n – 1)</span><span style="font-family: arial;"> = 9</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">n</span><span style="font-family: arial;"> = 9 + 1</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">n = 10.</span></div></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, here n</span><span style="font-family: arial;"> = 10.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b><span>v) </span></b><b><span>a = 3.5, d = 0, n</span><span> = 105, find </span><span>a</span><sub>n.</sub></b><b><span> </span></b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 3.5, d = 0, n = 105,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 3.5 + 0(105 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 3.5 + 0(104)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 3.5 + 0</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 3.5</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) So, here </span><span>a</span><sub>n</sub><span> = 3.5.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q2. Choose the correct choice in the following and justify :</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) 30th term of the AP: 10, 7, 4, . . . , is</b></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(A) 97 (B) 77 (C) –77 (D) – 87</b></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(ii) 11th term of the AP: – 3, - 1/2, 2</b><b>, . . ., is</b></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(A) 28 (B) 22 (C) –38 (D) – 48 and 1/2</b></span></div></div></blockquote></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) In general, </span><span face="Arial, sans-serif">a</span><sub>n </sub><span face="Arial, sans-serif">is known as the nth term of an AP, and </span><span face="Arial, sans-serif">a</span><sub>r</sub> is known as the general</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">term or the rth term of an AP.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Sometimes </span><span>a</span><sub>n </sub><span>is also called the last term of an AP and is denoted by 'l'.</span></span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) 30th term of the AP: 10, 7, 4, . . . , is</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(A) 97 (B) 77 (C) –77 (D) – 87</b></span></div></div></blockquote></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>d = </span><span>a</span><sub>2</sub><span> - </span><span>a</span><sub>1</sub></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = 7</span><span style="font-family: arial;"> - 10<br /></span><span style="font-family: arial;">d = - 3</span></span></blockquote><span style="font-family: arial; font-size: medium;"> <span>2) So here,</span><br /></span><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 10, d = - 3, n = 30,</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>30</sub> = 10 + (30 – 1) (- 3)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>30</sub> = 10 - 3(29)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>30</sub> = 10 - 87</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>30</sub><span> = - 77</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) So, the answer is B i.e. </span><span>a</span><sub>30</sub><span> = - 77.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 11th term of the AP: - 3, - 1/2, 2</b><b>, . . ., is</b></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><b>(A) 28 (B) 22 (C) –38 (D) – 48 and 1/2</b></span></div></div></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>d = </span><span>a</span><sub>2</sub><span> - </span><span>a</span><sub>1</sub></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = (- 1/2)</span><span style="font-family: arial;"> - (- 3)<br /></span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = - 1/2</span><span style="font-family: arial;"> + 3</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d = (6 - 1)/2</span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = 5/2</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>2) So here,</span><br /></span><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = - 3, d = 5/2, n = 11,</span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span> </span></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>11</sub> = - 3 + (11 – 1) (5/2)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>11</sub> = - 3 + (5/2)(10)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>11</sub> = - 3 + 25</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>11</sub><span> = 22</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) So, the answer is B i.e. </span><span>a</span><sub>11</sub><span> = 22.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>Q3. In the following APs, find the missing terms in the boxes :</b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 2, <span style="text-align: center;">囗, </span>26</b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(ii) <span style="text-align: center;">囗</span>, 13, <span style="text-align: center;">囗</span>, 3</b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(iii) 5, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, 9½</b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(iv) - 4, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, 6</b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(v) <span style="text-align: center;">囗</span>, 38, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, - 22</b></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) In general, </span><span face="Arial, sans-serif">a</span><sub>n </sub><span face="Arial, sans-serif">is known as the nth term of an AP, and </span><span face="Arial, sans-serif">a</span><sub>r</sub> is known as the general</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">term or the rth term of an AP.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Sometimes </span><span>a</span><sub>n </sub><span>is also called the last term of an AP and is denoted by 'l'.</span></span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 2, <span style="text-align: center;">囗, </span>26</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><div><span style="font-family: arial; font-size: medium;">1) Here,</span></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span face="Arial, sans-serif">a</span><sub>1</sub> = 2, <span face="Arial, sans-serif">a</span><sub>2</sub> = ?, <span face="Arial, sans-serif">a</span><sub>3</sub> = 26</div></div></span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d, we can find the value of d and all other terms.</span> </span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub> = a = 2 --------- equation 1</span></div></blockquote></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = a + (3 - 1)(d)</span></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = 2 + d(2)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">26 = 2 + 2d</span></div></blockquote></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;">2d = 26 - 2</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div></div></span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial;">2d = 24</span></div></div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = 24/2</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">d = 12</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 2</span></div></div></span></span></div></blockquote><div><span><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span><span><div></div></span></span></div></blockquote><div style="text-align: left;"><span>3) From equation 1 and equation 2, we have a = 2, d = 12</span>.<br /></div><div style="text-align: left;"><span>4) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d, we can find the value of </span><span>a</span><sub>2.</sub></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></div></div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = 2 + (2 – 1) (12)</span></div></div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2</sub><span> = 2 + 12(1)<br /></span><span>a</span><sub>2</sub><span> = 14</span> </span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;">5) So, the missing term </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> is 14.</span></div></div></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><b>(ii) <span style="text-align: center;">囗</span>, 13, <span style="text-align: center;">囗</span>, 3</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>1</sub> = ? (say a), a<sub>2</sub> = 13, a<sub>3</sub> = ?, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 3</span></div></blockquote><div><div><span style="font-family: arial;">2) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d, we can find the value of d and all other terms.</span> </div></div></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = a --------- equation 1</span></div></div></div></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a<sub>2</sub> = a + (2 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub> = a + d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>13 = a + d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a + d = 13</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, and </span><span>a</span><sub>4</sub><span> = 3 we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a<sub>4</sub> = a + (4 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub> = a + 3d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>3 = a + 3d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a + 3d = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Subtract equation 2 from equation 3, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + 3d = 3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a + d = 13</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> ( - ) ( - ) ( - ) </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> 2d = - 10 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = - 10/2</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = - 5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = - 5 from</span><span style="font-family: arial;"> equation 4 in equation 2, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + d = 13</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a + (- 5) = 13<br /></span><span style="font-family: arial;">a - 5 = 13<br /></span><span style="font-family: arial;">a = 13 + 5<br /></span><span style="font-family: arial;">a = 18</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 5</span></span></blockquote><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div><span><span><div><span style="font-family: arial;">6) Now we will find </span><span style="font-family: arial;">a</span><sub>3.</sub></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></div><div style="text-align: left;"><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)</span></div><div style="text-align: left;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = 18 + 2 (- 5)</span></div><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 18 - 10</span><br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 8</span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, the missing term </span><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = 18, and </span><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 8</span></div><div><span style="font-family: arial;"><br /></span></div></div><div><span style="font-family: arial;"><b>(iii) 5, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, 9½</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><div><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 5, a<sub>2</sub> = ?, a<sub>3</sub> = ?, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = </span>9½</blockquote><div><span style="font-family: arial;">2) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d, we can find the value of d and all other terms.</span> </div></div></span></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = a = 5 --------- equation 1</span></div></div></span></span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></span></div></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>4</sub> = 5 + (4 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>4</sub> = 5 + 3d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">9½<span style="font-family: arial;"><span> = 5 + 3d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">3d = </span></span>9½ - 5</blockquote></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3d = </span></span>19/2 - 5</div></span></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3d = (</span><span style="font-family: arial;">19 - 10)/2<br /></span><span style="font-family: arial;">3d = </span><span style="font-family: arial;">9/2<br /></span><span style="font-family: arial;">d = </span><span style="font-family: arial;">3/2 </span><span style="font-family: arial;">--------- equation 2</span></span></blockquote><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, a = 5, d = 3/2 </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>2</sub> = a + (2 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a<sub>2</sub> = a + d </div></span></div></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>2</sub> = 5 + 3/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">a</span><sub>2</sub> = (10 + 3)/2</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">a</span><sub>2</sub> = 13/2<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></span></blockquote><div><span style="font-family: arial;">4) Now we will find </span><span face="Arial, sans-serif">a</span><sub>2</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)<br /></span></div></blockquote></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a<sub>3</sub> = a + 2d </div></span></div></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = 5 + 2(3/2)<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub> = 5 + 3<br /><span face="Arial, sans-serif">a</span><sub>3</sub> = 8<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></div></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">5) So, the missing term </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = 13/2, and </span><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 8.</span></div></div></span></span></div></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><b>(iv) - 4, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, 6</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = - 4, a<sub>2</sub> = ?, a<sub>3</sub> = ?, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = </span>?, <span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> = </span>?, <span style="font-family: arial;">a</span><sub>6</sub><span style="font-family: arial;"> = 6.</span></blockquote><div><span style="font-family: arial;">2) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d, we can find the value of d and all other</span></div></div></span></span></span></div></span></div></span></span></div></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial;">terms.</span> </div></div></span></span></span></div></span></div></span></span></div></span></div></span></div></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = a = - 4 --------- equation 1<br /></span></div></div></span></span></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>6</sub> = - 4 + (6 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>6</sub> = - 4 + 5d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">6<span style="font-family: arial;"><span> = - 4 + 5d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">5d = </span></span>6 + 4</blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;">5d = </span></span>10</span></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = </span><span style="font-family: arial;">10/5<br /></span><span style="font-family: arial;">d = </span><span style="font-family: arial;">2 </span><span style="font-family: arial;">--------- equation 2</span></blockquote><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, a = - 4, d = 2 </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>2</sub> = a + (2 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>2</sub> = a + d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>2</sub> = - 4 + 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">a</span><sub>2</sub> = - 2<span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></blockquote><div><span style="font-family: arial;">4) Now we will find </span><span face="Arial, sans-serif">a</span><sub>3</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>3</sub> = a + 2d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = - 4 + 2(2)<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub> = - 4 + 4<br /><span face="Arial, sans-serif">a</span><sub>3</sub> = 0<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></div></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">5) Now we will find </span><span face="Arial, sans-serif">a</span><sub>4</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>4</sub> = a + (4 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>4</sub> = a + 3d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>4</sub> = - 4 + 3(2)<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub> = - 4 + 6<br /></div></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span face="Arial, sans-serif">a</span><sub>4</sub> = 2<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 5</span></span></div></span></div></span></span></div></span></div></span></div></span></div></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">6) Now we will find </span><span face="Arial, sans-serif">a</span><sub>5</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span><br /><div><span style="font-family: arial;">a<sub>5</sub> = a + (5 - 1)(d)<br /></span></div></span></div></span></span></div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>5</sub> = a + 4d </span></div></span></span><div><span style="font-family: arial;"></span></div></span></div></span></span></div></span></div></span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>5</sub> = - 4 + 4(2)<br /><span face="Arial, sans-serif">a</span><sub>5</sub> = - 4 + 8<br /><div><span style="font-family: arial;"></span>a<sub>5</sub> = 4<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 6</span></span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div></div></blockquote></span></div></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, the missing term </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = - 2, </span><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 0, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 2, </span><span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> = 4</span>.</div><div><br /></div></span></span></span></div></span></span></div></span></div><div><span style="font-family: arial;"><b>(v) <span style="text-align: center;">囗</span>, 38, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, <span style="text-align: center;">囗</span>, - 22</b></span></div></span></div><div><span style="font-family: arial;"><br /></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = ?, a<sub>2</sub> = 38, a<sub>3</sub> = ?, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = </span>?, <span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> = </span>?, <span style="font-family: arial;">a</span><sub>6</sub><span style="font-family: arial;"> = - 22.</span></blockquote><div><span style="font-family: arial;">2) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d, we can find the value of d and all other</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">terms.</span> </div></span></span></span></span></div></span></span></div></span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = a --------- equation 1<br /></span></div></div></span></span></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>2</sub> = a + (2 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>2</sub> = a + d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">38<span style="font-family: arial;"><span> = a + d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + d = </span></span>38 --------- equation 2</blockquote></span></div></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, </span></span><span style="font-family: arial;">a</span><sub>6</sub><span style="font-family: arial;"> = - 22</span><span style="font-family: arial;"><span> </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>6</sub> = a + (6 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>6</sub> = a + 5d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- 22 = a + 5d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a + 5d = - 22<span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></blockquote><div><span style="font-family: arial;">4) Subtract equation 2 from equation 3, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 5d = - 22</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + d = 38</span></blockquote><div><span style="font-family: arial;"> ( - ) ( - ) ( - ) </span></div><div><span style="font-family: arial;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"> 4d = - 60 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">d = - 60/4</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = - 15</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></blockquote></blockquote><div><span><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = - 15 from</span><span style="font-family: arial;"> equation 4 in equation 2, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + d = 38</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + (- 15) = 38<br /></span><span style="font-family: arial;">a - 15 = 38<br /></span><span style="font-family: arial;">a = 38 + 15<br /></span><span style="font-family: arial;">a = 53</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 5</span></span><span style="font-family: arial;"> </span></blockquote><div><span style="font-family: arial;">6) Now we will find </span><span face="Arial, sans-serif">a</span><sub>3</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>3</sub> = a + 2d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = 53 + 2(- 15)<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub> = 53 - 30<br /><span face="Arial, sans-serif">a</span><sub>3</sub> = 23<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 6</span></span></div></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">7) Now we will find </span><span face="Arial, sans-serif">a</span><sub>4</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>4</sub> = a + (4 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>4</sub> = a + 3d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>4</sub> = 53 + 3(- 15)<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub> = 53 - 45<br /></div></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>4</sub> = 8<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 7</span></span></span></div></span></span></div></span></div></span></span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">8) Now we will find </span><span face="Arial, sans-serif">a</span><sub>5</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span><br /><div><span style="font-family: arial;">a<sub>5</sub> = a + (5 - 1)(d)<br /></span></div></span></div></span></span></div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>5</sub> = a + 4d </span></div></span></span><div><span style="font-family: arial;"></span></div></span></div></span></span></div></span></div></span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>5</sub> = 53 + 4(- 15)<br /><span face="Arial, sans-serif">a</span><sub>5</sub> = 53 - 60<br /><div><span style="font-family: arial;"></span>a<sub>5</sub> = - 7<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 6</span></span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div></div></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, the missing term </span><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;"> = 53, </span><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = 23, </span><span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 8, </span><span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> = - 7</span>.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div><div><span style="font-family: arial;"><br /></span></div><div><span style="font-family: arial;"><b>Q4. Which term of the AP : 3, 8, 13, 18, . . . , is 78?</b></span></div><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 3, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span>8, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>13, <span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 18.</span></blockquote><div style="text-align: left;">2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = 8</span><span style="font-family: arial;"> - 3<br /></span><span style="font-family: arial;">d = 5</span><span style="font-family: arial;"> --------- equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>3) Here.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">d = a<sub>3</sub> - a<sub>2</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = 13</span><span style="font-family: arial;"> - 8<br /></span><span style="font-family: arial;">d = 5</span><span style="font-family: arial;"> --------- equation 2</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) From equation 1 and equation 2 as d = </span><span>a</span><sub>2</sub><span> - a</span><sub>1 </sub><span>= </span><span>a</span><sub>3</sub><span> - a</span><sub>2 </sub><span>= 5.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d,</span></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">78 = 3 + (n - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">78 = 3 + 5(n - 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1) = 78 - 3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1) = 75</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 75/5</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 15</blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n = </span><span style="font-family: arial;">15 + 1</span></blockquote></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;">n = </span><span style="font-family: arial;">16</span></span> --------- equation 3</div></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">6) So, the 16th term of the given AP is 78.</span></div></span></span></span></div></span></span></div></span></div></span></span></div></span></div><div><span style="font-family: arial;"><br /></span></div></span></div><div style="text-align: left;"><div><b><span style="font-family: arial; font-size: medium;">Q</span><span style="font-family: arial; font-size: medium;">5. Find the number of terms in each of the following APs :</span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(i) 7, 13, 19, . . . , 205 <span> </span>(ii) 18, </span><span>15½</span><span>, 13, . . . , – 47</span></span></b></div></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) In general, </span><span face="Arial, sans-serif">a</span><sub>n </sub><span face="Arial, sans-serif">is known as the nth term of an AP, and </span><span face="Arial, sans-serif">a</span><sub>r</sub> is known as the general</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">term or the rth term of an AP.</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Sometimes </span><span>a</span><sub>n </sub><span>is also called the last term of an AP and is denoted by 'l'.</span></span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></div><div style="text-align: left;"><b><span style="font-family: arial;"><span style="font-size: medium;">(i) 7, 13, 19, . . . , 205</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 7, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span>13, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>19, <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = 205.</span></blockquote><div>2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 13</span><span style="font-family: arial;"> - 7<br /></span><span style="font-family: arial;">d = 6</span><span style="font-family: arial;"> --------- equation 1</span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>3) Here.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>3</sub> - a<sub>2</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 19</span><span style="font-family: arial;"> - 13<br /></span><span style="font-family: arial;">d = 6</span><span style="font-family: arial;"> --------- equation 2</span></blockquote><div style="text-align: left;"><span style="font-family: arial;">4) From equation 1 and equation 2 as d = </span>a<sub>2</sub> - a<sub>1 </sub><span style="font-family: arial;">= </span>a<sub>3</sub> - a<sub>2 </sub><span style="font-family: arial;">= 6.</span> </div></span></div><div><div><span style="font-family: arial; font-size: medium;"><span>5) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d,</span></span></div><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">205 = 7 + (n - 1)(6)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">205 = 7 + 6(n - 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">6(n - 1) = 205 - 7</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">6(n - 1) = 198</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 198/6</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 33</blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n = </span><span style="font-family: arial;">33 + 1</span></blockquote></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-size: medium;">n = </span>34</span></blockquote><div><span style="font-family: arial; font-size: medium;">6) The number of terms in the given AP is 34.</span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span>(ii) 18, </span><span>15½</span><span>, 13, . . . , – 47</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 18, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span><span><span style="font-family: arial;"><span>15½</span></span></span>, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>13, <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = - 47.</span></blockquote><div>2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = </span><span style="font-family: arial;">15½</span><span style="font-family: arial;"> - 18<br /></span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">d = 31/2 - 18</div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d = (31 - 36)/2</span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = - 5/2</span><span style="font-family: arial;"> --------- equation 1</span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>3) Here.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>3</sub> - a<sub>2</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 13</span><span style="font-family: arial;"> - </span><span style="font-family: arial;">15½</span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">d = 13 - 31/2<br /></span><span style="font-family: arial;">d = (26 - 31)/2</span><div style="text-align: left;">d = - 5/2 --------- equation 2</div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">4) From equation 1 and equation 2 as d = </span>a<sub>2</sub> - a<sub>1 </sub><span style="font-family: arial;">= </span>a<sub>3</sub> - a<sub>2 </sub><span style="font-family: arial;">= - 5/2.</span> </div></span></div><div><div><span style="font-family: arial; font-size: medium;"><span>5) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d,</span></span></div><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">- 47 = 18 + (n - 1)(- 5/2)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- 47 = 18 - 5(n - 1)/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1)/2 = 18 + 47</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1)/2 = 65</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1) = 2(65)</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 2(65)/5</blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n - 1) = 2(13)</span></blockquote></span></div></span></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">(n - 1) = 26</div></span></div></span></span></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">n = 26 + 1</span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">n = </span><span style="font-family: arial;">27</span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">6) The number of terms in the given AP is 27.</span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>Q6. Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 11, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span><span><span style="font-family: arial;"><span>8</span></span></span>, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>5, <span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 2.</span></blockquote><div>2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = </span><span style="font-family: arial;">8</span><span style="font-family: arial;"> - 11<br /></span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = - 3</span></blockquote><div><div><span style="font-family: arial;"><span style="font-family: arial;">3) So d = </span>a<sub>2</sub> - a<sub>1 </sub><span style="font-family: arial;">= - 3.</span> </span></div><div><div><span style="font-family: arial;">4) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d,</span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">- 150 = 11 + (n - 1)(- 3)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- 150 = 11 - 3(n - 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3(n - 1) = 11 + 150</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3(n - 1) = 161</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 161/3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">n = (161/3) + 1</blockquote></span></div></span></span></div></span></div></span></span></div></span></span></div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">n = (161 + 3)/3</div></span></div></span></span></div></span></div></span></span></div></span></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">n = (164)/3</span><span style="font-family: arial;"> </span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">5) As, n = 164/3 is not an integer, (- 150) is not a term of this AP.</span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q</b></span><span style="font-family: arial; font-size: medium;"><b>7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.</b></span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial;">Explanation:</span></h3><div><span style="font-family: arial;">1) The nth term <span face="Arial, sans-serif">a</span><sub>n</sub><span face="Arial, sans-serif"> </span>of an AP with the first term 'a' and common difference 'd' is given</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">by <span face="Arial, sans-serif">a</span><sub>n</sub> = a + (n – 1) d.</span></blockquote><div><span style="font-family: arial;"><span>2) In general, </span><span face="Arial, sans-serif">a</span><sub>n </sub><span face="Arial, sans-serif">is known as the nth term of an AP, and </span><span face="Arial, sans-serif">a</span><sub>r</sub> is known as the general</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">term or the rth term of an AP.</span></blockquote><div><span style="font-family: arial;"><span>3) Sometimes </span><span>a</span><sub>n </sub><span>is also called the last term of an AP and is denoted by 'l'.</span></span></div><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>11</sub> = 38, a<sub>16</sub> = 73, find a<sub>31</sub></span><span style="font-family: arial;">.</span></blockquote><div><span style="font-family: arial;">2) Let the first term be a and the common difference be d.</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>11</sub> = a + (11 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>11</sub> = a + 10d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">38<span style="font-family: arial;"><span> = a + 10d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 10d = </span></span>38 --------- equation 1</blockquote></span></div></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, </span></span><span style="font-family: arial;">a</span><sub>16</sub><span style="font-family: arial;"> = 73</span><span style="font-family: arial;"><span> </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>16</sub> = a + (16 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>16</sub> = a + 15d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">73 = a + 15d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a + 15d = 73<span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 2</span></blockquote><div><span style="font-family: arial;">4) Subtract equation 1 from equation 2, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 15d = 73</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + 10d = 38</span></blockquote><div><span style="font-family: arial;"> ( - ) ( - ) ( - ) </span></div><div><span style="font-family: arial;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"> 5d = 35 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">d = 35/5</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = 7</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></span></blockquote></blockquote><div><span><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = 7 from</span><span style="font-family: arial;"> equation 3 in equation 1, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 10d = 38</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 10(7) = 38<br /></span><span style="font-family: arial;">a + 70 = 38<br /></span><span style="font-family: arial;">a = 38 - 70<br /></span><span style="font-family: arial;">a = - 32</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span><span style="font-family: arial;"> </span></blockquote><div><span style="font-family: arial;">6) Now we will find </span><span face="Arial, sans-serif">a</span><sub>31</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>31</sub> = a + (31 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>31</sub> = a + 30d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>31</sub> = - 32 + 30(7)<br /></span><span face="Arial, sans-serif">a</span><sub>31</sub> = - 32 + 210<br /><span face="Arial, sans-serif">a</span><sub>31</sub> = 178<span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 5</span></span></div></blockquote></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, here </span><span style="font-family: arial;">a</span><sub>31</sub><span style="font-family: arial;"> = 178</span>.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q</b></span><span><b>8. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th </b></span><b>term.</b></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">n = 50, a<sub>3</sub> = 12, a<sub>50</sub> = 106, find a<sub>29</sub></span><span style="font-family: arial;">.</span></blockquote><div><span style="font-family: arial;">2) Let the first term be "a" and the common difference be "d".</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = a + 2d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">12<span style="font-family: arial;"><span> = a + 2d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 2d = </span></span>12 --------- equation 1</blockquote></span></div></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, </span></span><span style="font-family: arial;">a</span><sub>50</sub><span style="font-family: arial;"> = 106</span><span style="font-family: arial;"><span> </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>50</sub> = a + (50 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>50</sub> = a + 49d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">106 = a + 49d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a + 49d = 106<span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 2</span></blockquote><div><span style="font-family: arial;">4) Subtract equation 1 from equation 2, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 49d = 106</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + 2d = 12</span></blockquote><div><span style="font-family: arial;"> ( - ) ( - ) ( - ) </span></div><div><span style="font-family: arial;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"> 47d = 94 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">d = 94/47</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></span></blockquote></blockquote><div><span><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = 2 from</span><span style="font-family: arial;"> equation 3 in equation 1, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 2d = 12</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 2(2) = 12<br /></span><span style="font-family: arial;">a + 4 = 12<br /></span><span style="font-family: arial;">a = 12 - 4<br /></span><span style="font-family: arial;">a = 8</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span><span style="font-family: arial;"> </span></blockquote><div><span style="font-family: arial;">6) Now we will find </span><span face="Arial, sans-serif">a</span><sub>29</sub>,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">a<sub>29</sub> = a + (29 - 1)(d)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>29</sub> = a + 28d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>29</sub> = 8 + 28(2)<br /></span><span face="Arial, sans-serif">a</span><sub>29</sub> = 8 + 56<br /><span face="Arial, sans-serif">a</span><sub>29</sub> = 64<span><span style="font-family: arial;">--------- equation 5</span></span></div></blockquote></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, here </span><span style="font-family: arial;">a</span><sub>29</sub><span style="font-family: arial;"> = 64</span>.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b><div>Q9. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?</div></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>3</sub> = 4, a<sub>9</sub> = - 8, find which term is 0</span><span style="font-family: arial;">.</span></blockquote><div><span style="font-family: arial;">2) Let the first term be "a" and the common difference be "d".</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>3</sub> = a + (3 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>3</sub> = a + 2d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">4<span style="font-family: arial;"><span> = a + 2d</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 2d = </span></span>4 --------- equation 1</blockquote></span></div></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, </span></span><span style="font-family: arial;">a</span><sub>9</sub><span style="font-family: arial;"> = - 8</span><span style="font-family: arial;"><span> </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>9</sub> = a + (9 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>9</sub> = a + 8d </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- 8 = a + 8d</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">a + 8d = - 8<span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 2</span></blockquote><div><span style="font-family: arial;">4) Subtract equation 1 from equation 2, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 8d = - 8</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + 2d = 4</span></blockquote><div><span style="font-family: arial;"> ( - ) ( - ) ( - ) </span></div><div><span style="font-family: arial;"> --------------------------- </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"> 6d = - 12 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">d = - 12/6</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">d = - 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></span></blockquote></blockquote><div><span><span style="font-family: arial;">5) Put </span><span style="font-family: arial;">d = - 2 from</span><span style="font-family: arial;"> equation 3 in equation 1, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a + 2d = 4</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">a + 2(- 2) = 4<br /></span><span style="font-family: arial;">a - 4 = 4<br /></span><span style="font-family: arial;">a = 4 + 4<br /></span><span style="font-family: arial;">a = 8</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span><span style="font-family: arial;"> </span></blockquote><div><span style="font-family: arial;">6) Now we will find n when </span><span face="Arial, sans-serif">a</span><sub>n </sub>= 0.</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div><span style="font-family: arial;">0 = 8 + (n - 1)(- 2)<br /></span></div></blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">0 = 8 - 2(n - 1) </span></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">2(n - 1) = 8<div>(n - 1) = 8/2<br /></div></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">(n - 1) = 4 </div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div>n = 5 <span style="font-size: medium;"><span style="font-family: arial;">--------- equation 5</span></span></div></blockquote></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">7) So, here the 5th term is 0</span>.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q</b></span><span style="font-family: arial; font-size: medium;"><b>10. The 17th term of an AP exceeds its 10th term by 7. Find the common difference.</b></span></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">1) Here,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>17</sub> = a<sub>10</sub> + 7 </span><span style="font-family: arial;"><span><span style="font-family: arial;">--------- equation 1</span></span>.</span></blockquote><div><span style="font-family: arial;">2) Let the first term be "a" and the common difference be "d".</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>17</sub> = a + (17 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>17</sub> = a + 16d</span> --------- equation 2</blockquote></span></div></span></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) Using formula a<sub>n</sub> = a + (n – 1) d, </span></span><span style="font-family: arial;">a</span><sub>9</sub><span style="font-family: arial;"> = - 8</span><span style="font-family: arial;"><span> </span><span>we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>a</span><sub>n</sub><span> = a + (n – 1) d</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>10</sub> = a + (10 - 1)(d)</span></blockquote></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">a<sub>10</sub> = a + 9d</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 3</span></div></span></span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">4) From equations 1, 2, and 3 we have,</span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>17</sub><span style="font-family: arial;"> = a</span><sub>10</sub><span style="font-family: arial;"> + 7</span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a + 16d = a + 9d + 7</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + 16d - a - 9d = 7</span></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">16d - 9d = 7 </div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7d = 7 </span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">d = 7/7</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">d = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">--------- equation 4</span></span></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">5) So, here the common difference is 1</span>.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q11. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?</b></span></div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 3, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span>15, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>27, <span style="font-family: arial;">a</span><sub>4</sub><span style="font-family: arial;"> = 39.</span></blockquote><div>2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 15</span><span style="font-family: arial;"> - 3<br /></span><span style="font-family: arial;">d = 12</span><span style="font-family: arial;"> --------- equation 1</span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;">3) Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d,</span></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div></span></div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a</span><sub>54</sub><span style="font-family: arial;"> = 3 + 12 (54 – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>54</sub><span style="font-family: arial;"> = 3 + 12 (53)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>54</sub><span style="font-family: arial;"> = 3 + 636</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a</span><sub>54</sub><span style="font-family: arial;"> = 639</span> --------- equation 2</blockquote></span></div></span></span></div></span></div></span></span></div></span></span></div><div><span style="font-family: arial;">4) </span><span style="font-family: arial;">Now we will find n for the term more by 132 than the 54th term.</span></div></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = </span><span style="font-family: arial;">a</span><sub>54</sub><span style="font-family: arial;"> + 132</span></div></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>n</sub><span> = 639</span><span> + 132<br /></span><span>a</span><sub>n</sub><span> = 771</span><span> --------- equation 3</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>5) Now we will find the value of n.</span> <br /></span><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">771 = 3 + 12 (n – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">12 (n – 1) = 771 - 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">12 (n – 1) = 768</blockquote></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">(n – 1) = 768/12</div></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(n – 1) = 64</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">n = 64 + 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">n = 65 --------- equation 4</span></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">6) So, 65th term is 132 more than 54th term.</span></div><div><span style="font-family: arial; font-size: medium;">7) Second method: our term is 132 more than the 54th term. As the difference is 12,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">132 will give us 132/12 = 11. So our term is 54 + 11 = 65. </span><span style="font-family: arial;">So, 65th term is 132 more than 54th term.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q12. Two APs have the same common difference. The difference between their 100th terms is </b></span><b>100, what is the difference between their 1000th terms?</b></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div>1) Let the first terms of two APs be "x" and "y", and the common difference be "d".</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div style="text-align: left;">2) The 100th term first AP of which the first term is x:</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>100</sub> = x + (100 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>100</sub> = x + 99d</span> --------- equation 1</blockquote></span></div></span></span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span>3) </span></span>The 100th term of the second AP of which the first term is y:</div></span><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>100</sub> = y + (100 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>100</sub> = y + 99d</span> --------- equation 2</blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></span></span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">4) As the difference between their 100th term 100, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(x + 99d) - (y + 99d) = 100</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x - y) = 100</span><span style="font-family: arial;"> --------- equation 3</span></blockquote><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>5) The 1000th term first AP of which the first term is x:</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>1000</sub> = x + (1000 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>1000</sub> = x + 999d</span> --------- equation 4</blockquote></span></div></span></span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span>6) </span></span>The 1000th term second AP of which the first term is y:</span><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>1000</sub> = y + (1000 - 1)(d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>1000</sub> = y + 999d</span> --------- equation 5</blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div><span style="font-family: arial;">7) Now we find the difference between their 1000th terms,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">The difference = (x + 999d) - (y + 999d)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">The difference = (x - y)</span><span style="font-family: arial;"> --------- equation 6</span></blockquote><div style="text-align: left;">8) From equations 3 and 6, we have, </div></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;">The difference = 100</div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">9) So, </span>the difference between their 1000th terms is 100.</div></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q13. How many three-digit numbers are divisible by 7?</b></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>1) Here the first 3-digit number which is divisible by 7 is a = 105 and the common</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;">difference d = 7.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>2) The greatest 3-digit number is 999. When 7 divides 999, the remainder is 5. </div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">So 999-5 divisible by 7. i.e 994 is divisible by 7. So here <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = 997.</span></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span>3) </span></span></span>Using the formula <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d, we have,</span></div></span></span><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">994 = 105 + 7 (n – 1)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">7 (n – 1) = 994 - 105</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">7 (n – 1) = 889</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n – 1) = 889/7</blockquote></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></div></span></span></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(n – 1) = 127</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">n = 127 + 1</span></div><span style="font-family: arial; font-size: medium;">n = 128</span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) </span><span>So, there are 128 3-digit numbers that are divisible by 7.</span><br /></span><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q14. How many multiples of 4 lie between 10 and 250?</b></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>1) Here the first number between 10 and 250 which is divisible by 4 is a = 12 and</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;">the common difference d = 4.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>2) When 4 divides 250, the remainder 2. So 250-2 is divisible by 4. i.e. 248 is</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">divisible by 4. So here <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = 248.</span></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>3) </span><span>Using the formula</span><span> </span><span>a</span><sub>n</sub><span> = a + (n – 1) d, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">248 = 12 + 4 (n – 1)</span></div><span style="font-family: arial;">4 (n – 1) = 248 - 12<br /></span>4 (n – 1) = 236<br />(n – 1) = 236/4</span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></div></span></span></div></span></div></span></div></div><span style="font-family: arial; font-size: medium;">(n – 1) = 59</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">n = 59 + 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">n = 60</span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">So, there are 60 numbers that are divisible by 4 between 10 and 250.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q15. For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?</b></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>1) For first AP, <span style="font-family: arial;">a<sub>1</sub> = a = 63, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span><span><span style="font-family: arial;"><span>65</span></span></span>, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>67.</div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>2) Here.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 65</span><span style="font-family: arial;"> - 63<br /></span><span style="font-family: arial;">d = 2</span></blockquote></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div>2) The nth term first AP will be,</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>n</sub> = 63 + (n - 1)(2)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 63 + 2(n - 1)</span> --------- equation 1</blockquote><div style="text-align: left;">3) For first AP, <span style="font-family: arial;">a<sub>1</sub> = a = 3, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span><span><span style="font-family: arial;"><span>10</span></span></span>, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>17.</div><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div>4) Here.</div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = 10</span><span style="font-family: arial;"> - 3<br /></span><span style="font-family: arial;">d = 7</span></blockquote></div></span></div></span></span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span>5) </span></span>The nth term second AP will be,</span><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a<sub>n</sub> = 3 + (n - 1)(7)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span face="Arial, sans-serif">a</span><sub>n</sub> = 3 + 7</span>(n - 1) --------- equation 2</blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></span></span></div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">6) As nth term of both the APs are the same, from equations 1 and 2 we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">63 + 2(n - 1)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">3 + 7</span><span style="font-family: arial;">(n - 1)</span></blockquote></span></div></span></span></div></span></div></span></span></div></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7</span>(n - 1) - 2(n - 1) = 63 - <span style="font-family: arial; font-size: medium;">3</span></div></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7</span><span style="font-family: arial;">n - 7 - </span><span style="font-family: arial;">2n + 2 = 60<br /></span><span style="font-family: arial;">5</span><span style="font-family: arial;">n - 5</span><span style="font-family: arial;"> = 60<br /></span><span style="font-family: arial;">5</span><span style="font-family: arial;">n = </span><span style="font-family: arial;">60 + 5<br /></span><span style="font-family: arial;">5</span><span style="font-family: arial;">n </span><span style="font-family: arial;">= 65<br /></span><span style="font-family: arial;">n </span><span style="font-family: arial;">= 65/5<br /></span><span style="font-family: arial;">n </span><span style="font-family: arial;">= 13</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #333333;">7) Therefore, the 13th</span><span style="background-color: white; color: #333333;"> terms of both these A.P.s are equal.</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q16. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.</b></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><h3><span style="font-family: arial;">Solution:</span></h3></span></div></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div>1) Let the first term of an AP be "a" and the common difference be "d".</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div>2) <span style="font-family: arial;">Using the formula </span><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d,</span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></span></div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = a + (3 – 1) d</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">16 = a + </span>2d</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a + 2d = 16</span> --------- equation 1</blockquote></span></div></span></span></div></span></div></span></span></div></span></span></div><div><span style="font-family: arial;">3) </span><span style="font-family: arial;">Now we will find the 5th term.<br /></span></div></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span></div></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>5</sub><span> = a + (5 – 1) d</span><span> <br /></span><span>a</span><sub>5</sub><span> = a + 4d</span><span> --------- equation 2</span></span></blockquote><div><div><div><div><span style="font-family: arial; font-size: medium;">4) Now we will find the 7th term.<br /></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><span style="font-family: arial; font-size: medium;">a<sub>n</sub> = a + (n – 1) d</span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>7</sub><span> = a + (7 – 1) d</span><span> <br /></span><span>a</span><sub>7</sub><span> = a + 6d</span><span> --------- equation 3</span> </span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) As </span><span>the 7th term exceeds the 5th term by 12, we have,</span><br /></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>7</sub><span style="font-family: arial;"> = </span><span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> + 12</span> --------- equation 4</div></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">6) So, from equations 2, 3, and 4, we have,</div></div></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">a</span><sub>7</sub><span style="font-family: arial;"> = </span><span style="font-family: arial;">a</span><sub>5</sub><span style="font-family: arial;"> + 12</span></div></div></span></span></span></div></span></span></div></span></div></span></span></div></span></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>a + 6d</span> = </span><span>a + 4d + 12<br /></span><span><span>a + 6d</span> - </span><span>a - 4d = 12</span><br /><span><span>6d</span> - </span><span>4d = 12<br /></span><span><span>2d</span></span><span> = 12<br /></span><span><span>d</span> =</span><span> 12/2<br /></span><span><span>d</span> </span><span>= 6</span><span> --------- equation 5</span> </span></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">7) Put d = 6 from equation 5 in equation 1, and we get</span><span style="font-family: arial;">.</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + 2d = 16</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + 2(6) = 16</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a + 12 = 16</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a = 16 - 12</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a = 4</span><span style="font-family: arial;"> --------- equation 6</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) So, the AP will be 4, 10, 16, 22...</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q17. Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.</b></span></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div>1) Here,</div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a<sub>1</sub> = a = 3, </span><span style="font-family: arial;">a</span><sub>2</sub><span style="font-family: arial;"> = </span><span><span style="font-family: arial;"><span>8</span></span></span>, <span style="font-family: arial;">a</span><sub>3</sub><span style="font-family: arial;"> = </span>13, <span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = 253.</span></blockquote><div>2) Here.</div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">d = a<sub>2</sub> - a<sub>1</sub></span></div></div></span></span></span></span></div></span></span></div></span></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">d = </span><span style="font-family: arial;">8</span><span style="font-family: arial;"> - 3<br /></span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">d = 5</span><span style="font-family: arial;"> --------- equation 1</span><span style="font-family: arial;"> </span></span></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><span>3) Using the formula </span><span>a</span><sub>n</sub><span> = a + (n – 1) d,</span></span></div><div><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><span style="font-family: arial;">a</span><sub>n</sub><span style="font-family: arial;"> = a + (n – 1) d</span> </div></div></span></span></blockquote></span></div></span></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">253 = 3 + (n - 1)(5)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">253 = 3 + 5(n - 1)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1) = 253 - 3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1)/2 = 65</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">5(n - 1) = 2(65)</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(n - 1) = 2(65)/5</blockquote></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(n - 1) = 2(13)</span></blockquote></span></div></span></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial;">(n - 1) = 26</span></div></span></span></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">n = 26 + 1</span><div><span style="font-size: medium;"><span style="font-family: arial;">n = </span><span style="font-family: arial;">27</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">6) The number of terms in the given AP is 27.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span style="background-color: white; color: #161719; white-space-collapse: break-spaces;">Need help with math? We're here to help! Our resources include NCERT textbooks, lessons on Arithmetic Progressions, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #ArithmeticProgressions #education #learning #students #teachers #math</span></span></div></div><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/10/162-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span> </span>⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.3</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-73262818631078482392023-09-13T11:13:00.002+05:302023-09-21T17:45:02.522+05:30160-NCERT-10-5-Arithmetic Progressions - Ex-5.1<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 5.1</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 5 Arithmetic Progressions</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/09/159-ncert-10-4-quadratic-equations-ex-44.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations-Ex- 4.4</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 5.1</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q1. In which of the following situations, does the list of numbers involved make an arithmetic </b></span><b>progression, and why?</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each </b></span><b>additional km.</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) The amount of air present in a cylinder when a vacuum pump removes </b></span><span><b>1/4 </b></span><span><b>of the </b></span><b>air remaining in the cylinder at a time.</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the </b></span><b>first metre and rises by Rs 50 for each subsequent metre.</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(iv) The amount of money in the account every year, when Rs 10000 is deposited at </b></span><b>compound interest at 8 % per annum.</b></span></div></blockquote><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) An arithmetic progression is a list of numbers in which each term is </span><span style="font-family: arial;">obtained by</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">adding a fixed number to the preceding term except the first </span><span style="font-family: arial;">term.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) The fixed number is known as the common difference of an AP.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) The common difference of may positive, negative, or zero.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) If the first of an AP is "a" and the common difference is "d" then the terms of an</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>AP will be:</span> <span>a, (a + d), (a + 2d), (a + 3d) ...</span></span></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><p><span style="font-family: arial; font-size: medium;"><span><b>(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each </b></span><b>additional km.</b></span></p><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Taxi fare for the first km = 15.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) Taxi fare for the first 2 km = 15 + 8 = 23.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">c) Taxi fare for the first 3 km = 15 + 8 + 8 = 31.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) As the terms increment by a constant 8, it forms an AP.</span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of </b></span><span><b>the </b></span><b>air remaining in the cylinder at a time.</b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) Let the amount of air in the cylinder be x.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the problem, every time the vacuum pump removes (1/4) of air</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">remaining in the cylinder, so</span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) The volume after first removal </span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= x - (x/4) </span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= x(1 - 1/4)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= 3x/4 ---------------- equation 1</span></div></blockquote></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">The volume after second removal</span><span style="font-family: arial;"> </span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= 3x/4 - 1/4(3x/4) </span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;">3x/4)(1 - 1/4)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;">3x/4)(4 - 1)/4</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;">3x/4)(3/4)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= </span><span style="font-family: arial;">9x/16</span><span style="font-family: arial;"> ---------------- equation 2</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="font-family: arial;">The volume after third removal</span><span style="font-family: arial;"> </span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 9x/16 - 1/4(9x/16)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;">9x/16)(1 - 1/4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;">9x/16)(3/4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= </span><span style="font-family: arial;">27x/64</span><span style="font-family: arial;"> ---------------- equation 3</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) Now we will check the terms x, 3x/4, 9x/16, 27x/64 are in AP or not.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Second term - first term = 3x/4 - x </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> = (3x - 4x)/4 </span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> = - x/4</span><span style="font-family: arial;"> ---------------- equation 4.</span></span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) Third term - Second term = 9x/16 - 3x/4 </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"> = (9x - 12x)/16</span></blockquote></blockquote></blockquote></blockquote></blockquote></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> = - 3x/16</span><span style="font-family: arial;"> ---------------- equation 5.</span></span></blockquote></blockquote></blockquote></blockquote></blockquote></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">From equation 4 and equation 5, as the differences between the terms are not</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">same, we can say that these terms are not in AP.</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the </b></span><b>first metre and rises by Rs 50 for each subsequent metre.</b> </span></div><div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;">1) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Cost of digging the well after 1 meter = 150.</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Cost of digging the well after 2 meter = 150 + 50 = 200.</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Cost of digging the well after 3 meter = 150 + 50 + 50 = 250.</span></blockquote><div><span style="font-family: arial; font-size: medium;">2) As the terms increment by a constant 50, it forms an AP.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>(iv) The amount of money in the account every year, when Rs 10000 is deposited at </b></span><b>compound interest at 8 % per annum.</b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;">1) We know that if Rs P is invested at the rate of r % per annum for n years, the</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">amount received will be A = P[1+(r/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>n</sup></span><span style="font-family: arial;">.</span></span></div></div></div></blockquote><div><div><div><span style="font-family: arial; font-size: medium;">2) According to the problem, for P = 10000, r = 8 %, </span></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) The amount after first year</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">A = P</span><span style="font-family: arial;">[1+(r/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>n</sup></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">A = 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>1</sup></span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">A = 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"> ---------------- equation 1</span></span></blockquote></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">The amount after second year</span><span style="font-family: arial;"> </span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">A = P</span><span style="font-family: arial;">[1+(r/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>n</sup></span></span></div></blockquote></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">A = 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;"> ---------------- equation 2</span></span></blockquote></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span style="font-family: arial;">The amount after third year</span><span style="font-family: arial;"> </span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">A = P</span><span style="font-family: arial;">[1+(r/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>n</sup></span></span></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">A = 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>3 </sup></span><span style="font-family: arial;">---------------- equation 3</span></span></blockquote></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) Now we will check the terms </span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;">,</span><span style="font-family: arial;"> </span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;">, </span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>3</sup></span><span style="font-family: arial;">, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>4</sup></span><span style="font-family: arial;"> are in AP or not.</span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Second term - first term </span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">= 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;"> - 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= 10000</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]{</span><span style="font-family: arial;">1+(8/100) - 1</span><span style="font-family: arial;">}</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>= 10000</span><span>[1+(8/100)</span><span>]</span><span>(8/100)</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;">= 10000</span><span style="font-family: arial;">(8/100)</span><span style="font-family: arial;">[1+(8/100)</span><span style="font-family: arial;">]</span><span style="font-family: arial;"> ---------------- equation 4.</span></span></div></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Third term - Second term </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>= 10000</span><span>[1+(8/100)</span><span>]</span><span><sup>3</sup></span><span> - 10000</span><span>[1+(8/100)</span><span>]</span><span><sup>2</sup></span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>= 10000</span><span>[1+(8/100)</span><span>]</span><span><sup>2</sup></span><span>{</span><span>1+(8/100) - 1</span><span>}</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>= 10000</span><span>[1+(8/100)</span><span>]</span><span><sup>2</sup></span><span>(8/100)</span> <br /><span>= 10000</span><span>(8/100)</span><span>[1+(8/100)</span><span>]</span><span><sup>2</sup></span><span> ---------------- equation 5.</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">4) From equation 4 and equation 5, as the differences between the terms are not</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">same, we can say that these terms are not in AP.</span></div></div></blockquote><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q2. Write the first four terms of the AP, when the first term a, and the common difference d are </b></span><b>given as follows:</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span><b>(i) a = 10, d = 10 <span> </span>(ii) a = –2, d = 0 <span> </span></b></span><b>(iii) a = 4, d = – 3</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(iv) a = – 1, d = 1/2<span> </span></b><b>(v) a = – 1.25, d = – 0.25</b></span></div></blockquote><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) An arithmetic progression is a list of numbers in which each term is </span><span style="font-family: arial;">obtained by</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">adding a fixed number to the preceding term except the first </span><span style="font-family: arial;">term.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) The fixed number is known as the common difference of an AP.</span></div><div><span style="font-family: arial; font-size: medium;">3) The common difference of may positive, negative, or zero.</span></div><div><span style="font-family: arial; font-size: medium;">4) If the first of an AP is "a" and the common difference is "d" then the terms of an</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>AP will be:</span> <span>a, (a + d), (a + 2d), (a + 3d) ...</span></span></blockquote><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div><b><span style="font-family: arial; font-size: medium;">(i) a = 10, d = 10</span></b></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>1) Let the terms of an AP be </span><span style="line-height: 107%;"><span>a<sub>1</sub>,
a<sub>2</sub>, a<sub>3</sub>, a<sub>4.</sub></span></span><span face=""Arial",sans-serif" style="line-height: 107%; mso-ansi-language: EN-IN; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"><sub><br /></sub></span><sub><span>2) Here, </span></sub><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 10 is the first term and d = 10 is the common difference.<br /></span><span>3) Now we will find the terms of an AP:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First term: </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 10</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 10</span><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">10<br /></span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 20</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Third term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 20 + 10<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 30</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Fourth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 30 + 10<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 40</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) So the first 4 terms of an AP with a =10 and d = 10 are 10, 20, 30, and 40.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) a = –2, d = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) Let the terms of an AP be </span><span style="line-height: 19.9733px;"><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4.</sub></span></span><span face="Arial, sans-serif" style="line-height: 19.9733px;"><sub><br /></sub></span><sub><span>2) Here, </span></sub><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 2 is the first term and d = 0 is the common difference.<br /></span><span>3) Now we will find the terms of an AP:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 2</span><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">0<br /></span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Third term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 2 + 0<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Fourth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 2 + 0<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">4) So first 4 terms of an AP with a = - 2 and d = 0 are - 2, - 2, - 2, and - 2.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) a = 4, d = – 3</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) Let the terms of an AP be </span><span style="line-height: 19.9733px;"><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4.</sub></span></span><span face="Arial, sans-serif" style="line-height: 19.9733px;"><sub><br /></sub></span><sub><span>2) Here, </span></sub><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 4 is the first term and d = - 3 is the common difference.<br /></span><span>3) Now we will find the terms of an AP:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 4</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4</span><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">3<br /></span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Third term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 1 - 3<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Fourth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 2 - 3<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 5</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">4) So first 4 terms of an AP with a = 4 and d = - 3 are 4, 1, - 2, and - 5.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iv) a = – 1, d = 1/2</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span>1) Let the terms of an AP be </span><span style="line-height: 19.9733px;"><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4.</sub></span></span><span face="Arial, sans-serif" style="line-height: 19.9733px;"><sub><br /></sub></span><sub><span>2) Here, </span></sub><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 1 is the first term and d = 1/2 is the common difference.<br /></span><span>3) Now we will find the terms of an AP:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1</span><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">1/2<br /></span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1/2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Third term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 1/2 + 1/2<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 0</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Fourth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 0 + 1/2<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 1/2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">4) So first 4 terms of an AP with a = - 1 and d = 1/2 are - 1, - 1/2, 0, and 1/2.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(v) a = – 1.25, d = – 0.25</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) Let the terms of an AP be </span><span style="line-height: 19.9733px;"><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4.</sub></span></span><span face="Arial, sans-serif" style="line-height: 19.9733px;"><sub><br /></sub></span><sub><span>2) Here, </span></sub><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 1.25 is the first term and d = - 0.25 is the common difference.<br /></span><span>3) Now we will find the terms of an AP:</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 1.25</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1.25</span><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">0.25<br /></span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1.50</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Third term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 1.50 - 0.25<br /></span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 1.75</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">d) Fourth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> + </span><span face="Arial, sans-serif">d</span><span face="Arial, sans-serif"><br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 1.75 - 0.25<br /></span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 2.00</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">4) So first 4 terms of an AP with a = - 1.25 and d = - 0.25 are - 1.25, - 1.50, - 1.75,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">and - 2.00.</span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q3. For the following APs, write the first term and the common difference:</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 3, 1, – 1, – 3, . . . <span> </span>(ii) – 5, – 1, 3, 7, . . .</b></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(iii) 1/3, 5/3, 9/3, 13/3, . . . <span> </span></b><b>(iv) 0.6, 1.7, 2.8, 3.9, . . .</b></span></div></blockquote><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) An arithmetic progression is a list of numbers in which each term is </span><span style="font-family: arial;">obtained by</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">adding a fixed number to the preceding term except the first </span><span style="font-family: arial;">term.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) The fixed number is known as the common difference of an AP.</span></div><div><span style="font-family: arial; font-size: medium;">3) The common difference of may positive, negative, or zero.</span></div><div><span style="font-family: arial; font-size: medium;">4) If the first of an AP is "a" and the common difference is "d" then the terms of an</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>AP will be:</span> <span>a, (a + d), (a + 2d), (a + 3d) ...</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) For the terms a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4, </sub></span><span face="Arial, sans-serif">when </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = d, then we say</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">that the terms </span><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4, </sub></span><span face="Arial, sans-serif">are in AP.</span></span></div></div></blockquote><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) 3, 1, – 1, – 3, . . .</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 3.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 3, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 1, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 1, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 3.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 1 - 3</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 2 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1 - 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 2 --------- equation 2</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) Here the first term a = 3 and </span><span face="Arial, sans-serif">the common difference d = - 2.</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) – 5, – 1, 3, 7, . . .</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = - 5.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 5, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 3, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 7.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 1 - (- 5)</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 1 + 5</span> </span></div></div></blockquote></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 4 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3 - (- 1)</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3 + 1</span> </span></div></div></blockquote></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4 --------- equation 2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) Here the first term a = - 5 and </span><span face="Arial, sans-serif">the common difference d = 4.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 1/3, 5/3, 9/3, 13/3, . . .</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 1/3.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 1/3, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 5/3, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 9/3, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 13/3.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 5/3 - 1/3</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (5 - 1)/3</span> </span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 4/3 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 9/3 - 5/3</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = (9 - 5)/3</span> </span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4/3 --------- equation 2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) Here the first term a = 1/3 and </span><span face="Arial, sans-serif">the common difference d = 4/3.</span></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iv) 0.6, 1.7, 2.8, 3.9, . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 0.6.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 0.6, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 1.7, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 2.8, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 3.9.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 1.7 - 0.6</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 1.1 --------- equation 1</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 2.8 - 1.7</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 1.1 --------- equation 2</span></span></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span>4) Here the first term a = 0.6 and </span><span face="Arial, sans-serif">the common difference d = 1.1.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q4. Which of the following are APs? If they form an AP, find the common difference d and </b></span><b>write three more terms.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) 2, 4, 8, 16, . . . <span> </span></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 2, 5/2, 3, 7/2, . . . <span> </span></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . </b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iv) – 10, – 6, – 2, 2, . . .</b></span></div><div><b><span style="font-family: arial; font-size: medium;">(v) 3, 3 + <span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , 3 + 2 <span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , 3 + 3 <span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , . . .</span></b></div><div><b><span style="font-family: arial; font-size: medium;">(vi) 0.2, 0.22, 0.222, 0.2222, . . .</span></b></div><div><b><span style="font-family: arial; font-size: medium;">(vii) 0, – 4, – 8, –12, . . .</span></b></div><div><b><span style="font-family: arial; font-size: medium;">(viii) - 1/2, - 1/2, - 1/2, - 1/2, . . .</span></b></div><div><b><span style="font-family: arial; font-size: medium;">(ix) 1, 3, 9, 27, . . .</span></b></div><div><b><span style="font-family: arial; font-size: medium;">(x) a, 2a, 3a, 4a, . . .</span></b></div><div><span style="font-family: arial; font-size: medium;"><b>(xi) a, a</b><span><sup><b>2</b></sup></span><b>, </b><b>a</b><span><sup><b>3</b></sup></span><b>, </b><b>a</b><span><sup><b>4</b></sup></span><b>, . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xii) </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>2, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>8, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>18 , </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>32, . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xiii) </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>3, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>6, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>9 , </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>12 , . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xiv) </b><b>1</b><span><sup><b>2</b></sup></span><b>, </b><b>3</b><span><sup><b>2</b></sup></span><b>, </b><b>5</b><span><sup><b>2</b></sup></span><b>, </b><b>7</b><span><sup><b>2</b></sup></span><b>, . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xv) 1</b><span><sup><b>2</b></sup></span><b>, </b><b>5</b><span><sup><b>2</b></sup></span><b>, </b><b>7</b><span><sup><b>2</b></sup></span><b>, </b><b>7</b><span><sup><b>3</b></sup></span><b>, . . .</b></span></div></blockquote><div style="text-align: left;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) An arithmetic progression is a list of numbers in which each term is </span><span style="font-family: arial;">obtained by</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">adding a fixed number to the preceding term except the first </span><span style="font-family: arial;">term.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) The fixed number is known as the common difference of an AP.</span></div><div><span style="font-family: arial; font-size: medium;">3) The common difference may be positive, negative, or zero.</span></div><div><span style="font-family: arial; font-size: medium;">4) If the first of an AP is "a" and the common difference is "d" then the terms of an</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>AP will be:</span> <span>a, (a + d), (a + 2d), (a + 3d) ...</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>5) For the terms a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4, </sub></span><span face="Arial, sans-serif">when </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = d, then we say</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">that the terms </span><span>a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4, </sub></span><span face="Arial, sans-serif">are in AP.</span></span></blockquote><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) 2, 4, 8, 16, . . . </span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 8, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 16.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 4 - 2</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 2 --------- equation 1</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 8 - 4</span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4 --------- equation 2</span></span></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 107%;"><span>≠</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 2, 5/2, 3, 7/2, . . .</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 5/2, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 3, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 7/2.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 5/2 - 2</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (5 - 4)/2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 1/2 --------- equation 1</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3 - 5/2</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = (6 - 5)/2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 1/2 --------- equation 2</span></span></div></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here the common difference d = 1/2.</span></div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 7/2</span><span face="Arial, sans-serif"> + 1/2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> (7 + 1)/2</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 8/2</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 4</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 4</span><span face="Arial, sans-serif"> + 1/2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> (8 + 1)/2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 9/2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 9/2</span><span face="Arial, sans-serif"> + 1/2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> (9 + 1)/2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 10/2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 5</span></span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are 4, 9/2, and 5.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = - 1.2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 1.2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 3.2, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 5.2, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 7.2.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (- 3.2) - (- 1.2)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 3.2 + 1.2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 2 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(- 5.2) - (- 3.2)</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 5.2 + 3.2</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 2 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here the common difference is d = - 2.</span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 7.2</span><span face="Arial, sans-serif"> + (- 2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 7.2 - 2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 9.2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 9.2</span><span face="Arial, sans-serif"> + (- 2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 9.2 - 2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 11.2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 11.2</span><span face="Arial, sans-serif"> + (- 2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 11.2 - 2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 13.2</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) So the next 3 terms are - 9.2, - 11.2, and - 13.2.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iv) – 10, – 6, – 2, 2, . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = - 10.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 10, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 6, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 2, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 2.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (- 6) - (- 10)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 6 + 10</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 4 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(- 2) - (- 6)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 2 + 6</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 4 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here the common difference is d = 4.</span></div></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 2</span><span face="Arial, sans-serif"> + 4</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6</span><span face="Arial, sans-serif"> + 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 8</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 8</span><span face="Arial, sans-serif"> + 2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 10</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are 6, 8, and 10.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(v) 3, 3 + <span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , 3 + 2<span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , 3 + 3 <span><span><span style="white-space: pre-wrap;">√</span></span></span>2 , . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 3.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 3, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>3 + <span><span><span style="white-space: pre-wrap;">√</span></span></span>2</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>3 + 2<span><span><span style="white-space: pre-wrap;">√</span></span></span>2</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span>3 + 3<span><span><span style="white-space: pre-wrap;">√</span></span></span>2</span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (</span><span>3 + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif">) - (3)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span>3 + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif"> - 3</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif"> --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(</span><span>3 + 2</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif">) - (</span><span>3 + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif">)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>3 + 2</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2 - </span><span>3 - </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif"> --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">5) So, here the common difference is d = </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span><span face="Arial, sans-serif">.</span></span></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> (</span><span>3 + 3</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif"> + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span>3 + 4</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">(</span><span>3 + 4</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif"> + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span>3 + 5</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">(</span><span>3 + 5</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif"> + </span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span>3 + 6</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2</span></span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) So next 3 terms are (</span><span>3 + 4</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">(</span><span>3 + 5</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif">, and </span><span face="Arial, sans-serif">(</span><span>3 + 6</span><span><span><span style="white-space: pre-wrap;">√</span></span></span><span>2)</span><span face="Arial, sans-serif">.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(vi) 0.2, 0.22, 0.222, 0.2222, . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 0.2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 0.2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 0.22, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 0.222, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 0.2222.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (0.22) - (0.2)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 0.22 - 0.2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 0.02 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(0.222) - (0.22)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 0.222 - 0.22</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 0.002 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>≠</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(vii) 0, – 4, – 8, –12, . . .</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 0.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 0, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 4, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 8, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 12.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (- 4) - (0)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 4 + 0</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 4 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(- 8) - (- 4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 8 + 4</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 4 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here the common difference is d = - 4.</span></div></div><div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 12</span><span face="Arial, sans-serif"> + (- 4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 16</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 16</span><span face="Arial, sans-serif"> + (- 4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 20</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 20</span><span face="Arial, sans-serif"> + (- 4)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 24</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are - 16, - 20, and - 24.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(viii) - 1/2, - 1/2, - 1/2, - 1/2, . . .</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = - 1/2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = - 1/2 , </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = - 1/2, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = - 1/2, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = - 1/2.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (- 1/2) - (- 1/2)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = - 1/2 + 1/2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 0 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(- 1/2) - (- 1/2)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">- 1/2 + 1/2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 0 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here the common difference is d = 0.</span></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span><span face="Arial, sans-serif"> + 0</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span><span face="Arial, sans-serif"> + 0</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span><span face="Arial, sans-serif"> + 0</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> - 1/2</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) So next 3 terms are - 1/2, - 1/2, and - 1/2.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(ix) 1, 3, 9, 27, . . .</span></b></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = 1.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = 1 , </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3 </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 9, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 27.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (3) - (1)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 3 - 1</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 2 --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">9 - 3</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 6 --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span>≠</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(x) a, 2a, 3a, 4a, . . .</span></b></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = a.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = a, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 2a, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = 3a, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = 4a.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (2a) - (a)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 2a - a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(3a) - (2a)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3a - 2a</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = a --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">5) So, here common difference d = a.</span></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 4a</span><span face="Arial, sans-serif"> + a</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 5a</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 5a</span><span face="Arial, sans-serif"> + a</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6a</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6a</span><span face="Arial, sans-serif"> + a</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 7a</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are 5a, 6a, and 7a.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xi) a, a</b><span><sup><b>2</b></sup></span><b>, </b><b>a</b><span><sup><b>3</b></sup></span><b>, </b><b>a</b><span><sup><b>4</b></sup></span><b>, . . .</b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here the first term is a = a.</span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = a, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>a</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>a</span><span><sup>3</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span>a</span><span><sup>4</sup></span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = (</span><span>a</span><span><sup>2</sup></span><span face="Arial, sans-serif">) - (a)</span></span></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a(a - 1)</span><span face="Arial, sans-serif"> --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">(</span><span>a</span><span><sup>3</sup></span><span face="Arial, sans-serif">) - (</span><span>a</span><span><sup>2</sup></span><span face="Arial, sans-serif">)</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>a</span><span><sup>2</sup></span><span face="Arial, sans-serif">(a - 1)</span><span face="Arial, sans-serif"> --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>≠</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b>(xii) </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>2, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>8, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>18 , </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>32, . . .</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here the first term is a = </span><span style="font-family: arial;"><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span style="font-family: arial;">2</span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>8</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>18</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>32</span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>8</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></blockquote></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 2</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2 </span><span face="Arial, sans-serif">- </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></div></blockquote></blockquote><div style="text-align: left;"><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√1</span></span></span></span><span>8</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>8</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = 3</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2 </span><span face="Arial, sans-serif">- 2</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> ---------</span><span face="Arial, sans-serif"> equation 2</span></span></blockquote></div></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">5) So, here the common difference is d = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif">.</span></span></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>32</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 4</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></blockquote></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 5</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>50</span> </span></blockquote></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d<br /></span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>50</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 5</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2<br /></span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>72</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d<br /></span><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>72</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 6</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> + </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></div></div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> 7</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2<br /></span><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>98</span> </span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>50</span><span face="Arial, sans-serif">, </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>72</span><span face="Arial, sans-serif">, and </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>98</span><span face="Arial, sans-serif">.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xiii) </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>3, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>6, </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>9 , </b><b><span><span><span style="white-space: pre-wrap;">√</span></span></span></b><b>12 , . . .</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here the first term is a = </span><span style="font-family: arial;"><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span style="font-family: arial;">3</span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>6</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>9</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>12</span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>6</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span></span></div></blockquote></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3(</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span face="Arial, sans-serif"> - </span><span><span style="white-space: pre-wrap;">1)</span></span><span face="Arial, sans-serif"> --------- equation 1</span></span></blockquote></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>9</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>6</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3(</span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>3</span><span face="Arial, sans-serif"> - </span><span><span><span><span style="white-space: pre-wrap;">√</span></span></span></span><span>2</span><span><span style="white-space: pre-wrap;">)</span></span><span face="Arial, sans-serif"> --------- equation </span><span face="Arial, sans-serif">2</span> </span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>≠</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xiv) </b><b>1</b><span><sup><b>2</b></sup></span><b>, </b><b>3</b><span><sup><b>2</b></sup></span><b>, </b><b>5</b><span><sup><b>2</b></sup></span><b>, </b><b>7</b><span><sup><b>2</b></sup></span><b>, . . .</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here the first term is a = </span><span style="font-family: arial;">1</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = </span><span>1</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>3</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>5</span><span><sup>3</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span>7</span><span><sup>4</sup></span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span>3</span><span><sup>2</sup></span><span face="Arial, sans-serif"> - </span><span>1</span><span><sup>2</sup></span></span></div></blockquote></blockquote></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span>9</span><span face="Arial, sans-serif"> - </span><span>1</span> </span></div></div></blockquote></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 8</span><span face="Arial, sans-serif"> --------- equation 1</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>5</span><span><sup>2</sup></span><span face="Arial, sans-serif"> - </span><span>3</span><span><sup>2</sup></span></span></blockquote></blockquote></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>25</span><span face="Arial, sans-serif"> - </span><span>9</span> </span></div></div></blockquote></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>16</span><span face="Arial, sans-serif"> --------- equation 2</span></span></blockquote></blockquote></div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1 and </span><span face="Arial, sans-serif">equation 2, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>≠</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif">, so given terms are not in AP.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(xv) 1</b><span><sup><b>2</b></sup></span><b>, </b><b>5</b><span><sup><b>2</b></sup></span><b>, </b><b>7</b><span><sup><b>2</b></sup></span><b>, 73</b><b>, . . .</b></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here the first term is a = </span><span style="font-family: arial;">1</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) Here </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = a = </span><span>1</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>5</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>7</span><span><sup>2</sup></span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> = </span><span>73</span><span face="Arial, sans-serif">.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First difference:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span>5</span><span><sup>2</sup></span><span face="Arial, sans-serif"> - </span><span>1</span><span><sup>2</sup></span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = </span><span>25</span><span face="Arial, sans-serif"> - </span><span>1</span> </span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> = 24</span><span face="Arial, sans-serif"> --------- equation 1</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">b) Second difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>7</span><span><sup>2</sup></span><span face="Arial, sans-serif"> - </span><span>5</span><span><sup>2</sup></span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>49</span><span face="Arial, sans-serif"> - </span><span>25</span> </span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> = </span><span>24</span><span face="Arial, sans-serif"> --------- equation 2</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">c) Third difference:</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>73</span><span face="Arial, sans-serif"> - </span><span>7</span><span><sup>2</sup></span></span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>73</span><span face="Arial, sans-serif"> - </span><span>49</span></span></blockquote></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">d = a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> = </span><span>24</span><span face="Arial, sans-serif"> --------- equation 3</span></span></blockquote></blockquote></div><div><div><span style="font-family: arial; font-size: medium;"><span>4) From </span><span face="Arial, sans-serif">equation 1, 2 and 3</span><span face="Arial, sans-serif">, </span><span face="Arial, sans-serif">a</span><sub>2</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>1</sub><span face="Arial, sans-serif"> </span><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>2 </sub><span style="line-height: 17.12px;"><span>=</span></span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> - </span><span face="Arial, sans-serif">a</span><sub>3</sub><span face="Arial, sans-serif">, so given terms are in AP.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">5) So, here the common difference is d = </span><span><span style="white-space: pre-wrap;">24</span></span><span face="Arial, sans-serif">.</span></span></div><div><div><span style="font-family: arial; font-size: medium;">6) So the next 3 terms are:</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) Fifth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span><span style="white-space: pre-wrap;">73</span></span><span face="Arial, sans-serif"> + </span><span><span style="white-space: pre-wrap;">24</span></span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>5</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">97</span> </span></div></blockquote></blockquote></div></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Sixth term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d<br /></span><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span><span style="white-space: pre-wrap;">97</span></span><span face="Arial, sans-serif"> + </span><span><span style="white-space: pre-wrap;">24</span></span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>6</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif">121</span> </span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">c) Seventh term:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">a</span><sub>4</sub><span face="Arial, sans-serif"> + d<br /></span><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = </span><span><span style="white-space: pre-wrap;">121</span></span><span face="Arial, sans-serif"> + </span><span><span style="white-space: pre-wrap;">24</span></span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">a</span><sub>7</sub><span face="Arial, sans-serif"> = 145</span> </span></div></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"> <span face="Arial, sans-serif">7) The next 3 terms are </span><span><span style="white-space: pre-wrap;">97</span></span><span face="Arial, sans-serif">, </span><span><span style="white-space: pre-wrap;">121</span></span><span face="Arial, sans-serif">, and </span><span><span style="white-space: pre-wrap;">145</span></span><span face="Arial, sans-serif">.</span></span></div></div></div></div><div><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif"><br /></span></span></div><div><span style="color: #161719; font-family: arial; font-size: medium;"><span style="background-color: white; white-space-collapse: break-spaces;">Need help speed math? We're here to help! Our resources include NCERT textbooks, lessons on Arithmetic Progressions, and more. Join our community of students and teachers today! #mathhelp #NCERT #studentsuccess #ArithmeticProgressions #education #learning #students #teachers #math</span></span></div></div><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/09/161-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.2</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-5781721292307159972023-09-06T16:16:00.001+05:302023-09-13T11:14:18.930+05:30159-NCERT-10-4-Quadratic Equations - Ex-4.4<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 4.4</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 4 Quadratic Equations</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/08/158-ncert-10-4-quadratic-equations-ex-43.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations-Ex- 4.3</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 4.4</span></h3></div><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">Q1. Find the nature of the roots of the following quadratic equations. If the real roots exist, </span><span style="font-family: arial;">find them:</span></span></b></div><div><span style="font-family: arial; font-size: medium;"><span><b>(i) 2x<span><sup>2</sup></span> – 3x + 5 = 0 (ii) 3x<span><sup>2</sup></span> – 4<span style="white-space: pre-wrap;">√</span>3 x + 4 = 0 </b></span><b>(iii) 2x<span><sup>2</sup></span> – 6x + 3 = 0 </b></span></div><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span>1) For </span><span>the </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span>(b</span><sup>2</sup><span> - 4ac) is known as </span></span><span style="font-family: arial;">discriminant.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) The </span><span>quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0 has</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(a) two distinct real roots, if </span><span>b</span><sup>2</sup><span> – 4ac > 0,</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(b) two equal real roots, if </span><span>b</span><sup>2</sup><span> – 4ac = 0,</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(c) no real roots, if </span><span>b</span><sup>2</sup><span> – 4ac < 0.</span></span></div></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) 2x<span><sup>2</sup></span> – 3x + 5 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x</span><sup>2</sup><span> - 3x + 5 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span><span>2) </span></span><span>Equate the coefficient of equation </span><span>2x</span><sup>2</sup><span> - 3x + 5 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, we have,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 2, b = - 3, c = 5.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>- 3)</span><sup>2 </sup><span>- 4(2)(5)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 9</span><sup> </sup><span>- 40</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = - 31</span><span> ------------------ equation 2.</span><span> </span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span><span>4) </span><span>As </span><span><span>b</span><sup>2 </sup><span>- 4ac <span style="background-color: white;">< 0</span></span>, it has no real roots of the quadratic equation </span></span><span>2x</span><sup>2</sup><span> - 3x + 5 = 0</span><span>.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 3x<span><sup>2</sup></span> – 4<span style="white-space: pre-wrap;">√</span>3 x + 4 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>3x<span><sup>2</sup></span> – 4<span style="white-space: pre-wrap;">√</span>3 x + 4 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>Equate the coefficient of equation </span><span>3x<span><sup>2</sup></span> – 4<span style="white-space: pre-wrap;">√</span>3 x + 4 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">we have,</span></div></div></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 3, b = - 4<span style="white-space: pre-wrap;">√</span>3, c = 4.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>- 4<span style="white-space: pre-wrap;">√</span>3)</span><sup>2 </sup><span>- 4(3)(4)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 48</span><sup> </sup><span>- 48</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 0</span><span> ------------------ equation 2.</span><span> </span></span></blockquote><div><span style="font-size: medium;"><span><span style="font-family: arial;">4) </span></span><span style="font-family: arial;">As, </span><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 0</span>, it has two equal real roots, so </span><span style="font-family: arial;">from equation 2 and equation 3,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">we </span><span style="font-family: arial;">have,</span></span></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(- 4<span style="white-space: pre-wrap;">√</span>3)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>0</span></span><span>]</span><span>/2(3)<br /></span><span>x</span><span> = (</span><span>4<span style="white-space: pre-wrap;">√</span>3 </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">0</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/2(3)<br /></span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 2(</span><span style="font-family: arial;"><span style="white-space: pre-wrap;">√</span>3</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/3</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 2(</span><span style="font-family: arial;"><span style="white-space: pre-wrap;">√</span>3</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/(</span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3</span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 2</span><span style="font-family: arial;">/</span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3</span></span></blockquote><span style="font-family: arial; font-size: medium;"> <span>5) So, </span><span> </span><span>x</span><span> = </span><span><span>2</span><span>/</span><span style="white-space: pre-wrap;">√</span>3 or </span><span>x</span><span> = </span><span><span>2</span><span>/</span><span style="white-space: pre-wrap;">√</span>3.</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 2x<span><sup>2</sup></span> – 6x + 3 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x<span><sup>2</sup></span> – 6x + 3 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>Equate the coefficient of equation </span><span>2x<span><sup>2</sup></span> – 6x + 3 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">we have,</span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 2, b = - 6, c = 3.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>- 6)</span><sup>2 </sup><span>- 4(2)(3)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 36</span><sup> </sup><span>- 24</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 12</span><span> ------------------ equation 2.</span><span> </span></span></blockquote><div><span style="font-size: medium;"><span><span style="font-family: arial;">4) </span></span><span style="font-family: arial;">As, </span><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">≥</span> 0</span>, it has two distinct real roots, so </span><span style="font-family: arial;">from equation 2 and equation 3,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">we </span><span style="font-family: arial;">have,</span></span></div></div></div></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(- 6)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>12</span></span><span>]</span><span>/2(2)<br /></span><span>x</span><span> = (6</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;"><span><b>√</b>12)</span></span><span>/6</span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (6</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± 2</span><span style="font-family: arial; white-space: pre-wrap;"><b>√</b>3)</span><span style="font-family: arial;">/6</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial; white-space: pre-wrap;"><b>√</b>3)</span><span style="font-family: arial;">/3</span></span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">5) So, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = </span><span style="font-family: arial;"><span>(3</span><span> </span><span face="Arial, sans-serif">+ </span><span style="white-space: pre-wrap;"><b>√</b>3)</span><span>/3</span> or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = </span><span style="font-family: arial;"><span>(3</span><span> </span><span face="Arial, sans-serif">- </span><span style="white-space: pre-wrap;"><b>√</b>3)</span><span>/3</span>.</span></span><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. Find the values of k for each of the following quadratic equations, so that they have two </b></span><b>equal roots.</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>(i) 2</b></span><b>x<span><sup>2</sup></span></b><span><b> + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0</b></span></span></div><div style="text-align: left;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span>1) For </span><span>the </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span>(b</span><sup>2</sup><span> - 4ac) is known as </span></span><span style="font-family: arial;">discriminant.</span></span></div></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) The </span><span>quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0 has</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(a) two equal real roots, if b2 – 4ac = 0,</span></blockquote><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(i) 2</b></span><b>x<span><sup>2</sup></span></b><span><b> + kx + 3 = 0</b></span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x<span><sup>2</sup></span> + kx + 3 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>Equate the coefficient of equation </span><span>2x<span><sup>2</sup></span> + kx + 3 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">we have,</span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 2, b = k, c = 3.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (k</span><span>)</span><sup>2 </sup><span>- 4(2)(3)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = </span><span>k</span><sup>2</sup><sup> </sup><span>- 24</span></span><span style="font-family: arial;"> ------------------ equation 2.</span><span style="font-family: arial;"> </span></span></div></blockquote><div><span style="font-size: medium;"><span><span style="font-family: arial;">4) </span></span><span style="font-family: arial;">As the quadratic equation has two equal roots, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 0</span></span></div></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-size: medium;"><span>k</span><sup>2</sup><sup> </sup><span>- 24 = 0</span></span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>k</span><sup>2</sup><sup> </sup><span>= 24</span> </span></div></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"><span>k</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>24 = </span></span></span><span style="font-family: arial;">± 2</span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b>6</span></span></span></blockquote></div></div><span style="font-family: arial; font-size: medium;"> <span>5) So, </span><span> </span><span>k</span><span> = 2</span><span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">6</span> or </span><span>k</span><span> = - 2</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">6</span><span>.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) kx (x – 2) + 6 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">kx (x – 2) + 6 = 0</span></div></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">k</span><span style="font-family: arial;">x</span><span style="font-family: arial;"><sup>2</sup></span><span style="font-family: arial;"> – 2kx + 6 = 0</span><span style="font-family: arial;"> ------------------ equation 1.</span></span></div></div></div></blockquote><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>Equate the coefficient of equation </span><span>kx<span><sup>2</sup></span> – 2kx + 6 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">we have,</span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = k, b = - 2k, c = 6.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (- 2k</span><span>)</span><sup>2 </sup><span>- 4(k)(6)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 4</span><span>k</span><sup>2</sup><sup> </sup><span>- 24k</span></span><span style="font-family: arial;"> ------------------ equation 2.</span><span style="font-family: arial;"> </span></span></div></blockquote><div><span style="font-size: medium;"><span><span style="font-family: arial;">4) </span></span><span style="font-family: arial;">As the quadratic equation has two equal roots, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 0</span></span></div></div></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span><span style="font-size: medium;"><span>4</span><span>k</span><sup>2</sup><sup> </sup><span>- 24k</span> = 0</span></span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>4k(</span><span>k</span><sup> </sup><span>- 6) = 0<br /></span><span>k(k</span><sup> </sup><span>- 6) = 0</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"> <span>5) So, </span><span> </span><span>k</span><span> = 0</span><span> or </span><span>k</span><span> = 6</span><span>.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;"><span>Q</span><span>3. Is it possible to design a rectangular mango grove whose length is twice its breadth </span><span>and the area is 800 </span><span>m</span><sup>2</sup><span>? If so, find its length and breadth.</span></span></b></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Let the breadth of a rectangular mango grove be x m.</span></div><div><span style="font-family: arial; font-size: medium;">2) So, the length of a rectangular mango grove will be 2x m </span></div><div><span style="font-family: arial; font-size: medium;">3) According to the problem, the area is 800, so</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2x(x) = 800</span></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">2x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= 800</span></span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= 400</span></blockquote></span></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 400 = 0</span> ------------------ equation 1.</div></span></span></span></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>4) </span><span>Equate the coefficient of equation </span><span>x<span><sup>2</sup></span> – 400 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">we have,</span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a = 1, b = 0, c = - 400.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">5) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (0</span><span>)</span><sup>2 </sup><span>- 4(1)(- 400)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 1600</span></span><span style="font-family: arial;"> ------------------ equation 2.</span></span></div></blockquote></div></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">6) As b<sup>2 </sup>- 4ac = 1600 > 0, it has real roots, so from equation 1 and equation 2, we</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">have,</span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">b</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b></span></span><span style="font-family: arial;">(</span><span style="font-family: arial;">b</span><sup>2 </sup><span style="font-family: arial;">- 4ac)]</span><span style="font-family: arial;">/2a<br /></span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">(0)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b>1600]/2</span></span><span style="font-family: arial;"><br /></span><div><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial; white-space: pre-wrap;">40</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2</span> <br /></div></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"></span>7) So, x = (40<span style="font-family: arial; white-space: pre-wrap;">)</span>/2 or x = (- 40<span style="font-family: arial; white-space: pre-wrap;">)</span>/2, i.e. x = 20, or x = - 20.</span></span></div><div><span style="font-family: arial; font-size: medium;">8) As length is always positive, ignore x = - 20.</span></div><div><span style="font-family: arial; font-size: medium;"><span>9) So, the breadth of the </span>rectangular mango grove is 20 m and its length is 40 m.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q4. Is the following situation possible? If so, determine their present ages.</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>The sum of the ages of two friends is 20 years. Four years ago, the product of their age </b></span><b>in years was 48.</b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the present age of the first friend be x.</span></div><div><span style="font-family: arial; font-size: medium;"><span><span>2) So, the </span></span>present age of the second friend will be (20 - x).</span></div></div><div><span style="font-family: arial; font-size: medium;">3) 4 years ago, their ages will be (x - 4) and (20 - x - 4).</span></div></div><div><span style="font-family: arial; font-size: medium;">4) According to the problem,</span></div></div></div></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 4)(16 - x) = 48</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x(16 - x) - 4(16 - x) = 48</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>16x - </span><span>x</span><sup>2</sup><span> </span><span>- 64 + 4x </span>= 48</span></div><span style="font-family: arial; font-size: medium;"><span><span>20x - </span><span>x</span><sup>2</sup><span> </span><span>- 64 - 48 </span>= 0<br /></span><span><span>20x - </span><span>x</span><sup>2</sup><span> </span><span>- 112 </span>= 0<br /></span><span>x</span><sup>2</sup><span> </span><span>-</span><span> 20x + 112 = 0</span><span> ------------------ equation 1.</span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Equate the coefficient of equation <span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">-</span><span style="font-family: arial;"> 20x + 112 = 0</span> with ax<sup>2</sup> + bx + c = 0, so</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 1, b = - 20, c = 112</span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">6) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">20</span><span>)</span><sup>2 </sup><span>- 4(1)(112)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 400</span><sup> </sup><span>- 448</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = - 48</span><span> ------------------ equation 2.</span></span></blockquote></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As b<sup>2 </sup>- 4ac = - 48 < 0, </span><span style="font-family: arial;">it has no real roots, so the given situation is not possible.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q5. Is it possible to design a rectangular park of perimeter 80 m and an area of 400 m2? If so, find </b></span><b>its length and breadth.</b></span></div></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Let the breadth of a rectangular park be x m.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) As, the perimeter of </span><span style="font-family: arial;">a rectangular park = 80 m.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) So, length = [(perimeter/2) - breadth]</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">length = [(80/2) - x]</span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">length = (40 - x)</span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">4) According to the problem,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x(40 - x) = 400</span></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;">40x - x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= 400</span></span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> - 40x + 400 = 0</span> ------------------ equation 1.</blockquote><div style="text-align: left;"><div><div><span style="font-family: arial;">5) Equate the coefficient of equation <span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">-</span><span style="font-family: arial;"> 40x + 400 = 0</span> with ax<sup>2</sup> + bx + c = 0, so</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a = 1, b = - 40, c = 400</span></blockquote></div><div><div><span style="font-family: arial;">6) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">40</span><span>)</span><sup>2 </sup><span>- 4(1)(400)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 1600</span><sup> </sup><span>- 1600</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 0</span><span> ------------------ equation 2.</span></span></blockquote></div><div><div><div><div><span><span style="font-family: arial;">7) </span></span><span style="font-family: arial;">As, </span><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 0</span>, it has two equal real roots, so </span><span style="font-family: arial;">from equation 1 and equation 2,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">we </span><span style="font-family: arial;">have,</span></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(- 4<span style="white-space: pre-wrap;">0</span>)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>0</span></span><span>]</span><span>/2(1)<br /></span><span>x</span><span> = (</span><span>4<span style="white-space: pre-wrap;">0</span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">0</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/2<br /></span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">(20</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial; white-space: pre-wrap;">0</span><span style="font-family: arial; white-space: pre-wrap;">)</span></div></blockquote><span style="font-family: arial;">8) So, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 20, the breadth = 20 m and the length = 40 - 20 = 20 m.</span></div></div><div><span style="font-family: arial;"><br /></span></div><div><span style="font-family: arial;"><span style="background-color: white; color: #161719; white-space-collapse: break-spaces;">#mathhelp #NCERT #education #studentsuccess </span></span>#math #quadraticequations #learning #students #teachers</div></div></span></span></div></span></span></div></div></div><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/09/160-ncert-10-5-arithmetic-progressions.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-5-Arithmetic Progressions - Ex- 5.1</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-12797213115880552532023-08-31T18:25:00.005+05:302023-09-06T16:18:56.731+05:30158-NCERT-10-4-Quadratic Equations - Ex-4.3<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 4.3</span></span></div><div style="color: black; font-family: "Times New Roman"; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 4 Quadratic Equations</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/07/157-ncert-10-4-quadratic-equations-ex-42.html" rel="nofollow" target="_blank">To access the NCERT-10-4-Quadratic Equations-Ex-4.2, please click on the provided link.</a></span></h2><h2 style="clear: both;"><span style="font-family: arial; font-size: large;">EXERCISE 4.3</span></h2><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Q1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:</span></b></div><div><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i)
2x<sup>2</sup> – 7x + 3 = 0 <span> </span><span> </span>(ii) 2x<sup>2</sup> + x – 4 = 0</b></span></div>
<div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span face=""Arial",sans-serif" style="line-height: 107%;">(iii) 4x<sup>2</sup> + 4</span><span><span style="white-space: pre-wrap;">√</span></span><span face="Arial, sans-serif">3x + 3 = 0 </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">(iv) 2x</span><sup>2</sup><span face="Arial, sans-serif"> + x
+ 4 = 0</span></span></b></div></div><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span>1) </span><span>The </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + (c/a) = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x = - (c/a)<br /></span><span>x</span><sup>2</sup><span> + (b/a)x = - (c/a) ------------------ equation 1.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) In the quadratic equation, for getting the perfect square, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>last term = [(middle term)</span><sup>2</sup><span>]/4</span><span> ------------------ equation 2.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) From equation 1 and equation 2, last term = </span><span>[(b/a)</span><sup>2</sup><span>]/4.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) Adding </span><span>[(b/a)</span><sup>2</sup><span>]/4 to both sides of equation 1, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>[(b/a)</span><sup>2</sup><span>]/4 </span><span>= </span><span>[(b/a)</span><sup>2</sup><span>]/4 </span><span>- (c/a)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>(b/2a)</span><sup>2</sup><span> </span><span>= </span><span>(b/2a)</span><sup>2</sup><span> </span><span>- (c/a)<br /></span><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>(b/2a)</span><sup>2</sup><span> </span><span>= </span><span>(b</span><span>)</span><sup>2</sup><span>/4a</span><sup>2</sup><span> </span><span>- (c/a)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><sup>2</sup><span> </span><span>= (</span><span>b</span><sup>2 </sup><span>- 4ac)</span><span>/4a</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><span> </span><span>= </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)</span><span>/4a</span><sup>2</sup><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><span> </span><span>= </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><span> = - </span><span>b/2a</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 3.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) This is the method of the square.</span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>(i) 2x<sup>2</sup> – 7x + 3 = 0</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) The given equation is </span>2x<sup>2</sup> – 7x + 3 = 0 <span style="font-family: arial;">------------------ equation 1.</span></div><div>2) Divide equation 1 by 2 we get,</div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">2x<sup>2</sup> – 7x + 3 = 0</div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x<sup>2</sup> – (7/2)x + (3/2) = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> – (7/2)x = - (3/2)</span><span> </span><span>------------------ equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) From </span><span style="font-family: arial;">equation 1, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(middle term)</span><sup>2</sup><span>]/4</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(7/2)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">last term = </span><span style="font-family: arial;">49/16</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) Add 49/16 to both sides of equation 2, and we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> - (7/2)x + 49/16 = 49/16 - 3/2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> - (7/2)x + 49/16 = 49/16 - 24/16</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> - (7/2)x + 49/16 = (49 - 24)/16<br /></span><span>(x</span><span> - 7/4)</span><sup>2</sup><span> = 25/16</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> - 7/4</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>(</span></span><span>25/16)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> - 7/4)</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;">5/4<br /></span></span><span>x</span><span> = 7/4</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;">5/4<br /></span></span><span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;">5)/4<br /></span></span><span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">+ </span><span><span style="white-space: pre-wrap;">5)/4, or </span></span><span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">- </span><span><span style="white-space: pre-wrap;">5)/4<br /></span></span><span>x</span><span> = 12</span><span><span style="white-space: pre-wrap;">/4, or </span></span><span>x</span><span> = 2</span><span><span style="white-space: pre-wrap;">/4<br /></span></span><span>x</span><span> = 3</span><span><span style="white-space: pre-wrap;">, or </span></span><span>x</span><span> = 1.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) The roots of the quadratic equation </span><span>2x</span><sup>2</sup><span> – 7x + 3 = 0 are 3 and 1.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 2x<sup>2</sup> + x – 4 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) The given equation is </span>2x<sup>2</sup> + x - 4 = 0 ------------------ equation 1.</div><div>2) Divide equation 1 by 2 we get,</div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x<sup>2</sup> + x - 4 = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x<sup>2</sup> + (1/2)x - (4/2) = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x = 2</span><span> </span><span>------------------ equation 2.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) From </span><span style="font-family: arial;">equation 1, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(middle term)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(1/2)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">last term = </span><span style="font-family: arial;">1/16</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) Add 1/16 to both sides of equation 2, and we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = 1/16 + 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = 1/16 + 32/16</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = (1 + 32)/16<br /></span><span>(x</span><span> + 1/4)</span><sup>2</sup><span> = 33/16</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + 1/4)</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>(</span></span><span>33/16)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + 1/4)</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>33/4</span><span><span style="white-space: pre-wrap;"><br /></span></span><span>x</span><span> = - (1/4</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>33/4)</span><span><span style="white-space: pre-wrap;"><br /></span></span><span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>33</span><span><span style="white-space: pre-wrap;">)/4<br /></span></span><span>x = (- 1 + <span style="white-space: pre-wrap;"><b>√</b></span>33<span style="white-space: pre-wrap;">)/4, or </span>x = (- 1 - <span style="white-space: pre-wrap;"><b>√</b></span>33</span><span><span style="white-space: pre-wrap;">)/4.</span></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>5) The roots of the quadratic equation </span><span>2x</span><sup>2</sup><span> + x - 4 = 0</span><span> are </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(- 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">+ </span><span style="font-family: arial; white-space: pre-wrap;"><b>√</b></span><span style="font-family: arial;">33</span><span style="font-family: arial; white-space: pre-wrap;">)/4</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">(- 1 - <span style="white-space: pre-wrap;"><b>√</b></span>33</span><span style="font-family: arial;"><span style="white-space: pre-wrap;">)/4.</span></span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif" style="line-height: 17.12px;">(iii) 4x<sup>2</sup> + 4</span><span><span style="white-space: pre-wrap;">√</span></span><span face="Arial, sans-serif">3x + 3 = 0</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) The given equation is </span><span style="line-height: 17.12px;">4x<sup>2</sup> + 4</span><span style="white-space: pre-wrap;">√</span>3x + 3 = 0 ------------------ equation 1.</div><div>2) Divide equation 1 by 4 we get,</div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="line-height: 17.12px;">4x<sup>2</sup> + 4</span><span style="white-space: pre-wrap;">√</span>3x + 3 = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="line-height: 17.12px;">x<sup>2</sup> + </span><span style="white-space: pre-wrap;">√</span>3x + (3/4) = 0</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; line-height: 17.12px;">x<sup>2</sup> + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3x = - (3/4)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------------ equation 2.</span></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) From </span><span style="font-family: arial;">equation 1, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(middle term)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(</span><span style="white-space: pre-wrap;">√</span><span>3</span><span>)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">last term = </span><span style="font-family: arial;">3/4</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) Add 3/4 to both sides of equation 2, and we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; line-height: 17.12px;">x<sup>2</sup> + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3x + (3/4) = (3/4) - (3/4)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; line-height: 17.12px;">x<sup>2</sup> + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3x + (3/4) = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>3</span><span>/2)</span><sup>2</sup><span> = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + 3/2)</span><span> = </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;">0</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = - (3/2</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;"><span>0</span></span><span>)</span><span><span style="white-space: pre-wrap;"><br /></span></span><span>x</span><span> = - 3/2</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">0</span><span><span style="white-space: pre-wrap;"><br /></span></span><span>x = - 3/2</span><span><span style="white-space: pre-wrap;">, or </span>x = - 3/2</span><span><span style="white-space: pre-wrap;">.</span></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) The roots of the quadratic equation </span><span style="font-family: arial; line-height: 17.12px;">4x<sup>2</sup> + 4</span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3x + 3 = 0</span><span style="font-family: arial;"> are -3/2 and -3/2.</span></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">(iv) 2x</span><sup>2</sup><span face="Arial, sans-serif"> + x + 4 = 0</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">1) The given equation is </span>2x<sup>2</sup> + x + 4 = 0 ------------------ equation 1.</div><div>2) Divide equation 1 by 2 we get,</div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x<sup>2</sup> + x + 4 = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x<sup>2</sup> + (1/2)x + (4/2) = 0</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x = - 2</span><span> </span><span>------------------ equation 2.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) From </span><span style="font-family: arial;">equation 1, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(middle term)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = </span><span>[(1/2)</span><sup>2</sup><span>]/4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">last term = </span><span style="font-family: arial;">1/16</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) Add 1/16 to both sides of equation 2, and we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = 1/16 - 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = 1/16 - 32/16</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (1/2)x + 1/16 = (1 - 32)/16<br /></span><span>(x</span><span> + 1/4)</span><sup>2</sup><span> = - 33/16</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5) As the square of any real number can't be negative, so </span><span>2x</span><sup>2</sup><span> + x + 4 = 0 has no</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">real roots.</span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic </b></span><b>formula.</b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: arial; font-size: medium;"><b>(i) 2x<sup>2</sup> – 7x + 3 = 0 (ii) 2x<sup>2</sup> + x – 4 = 0</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif" style="line-height: 17.12px;">(iii) 4x<sup>2</sup> + 4</span><span><span style="white-space: pre-wrap;">√</span></span><span face="Arial, sans-serif">3x + 3 = 0 </span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">(iv) 2x</span><sup>2</sup><span face="Arial, sans-serif"> + x + 4 = 0</span></span></b></div></div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span>1) </span><span>The </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + (c/a) = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x = - (c/a)<br /></span><span>x</span><sup>2</sup><span> + (b/a)x = - (c/a) ------------------ equation 1.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) In the quadratic equation, for getting the perfect square, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>last term = [(middle term)</span><sup>2</sup><span>]/4</span><span> ------------------ equation 2.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From equation 1 and equation 2, last term = </span><span>[(b/a)</span><sup>2</sup><span>]/4.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>4) Adding </span><span>[(b/a)</span><sup>2</sup><span>]/4 to both sides of equation 1, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>[(b/a)</span><sup>2</sup><span>]/4 </span><span>= </span><span>[(b/a)</span><sup>2</sup><span>]/4 </span><span>- (c/a)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>(b/2a)</span><sup>2</sup><span> </span><span>= </span><span>(b/2a)</span><sup>2</sup><span> </span><span>- (c/a)<br /></span><span>x</span><sup>2</sup><span> + (b/a)x + </span><span>(b/2a)</span><sup>2</sup><span> </span><span>= </span><span>(b</span><span>)</span><sup>2</sup><span>/4a</span><sup>2</sup><span> </span><span>- (c/a)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><sup>2</sup><span> </span><span>= (</span><span>b</span><sup>2 </sup><span>- 4ac)</span><span>/4a</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><span> </span><span>= </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)</span><span>/4a</span><sup>2</sup><span>]</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>(x</span><span> + </span><span>b/2a)</span><span> </span><span>= </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><span> = - </span><span>b/2a</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>[</span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 3.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) This is the method of the square.</span></div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>(i) 2x<sup>2</sup> – 7x + 3 = 0</b></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x</span><sup>2</sup><span> - 7x + 3 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">using the formula:</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) Equate the coefficient of equation </span><span>2x</span><sup>2</sup><span> - 7x + 3 = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a = 2, b = - 7, c = 3.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>- 7)</span><sup>2 </sup><span>- 4(2)(3)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 49</span><sup> </sup><span>- 24</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 25</span><span> ------------------ equation 3.</span><span> </span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">As </span><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 25</span>, it has real roots, so </span><span style="font-family: arial;">from equation 2 and equation 3, we</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(- 7)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>25</span></span><span>]</span><span>/2(2)<br /></span><span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;"><span>5)</span></span><span>/4</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>6) So, </span> <span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">+ </span><span style="white-space: pre-wrap;"><span>5)</span></span><span>/4 or </span><span>x</span><span> = (7</span><span> </span><span face="Arial, sans-serif">- </span><span style="white-space: pre-wrap;"><span>5)</span></span><span>/4.</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (12</span><span style="white-space: pre-wrap;"><span style="font-family: arial;">)</span></span><span style="font-family: arial;">/4 or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (2</span><span style="white-space: pre-wrap;"><span style="font-family: arial;">)</span></span><span style="font-family: arial;">/4</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 3</span><span style="font-family: arial;"> or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 1/2</span></span></div></blockquote><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7) So, x</span><span style="font-family: arial;"> = 3</span><span style="font-family: arial;"> or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 1/2.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 2x<sup>2</sup> + x – 4 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x</span><sup>2</sup><span> + x - 4 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">using the formula:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Equate the coefficient of equation </span><span>2x</span><sup>2</sup><span> + x - 4</span><span> = 0 with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 2, b = 1, c = - 4.</span></blockquote><div><span style="font-family: arial; font-size: medium;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>1)</span><sup>2 </sup><span>- 4(2)(- 4)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 1</span><sup> </sup><span>+ 32</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 33</span><span> ------------------ equation 3.</span><span> </span></span></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;">5) As </span><span><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">=</span> 33</span>, it has real roots, so </span>from equation 2 and equation 3, we</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">have,</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(1)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>33</span></span><span>]</span><span>/2(2)<br /></span><span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">± </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">33</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>6) So, </span> <span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">+ </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">33</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4 or </span><span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">- </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">33</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4.</span></span><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif" style="line-height: 17.12px;">(iii) 4x<sup>2</sup> + 4</span><span><span style="white-space: pre-wrap;">√</span></span><span face="Arial, sans-serif">3x + 3 = 0</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>4x</span><sup>2</sup><span> + 4</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>x + 3 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">using the formula:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Equate the coefficient of equation </span><span>4x</span><sup>2</sup><span> + 4</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>x + 3 = 0</span><span> with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, so</span></span></div></div></div><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a = 4, b = </span><span>4</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>, c = 3.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (</span><span>4</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>)</span><sup>2 </sup><span>- 4(4)(3)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 48</span><sup> </sup><span>- 48</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 0</span><span> ------------------ equation 3.</span><span> </span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) <span style="font-family: arial;">As, </span><span><span>b</span><sup>2 </sup><span>- 4ac <span face="Arial, Verdana, sans-serif" style="background-color: white;">≥</span> 0</span>, it has real roots, so </span>from equation 2 and equation 3, we</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">have,</span></div></div></div></blockquote><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(1)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>0</span></span><span>]</span><span>/2(4)<br /></span><span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">0</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>6) So, </span> <span>x</span><span> = (- 1</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4 or </span><span>x</span><span> = (- 1</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">(iv) 2x</span><sup>2</sup><span face="Arial, sans-serif"> + x + 4 = 0</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) The given equation is </span><span>2x</span><sup>2</sup><span> + </span><span>x + 4 = 0 ------------------ equation 1.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">using the formula:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) Equate the coefficient of equation </span><span>2x</span><sup>2</sup><span> + </span><span>x + 4 = 0</span><span> with </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, so</span></span></div></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a = 2, b = 1</span><span style="font-family: arial; font-size: medium;">, c = 4.</span></blockquote><div><span style="font-family: arial; font-size: medium;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (1</span><span>)</span><sup>2 </sup><span>- 4(2)(4)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 1</span><sup> </sup><span>- 32</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = - 31</span><span> ------------------ equation 3.</span><span> </span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">As </span><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = - 31 <</span> 0, <span>so </span><span>2x</span><sup>2</sup><span> + x + 4 = 0</span> has no real roots</span><span style="font-family: arial;">.</span></span></div></div></div></div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><b>Q3. Find the roots of the following equations:</b></div><div><b>(i) x - (1/x) = 3, x ≠ 0 (ii) 1/(x + 4) - 1/(x - 7) = 11/30, x ≠ - 4, x ≠ 7</b></div><div><div><h3><span style="font-family: arial;">Explanation:</span></h3></div><div><span style="font-family: arial;">1) Convert your equation in to </span><span style="font-family: arial;">the </span><span style="font-family: arial;">quadratic form </span><span style="font-family: arial;">a</span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></div></div><div><span style="font-family: arial;">2) Solve this equation to get the values of the variable x.</span></div><div><h3><span style="font-family: arial;">Solution:</span></h3></div><div><b>(i) x - (1/x) = 3, x ≠ 0</b></div><div><br /></div><div><div><div><div><div><span style="font-family: arial;"><span>1) The given equation is </span></span></div></div></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">x - (1/x) = 3</span></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x</span><sup>2</sup><span> </span><span>- 1)/x = 3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(x</span><sup>2</sup><span> </span><span>- 1) = 3x</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>- 3x -1 = 0</span><span> ------------------ equation 1.</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">using the formula:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial;"><span>3) </span></span><span style="font-family: arial;">Equate the coefficient of equation </span><span style="font-family: arial;"><span>x</span><sup>2</sup><span> </span>- 3x -1 = 0 with </span><span style="font-family: arial;">a</span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> + bx + c = 0, so</span></div></div></div></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>a = 1, b = - 3</span><span>, c = 1.</span></span></blockquote><div><span style="font-family: arial;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">3</span><span>)</span><sup>2 </sup><span>- 4(1)(1)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 9</span><sup> </sup><span>- 4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 5</span><span> ------------------ equation 3.</span></span></blockquote><div><span style="font-family: arial;">5) As </span><span style="font-family: arial;">b</span><sup>2 </sup><span style="font-family: arial;">- 4ac = 5 > 0, it has real roots, so </span><span style="font-family: arial;">from equation 2 and equation 3, we</span></div></div></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">have,</span></div></div></div></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(1)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>0</span></span><span>]</span><span>/2(4)<br /></span><span>x</span><span> = (- 1</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">0</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4</span></span></blockquote><span style="font-family: arial;"><span>6) So, </span> <span>x</span><span> = (- 1</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4 or </span><span>x</span><span> = (- 1</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/4.</span></span></div><div><span style="font-family: arial;"><span><br /></span></span></div><div><b>(ii) 1/(x + 4) - 1/(x - 7) = 11/30, x ≠ - 4, x ≠ 7</b></div></div></div><div><br /></div><div><div><div><span style="font-family: arial;">1) The given equation is </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">1/(x + 4) - 1/(x - 7) = 11/30</span></div></blockquote></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">[1(x - 7) - 1(x + 4)]/(x + 4)(x - 7) = 11/30</div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x - 7 - x - 4)/(x + 4)(x - 7) = 11/30</span></div><span style="font-family: arial; font-size: medium;"><span>30(x - 7 - x - 4) = 11 (x + 4)(x - 7)<br /></span><span>30(- 11) = 11 (x + 4)(x - 7)<br /></span><span>- 30 = (x + 4)(x - 7)<br /></span><span>- 30 = x(x - 7) + 4 (x - 7)<br /></span><span>- 30 = </span><span>x</span><sup>2</sup><span> </span><span>-</span><span> 7x + 4x - 28<br /></span><span>- 30 = </span><span>x</span><sup>2</sup><span> </span><span>-</span><span> 3x - 28<br /></span><span>x</span><sup>2</sup><span> </span><span>-</span><span> 3x - 28 + 30 = 0<br /></span><span>x</span><sup>2</sup><span> </span><span>-</span><span> 3x + 2 = 0</span><span> ------------------ equation 1.</span></span></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><span>2) We know that the quadratic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 can be solved</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">using the formula:</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a</span><span> ------------------ equation 2.</span></span></blockquote></span></div><div><span style="font-family: arial;"></span><span style="font-family: arial;">3) Equate the coefficient of equation <span>x</span><sup>2</sup><span> </span>- 3x + 2 = 0 with ax<sup>2</sup> + bx + c = 0, so</span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">a = 1, b = - 3</span><span style="font-family: arial;">, c = 2.</span></div></div></div></span></div></div></div></blockquote><div><span style="font-family: arial; font-size: medium;">4) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">3</span><span>)</span><sup>2 </sup><span>- 4(1)(2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 9</span><sup> </sup><span>- 8</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2 </sup><span>- 4ac = 1</span><span> ------------------ equation 3.</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) As b<sup>2 </sup>- 4ac = 1 > 0, it has real roots, so from equation 2 and equation 3, we</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">have,</span></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>x</span><span> = [- </span><span>b</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b></span></span><span>(</span><span>b</span><sup>2 </sup><span>- 4ac)]</span><span>/2a<br /></span><span>x</span><span> = [- </span><span>(- 3)</span><span> </span><span face="Arial, sans-serif">± </span><span><span style="white-space: pre-wrap;"><b>√</b>1</span></span><span>]</span><span>/2(1)<br /></span><span>x</span><span> = (3</span><span> </span><span face="Arial, sans-serif">± </span><span style="white-space: pre-wrap;">1</span><span style="white-space: pre-wrap;"><span>)</span></span><span>/2</span></span></blockquote><span style="font-family: arial;">6) So, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (4</span><span style="font-family: arial; white-space: pre-wrap;"><span>)</span></span><span style="font-family: arial;">/2 or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (2</span><span style="font-family: arial; white-space: pre-wrap;"><span>)</span></span><span style="font-family: arial;">/2, i.e. x = 2, or x = 1.<br /></span></div></div></span></span></div></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><br /></div><div><b>Q4. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3 Find his present age.</b></div><div><br /></div><div><div><div><div><div><span style="font-family: arial;">1) Let Rehman's present age be x.</span></div><div><span><span style="font-family: arial;">2) So, 3 years ago, his age </span><span style="font-family: arial;">was (x - 3).</span></span></div></div><div><span style="font-family: arial;">3) 5 years later, his age will be (x + 3).</span></div></div><div><span style="font-family: arial;">4) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">[1/(x - 3)] + [1/(x + 5)] = 1/3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">[1</span><span style="font-family: arial;">(x + 5) + 1</span><span style="font-family: arial;">(x - 3)]/[</span><span style="font-family: arial;">(x + 5)</span><span style="font-family: arial;">(x - 3)]</span><span style="font-family: arial;"> = 1/3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(x + 5 + x - 3)/(x + 5)(x - 3) = 1/3</span></div><span style="font-family: arial;">3(2x + 2) = (x + 5)(x - 3)<br /></span><span style="font-family: arial;">6x + 6 = </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">+</span><span style="font-family: arial;"> 5x - 3x - 15<br /></span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">+</span><span style="font-family: arial;"> 2x - 15 - 6x - 6 = 0<br /></span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">-</span><span style="font-family: arial;"> 4x - 21 = 0</span><span style="font-family: arial;"> ------------------ equation 1.</span></blockquote><div><div><span style="font-family: arial;">5) Equate the coefficient of equation <span>x</span><sup>2</sup><span> </span>- 4x - 21 = 0 with ax<sup>2</sup> + bx + c = 0, so</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a = 1, b = - 4, c = - 21</span></div></blockquote><div><span style="font-family: arial;">6) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">4</span><span>)</span><sup>2 </sup><span>- 4(1)(- 21)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 16</span><sup> </sup><span>+ 84</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 100</span><span> ------------------ equation 2.</span></span></blockquote><div><div><span style="font-family: arial;">7) As b<sup>2 </sup>- 4ac = 100 > 0, it has real roots, so from equation 1 and equation 2, we</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">have,</span></div></blockquote></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">b</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b></span></span><span style="font-family: arial;">(</span><span style="font-family: arial;">b</span><sup>2 </sup><span style="font-family: arial;">- 4ac)]</span><span style="font-family: arial;">/2a<br /></span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">(- 4)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b>100</span></span><span style="font-family: arial;">]</span><span style="font-family: arial;">/2(1)<br /></span><div style="text-align: left;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (4</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial; white-space: pre-wrap;">10</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2</span> <br /></div></div></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial;">8) So, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (14</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2 or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (- 6</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2, i.e. x = 7, or x = - 3.</span></div><div><span style="font-family: arial;">9) As age can't be negative, x = 7, so Rehman's present age is 7 years.</span></div></div></div><div><br /></div><div><b>Q5. In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.</b></div><div><br /></div><div><div><div><div><span style="font-family: arial;">1) Let Shefali's marks in Mathematics be x.</span></div><div><span><span style="font-family: arial;">2) Her English marks</span></span> will be (30 - x).</div></div></div><div><span style="font-family: arial;">3) According to the problem,</span></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">(x + 2)(30 - x - 3) = 210</div></span></div></blockquote><div><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(x + 2)(27 - x) = 210</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><span style="font-family: arial;">(27 - x) + 2</span><span style="font-family: arial;">(27 - x)</span><span style="font-family: arial;"> = 210</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">27x - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">+ </span>54 - 2x = 210</div><span style="font-family: arial;"><span style="font-family: arial;">27x</span></span> - 2x<span style="font-family: arial;"> - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">+ </span>54 = 210</blockquote></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial;">25x</span></span> <span style="font-family: arial;">- </span><span style="font-family: arial;">x</span><sup>2</sup> = 210 - 54</div></span></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial;">25x</span></span> <span style="font-family: arial;">- </span><span style="font-family: arial;">x</span><sup>2</sup> = 156</span></blockquote></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;"><span style="font-family: arial;">x</span><sup>2</sup> - 25x + 156 = 0</span> ------------------ equation 1.</div></span></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">4) Equate the coefficient of equation <span style="font-family: arial;">x</span><sup>2</sup> - 25x + 156 = 0 with ax<sup>2</sup> + bx + c = 0, so</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">a = 1, b = - 25, c = 156</span></blockquote><div><span style="font-family: arial;">6) First we will find:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = (- </span><span style="white-space: pre-wrap;">25</span><span>)</span><sup>2 </sup><span>- 4(1)(156)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 625</span><sup> </sup><span>- 624</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span>b</span><sup>2 </sup><span>- 4ac = 1</span><span> ------------------ equation 2.</span></span></blockquote><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">7) As b<sup>2 </sup>- 4ac = 1 > 0, it has real roots, so from equation 1 and equation 2, we</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">have,</span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">b</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b></span></span><span style="font-family: arial;">(</span><span style="font-family: arial;">b</span><sup>2 </sup><span style="font-family: arial;">- 4ac)]</span><span style="font-family: arial;">/2a<br /></span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = [- </span><span style="font-family: arial;">(- 25)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b>1</span></span><span style="font-family: arial;">]</span><span style="font-family: arial;">/2(1)<br /></span><div><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (25</span><span style="font-family: arial;"> </span><span style="font-family: arial;">± </span><span style="font-family: arial; white-space: pre-wrap;">1</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2</span> <br /></div></span></div></blockquote><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"></span><span style="font-family: arial;">8) So, </span><span style="font-family: arial;"> </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (26</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2 or </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = (24</span><span style="font-family: arial; white-space: pre-wrap;">)</span><span style="font-family: arial;">/2, i.e. x = 13, or x = 12.</span></div></span></div></div><div style="text-align: left;">9) If x = 13, then </div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div style="text-align: left;">she got 13 marks in Mathematics and 30 - 13 = 17 marks in English.</div></span></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div>10) If x = 12, then </div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">she got 12 marks in Mathematics and 30 - 12 = 18 marks in English.</div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><br /></div><div><b>Q6. <span style="background-color: white; color: #222222;">The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.</span></b></div><div><br /></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the shorter side of a rectangle be x m.</span></div><div><span style="font-family: arial; font-size: medium;">2) So, the longer side of a rectangle will be (x + 30) m </span></div><div><span style="font-family: arial; font-size: medium;">3) According to the problem,</span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">+ (</span><span style="font-family: arial;">x + 30)</span><sup>2</sup><span style="font-family: arial;"> = (</span><span style="font-family: arial;">x + 60)</span><sup>2</sup></div></div></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= (</span><span style="font-family: arial;">x + 60)</span><sup>2</sup><span style="font-family: arial;"> - (</span><span style="font-family: arial;">x + 30)</span><sup>2</sup> <span> </span><span> use </span><span style="font-family: arial;">a</span><sup>2</sup><span style="font-family: arial;"> - </span><span style="font-family: arial;">b</span><sup>2</sup><span style="font-family: arial;"> = (a - b)(a + b)</span></div></span></div></blockquote><div><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= [(</span><span style="font-family: arial;">x + 60) - </span><span style="font-family: arial;">(</span><span style="font-family: arial;">x + 30)] </span><span style="font-family: arial;">[(</span><span style="font-family: arial;">x + 60) + </span><span style="font-family: arial;">(</span><span style="font-family: arial;">x + 30)]</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= (</span><span style="font-family: arial;">x + 60 - </span><span style="font-family: arial;">x - 30) </span><span style="font-family: arial;">(</span><span style="font-family: arial;">x + 60 + </span><span style="font-family: arial;">x + 30)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= 30</span><span style="font-family: arial;">(2</span><span style="font-family: arial;">x + 90</span><span style="font-family: arial;">)</span></div><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= 60x</span><span style="font-family: arial;"> + 2700</span></blockquote></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 60x</span><span style="font-family: arial;"> - 2700 = 0</span><span style="font-family: arial;"> ------------------ equation 1.</span></div></blockquote><div><span style="font-family: arial;">4) </span><span style="font-family: arial;">Quadratic equation: </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 60x</span><span style="font-family: arial;"> - 2700 = 0</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 2700 and its sign is minus, </span><span style="text-align: right;">factorise 1 x 2700 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their difference will be 60.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">1 x 2700 = (30) x (- 90) (30 - 90 = - 60).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 30x - 90x - 2700 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x(x + 30) - 90(x + 30) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + 30)(x - 90) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span><div>5) So, (x + 30) = 0, or (x - 90) = 0</div></span></span></div><div style="text-align: left;">6) So, x = - 30 or x = 90.</div><div><span style="font-family: arial;"><span style="font-family: arial;"><div>7) As the length of any side can't be negative, ignore x = -30, so we have x = 90.</div></span></span></div><div><div><span style="font-family: arial;">8) The length of the shorter side is 90 m and the longer side is 90 + 30 = 120 m.</span></div><div><span style="font-family: arial;"><br /></span></div><div><b><span style="font-family: arial;">Q</span>7. The difference of the squares of the two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.</b></div></div></div><div><br /></div><div><div><div><span style="font-family: arial;">1) Let the larger number be x.</span></div></div><div><span style="font-family: arial;">2) So, the square of the smaller number will be 8</span><span style="font-family: arial;">x</span></div><div><span style="font-family: arial;">3) According to the problem,</span></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 8x = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 8x - 180 = 0</span> ------------------ equation 1.</blockquote><div style="text-align: left;">4) Quadratic equation: <span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 8x</span><span style="font-family: arial;"> - 180 = 0</span> </div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 180 and its sign is minus, </span><span style="text-align: right;">factorise 1 x 180 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their difference will be 8.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">1 x 180 = (10) x (- 18) (10 - 18 = - 8).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 10x - 18x - 180 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x(x + 10) - 18(x + 10) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + 10)(x - 18) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span>5) So, (x + 10) = 0, or (x - 18) = 0</span></span></div><div style="text-align: left;">6) So, x = - 10 or x = 18.</div>7) If x = - 10, the smaller number will be - 10 (8) = - 80, as a square of any</span></div></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">number can't be negative, we can't take x = -10 as the lager number.</span></div></div></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;">8) So, our larger number will be 18. </span></div><div><span style="font-family: arial; font-size: medium;">9) According to the problem, a square of a smaller number = 8x = 8(18).</span></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">square of a smaller number = 8(18)</span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">square of a </span><span style="font-family: arial;">smaller number = 2 x 2 x 2 x 2 x 3 x 3, so,</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">smaller number = </span><span style="font-family: arial;">± (</span><span style="font-family: arial;">2 x 2 x 3)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">smaller number = </span><span style="font-family: arial;">± 1</span><span style="font-family: arial;">2.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">10) The numbers are 12 and 18 or - 12 and 18.</span></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b>Q8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.</b></div><div><b><br /></b></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the uniform speed of a train be x km/h.</span></div><div><span style="font-family: arial; font-size: medium;">2) To cover 360 km, the time will be (360/x) hrs.</span></div><div><span style="font-family: arial; font-size: medium;">3) If the speed is increased by 5 km/h, then time will be decreased by 1 hr, so </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">new speed = (x + 5) km/h, and new time = [</span><span style="font-family: arial;">(360/x) -1] hrs.</span></span></div></div></div></div></div></div></div></div></blockquote><div><span style="font-family: arial; font-size: medium;">4) According to the condition,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">(x + 5)[(360/x) -1] = 360</span></div></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 5)[(360 - x)/x] = 360</span></blockquote></div></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><div style="text-align: left;">(x + 5)(360 - x) = 360x</div></div></div></div></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x(360 - x) + 5(360 - x) = 360x</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>360x - </span><span>x</span><sup>2</sup><span> + 1800 - 5x = 360x</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>- </span><span>x</span><sup>2</sup><span> + 1800 - 5x = 0</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span>x</span><sup>2</sup><sup><span> </span></sup><span>+ 5x - 1800 = 0</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------- equation 1</span></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div></div></div></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">4) Quadratic equation: <span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> +</span><span style="font-family: arial;"> 5x</span><span style="font-family: arial;"> - 1800 = 0</span> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 1800 and its sign is minus, </span><span style="text-align: right;">factorise 1 x 1800 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their difference will be 5.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">1 x 1800 = (45) x (- 40) (45 - 40 = 5).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 45x - 40x - 1800 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x(x + 45) - 40(x + 45) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + 45)(x - 40) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span>5) So, (x + 45) = 0, or (x - 40) = 0</span></span></div><div>6) So, x = - 45 or x = 40.</div>7) As x is the speed, it can't be negative, so we have x = 40.</span></div><div><span style="font-family: arial;">8) The speed of the train is 40 km/h.</span></div><div><div class="separator" style="clear: both; text-align: center;"><span><br /></span></div><span><b>Q9.Two water taps together can fill a tank in </b><b>9 and 3/8 hours. The tap of a larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.</b></span></div></div><div><br /></div><div><div><div><span style="font-family: arial;">1) Let the time taken by the smaller pipe to fill the tank be x hrs.</span></div><div><div><div><span style="font-family: arial;">2)</span> Time taken by the larger pipe will be (x - 10) hrs.</div><div><span style="font-family: arial;">3) The part of a tank filled by a smaller pipe in 1 hr will be 1/x.</span></div></div></div><div><span>4) The part of a tank filled by a larger pipe in 1 hr will be 1/(x - 10).</span></div><div><span>5) Total time taken by both the pipes to fill the tank = 9 and 3/8 = (72+3)/8 = 75/8.</span></div><div><span>6) According to the problem, </span><span>the part of a tank filled by both pipes in 1 hour is 8/75.</span></div><div><span>7) So, we have,</span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-size: medium;">1/x + 1/(x - 10) = 8/75</span></div></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">[(x - 10) + x]/x(x - 10) = 8/75</span></div><span style="font-size: medium;"><span style="font-family: arial;">75(2x - 10) = 8</span><span style="font-family: arial;">x(x - 10)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>150x - 750 = 8</span><span>x</span><sup>2</sup><span> </span><span>- 80x</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>8</span><span>x</span><sup>2</sup><span> </span><span>- 80x - 150x + 750 = 0</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">8</span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">- 230x + 750 = 0</span><span> </span><span>---------- equation 1</span></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><div></div></div></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">8) Quadratic equation: 8<span>x</span><sup>2</sup><span> </span>- 230x + 750 = 0 </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 750 and its sign is plus, </span><span style="text-align: right;">factorise 8 x 750 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their sum will be - 230. </span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">8 x 750 = (- 200) x (- 30) </span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></span></div></div></span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="text-align: right;">(- 200 - 30 = - 230).</span></div></span></div></span></div></div></div></span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">8x<span><sup>2</sup></span> - 200x - 30x + 750 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8x(x - 25) - 30(x - 25) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x - 25)(8x - 30) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span>9) So, (x - 25) = 0, or (8x - 30) = 0</span></span></div><div>10) So, x = 25 or x = 30/8.</div>11) If we consider x = 30/8, the time taken by a larger pipe will be </span></div></span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(30/8 - 10) = (30 - 80)/8 = - 50/8 which negative, so ignore x = 30/8.</span></div></span></div></div></div></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">12) So consider x = 25. i.e. time taken by smaller pipe to fill the tank will be 25 hrs</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">and the </span><span style="font-family: arial;">time taken by larger pipe to fill the tank will be 15 hrs.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q10. An express train takes 1 hour less than a passenger train to travel 132 km between </b></span><b>Mysore and Bangalore (without taking into consideration the time they stop at </b><b>intermediate stations). If the average speed of the express train is 11km/h more than that </b><b>of the passenger train, find the average speed of the two trains. </b></span></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><br /></div><div><div><div><span style="font-family: arial;">1) Let the average speed of a passenger train be x km/h.</span></div><div><span style="font-family: arial;">2) So the average speed of the express train will be (x + 11) km/h.</span></div><div><span style="font-family: arial;">3) We know that time = distance/speed, so to cover 132 km,</span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;">a) for passenger train, time = 132/x hrs.</div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) for express train, time = 132/(x + 11) hrs.</span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">4) As the express train takes 1 hour less than the passenger train to cover 132 km,</span></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">so</span></div></div></span></div></div></div></blockquote><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">[132/x] - [132/(x + 11)] = 1</span></div></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">132[1/x - 1/(x + 11)] = 1</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">132[(x + 11) - x ]/x(x + 11)] = 1</span></blockquote></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">132(11)/x(x + 11) = 1</span></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">132(11) = x(x + 11)</span></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>1452 = x</span><sup>2</sup><span> </span><span>+ 11x</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span><span>x</span><sup>2</sup><span> </span><span>+ 11x - 1452 = 0</span></span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></blockquote></div></div><div><div><div><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><span style="font-family: arial;">5) Quadratic equation: <span>x</span><sup>2</sup><span> </span><span>+ 11x - 1452 = 0</span> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 1452 and its sign is minus, </span><span style="text-align: right;">factorise 1 x 1452 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their difference will be 11. </span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">1 x 1452 = (44) x (- 33) </span><span style="text-align: right;">(44 - 33 = 11).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></span></div></div></span></div></div></div></span></div></div></div><div><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 44x - 33x - 1452 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x(x + 44) - 33(x + 1452) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + 44)(x - 33) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span>6) So, (x + 44) = 0, or (x - 33) = 0</span></span></div><div>7) So, x = - 44 or x = 33.</div>8) As the speed can't be negative, so ignore x = - 44, so the average speed of</span></div></span></div></div></div></span></div></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">passenger train will be 33 km/h and the average speed of the express train will be 44 km/h.</span></div></span></div></div></div></span></div></div></div></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q11. The sum of the areas of the two squares is 468 </b></span><b><span>m</span><span><sup>2</sup></span></b><span><b>. If the difference of their perimeters is 24 m, </b></span><b>find the sides of the two squares. </b></span></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><br /></div><div><div><div><span style="font-family: arial;">1) Let the side of the first square be x m.</span></div><div><span style="font-family: arial;">2) So, its area will be </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">.</span><span style="font-family: arial;"> </span></div><div><span style="font-family: arial;">3) According to the problem, the area of the other square will be (468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">). the side</span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">of this square will be </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><b>√</b>(</span></span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">).</span></div></div></div></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) The perimeter of the first square will be 4x and that of the other will be </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4</span><span><span style="white-space: pre-wrap;"><b>√</b>(</span></span><span>468 - </span><span>x</span><sup>2</sup><span>)</span> </span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) As the difference between their perimeters is 24,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div><div><span><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">4x - 4<span style="white-space: pre-wrap;"><b>√</b>(</span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">) = 24</span></div></span></span></div></div></div></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><div><span><span style="font-family: arial;"></span><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">4[x - <span style="white-space: pre-wrap;"><b>√</b>(</span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">)] = 24</span></blockquote><div><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">x - <span style="white-space: pre-wrap;"><b>√</b>(</span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">) = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">x - 6 = <span style="white-space: pre-wrap;"><b>√</b>(</span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">) squaring both side, we get</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(x - 6)</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">= </span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;">)</span></div></blockquote></span></span></div></span></span></div></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div><div><span><span style="font-family: arial;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">x</span><sup>2</sup><span style="font-family: arial;"> - 12x + 36 </span><span style="font-family: arial;">= </span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup></div></span></span></span></span></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span><span>x</span><sup>2 </sup>+ x</span><sup>2</sup><span> - 12x + 36 </span><span>- </span><span>468 = 0<br /></span><span><span>2x</span><sup>2</sup></span><span> - 12x </span><span>- </span><span>432 = 0<br /></span><span><span>x</span><sup>2</sup></span><span> - 6x </span><span>- </span><span>216 = 0</span><span> ------------------ equation 1.</span></span></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span><span style="font-family: arial;"><div><span style="font-family: arial;">6) </span><span style="font-family: arial;">Quadratic equation: </span><span style="font-family: arial;"><span style="font-family: arial;">x</span><sup>2</sup></span><span style="font-family: arial;"> - 6x </span><span style="font-family: arial;">- </span><span style="font-family: arial;">216 = 0</span></div></span></span><span><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 216 and its sign is minus, </span><span style="text-align: right;">factorise 1 x 216 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">that their difference will be - 6.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; text-align: right;">1 x 216 = (12) x (- 18) (12 - 18 = - 6).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 12x - 18x - 216 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x(x + 12) - 18(x + 12) = 0</span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + 12)(x - 18) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial;"><span></span><span></span><span>5) So, (x + 12) = 0, or (x - 18) = 0</span></span></div><div>6) So, x = - 12 or x = 18.</div><div><span style="font-family: arial;"><span style="font-family: arial;">7) As the length of any side can't be negative, ignore x = - 12, so we have x = 18.</span></span></div><div><span style="font-family: arial;">8) The length of the side of the first square will be 18 m.</span></div><div><span style="font-family: arial;">9) So the Area of the other square is</span></div></span></span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><div><span><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">= </span><span style="font-family: arial;">468 - </span><span style="font-family: arial;">x</span><sup>2</sup></div></span></span></div></div></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>= </span><span>468 - </span><span>18</span><sup>2<br /></sup><span>= </span><span>468 - 324<br /></span><span>= 144. Therefore side of other square will be </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">144 = 12 m.</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>10) The sides of the squares are 12 m and 18 m.</span></span><div><span style="color: #0400ff; font-family: arial; font-size: large;"><br /></span></div><div><span style="color: #0400ff; font-family: arial; font-size: large;"> </span><span style="font-family: arial; font-size: medium;">#NCERT #math #quadraticequations #education #learning #students #teachers</span></div><div><div><div style="text-align: left;"><div><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/09/159-ncert-10-4-quadratic-equations-ex-44.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations - Ex-4.4</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-85141069590698133102023-07-10T19:38:00.003+05:302023-08-31T18:30:09.708+05:30157-NCERT-10-4-Quadratic Equations - Ex-4.2<h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><span style="color: #0400ff;"><div style="clear: both; color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="clear: both; color: black; font-weight: 400;"><span style="font-family: arial;">10th Mathematics</span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Exercise 4.2</span></span></div><div style="color: black; font-weight: 400;"><span style="font-family: arial; font-size: medium;"><span>Topic: 4 Quadratic Equations</span></span></div></span></span></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/07/156-ncert-10-4-quadratic-equations-ex-41.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations-Ex-4.1</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 4.2</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>Q1. Find the roots of the following quadratic equations by factorisation:</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(i) x<sup>2</sup> – 3x – 10 = 0 <span> </span>(ii) 2x<sup>2</sup> + x – 6 = 0</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>(iii) </b></span><b><span style="white-space: pre-wrap;">√</span></b><b>2 x<sup>2</sup> + 7 x + 5</b><b><span style="white-space: pre-wrap;">√</span></b><b>2 = 0 <span> </span>(iv) 2x<sup>2</sup> – x + 1/8 = 0</b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(v) 100x<sup>2</sup> – 20x + 1 = 0 </b></span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span>1) </span><span>The </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) If α and β are the roots of the equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, we have, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><span>α</span><sup>2</sup><span> + b</span><span>α</span><span> + c = </span><span>0 and </span><span>a</span><span>β</span><sup>2</sup><span> + b</span><span>β</span><span> + c = 0. i.e. </span><span>we say that, </span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>α and β satisfy the equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) We can solve this qardatic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0 using factorisation method.</span> </span></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) x<sup>2</sup> – 3x – 10 = 0</span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><div><span>1) </span>Quadratic equation: x<span><sup>2</sup></span> – 3x – 10 = 0</div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><div><span style="text-align: right;">here last term is 10 and its sign is minus, </span><span style="text-align: right;">factorise 10 in such a way</span></div></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium; text-align: right;">that their difference will be - 3.</span></div></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium; text-align: right;">10 = (- 5) x (2) (- 5 + 2 = - 3).</span></div></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> – 5x + 2x – 10 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x – 5) + 2(x – 5) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(x – 5)(x + 2) = 0</div></span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><div>2) So, (x – 5) = 0 or (x + 2) = 0.</div><div>3) So, x = 5 or x = - 2.</div></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 2x<sup>2</sup> + x – 6 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span>1) </span>Quadratic equation: 2x<span><sup>2</sup></span> + x – 6 = 0</div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 6 and its sign is minus, </span><span style="text-align: right;">factorise 2 x 6 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; text-align: right;">that their difference will be 1.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; text-align: right;">2 x 6 = (3) x (- 4) (4 - 3 = 1).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">2x<span><sup>2</sup></span> – 3x + 4x – 6 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(2x – 3) + 2(2x – 3) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(2x – 3)(x + 2) = 0</div></span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>2) So, (2x – 3) = 0, or (x + 2) = 0</div><div>3) So, x = 3/2 or x = – 2.</div><div><br /></div><div><span><b>(iii) </b></span><b><span style="white-space: pre-wrap;">√</span></b><b>2x<sup>2</sup> + 7x + 5</b><b><span style="white-space: pre-wrap;">√</span></b><b>2 = 0</b></div></span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span>1) </span>Quadratic equation: <b><span style="white-space: pre-wrap;">√</span></b>2x<span><sup>2</sup></span> + 7x + 5<b><span style="white-space: pre-wrap;">√</span></b>2 = 0</div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is <span style="text-align: left;">5</span><b style="text-align: left;"><span style="white-space: pre-wrap;">√</span></b><span style="text-align: left;">2</span> and its sign is plus, </span><span style="text-align: right;">factorise </span><b><span style="white-space: pre-wrap;">√</span></b>2<span style="text-align: right;"> x </span>5<b><span style="white-space: pre-wrap;">√</span></b>2<span style="text-align: right;"> in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; text-align: right;">that their sum will be 7.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium; text-align: right;"><b style="text-align: left;"><span style="white-space: pre-wrap;">√</span></b><span style="text-align: left;">2</span> x <span style="text-align: left;">5</span><b style="text-align: left;"><span style="white-space: pre-wrap;">√</span></b><span style="text-align: left;">2</span> = 5 x 2 (5 + 2 = 7).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><b><span style="white-space: pre-wrap;">√</span></b>2x<span><sup>2</sup></span> + 5x + 2x + 5<span style="white-space: pre-wrap;"><b>√</b>2</span> = 0</div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span style="white-space: pre-wrap;">x(<b>√</b></span></span><span><span>2x</span><span> + 5) + </span></span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">2(</span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">2</span><span><span>x + 5)</span></span><span> = 0</span></span></blockquote></blockquote></blockquote></blockquote></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(<b><span style="white-space: pre-wrap;">√</span></b>2x + 5)(x + <b style="white-space: pre-wrap;">√</b>2) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">2) So, (<b><span style="white-space: pre-wrap;">√</span></b>2x + 5) = 0, or (x + <b style="white-space: pre-wrap;">√</b>2) = 0</div></span><div><span style="font-family: arial; font-size: medium;"><div>3) So, x = (- 5/<b><span style="white-space: pre-wrap;">√</span></b>2) or x = - <b style="white-space: pre-wrap;">√</b>2.</div><div><br /></div><div><b>(iv) 2x<sup>2</sup> – x + 1/8 = 0</b></div><div><div><span><div><br /></div><div><div><span>1) </span>Quadratic equation: 2x<span><sup>2</sup></span> – x + 1/8 = 0</div><div>2) Multiply by 8, and we get: 16x<span><sup>2</sup></span> – 8x + 1 = 0</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 1 and its sign is plus, </span><span style="text-align: right;">factorise 16 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their sum will be </span>– 8<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">16 x 1 = (<span style="text-align: left;">– 4)</span> x (</span>–<span style="text-align: right;">4) (</span>– <span style="text-align: right;">4 </span>–<span style="text-align: right;"> 4 = </span>– <span style="text-align: right;">8).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;">16x<span><sup>2</sup></span> – 4x – 4x + 1 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div>4x(4x – 1) – (4x – 1) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>(4x – 1)</span>(4x – 1) = 0</blockquote></blockquote></blockquote></blockquote></blockquote><div><span></span><span></span><span><div>3) So, (4x – 1) = 0, or (4x – 1) = 0</div><div>4) So, x = 1/4 or x = 1/4.</div><div><br /></div><div><b>(v) 100x<sup>2</sup> – 20x + 1 = 0</b><br /><br /></div></span></div></div></span></div></div><div><div><span>1) </span><span> </span>Quadratic equation: 100x<span><sup>2</sup></span> – 20x + 1 = 0</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 1 and its sign is plus, </span><span style="text-align: right;">factorise 100 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their sum will be </span>– 20<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">100 x 1 = (<span style="text-align: left;">– 10)</span> x (</span>– <span style="text-align: right;">10) (</span>– <span style="text-align: right;">10 </span>–<span style="text-align: right;"> 10 = </span>– <span style="text-align: right;">20).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div style="text-align: center;"><div style="text-align: left;">100x<span><sup>2</sup></span> – 10x – 10x + 1 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><div>10x(10x – 1) – (10x – 1) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span>(10x – 1)</span>(10x – 1) = 0</blockquote></blockquote></blockquote></blockquote></blockquote><div><span></span><span></span><span><div>2) So, (10x – 1) = 0, or (10x – 1) = 0</div><div>3) So, x = 1/10 or x = 1/10.</div><div><br /></div></span></div></div><div><div><span><b>Q2. Solve the problems given in Example 1.</b></span></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div style="font-weight: bold; text-align: left;">(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.</div></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div style="font-weight: bold; text-align: left;">(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.</div></div></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><h3><span>Explanation:</span></h3></div><div><span>1) </span><span>The </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></div><div><span>2) Frame the quadratic equation from the given condition.</span></div></div><div><span>3) Solve this qardatic equation </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0 using factorisation method.</span> </div><div><h3><span>Solution:</span></h3></div></div><div><span style="font-weight: 700;">(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.</span></div><div><span style="font-weight: 700;"><br /></span></div><div><div><div><div><div><div><span>1) Let the number of marbles with John be x.</span></div><div><span>2) So, according to the problem, the number of marbles with Jivanti is (45 - x).</span></div><div><span>3) Both have lost 5 marbles, so now they have (x - 5) and (40 - x).</span></div><div><span>4) According to the problem,</span></div></div></div></div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><div><div style="text-align: left;">(x - 5)(40 - x) = 124</div></div></div></div></div></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x(40 - x) - 5(40 - x) = 124<br /></span><span>40x - </span><span>x</span><sup>2</sup><span> - 200 + 5x = 124<br /></span><span>- </span><span>x</span><sup>2</sup><span> + 40x + 5x - 200 - 124 = 0<br /></span><span>- </span><span>x</span><sup>2</sup><span> + 45x - 324 = 0<br /></span><span>x</span><sup>2</sup><span> - 45x + 324 = 0</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>5) </span><span> </span>Quadratic equation: x<span><sup>2</sup></span> – 45x + 324 = 0</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 324 and its sign is plus, </span><span style="text-align: right;">factorise 324 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their sum will be </span>– 45<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">324 x 1 = (<span style="text-align: left;">– 36)</span> x (</span>– <span style="text-align: right;">9) (</span>– <span style="text-align: right;">36 </span>–<span style="text-align: right;"> 9 = </span>– <span style="text-align: right;">45).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> – 36x – 9x + 324 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x – 36) – 9(x – 36) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x – 36)</span>(x – 9) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>6) So, (x – 36) = 0, or (x – 9) = 0</div></span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) So, x = 36 or x = 9.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) John has 36 marbles and Jivanti has 9 marbles.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span>1) Let the number of toys produced in a day be x.</span></div><div><span>2) So, each toy's production cost on a day will be Rs (55 - x).</span></div><div><span>3) On one particular day, the total cost was Rs 750.</span></div><div>4) According to the problem,</div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div>x(55 - x) = 750</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>55x - </span><span>x</span><sup>2</sup><span> = 750<br /></span><span>55x - </span><span>x</span><sup>2</sup><span> - 750 = 0</span><span><br /></span><span>- </span><span>x</span><sup>2</sup><span> + </span><span>55x</span><span> </span><span>- 750 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> - </span><span>55x</span><span> </span><span>+ 750 = 0</span><br /></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>5) </span><span> </span>Quadratic equation: x<sup>2</sup> - 55x + 750 = 0</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 750 and its sign is plus, </span><span style="text-align: right;">factorise 750 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their sum will be </span>– 55<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">750 x 1 = (<span style="text-align: left;">– 25)</span> x (</span>– <span style="text-align: right;">30) (</span>– <span style="text-align: right;">25 </span>–<span style="text-align: right;"> 30 = </span>– <span style="text-align: right;">55).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> – 25x – 30x + 750 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x – 25) – 30(x – 25) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x – 25)</span>(x – 30) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>6) So, (x – 25) = 0, or (x – 30) = 0</div></span></span></div><div><span style="font-family: arial; font-size: medium;">7) So, x = 25 or x = 30.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) Hence, the </span><span style="font-family: arial;">number of toys produced that day is 25 or 30.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q3. Find two numbers whose sum is 27 and whose product is 182.</b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span>1) Let the first number be x.</span></div><div><span>2) The second number will be (27 - x).</span></div><div><span>3) </span>According to the problem,</div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div>x(27 - x) = 182</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>27x - </span><span>x</span><sup>2</sup><span> = 182<br /></span><span>27x - </span><span>x</span><sup>2</sup><span> - 182 = 0</span><span><br /></span><span>- </span><span>x</span><sup>2</sup><span> + </span><span>27x</span><span> </span><span>- 182 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> - </span><span>27x</span><span> </span><span>+ 182 = 0</span><br /></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) </span><span> </span>Quadratic equation: x<sup>2</sup> - 27x + 182 = 0</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 182 and its sign is plus, </span><span style="text-align: right;">factorise 182 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their sum will be </span>– 27<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">182 x 1 = (<span style="text-align: left;">– 14)</span> x (</span>– <span style="text-align: right;">13) (</span>– <span style="text-align: right;">14 </span>–<span style="text-align: right;"> 13 = </span>– <span style="text-align: right;">27).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> – 14x – 13x + 182 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x – 14) – 13(x – 14) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x – 14)</span>(x – 13) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>5) So, (x – 14) = 0, or (x – 13) = 0</div></span></span></div><div><span style="font-family: arial; font-size: medium;">6) So, x = 14 or x = 14.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) Hence, the </span><span style="font-family: arial;">numbers are 13 and 14.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q4. Find two consecutive positive integers, the sum of whose squares is 365.</b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span>1) Let the first positive integer be x.</span></div><div><span>2) The second number will be (x + 1).</span></div><div><span>3) </span>According to the problem,</div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div>x<sup>2 </sup>+ (x + 1)<sup>2</sup> = 365</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + </span><span>x</span><sup>2 </sup><span>+ 2x + 1 </span><span>= 365</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + 2</span><span>x </span><span>- 365 + 1 = 0</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + 2</span><span>x </span><span>- 364 = 0</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + </span><span>x </span><span>- 182 = 0</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>4) </span><span> </span>Quadratic equation: x<sup>2</sup> + x - 365 = 0</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is <span style="text-align: left;">182</span> and its sign is minus, </span><span style="text-align: right;">factorise 182 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their difference will be </span>1<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">182 x 1 = (<span style="text-align: left;">14)</span> x (</span>– <span style="text-align: right;">13) (</span><span style="text-align: right;">14 </span>–<span style="text-align: right;"> 13 = 1</span><span style="text-align: right;">).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 14x – 13x – 182 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x + 14) – 13(x + 14) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x + 14)</span>(x – 13) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>6) So, (x + 14) = 0, or (x – 13) = 0</div></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) So, x = </span><span style="font-family: arial;">–</span><span style="font-family: arial;">14 or x = 13.</span></span></div><div><span style="font-family: arial; font-size: medium;">8) As our integer is positive, we have x = 13. </span></div><div><span style="font-size: medium;"><span style="font-family: arial;">9) So the </span><span style="font-family: arial;">integers are 13 and 14.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-size: medium;"><span style="font-family: arial;">Q 5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find </span><span style="font-family: arial;">the other two sides.</span></span></b></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;">1) Here, a right triangle's altitude depends on its base, so let the base be x.</span></div><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span>2) So, altitude will be (x - 7).</span></div><div><span>3) </span>According to the problem,</div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div>x<sup>2 </sup>+ (x - 7)<sup>2</sup> = 13<sup>2</sup></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2 </sup><span>+ </span><span>x</span><sup>2 </sup><span>- 14</span><span>x + 49</span><span> = 13</span><sup>2</sup></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2 </sup><span>- 14</span><span>x + 49</span><span> = 169</span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2 </sup><span>- 14</span><span>x + 49</span><span> - 169 = 0</span></span></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2 </sup><span>- 14</span><span>x - 120</span><span> = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2 </sup><span>- 7</span><span>x - 60</span><span> = 0</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>4) </span><span> </span><span>Quadratic equation: x</span><sup>2</sup><span> - 7x - 60 = 0</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is <span style="text-align: left;">60</span> and its sign is minus, </span><span style="text-align: right;">factorise 60 x 1 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their difference will be </span>- 7<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">60 x 1 = (<span style="text-align: left;">12)</span> x (5</span><span style="text-align: right;">) (</span><span style="text-align: right;">5 </span>–<span style="text-align: right;"> 12 = - 7</span><span style="text-align: right;">).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">x<span><sup>2</sup></span> + 5x – 12x – 60 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(x + 5) – 12(x + 5) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(x + 5)</span>(x – 12) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>6) So, (x + 5) = 0, or (x – 12) = 0</div></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) So, x = </span><span style="font-family: arial;">– </span><span style="font-family: arial;">5 or x = 12.</span></span></div><div><span style="font-family: arial; font-size: medium;">8) As the sides of a triangle can't be negative, ignore x = - 5, so we have x = 12. </span></div><div><span style="font-family: arial; font-size: medium;">9) So the base of a triangle is 12 and the altitude is 7.</span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q6. A cottage industry produces a certain number of pottery articles in a day. It was observed </b></span><b>on a particular day that the cost of production of each article (in rupees) was 3 more than </b><b>twice the number of articles produced on that day. If the total cost of production on that </b><b>day was Rs 90, find the number of articles produced and the cost of each article.</b></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span>1) Let the number of pottery articles produced in a day be x.</span></div><div><span>2) So, the cost of production of each article will be Rs (2x + 3).</span></div><div><span>3) On one particular day, the total cost was Rs 90.</span></div><div>4) According to the problem,</div></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div>x(2x + 3) = 90</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + 3x = 90</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + 3x - 90 = 0</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + 3x - 90 = 0</span></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>5) </span>Quadratic equation: 2x<sup>2</sup> + 3x - 90 = 0</span></div><div><div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">here last term is 90 and its sign is minus, </span><span style="text-align: right;">factorise 90 x 2 in such a way</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">that their difference will be </span>3<span style="text-align: right;">.</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;"><span style="text-align: right;">90 x 2 = (<span style="text-align: left;">15)</span> x (</span>– <span style="text-align: right;">12) (1</span><span style="text-align: right;">5 </span>–<span style="text-align: right;"> 12 = 3</span><span style="text-align: right;">).</span></div></div></span></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: center;"><div style="text-align: left;">2x<span><sup>2</sup></span> + 15x – 12x + 90 = 0</div></div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x(2x + 15) – 6(2x + 15) = 0</div></span></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(2x + 15)</span>(x – 6) = 0</span></blockquote></blockquote></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span></span><span></span><span><div>6) So, (2x + 15) = 0, or (x – 6) = 0</div></span></span></div><div><span style="font-family: arial; font-size: medium;">7) So, x = - 15/2 or x = 6.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) As the </span><span style="font-family: arial;">number of articles produced can't be negative, so, x can't be - 15/2.</span></span></div><div><span style="font-family: arial; font-size: medium;">9) So the number of articles produced on that day is 6 and the cost of each article is</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">Rs 15.</span></div></div></div></blockquote><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/08/158-ncert-10-4-quadratic-equations-ex-43.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations - Ex-4.3</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-69197029998880651302023-07-06T17:03:00.002+05:302023-07-10T19:39:26.016+05:30156-NCERT-10-4-Quadratic Equations - Ex-4.1<div style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>NCERT</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>10th Mathematics</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>Exercise 4.1</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>Topic: 4 Quadratic Equations</span></span></div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/154-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables-Ex-3.7</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 4.1</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><div>1. Check whether the following are quadratic equations :</div><div style="text-align: left;"><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="line-height: 107%;">(i) (x + 1)<sup>2</sup> = 2(x – 3) <span> </span>(ii) x<sup>2</sup> – 2x = (–2) (3 –
x)</span></div><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="line-height: 107%;"><span style="line-height: 107%;">(iii) (x – 2)(x + 1) = (x – 1)(x + 3) <span> </span>(iv) (x – 3)(2x +1) = x(x + 5)</span></span></div><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="line-height: 107%;"><span style="line-height: 107%;"><span style="line-height: 107%;">(v) (2x – 1)(x – 3) = (x + 5)(x – 1)<span> </span> (vi) x<sup>2</sup> + 3x + 1 = (x –
2)<sup>2</sup></span></span></span></div><div style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="line-height: 107%;"><span style="line-height: 107%;"><span style="line-height: 107%;"><div style="line-height: normal; margin-bottom: 0cm; text-align: left;">(vii) (x + 2)<sup>3</sup>
= 2x (x<sup>2</sup> – 1)<span> </span> (viii) x<sup>3</sup> – 4x<sup>2</sup> – x + 1 = (x –
2)<sup>3</sup></div></span></span></span></div></div></b></span></div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>1) The quadratic polynomial is of the form a</span><span>x</span><sup>2</sup><span> + bx + c, where a <span style="line-height: 107%;">≠ </span>0.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) The </span><span>quadratic equation is </span><span>of the form a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">Equation of the form p(x) = 0, where p(x) is a polynomial of degree </span><span style="font-family: arial;">2, is a</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">quadratic equation.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) The equation of the form </span><span>a</span><span>x</span><sup>2</sup><span> + bx + c = 0, where a <span style="line-height: 19.26px;">≠ </span>0 is known as the standard</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">form of a quadratic equation.</span><span style="font-family: arial; font-size: medium;"> The degree of this equation is 2.</span></div></blockquote><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(i) (x + 1)</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> = 2(x – 3)</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(x + 1)</span><sup>2</sup><span> = 2(x – 3)</span></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 2x +1</span><span> = 2x – 6</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 2x +1</span><span> - 2x + 6 = 0</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 7</span><span> = 0 -------------------equation 1</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 2, it is the quadratic equation. </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(ii) x</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> – 2x = (–2) (3 – x)</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> – 2x = (–2) (3 – x)</span></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> – 2x = – 6 + 2x</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> – 2x + 6 </span><span>–</span><span> 2x = 0</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> – 4x</span><span> + 6 = 0</span><span> -------------------equation 1</span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 2, it is the quadratic equation. </span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(iii) (x – 2)(x + 1) = (x – 1)(x + 3)</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x – 2)(x + 1) = (x – 1)(x + 3)</span></div><span style="font-size: medium;"><span style="font-family: arial;">x(x + 1) </span><span style="font-family: arial;">– 2(x + 1)</span><span style="font-family: arial;"> = x(x + 3) </span><span style="font-family: arial;">– (x + 3)</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + x </span><span>– 2x </span><span>–</span><span> 2</span><span> = </span><span>x</span><sup>2</sup><span> + 3x </span><span>– x </span><span>–</span><span> 3</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>– x </span><span>–</span><span> 2</span><span> = </span><span>x</span><sup>2</sup><span> + 2x</span><span> </span><span>–</span><span> 3<br /></span><span>x</span><sup>2</sup><span> </span><span>– x </span><span>–</span><span> 2 </span><span>–</span><span> </span><span>x</span><sup>2</sup><span> </span><span>–</span><span> </span><span>2x</span><span> </span><span>+</span><span> 3 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">–</span><span style="font-family: arial;"> </span><span style="font-family: arial;">3x</span><span style="font-family: arial;"> </span><span style="font-family: arial;">+</span><span style="font-family: arial;"> 1 = 0<br /></span><span style="font-family: arial;">3x</span><span style="font-family: arial;"> </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 1 = 0</span><span style="font-family: arial;"> -------------------equation 1</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 1, it is not the quadratic equation. </span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div><span style="font-family: arial; font-size: medium; font-weight: 700;">(iv) (x – 3)(2x +1) = x(x + 5)</span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">(x – 3)(2x +1) = x(x + 5)</span></div></div><div><span style="font-size: medium;"><span style="font-family: arial;">x(2x + 1) </span><span style="font-family: arial;">– 3(2x + 1)</span><span style="font-family: arial;"> = x(x + 5)</span></span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + x </span><span>– 6x </span><span>–</span><span> 3</span><span> = </span><span>x</span><sup>2</sup><span> + 5x</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> </span><span>+ x </span><span>– 6x </span><span>–</span><span> 3</span><span> </span><span>–</span><span> </span><span>x</span><sup>2</sup><span> </span><span>–</span><span> 5x = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>–</span><span> 10x </span><span>–</span><span> 3</span><span> = 0</span><span> -------------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 2, it is the quadratic equation.</span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(v) (2x – 1)(x – 3) = (x + 5)(x – 1)</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">(2x – 1)(x – 3) = (x + 5)(x – 1)</span></div></div><div><span style="font-size: medium;"><span style="font-family: arial;">2x</span><span style="font-family: arial;">(x – 3)</span><span style="font-family: arial;"> – (x – 3) = x</span><span style="font-family: arial;">(x – 1)</span><span style="font-family: arial;"> + 5(x – 1)</span></span></div><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> – 6x</span><span> – x + 3 = </span><span>x</span><sup>2</sup><span> – x</span><span> + 5x – 5</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> – 6x</span><span> – x + 3 </span><span>–</span><span> </span><span>x</span><sup>2</sup><span> + x</span><span> </span><span>–</span><span> 5x + 5 = 0</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> </span><span>–</span><span> </span><span>x</span><sup>2 </sup><span>– 6x</span><span> – x </span><span>+ x</span><span> </span><span>–</span><span> 5x</span><span> </span><span>+ 3 </span><span>+ 5 = 0</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>– 11x</span><span> </span><span>+ 8</span><span> = 0</span><span> -------------------equation 1</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 2, it is the quadratic equation.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(vi) x</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> + 3x + 1 = (x – 2)</span><sup style="font-weight: 700;">2</sup></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;">1) The given equation is:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 3x + 1 = (x – 2)</span><sup>2</sup></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 3x + 1 = </span><span>x</span><sup>2</sup><span> </span><span>–</span><span> 4x + 4</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 3x + 1 </span><span>–</span><span> </span><span>x</span><sup>2</sup><span> </span><span>+</span><span> 4x </span><span>–</span><span> 4 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2 </sup><span>–</span><span> </span><span>x</span><sup>2</sup><span> + 3x </span><span>+</span><span> 4x</span><span> </span><span>+ 1</span><span> </span><span>–</span><span> 4 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7x </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 3 = 0</span><span style="font-family: arial;">-------------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 1, it is <b>not</b> the quadratic equation.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(vii) (x + 2)</span><sup style="font-weight: 700;">3</sup><span style="font-weight: 700;"> = 2x (x</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> – 1)</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) We know that (a + b</span><span>)</span><sup>3</sup><span> = </span><span>a</span><sup>3</sup><span> + 3a</span><sup>2</sup><span> b + </span><span>3a b</span><sup>2</sup><span> + </span><span>b</span><sup>3</sup><span> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>(x + 2)</span><sup>3</sup><span> = 2x (x</span><sup>2</sup><span> – 1)</span></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> + 3x</span><sup>2</sup><span> (2) + </span><span>3x (2)</span><sup>2</sup><span> + </span><span>2</span><sup>3</sup><span> = 2x</span><sup>3</sup><span> </span><span>– 2x</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> + 6x</span><sup>2</sup><span> + </span><span>12x </span><span>+ 8</span><span> = 2x</span><sup>3</sup><span> </span><span>– 2x</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> + 6x</span><sup>2</sup><span> + </span><span>12x </span><span>+ 8</span><span> </span><span>–</span><span> 2x</span><sup>3</sup><span> </span><span>+ 2x = 0</span><span><br /></span><span>x</span><sup>3</sup><span> </span><span>–</span><span> 2x</span><sup>3</sup><span> + 6x</span><sup>2</sup><span> + </span><span>12x </span><span>+ 2x + 8 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>–</span><span> x</span><sup>3</sup><span> + 6x</span><sup>2</sup><span> + </span><span>14x </span><span>+ 8 = 0</span><span> -------------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 3, it is <b>not</b> the quadratic equation.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(viii) x</span><sup style="font-weight: 700;">3</sup><span style="font-weight: 700;"> – 4x</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> – x + 1 = (x – 2)</span><sup style="font-weight: 700;">3</sup></span></div><div><sup style="font-weight: 700;"><span style="font-family: arial; font-size: medium;"><br /></span></sup></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) We know that (a - b</span><span>)</span><sup>3</sup><span> = </span><span>a</span><sup>3</sup><span> </span><span>–</span><span> 3a</span><sup>2</sup><span> b + </span><span>3a b</span><sup>2</sup><span> </span><span>–</span><span> </span><span>b</span><sup>3</sup><span> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> – 4x</span><sup>2</sup><span> – x + 1 = (x – 2)</span><sup>3</sup></span></div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> – 4x</span><sup>2</sup><span> – x + 1</span><span> = </span><span>x</span><sup>3</sup><span> </span><span>–</span><span> 3x</span><sup>2</sup><span> (2) + </span><span>3x (2)</span><sup>2</sup><span> </span><span>–</span><span> </span><span>2</span><sup>3</sup></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> – 4x</span><sup>2</sup><span> – x + 1</span><span> = </span><span>x</span><sup>3</sup><span> </span><span>–</span><span> 6x</span><sup>2</sup><span> + </span><span>12x </span><span>–</span><span> 8</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>3</sup><span> – 4x</span><sup>2</sup><span> – x + 1</span><span> </span><span>–</span><span> </span><span>x</span><sup>3</sup><span> </span><span>+</span><span> 6x</span><sup>2</sup><span> </span><span>–</span><span> </span><span>12x </span><span>+</span><span> 8 = 0</span><span><br /></span><span>x</span><sup>3</sup><span> </span><span>–</span><span> </span><span>x</span><sup>3</sup><span> – 4x</span><sup>2</sup><span> </span><span>+</span><span> 6x</span><sup>2</sup><span> – x</span><span> </span><span>–</span><span> </span><span>12x</span><span> + 1</span><span> </span><span>+</span><span> 8 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> </span><span>–</span><span> </span><span>13x</span><span> + 9</span><span> = 0</span><span> -------------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) As the highest power of the variable x is 2, it is the quadratic equation.</span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><b><span style="font-family: arial;"><div>Q2. Represent the following situations in the form of quadratic equations :</div></span></b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span><b><span style="font-family: arial; font-size: medium;"><div>(i) The area of a rectangular plot is 528 m<sup>2</sup>. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.</div></span></b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div>(ii) The product of two consecutive positive integers is 306. We need to find the integers.</div></b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div>(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.</div></b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div>(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.</div></b></span></div></div></blockquote><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x be any suitable variable.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the given equation, frame the quadratic equation.</span></div><div><span style="font-family: arial; font-size: medium;"><span>3) Then solve this </span>equation to get the required solutions. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(i) The area of a rectangular plot is 528 m</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;">. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.</span></span></div><div><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here, length is depending on breadth, let the breadth be x m.</span></div><div><span style="font-family: arial; font-size: medium;">2) So, according to the problem, the length is more than twice its breadth,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">breadth = (2x + 1) m</span></blockquote></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3) The area of the plot is 528</span> <span>m</span><sup>2</sup><span>, so we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x(2x + 1) = 528</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + x = 528</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2x</span><sup>2</sup><span> + x </span><span>–</span><span> 528 = 0 </span><span>------------ equation 1</span></span></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>4) So, </span><span>2x</span><sup>2</sup><span> + x </span><span>–</span><span> 528 = 0 is the required</span><span> quadratic equation.</span></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium; font-weight: 700;">(ii) The product of two consecutive positive integers is 306. We need to find the integers.</span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the first integer be x.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, </span><span style="font-family: arial;">the next integer will be (x + 1).</span></span></div></div></div></div><div><span style="font-size: medium;"><span style="font-family: arial;">3) The product of the integers is 306</span><span style="font-family: arial;">, so we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x(x + 1) = 306</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + x = 306</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + x </span><span>–</span><span> 306 = 0 </span><span>------------ equation 1</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span>4) So, </span><span>x</span><sup>2</sup><span> + x </span><span>–</span><span> 306 = 0</span><span> is the required</span><span> quadratic equation.</span></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Here, the age of Rohan's mother is depending on him, so let Rohan's age be x.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, according to the problem, the </span><span style="font-family: arial;">age of Rohan's mother</span><span style="font-family: arial;"> will be (x + 26).</span></span></div></div><div><span style="font-family: arial; font-size: medium;">3) 3 years later, Rohan's age will be (x + 3), and his mother's age will be (x + 29). </span></div></div></div><div><span style="font-family: arial; font-size: medium;">4) According to the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 3)(x + 29) = 360</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x(x + 29) + 3</span><span style="font-family: arial;">(x + 29)</span><span style="font-family: arial;"> = 360</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 29x + 3</span><span>x + 87</span><span> = 360</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> + 32x </span><span>+ 87</span><span> = 360<br /></span><span>x</span><sup>2</sup><span> + 32x </span><span>+ 87</span><span> </span><span>–</span><span> 360 = 0<br /></span><span>x</span><sup>2</sup><span> + 32x </span><span>–</span><span> 273 = 0</span><span> </span><span>------------ equation 1</span></span></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>4) So, </span><span>x</span><sup>2</sup><span> + 32x </span><span>–</span><span> 273 = 0</span><span> is the required</span><span> quadratic equation.<br /></span></span></div></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><div><div><div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the uniform speed of a train be x km/h.</span></div><div><span style="font-family: arial; font-size: medium;">2) To cover 480 km, the time taken will be (480/x) hrs.</span></div><div><span style="font-family: arial; font-size: medium;">3) If the speed is reduced by 8 km/h, then time will be increased by 3 hrs, so </span></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">new speed = (x - 8) km/h, and new time = [</span><span style="font-family: arial;">(480/x) + 3] hrs.</span></span></div></div></div></div></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">4) According to the condition,</span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x - 8)[(480/x) + 3] = 480</span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 8)[(480 + 3x)/x] = 480</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x(x - 8) = (480 + 3x)</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>- 8x = 480 + 3x</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>- 8x - 480 - 3x = 0</span><span><br /></span><span>x</span><sup>2</sup><span> </span><span>- 8x - </span><span>3x - </span><span>480 = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x</span><sup>2</sup><span> </span><span>- 11x </span><span>- </span><span>480 = 0</span><span> </span><span>------------ equation 1</span></span></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><span>5) So, </span><span>x</span><sup>2</sup><span> </span><span>- 11x </span><span>- </span><span>480 = 0</span><span> is the required</span><span> quadratic equation.</span></span></div></div></div></div><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/07/157-ncert-10-4-quadratic-equations-ex-42.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations - Ex-4.2</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com2tag:blogger.com,1999:blog-2945240619290990604.post-18990019842158105982023-06-26T20:28:00.005+05:302023-06-26T20:39:15.343+05:30155 Suggestion to exclude certain letters and numbers from captchas<div style="text-align: left;"><span style="font-family: arial; font-size: medium;">CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart)</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) The main function of captchas is to distinguish between real people and artificial</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">bots.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) </span><span style="font-family: arial;">The design of captchas should be approached holistically, with an emphasis on</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">striking a balance between usability, accessibility, security, and efficacy. Combining many tactics, such as using warped letters, extra obstacles, or other strategies like picture recognition or mathematically straightforward operations like addition, subtraction, or multiplication, may result in a more effective strategy.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) It's important to keep in mind that captcha technology is always developing, and</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">researchers and developers are continuously looking for substitute ways to confirm that users are human, including invisible captchas that depend on behavioural analysis or biometrics. With these developments, usability and security are being balanced.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Additionally, it's critical to continually review and adapt captcha tactics since new</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>developments in machine learning and artificial intelligence may eventually make some solutions less effective.</span> </span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Here, we're trying to make the point that certain characters might not be utilized</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">while creating captcha software since they are unclear.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) Following characters are confusing:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) Looks the same in both upper and lower case:</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">i) 'c', 'j', 'k', 'o', 'p', 's', 'u', 'v', 'w', 'x', 'y', and 'z'</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">ii) 'C', 'J', 'K', 'O', 'P', 'S', 'U', 'V', 'W', 'X', 'Y', and 'Z'</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><p style="background: white; margin: 0cm;"><span style="font-family: "Segoe UI", sans-serif; font-size: 15pt;">Using
these characters for making captchas is not recommended since they are nearly
identical in upper- and lower-case.</span><span style="font-family: "Segoe UI",sans-serif; font-size: 15.0pt;"><o:p></o:p></span></p></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) Certain capital and lowercase letters are difficult to distinguish:</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">i) </span><span style="font-family: arial;"><span style="font-family: "Segoe UI",sans-serif; font-size: 15.0pt; line-height: 107%; mso-ansi-language: EN-IN; mso-bidi-language: HI; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-font-kerning: 0pt;">) In specific fonds</span>, the uppercase letter 'I' (capital 'i') occasionally</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">appears </span></span><span style="font-family: arial; font-size: large;">as the </span><span style="font-family: arial; font-size: large;">lowercase letter 'l' (lower 'L'). The user is confused about which one has to be taken. Therefore, while designing captchas, upper case letter 'I' and lower case letter 'l' must be eliminated.</span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">c) Uncertainty over the characters for the numbers '0' (zero) and the upper and</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">lower case </span><span style="font-family: arial;">'O' ('o')</span><span style="font-family: arial;">:</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">i) </span><span style="font-family: arial;">When the letter 'o' is typed in lower case, higher case, or the number '0'</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">in the captcha, there is a lot of misunderstanding. As a result, they might not be included when making captchas.</span></div></blockquote></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) Because of their closeness in appearance, several characters—including the</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">capital and lowercase versions of the letters 'c', 'k', 'o', 'p', 'u', 'v', 'w', 'x', </span><span style="font-family: arial;">'y', and 'z' </span><span style="font-family: arial;">— are excluded. </span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) This tactic must be carefully implemented, striking a balance between user ease</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">and security. Before implementing any substantial modifications to captcha systems, it is usually a good idea to undertake user testing and evaluate the impact on security.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">9) </span><span style="font-family: arial; font-size: medium;">We want to make it simpler for users to understand the CAPTCHA and decrease</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">the probability of mistakes or annoyance by getting rid of these potentially perplexing characters. However, it's crucial to keep in mind that CAPTCHAs should strike a balance between being difficult enough to stave off automated bots while still being clear to the majority of human users. A CAPTCHA may lose its effectiveness if it gets too simple to solve.</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">Here are my opinions. Everyone is welcome to share their worthwhile thoughts.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-87806555065424291842023-06-26T14:45:00.004+05:302023-07-06T17:05:16.799+05:30154-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.7<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 3.7</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/153-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables-Ex-3.6</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.7</span></h3></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Convert the variables suitably to get linear equations in two variables.</span></div><div><span style="font-family: arial; font-size: medium;"><span>3) Then solve these </span>equations to get the values of our variables. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q1. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old </b></span><b>as Ani, and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ </b><b>by 30 years. Find the ages of Ani and Biju.</b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) Let Ani's age be x and Biju's age be y.</span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">2) The age of Ani's father Dharam is 2x.</span></div><div><span style="font-family: arial; font-size: medium;"><span>3) </span>The age of Biju's sister Cathy is y/2.</span></div><div><span style="font-family: arial; font-size: medium;"><span>4) As </span>the ages of Cathy and Dharam differ by 30 years, as Dharam is older than</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">Cathy, we have,</span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x - y/2 = 30</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4x - y = 60 ---------------------- equation 1</span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span>5) Age difference between Ani and Biju is 3 years. Here we need to take 2 cases.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span><span>6) </span></span><span><b>Case 1: </b></span>a) Ani is older than Biju.</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x - y = 3</span><span style="font-family: arial;"> ---------------------- equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7) Subtract </span><span style="font-family: arial;">equation 2 from </span><span style="font-family: arial;">equation 1, and we get,</span></span></div><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x - y = 60</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"> x - y = 3</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> (-) (+) (-)</span></div><div><span style="font-family: arial; font-size: medium;"><span> -----------------------</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>3x = 57</span></span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = 57/3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 19</span><span style="font-family: arial;"> ---------------------- equation 3.</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">8) </span><span style="font-family: arial;">Put the value of x = 19 from equation 3 in equation 2, and we get,</span></span></div><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x - y = 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">19 - y = 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">y = 19 - 3</span></div><span style="font-family: arial; font-size: medium;">y = 16 ---------------------- equation 4</span></blockquote></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial;">If Ani is older than Biju, Ani's age is 19 years and Biju's age is 16 years.</span></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><span>10) </span></span><span style="font-family: arial;"><b>Case 2: </b></span>b) Biju is older than Ani.</div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y - x = 3</span></blockquote></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">- x + y = 3 ---------------------- equation 5.</div></span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">11) Add </span><span style="font-family: arial;">equation 5 to </span><span style="font-family: arial;">equation 1, and we get,</span></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">4x - y = 60</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">- x + y = 3</span></div></blockquote><div><span style="font-family: arial;"><span> -----------------------</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><span>3x = 63</span></span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 63/3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 21</span><span style="font-family: arial;"> ---------------------- equation 6.</span></blockquote><div><span style="font-family: arial;">12) </span><span style="font-family: arial;">Put the value of x = 21 from equation 6 in equation 5, and we get,</span></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">- x + y = 3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">-21 + y = 3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">y = 21 + 3</span></div><span style="font-family: arial;">y = 24 ---------------------- equation 4</span></blockquote></blockquote><div><span style="font-family: arial;">13) </span><span style="font-family: arial;">If Biju is older than Ani, Ani's age is 21 years and Biju's age is 24 years.</span></div><div><span style="font-family: arial;">14) </span>If Ani is older than Biju, Ani's age is 19 years and Biju's age is 16 years.</div></div></div></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The </b></span><b>other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the </b><b>amount of their (respective) capital? [From the Bijaganita of Bhaskara II] </b><b>[Hint : x + 100 = 2(y – 100), y + 10 = 6(x – 10)].</b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the first friend has Rs x and the other friend has Rs y.</span></div><div><span style="font-family: arial; font-size: medium;">2) If the first friend gets Rs 100 from the second friend, the new amount with them,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First friend has (x + 100) </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second friend has (y - 100), so, according to first condition, we have,</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + 100 = 2(y - 100)</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x + 100 = 2y - 200<br /></span><span style="font-family: arial;">x - 2y = - 200 - 100<br /></span><span style="font-family: arial;">x - 2y = - 300</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 2y - 300</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></div></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">If the second friend gets Rs 10 from the first friend, the new amount with them,</span></span></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) First friend has (x - 10) </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Second friend has (y + 10), so, according to second condition, we have,</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">y + 10 = 6(x - 10)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y + 10 = 6x - 60<br /></span><span style="font-family: arial;">6x - y = 10 + 60<br /></span><span style="font-family: arial;">6x - y = 70</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">4) Put the value of x = 2y - 300 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6x - y = 70</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">6(</span><span style="font-family: arial;">2y - 300)</span><span style="font-family: arial;"> - y = 70</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">12y - 1800</span><span style="font-family: arial;"> - y = 70</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">12y - </span><span style="font-family: arial;">y = 70 + </span><span style="font-family: arial;">1800</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div></div></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">11y</span><span style="font-family: arial;"> = 1870</span></span></div></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 1870/11</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">y = 170 </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 170 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6x - y = 70</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6x - 170 = 70</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">6x = 70 + 170</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>6x = 240</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">x = 240/6</span><span style="font-family: arial;"> </span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 40</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial;">The first friend has Rs 40 and the second friend has Rs 170</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><b><span><span>Q</span></span><span>3. A train covered a certain distance at a uniform speed. If the train would have been </span></b><b>10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train </b><b>were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find </b><b>the distance covered by the train.</b></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let uniform speed be x km/h and the time taken to cover the distance be y hrs.</span></div><div><span style="font-family: arial; font-size: medium;">2) We know that distance = speed x time, so the total distance = xy km.</span></div><div><span style="font-family: arial; font-size: medium;">3) When the train travels 10 km/h faster, then the time will be 2 hrs less, so,</span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x + 10)(y - 2) = xy</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">xy - 2x + 10y - 20 = xy</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">- 2x + 10y - 20 = 0</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">- 2x + 10y = 20</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">When the train travels 10 km/h slower, then the time will be 3 hrs more, so,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x - 10)(y + 3) = xy</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">xy + 3x - 10y - 30 = xy</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">3x - 10y - 30 = 0</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">3x - 10y = 30</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Add</span><span style="font-family: arial;"> </span><span style="font-family: arial;">equation 1 to </span><span style="font-family: arial;">equation 2, and we get,</span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> 3x - 10y = 30</span></div></div></div></blockquote><div style="text-align: left;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">- 2x + 10y = 20</span></blockquote><span style="font-size: medium;"><span style="font-family: arial;"> --------------------------</span><span style="font-family: arial;"> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"> x = 50</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 50</span><span style="font-family: arial;"> ---------------------- equation 3.</span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">6) Put the value of <span style="font-family: arial; font-size: medium;">x = 50</span> from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x - 10y = 30</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3(50) - 10y = 30</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">150 - 10y = 30</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>10y = 150 - 30</span></span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">10y = 120</span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">y = 120/10</span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 12</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">7) </span><span style="font-family: arial;">The uniform speed is 50 km/h and the time taken to cover the distance is 12 hrs</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) So, the total distance traveled by the train is 50 x 12 = 600 km.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><span style="font-family: arial; font-size: medium;"><b><span><span>Q</span></span><span>4. The students of a class are made to stand in rows. If 3 students are extra in a row, there </span></b><b>would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the </b><b>number of students in the class.</b></span></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial;"><span style="font-size: medium;"><div><div><div><div><span style="font-family: arial;">1) Let the number of rows be x and the number of students in a row is y.</span></div><div><span style="font-family: arial;">2) <span style="background-color: white; color: #333333;">Total number of students = Number of rows x Number of students in a row </span><span style="background-color: white; color: #333333;">= xy</span>.</span></div><div><span style="font-family: arial;">3) When 3 students are more in a row, then the number of rows will be less by 1,</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(x - 1)(y + 3) = xy</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">xy + 3x - y - 3 = xy<br />3x - y - 3 = 0<br />3x - y = 3<span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></blockquote><div><div><div><span style="font-family: arial;">4) </span>When 3 students are less in a row, then the number of rows will be more by 2,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">(x + 2)(y - 3) = xy</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">xy - 3x + 2y - 6 = xy<br />-3x + 2y - 6 = 0<br /><span style="font-family: arial;">-3x + 2y = 6 </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div><div><span style="font-family: arial;">5) </span><span style="font-family: arial;">Add</span><span style="font-family: arial;"> </span>equation 2 to equation 1, and we get, </div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"> 3x - y = 3</blockquote><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">- 3x + 2y = 6</blockquote><span> --------------------------</span> </div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"> y = 9</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 9</span> ---------------------- equation 3.</blockquote></div><div><div><span style="font-family: arial;">6) Put the value of y = 9 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3x - y = 3</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3x - 9 = 3</blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div>3x = 9 + 3</div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div>3x = 12</div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">x = 12/3</blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = 4</span><span style="font-family: arial;"> -------------------------- equation 4.</span></div></blockquote><div><div><span><span style="font-family: arial;">7) </span><span style="font-family: arial;">The number of rows is 4 and the number of students in a row is 9</span><span style="font-family: arial;">.</span></span></div><div><span><span style="font-family: arial;">8) So, the number of students in the class = 9 x 4 = 36.</span></span></div><div style="font-weight: bold;"><span><span style="font-family: arial;"><br /></span></span></div></div></div></span></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q5. In a <span style="line-height: 107%;">∆</span> ABC, < C = 3 < B = 2 (< A + < B). Find the three angles.</b></span></div><div><span style="font-family: arial;"><div><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div>1) Let the < A be x, and < B be y.</div><div>2) According to the first condition, </div></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">< C = 3y -------------------------- equation 1.</span></blockquote><div><span style="font-family: arial; font-size: medium;">3) We know that the sum of the angles of a triangle is 180 degrees, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">< A + < B + < C = 180</span></blockquote></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + y + 3y = 180</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x + 4y = 180 </span><span style="font-family: arial;">-------------------------- equation 2.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) According to the given condition,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< C = </span><span style="font-family: arial;">2 (< A + < B)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3y = 2(x + y)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>3y = 2x + 2y<br /></span><span>3y - 2y = 2x<br /></span><span>y = 2x</span><span> </span><span>-------------------------- equation 3.</span><span> </span> </span></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">5) Put the value of y = 2x from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 4y = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 4(2x) = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 8x = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9x = 180</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span>x = 20</span></span><sup><span style="font-family: arial;"> </span></sup><span style="font-family: arial;">-------------------------- equation 4.</span></div></div></blockquote><div><div><div><span style="font-family: arial;">6) Put the value of x = 20 from equation 4 in equation 3, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 2x</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 2(20)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">y = 40</span><span style="font-family: arial;"> -------------------------- equation 5.</span></span></blockquote><div><span><span style="font-family: arial;">7) The angles are < A = </span><span style="font-family: arial;">20</span><sup><span style="font-family: arial;">0</span></sup><span style="font-family: arial;">, < B = </span></span><span style="font-family: arial;">40</span><sup><span style="font-family: arial;">0</span></sup><span style="font-family: arial;">, </span><span style="font-family: arial;">and < C = 120</span><sup><span style="font-family: arial;">0</span></sup><span style="font-family: arial;">.</span></div></div><div><span style="font-family: arial;"><br /></span></div><div><span><b><span style="font-family: arial;">Q</span>6. Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the coordinates of </b></span><b>the vertices of the triangle formed by these lines and the y-axis.</b></div><div><span><span style="font-family: arial;"><br /></span></span></div><div><span><span style="font-family: arial;">1) First equation: 5x - y = 5, so, y = 5x - 5.</span></span></div><div><span><span style="font-family: arial;">2) Take ant two points as follows:</span></span></div><div><span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhf5nPvTJGwDqgFiRh4EW6ONcVy2lJ0GpdCu4D3O3-wccB9xkRfR6JYPq_mF3LtHoeKQfwlXr9l2Cn-veuKbcHfp_gpmoZFlxP3CisYM6qqKruQ5xMYxB_X30vCIP_qaDpngkfnXXRBoJMMq09LwxY66LOfLBxD5r6BmuwWZQynGdTNscApwCC8Eshq/s287/6-1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="69" data-original-width="287" height="69" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhf5nPvTJGwDqgFiRh4EW6ONcVy2lJ0GpdCu4D3O3-wccB9xkRfR6JYPq_mF3LtHoeKQfwlXr9l2Cn-veuKbcHfp_gpmoZFlxP3CisYM6qqKruQ5xMYxB_X30vCIP_qaDpngkfnXXRBoJMMq09LwxY66LOfLBxD5r6BmuwWZQynGdTNscApwCC8Eshq/s1600/6-1.png" width="287" /></a></div><span style="font-family: arial;"><div><span><span style="font-family: arial;">3) Second equation: 3x - y = 3, so, y = 3x - 3.</span></span></div><div><span><span style="font-family: arial;">4) Take ant two points as follows:</span></span></div></span></span></div><div><span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivhjXPQ9Qb1oukhiLxv7U1uxUS4ShxQH0-c9nPOyMTQ77EdDFLS_VpVT5TzJKtRYO3JhPyTikfCyTl9JIqUqG0OY03DcfzvPqfDsmbM-rhBsxBbRVe51_5rfy6kiFtr2BNBLE4YQeft3z7uTg4LOwPJQ1FDVLPdHrZn0Y3Zkqd2HZ7FCpHJf1jHzrr/s288/6-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="70" data-original-width="288" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivhjXPQ9Qb1oukhiLxv7U1uxUS4ShxQH0-c9nPOyMTQ77EdDFLS_VpVT5TzJKtRYO3JhPyTikfCyTl9JIqUqG0OY03DcfzvPqfDsmbM-rhBsxBbRVe51_5rfy6kiFtr2BNBLE4YQeft3z7uTg4LOwPJQ1FDVLPdHrZn0Y3Zkqd2HZ7FCpHJf1jHzrr/s1600/6-2.png" width="288" /></a></div><div><span><span style="font-family: arial;">5) Now we will draw the graph:<br /></span></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiet6Xf36M3FkFwKAwi6DDWWlPvP5kJiNmk1_u8W7GQErzb7tPTJLN6o2yKixoqgTJThGtPcKinCmriqBizyFRPMM6rrb_uaT4zUYJ2oQbNh_W8Zgs66i67EIROn85F2N5TiBeCib52O7F--ItXVIVdc2yMGP1WCPUEXY8Ud7RTOFGMGD3CchsQuTuO/s1229/6.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="903" data-original-width="1229" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiet6Xf36M3FkFwKAwi6DDWWlPvP5kJiNmk1_u8W7GQErzb7tPTJLN6o2yKixoqgTJThGtPcKinCmriqBizyFRPMM6rrb_uaT4zUYJ2oQbNh_W8Zgs66i67EIROn85F2N5TiBeCib52O7F--ItXVIVdc2yMGP1WCPUEXY8Ud7RTOFGMGD3CchsQuTuO/s320/6.png" width="320" /></a></div>6) So, the coordinates of the vertices of a triangle ABC are A(1, 0), B(0, -3), and C(0, -5).<br /><span style="font-family: arial;"><br /></span></span></div><div><span><span style="font-family: arial;"><b><div style="text-align: left;">Q7. Solve the following pair of linear equations:<br /></div></b></span></span></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(i) px + qy = p – q; <span> </span>qx – py = p + q</div></b></span></span></div></div></span></div></div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(ii) ax + by = c; bx + ay = 1 + c</div></b></span></span></div></div></span></div></div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(iii) (x/a) - (y/b) = 0; ax
+ by = a<sup>2</sup> + b<sup>2<br /></sup></div></b></span></span></div></div></span></div></div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(iv) (a – b)x + (a + b) y = a<sup>2</sup> – 2ab – b<sup>2</sup>; <span> </span>(a + b)(x + y) = a<sup>2</sup> + b<sup>2</sup></div></b></span></span></div></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial;"><div><div><span><span style="font-family: arial; font-size: medium;"><b><div style="text-align: left;">(v) 152x – 378y = – 74; <span> </span>–378x + 152y = – 604</div></b></span></span></div></div></span></div></div></blockquote><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700;">(i) px + qy = p – q; </span><span style="font-family: arial; font-weight: 700;"> </span><span style="font-family: arial; font-weight: 700;">qx – py = p + q</span></span></div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span><span style="font-family: arial;"><b><p class="MsoNormal"><span style="line-height: 107%;"><o:p></o:p></span></p></b></span></span></div><div><span><span style="font-family: arial;">1) The given equations are</span></span><span style="font-family: arial; font-weight: 700;"> </span></div></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">px + qy = p – q</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 1</span></div></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">qx – py = p + q</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 2</span></div></div></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span><span style="font-family: arial;">2) Multiply </span></span><span style="font-family: arial;">equation 1 by p</span><span style="font-family: arial;">, we get,</span></div></div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p(px + qy) = p(p – q)</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><span style="font-family: arial;">p</span><sup>2</sup><span style="font-family: arial;">x + pqy = </span><span style="font-family: arial;">p</span><sup>2</sup><span style="font-family: arial;"> – pq</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3</span></div></div></div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><span><span style="font-family: arial;">3) Multiply </span></span><span style="font-family: arial;">equation 2 by q</span><span style="font-family: arial;">, we get,</span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">q(qx – py) = q(p + q)</span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">q</span><sup>2</sup><span style="font-family: arial;">x - pqy = pq </span><span style="font-family: arial;">+</span><span style="font-family: arial;"> </span><span style="font-family: arial;">q</span><sup>2</sup><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 4</span></div></div></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>4) Add </span></span><span>equation 4 to equation 3</span><span>, we get,</span> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>p</span><sup>2</sup><span>x + pqy = </span><span>p</span><sup>2</sup><span> – pq</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>q</span><sup>2</sup><span>x - pqy = pq </span><span>+</span><span> </span><span>q</span><sup>2</sup></span></div></blockquote><span style="font-family: arial; font-size: medium;"><span> ---</span><span>-------</span>------------------------------ <br /></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"> (p<sup>2 </sup>+ q<sup>2</sup>) x = (p<sup>2 </sup>+ q<sup>2</sup>)</div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">x = (p<sup>2 </sup>+ q<sup>2</sup>)/(p<sup>2 </sup>+ q<sup>2</sup>)</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 5</span></span></div></blockquote><span style="font-family: arial; font-size: medium;"><span>5) Put the value of </span><span>x = 1</span><span> from equation 5 in equation 1, and we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">px + qy = p – q</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">p(1) + qy = p – q</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"> p + qy = p – q</div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">qy = – q</div></div></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"> y = – q/q</div></div></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"> y = – 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 6</span></span></blockquote></blockquote><span style="font-family: arial; font-size: medium;"><span>6) So, x = 1</span><span> and y = - 1</span><span>.</span> </span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;">(ii) ax + by = c; bx + ay = 1 + c</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><span><span style="font-family: arial;">1) The given equations are</span></span><span style="font-family: arial; font-weight: 700;"> </span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">ax + by = c</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 1</span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">bx + ay = 1 + c</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 2</span></div></div></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span><span style="font-family: arial;">2) Multiply </span></span><span style="font-family: arial;">equation 1 by a</span><span style="font-family: arial;">, we get,</span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">a(ax + by) = ac</span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">a</span><sup>2</sup><span style="font-family: arial;">x + aby = ac</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3</span></div></div></span></blockquote><div><div><span style="font-family: arial; font-size: medium;"><div><div><span><span style="font-family: arial;">3) Multiply </span></span><span style="font-family: arial;">equation 2 by b</span><span style="font-family: arial;">, we get,</span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">b(bx + ay) = b(c + 1)</span></div></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">b</span><sup>2</sup><span style="font-family: arial;">x + aby = </span><span style="font-family: arial;">bc + b</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 4</span></div></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span><span>4) Subtract </span></span><span>equation 4 from equation 3</span><span>, we get,</span> </span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sup>2</sup><span>x + aby = ac</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b</span><sup>2</sup><span>x + aby = </span><span>bc + b</span></span></div></blockquote><span style="font-family: arial; font-size: medium;"><span> (-) (-) (-) </span><br /> ---------------------------------------- <br /></span><div><span style="font-family: arial; font-size: medium;"><div> (a<sup>2 </sup>- b<sup>2</sup>) x = (ac - bc - b)</div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x = [c(a - b) - b]/(a<sup>2 </sup>- b<sup>2</sup>) -------------------------- equation 5</div></span></blockquote><span style="font-family: arial; font-size: medium;"><span>5) Put the value of </span><span>x = [c(a - b) - b]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span><span> from equation 5 in equation 1, we get,</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>ax + by = c</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>a[c(a - b) - b]/(a<sup>2 </sup>- b<sup>2</sup>) + by = c</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>[ac(a - b) - ab]/(a<sup>2 </sup>- b<sup>2</sup>) + by = c</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>by = c - [ac(a - b) - ab]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span><br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>by = {c</span><span>(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span><span> - [ac(a - b) - ab]}/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>by = [<span style="color: red;">c</span></span><span><span style="color: red;">a</span></span><sup><span style="color: red;">2</span> </sup><span>- cb</span><sup>2</sup><span> - </span><span style="color: red;"><span>a</span><sup>2</sup></span><span><span style="color: red;">c</span> + abc + ab]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>by = [abc</span><sup> </sup><span>- cb</span><sup>2</sup><span> </span><span>+ ab]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>by = [bc(a</span><sup> </sup><span>- b)</span><span> </span><span>+ ab]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>by = b[c(a</span><sup> </sup><span>- b)</span><span> </span><span>+ a]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)<br /></span><span> y = [c(a</span><sup> </sup><span>- b)</span><span> </span><span>+ a]/(a</span><sup>2 </sup><span>- b</span><sup>2</sup><span>)</span><span> </span><span>-------------------------- equation 6</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>6) So, x = </span><span>[c(a - b) - b]/(a<sup>2 </sup>- b<sup>2</sup>) and y = </span><span>[c(a<sup> </sup>- b) + a]/(a<sup>2 </sup>- b<sup>2</sup>).</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; font-weight: 700;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(iii) (x/a) - (y/b) = 0; ax + by = a</span><sup style="font-weight: 700;">2</sup><span style="font-weight: 700;"> + b</span><sup style="font-weight: 700;">2</sup></span></div><div style="text-align: left;"><span style="font-family: arial;"><span style="font-size: medium;"><b><br /></b></span></span><div><span style="font-family: arial; font-size: medium;"><div><div><span><span style="font-family: arial;">1) The given equations are</span></span><span style="font-family: arial; font-weight: 700;"> </span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">(x/a) - (y/b) = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 1</span></div></div></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>ax + by = a</span><sup>2</sup><span> + b</span><sup>2</sup><span> </span><span>-------------------------- equation 2</span> </span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, from equation 1, we have</span></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">(x/a) - (y/b) = 0</span></div></div></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x/a) = (y/b)</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = (ay/b)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3</span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;">3) Put the value of </span><span style="font-family: arial;">x = (ay/b)</span><span style="font-family: arial;"> from equation 3 in equation 2, we get,</span> </div></div></span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>ax + by = a<sup>2</sup> + b<sup>2</sup></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>a(ay/b) + by = a<sup>2</sup> + b<sup>2</sup></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(a<sup>2</sup>y/b) + by = a<sup>2</sup> + b<sup>2</sup></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(a</span><sup>2</sup><span>y + </span><span>b</span><sup>2</sup><span>y)/b = a</span><sup>2</sup><span> + b</span><sup>2</sup></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(a</span><sup>2</sup><span> + </span><span>b</span><sup>2</sup><span>)y/b = a</span><sup>2</sup><span> + b</span><sup>2</sup></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(a</span><sup>2</sup><span> + </span><span>b</span><sup>2</sup><span>)y = b(a</span><sup>2</sup><span> + b</span><sup>2</sup><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = b(a</span><sup>2</sup><span> + b</span><sup>2</sup><span>)/</span><span>(a</span><sup>2</sup><span> + b</span><sup>2</sup><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = b</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 4</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) Put the value of </span><span style="font-family: arial;">y = b</span><span style="font-family: arial;"> from equation 4 in equation 3, we get,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = (ay/b)</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = (ab/b)<br /></span><span>x = a</span> </span></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>5) So, x = a</span><span> and y = b</span><span>.</span> </span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span><span style="font-family: arial;"><b>(iv) (a – b)x + (a + b) y = a<sup>2</sup> – 2ab – b<sup>2</sup>; <span> </span></b></span></span><b>(a + b)(x + y) = </b><b><span>a</span><sup>2</sup><span> + b</span><sup>2</sup></b></div></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span><span style="font-family: arial;">1) The given equations are</span></span><span style="font-family: arial; font-weight: 700;"> </span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">(a – b)x + (a + b) y = a<sup>2</sup> – 2ab – b<sup>2</sup></span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 1</span></div></div></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span>(a + b)(x + y) = </span><span>a</span><sup>2</sup><span> + b</span><sup>2</sup><span> </span><span>-------------------------- equation 2</span> </div></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) So, from equation 2, we have,<span> </span></span></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;">(a + b)(x + y) = </span><span style="font-family: arial;">a</span><sup>2</sup><span style="font-family: arial;"> + b</span><sup>2</sup></div></div></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(a + b)x + </span><span>(a + b)</span><span>y = </span><span>a</span><sup>2</sup><span> + b</span><sup>2</sup><span> </span><span>-------------------------- equation 3</span></span></blockquote><div><span style="font-size: medium;"><span><span style="font-family: arial;">3) Subtract </span></span><span style="font-family: arial;">equation 1 from equation 3</span><span style="font-family: arial;">, we get,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(a + b)x + </span><span>(a + b)</span><span>y = </span><span>a</span><sup>2</sup><span> + b</span><sup>2</sup></span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(a – b)x + (a + b)y = a</span><sup>2</sup><span> – 2ab – b</span><sup>2</sup></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span> (-) (-) (-) <br /></span> --------------------------------------------------------- <br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"> 2bx = 2b<sup>2 </sup>+ 2ab</div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>2bx = 2</span><span>b</span><span>(b + a)<br /></span><span>x = </span><span>(a + b)</span><span> -------------------------- equation 4</span> </span></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;">4) Put the value of </span><span style="font-family: arial;">x = </span><span style="font-family: arial;">(a + b)</span><span style="font-family: arial;"> from equation 4 in equation 1, we get,</span> </div></div></span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(a – b)x + (a + b) y = a<sup>2</sup> – 2ab – b<sup>2</sup></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(a – b)(a + b) + (a + b) y = a<sup>2</sup> – 2ab – b<sup>2</sup></div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(a<sup>2</sup> – b<sup>2</sup>) + (a + b) y = (a<sup>2</sup> – b<sup>2</sup>) – 2ab</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(a + b) y = </span><span style="font-family: arial;">– 2ab</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = </span><span style="font-family: arial;">– 2ab/(a + b)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 5.</span></span></blockquote></div></div><div><span style="font-family: arial; font-size: medium;"><span>5) So, x = </span><span>(a + b) and y = </span><span>– 2ab/(a + b).</span> </span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-weight: 700;">(v) 152x – 378y = – 74; </span><span style="font-weight: 700;"> </span><span style="font-weight: 700;">–378x + 152y = – 604</span> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div><div><span style="font-family: arial;">1) The given equations are</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">152x – 378y = – 74</span> -------------------------- equation 1 </blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">–378x + 152y = – 604</span> -------------------------- equation 2</blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">2) Add </span></span><span style="font-family: arial;">equation 2 to equation 1</span><span style="font-family: arial;">, we get,</span> </div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">152x – 378y = – 74</span></div><div><span style="font-family: arial;">–378x + 152y = – 604</span></div></blockquote><span style="font-size: medium;"> ---</span><span style="font-size: medium;">-------</span><span style="font-size: medium;">------------------------------------ </span><br /><div><span style="font-family: arial; font-size: medium;"><div> – 226 x – 226 y = – 678</div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">x + y = (-678)/(-226)</div></span></div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x + y = 3 -------------------------- equation 3</div></span></blockquote><div><div><span style="font-size: medium;">3) Subtract </span>equation 2 from equation 1, we get,</div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">152x – 378y = – 74</blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">–378x + 152y = – 604</blockquote><div> (+) (-) (+) <br /> --------------------------------------------------------- <br /></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>530x – 530y = 530</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">x – y = 530/530<br /></blockquote></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div style="text-align: left;">x – y = 1 </div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">x = y + 1 -------------------------- equation 4 </blockquote><div><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;">4) Put the value of </span>x = y + 1 from equation 4 in equation 3, we get,</div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>x + y = 3</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>(y + 1) + y = 3</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div>2y + 1 = 3</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">2y = 3 - 1</blockquote></div></div></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div><div>y = 2/2</div></div></div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">y = 1 -------------------------- equation 5.</blockquote><div><span style="font-family: arial;">5) <span style="font-family: arial;">Put the value of </span>y = 1 from equation 5 in equation 4, we get,<div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div>x = y + 1</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div>x = 1 + 1</div></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><div>x = 2 -------------------------- equation 6.</div></span></blockquote></div></span></div><div style="text-align: left;"><span style="font-family: arial;">6) So, x = </span><span style="font-family: arial;">2 and y = </span><span style="font-family: arial;">1.</span></div></div><div><br /></div></div><div><div><b>Q8. ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.</b></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJoyMyZu6MPh1yxAdbCYo3xwtzX0mfRB3GMUtw-PeVbHP3w4SgAsn1KDAOabdD74HMjYiWmW8NgLOdcsUgytCA9fyje7xkEfz4D4LbFOnLFB6UMlxiOZK2j3fMfawN-Nd7l9ZQLkMAr3VQHIkWwPriVTdClmQaRVrOO7_t6A1SMfUyhi1ZusnfxoeP/s428/8.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="385" data-original-width="428" height="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJoyMyZu6MPh1yxAdbCYo3xwtzX0mfRB3GMUtw-PeVbHP3w4SgAsn1KDAOabdD74HMjYiWmW8NgLOdcsUgytCA9fyje7xkEfz4D4LbFOnLFB6UMlxiOZK2j3fMfawN-Nd7l9ZQLkMAr3VQHIkWwPriVTdClmQaRVrOO7_t6A1SMfUyhi1ZusnfxoeP/w164-h147/8.png" width="164" /></a></div><div><div><span style="font-family: arial;"><div>1) We know that the sum of the opposite angles of a cyclic quadrilateral is <span style="font-family: arial;">180</span><sup><span style="font-family: arial;">0</span></sup>.</div><div>2) So according to the diagram, we have, <br /></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(-7x + 5) + </span><span style="font-family: arial;">(3y - 5)</span><span style="font-family: arial;"> = 180 -------------------------- equation 1.</span></blockquote></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial;">(-4x) + (4y + 20) = 180 -------------------------- equation 2.</span> </div></div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">3) From equation 1 we get, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(-7x + 5) + </span><span style="font-family: arial;">(3y - 5)</span><span style="font-family: arial;"> = 180</span></blockquote></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">-7x + </span><span style="font-family: arial;">3y</span><span style="font-family: arial;"> = 180</span></div></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7x - </span><span style="font-family: arial;">3y</span><span style="font-family: arial;"> = -180</span><span style="font-family: arial;"> -------------------------- equation 3.</span></span></blockquote><span style="font-family: arial; font-size: medium;">4) From equation 2 we get, </span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(-4x) + (4y + 20) = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(-x) + (y + 5) = 45<br /></span><span style="font-family: arial;">-x + y = 45 - 5<br /></span><span style="font-family: arial;">-x + y = 40<br /></span><span style="font-family: arial;">y = x + 40</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = x + 40 from equation 4 in equation 3, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">7x - </span><span style="font-family: arial;">3y</span><span style="font-family: arial;"> = -180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">7x - </span><span style="font-family: arial;">3(</span><span style="font-family: arial;">x + 40)</span><span style="font-family: arial;"> = -180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">7x - </span><span style="font-family: arial;">3</span><span style="font-family: arial;">x - 120</span><span style="font-family: arial;"> = -180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x </span><span style="font-family: arial;">= 120 -180</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>4x = - 60</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x = - 60/4</span></span></div></blockquote></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial;">x = - 15 </span><span style="font-family: arial;">-------------------------- equation 5.</span></div></div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">6) Put the value of x = - 15 from equation 5 in equation 4, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = x + 40</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = - 15 + 40</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 25</span><span style="font-family: arial;"> -------------------------- equation 6.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">7) So, here, x = - 15 and y = 25, therefore we have,</span></span></div></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< A = </span></span><span style="font-family: arial;">(4y + 20) = </span><span style="font-family: arial;">(4(25) + 20)</span><span style="font-size: medium;"><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">(100 + 20)</span><span style="font-size: medium;"><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">120, so , < A =120.</span></div></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< B = (3y - 5)</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">(3(25) - 5)</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">(75 - 5)</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">70, so, < B = 70.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">< C = (- 4x)</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">(- 4(- 15))</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">60, so, < C = 60.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">< D = (- 7x + 5)</span><span style="font-family: arial;"> = (- 7(- 15) + 5)</span><span><span style="font-family: arial;"> = </span></span><span style="font-family: arial;">(105 + 5)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">110, < D = 110.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/07/156-ncert-10-4-quadratic-equations-ex-41.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-4-Quadratic Equations - Ex-4.1</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial;">ANIL SATPUTE</span></a></div></div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-34162542149781343552023-06-14T13:05:00.001+05:302023-06-26T14:46:44.329+05:30153-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.6<div style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">Exercise 3.6</span></div><div style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/152-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables-Ex-3.5</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.6</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>Q1. Solve the following pairs of equations by reducing them to a pair of linear equations: </b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><b><span style="font-family: arial; font-size: medium;">(i) 1/(2x) + 1/(3y) = 2; <span> </span>1/(3x) + 1/(2y) = 13/6,</span></b></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><b><span style="font-family: arial; font-size: medium;">(ii) (2/<span style="line-height: 107%;">√</span>x) + (3/<span style="line-height: 107%;">√</span>y) = 2; <span> </span>(4/<span style="line-height: 22.8267px;">√</span>x) - (9/<span style="line-height: 22.8267px;">√</span>y) = - 1,</span></b></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 4/x + 3y = 14; 3/x - 4y = 23,</span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iv) 5/(x - 1) + 1/(y - 2) = 2; 6/(x - 1) - 3/(y - 2) = 1,</span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(v) (7x - 2y)/xy = 5; (8x + 7y)/xy = 15,</b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(vi) 6x + 3y = 6xy; 2x + 4y = 5xy,</b></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(vii) 10/(x + y) + 2/(x - y) = 4; 15/(x + y) - 5/(x - y) = -2</b></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(viii) 1/(3x + y) + 1/(3x - y) = 3/4; 1/[2(3x + y)] - 1/[2(3x - y)] = -1/8.</span></b></div></blockquote><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Convert the variables suitably to get linear equations in two variables.</span></div><div><span style="font-family: arial; font-size: medium;"><span>3) Then solve these </span>equations to get the values of our variables. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;">(i) 1/(2x) + 1/(3y) = 2; </span>1/(3x) + 1/(2y) = 13/6.</b></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1/(2x) + 1/(3y) = 2; </span></span></span>1/(3x) + 1/(2y) = 13/6</blockquote></span></span></div></div></div></div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">2) In the equation, 1/(2x) + 1/(3y) = 2, put 1/x = p, and 1/y = q, so we have,</div></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">1/(2x) + 1/(3y) = 2</div></span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p/2 + q/3 = 2</span></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">(3p + 2q)/6 = 2<br /></span></span></blockquote></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">3p + 2q = 2(6)</div></span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3p + 2q = 12 ---------------equation 1</span></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, 1/(3x) + 1/(2y) = 13/6, put 1/x = p, and 1/y = q, so we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1/(3x) + 1/(2y) = 13/6 </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">p/3 + q/2 = 13/6</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(2p + 3q)/6 = 13/6</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">2p + 3q = 13</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">2p = 13 - 3q</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = (13 - 3q)/2 ---------------equation 2</span></div></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of </span></span>p = (13 - 3q)/2<span style="font-size: medium;"><span style="font-family: arial;"> from equation 2 in equation 1, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3p + 2q = 12</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">3(13 - 3q)/2 + 2q = 12</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(39 - 9q)/2 + 2q = 12</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(39 - 9q + 4q)/2 = 12</blockquote></div></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;">(39 - 5q) = 12 (2)</div></blockquote></div></div></div></span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;">(39 - 5q) = 24</div></blockquote></div></div></div></span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">5q = 39 - 24</span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">q = 15/5</span></blockquote></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">q = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 3</span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of q = 3 from equation 3 in equation 2, and we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div>p = (13 - 3q)/2</div>p = (13 - 3(3))/2</blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div>p = (13 - 9)/2</div></div></blockquote></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div style="text-align: left;">p = (4)/2</div></div></div></span></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></blockquote><div><span style="font-family: arial; font-size: medium;">6) The value of p = 2 and the value of q = 3.</span></div></div></div></div></span></span></div></div><div><span style="font-family: arial; font-size: medium;">7) As, p = 1/x and p = 2, we have x = 1/2</span></div><div><span style="font-family: arial; font-size: medium;">8) As, q = 1/y and q = 3, we have y = 1/3.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(ii) (2/<span style="line-height: 22.8267px;">√</span>x) + (3/<span style="line-height: 22.8267px;">√</span>y) = 2; (4/<span style="line-height: 22.8267px;">√</span>x) - (9/<span style="line-height: 22.8267px;">√</span>y) = - 1</span></b></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">(2/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">x) + (3/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">y) = 2</span><span style="font-family: arial; font-size: medium;">; </span></span></span><span style="font-family: arial; font-size: medium;">(4/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">x) - (9/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">y) = - 1</span></blockquote></span></span></div></div><div><div><span style="font-family: arial; font-size: medium;"><span><span>2) In the equation, (2/<span style="line-height: 22.8267px;">√</span>x) + (3/<span style="line-height: 22.8267px;">√</span>y) = 2, put 1/</span></span>√x = p, and 1/√y = q, so we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(2/<span style="line-height: 22.8267px;">√</span>x) + (3/<span style="line-height: 22.8267px;">√</span>y) = 2</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2p + 3q = 2 ---------------equation 1</span></blockquote></div></div><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, <span style="font-family: arial; font-size: medium;">(4/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">x) - (9/</span><span style="line-height: 22.8267px;">√</span><span style="font-family: arial; font-size: medium;">y) = - 1</span>, put <span style="font-size: medium;"><span style="font-family: arial;">1/</span></span>√x = p, and 1/√y = q, so we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(4/<span style="line-height: 22.8267px;">√</span>x) - (9/<span style="line-height: 22.8267px;">√</span>y) = - 1 </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">4p - 9q = - 1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4p = 9q - 1</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = (9q - 1)/4 ---------------equation 2</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) </span><span>Substitute the value of </span>p = (9q - 1)/4<span> from equation 2 in equation 1, we get,<br /></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2p + 3q = 2</span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2(9q - 1)/4 + 3q = 2</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">(9q - 1)/2 + 3q = 2</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(9q - 1 + 6q)/2 = 2 </span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">15q = 4 + 1</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">q = 5/15</span></blockquote></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">q = 1/3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 3</span></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Put the value of q = 1/3 from equation 3 in equation 2, and we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = (9q - 1)/4</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = (9(1/3) - 1)/4</span></div><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">p = (3 - 1)/4</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">p = (2)/4</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span><span style="font-family: arial;">p = 1/2</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) The value of p = 1/2 and the value of q = 1/3.</span></div><div><span style="font-family: arial; font-size: medium;">7) As, p = 1/√x and p = 1/2, we have √x = 2, so x = 4.</span></div><div><span style="font-family: arial; font-size: medium;">8) As, q = 1/√y and q = 1/3, we have √y = 3, so y = 9.</span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(iii) 4/x + 3y = 14; 3/x - 4y = 23</span></b></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4</span><span style="font-family: arial; font-size: medium;">/x + 3y = 14; </span>3/x - 4y = 23</blockquote></span></span></div></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) In the equation, <span style="font-family: arial; font-size: medium;">4</span><span style="font-family: arial; font-size: medium;">/x + 3y = 14</span>, put 1/x = p, so we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">4</span><span style="font-family: arial; font-size: medium;">/x + 3y = 14</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4p + 3y = 14 ---------------equation 1</span></blockquote></div></div><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, 1/(3x) + 1/(2y) = 13/6, put 1/x = p, and 1/y = q, so we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3/x - 4y = 23 </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">3p - 4y = 23</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3p = 23 + 4y</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = (23 + 4y)/3 ---------------equation 2</span></blockquote></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of </span></span>p = (23 + 4y)/3<span style="font-size: medium;"><span style="font-family: arial;"> from equation 2 in equation 1, we get,</span></span></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4</span><span style="font-family: arial; font-size: medium;">p + 3y = 14</span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(4(23 + 4y)/3) + 3y = 14</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">(92 + 16y)/3 + 3y = 14</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(92 + 16y + 9y)/3 = 14</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(92 + 25y)/3 = 14 </span></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(92 + 25y) = 14 (3) </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">(92 + 25y) = 42</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(92 + 25y) = 42</span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">25y = 42 - 92</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">25y = - 50</span></span></div></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = - 50/25</span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = - 2<span> </span><span>------------ equation 3</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Put the value of y = - 2 from equation 3 in equation 2, and we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = (23 + 4y)/3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = (23 + 4(-2))/3</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = (23 - 8)/3<br />p = (15)/3<br />p = 5<span> </span><span>------------ equation 4</span></span></blockquote><span style="font-family: arial; font-size: medium;">6) The value of p = 5 and the value of y = - 2. <br /></span><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">7) As, p = 1/x and p = 5, we have x = 1/5 and y = - 2.</span></div></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><b><span style="font-family: arial; font-size: medium;">(iv) 5/(x - 1) + 1/(y - 2) = 2; </span></b><b><span style="font-family: arial; font-size: medium;">6/(x - 1) - 3/(y - 2) = 1</span></b></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">5/(x - 1) + 1/(y - 2) = 2; </span></span><span style="font-family: arial; font-size: medium;">6/(x - 1) - 3/(y - 2) = 1</span></blockquote></span></span></div></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) In the equation, 5/(x - 1) + 1/(y - 2) = 2, put 1/(x - 1) = p, and 1/(y - 2) = q, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so we have,</span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">5/(x - 1) + 1/(y - 2) = 2</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">5p + q = 2</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">q = 2 - 5p</span></span><span style="font-family: arial;"> ---------------equation 1</span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, 6/(x - 1) - 3/(y - 2) = 1, put 1/(x - 1) = p, and 1/(y - 2) = q, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so we have,</span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">6/(x - 1) - 3/(y - 2) = 1</span><span style="font-family: arial;"> </span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span><span style="font-family: arial;">6p - 3q = 1</span></span><span style="font-family: arial;"> ---------------equation 2</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of </span></span><span style="font-family: arial;">q = (2 - 5p)</span><span style="font-family: arial;"> from equation 1 in equation 2, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div>6p - 3q = 1</div></div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div>6p - 3(2 - 5p) = 1</div></div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">6p - 6 + 15p = 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">6p + 15p = 1 + 6</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">21p = 7</span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div>p = 7/21</div></div></span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">p = 1/3</span></span><span style="font-family: arial;"> ---------------equation 3</span></span></div></div></div></div></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) Put the value of </span><span style="font-family: arial;">p = 1/3</span><span style="font-family: arial;"> from equation 3 in equation 1, and we get</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">q = 2 - 5p</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">q = 2 - 5(1/3)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">q = (6 - 5)/3</span><span style="font-family: arial;"><br />q = 1/3<span> </span><span>------------ equation 4</span></span></span></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) The value of p = 1/3 and the value of q = 1/3.</span></span></div></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As, p = 1/</span><span style="font-family: arial;">(x - 1)</span><span style="font-family: arial;"> and p = 1/3, we have </span><span style="font-family: arial;">(x - 1)</span><span style="font-family: arial;"> = 3, so x = 4.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">8) As, q = 1/</span><span style="font-family: arial;">(y - 2)</span><span style="font-family: arial;"> and q = 1/3, we have </span><span style="font-family: arial;">(y - 2)</span><span style="font-family: arial;"> = 3, so y = 5.</span></span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;"><br /></span></span></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(v) (7x - 2y)/xy = 5; (8x + 7y)/xy = 15</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) First we will simplify the first equation </span></span><span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">(7x - 2y)/xy = 5</span></span>:</div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">(7x - 2y)/xy = 5</span></span></div></span></span></div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(7x)/xy - (2y)/xy = 5</blockquote></span></span></div></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">7/y - 2/x = 5</div></span></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">- 2/x + 7/y = 5, here, put 1/x = p and 1/y = q, we get,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">- 2p + 7q = 5<br /></span><span style="font-family: arial;">2p - 7q = - 5<br /></span><span style="font-family: arial;">2p = 7q - 5<br /></span><span style="font-family: arial;"> p = (7q - 5)/2</span><span style="font-family: arial;"> ---------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) First we will simplify the second equation <span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">(8x + 7y)/xy = 15</span></span>:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(8x + 7y)/xy = 15</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(8x)/xy + (7y)/xy = 15</blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8/y + 7/x = 15</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7/x + 8/y = 15, </span><span style="font-family: arial;">here, put 1/x = p and 1/y = q, we get,<br /></span><span style="font-family: arial;">7p + 8q = 15</span><span style="font-family: arial;"> ---------------equation 2</span><span style="font-family: arial;"> </span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">Substitute the value of </span></span><span style="font-family: arial;">p = (7q - 5)/2 from equation 1 in equation 2, we get,</span></div></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">7p + 8q = 15</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">7</span><span style="font-family: arial;">(7q - 5)/2</span><span style="font-family: arial;"> + 8q = 15</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(49q - 35)/2</span><span style="font-family: arial;"> + 8q = 15</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(49q - 35 + 16q)/2</span><span style="font-family: arial;"> = 15</span></blockquote></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">(49q + 16q - 35)</span><span style="font-family: arial;"> = 15 (2)</span> </div></div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">(65q - 35)</span><span style="font-family: arial;"> = 30</span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">65q = 35</span><span style="font-family: arial;"> + 30</span></blockquote></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">65q = 65</span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">q = 1</span></span><span style="font-family: arial;"> ---------------equation 3</span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;">4) Put the value of </span></span><span style="font-family: arial;">q = 1</span><span style="font-size: medium;"><span style="font-family: arial;"> from equation 3 in equation 1, and we get</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = (7q - 5)/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = (7(1) - 5)/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = (7 - 5)/2</span></blockquote></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">p = 2/2 </div></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p = 1<span> </span><span>------------ equation 4</span></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) The value of p = 1 and the value of q = 1.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">6) As, p = 1/</span><span style="font-family: arial;">x</span><span style="font-family: arial;"> and p = 1, we have </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 1.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As, q = 1/</span><span style="font-family: arial;">y</span><span style="font-family: arial;"> and q = 1, we have </span><span style="font-family: arial;">y = 1.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><b>(vi) 6x + 3y = 6xy; 2x + 4y = 5xy</b></div></span></div><div><div><span style="font-family: arial; font-size: medium;">1) First we will simplify the first equation 6x + 3y = 6xy:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">6x + 3y = 6xy</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(6x)/xy + (3y)/xy = 6</blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;">6/y + 3/x = 6</span></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3/x + 6/y = 6, here, put 1/x = p and 1/y = q, we get,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3p + 6q = 6<br /></span><span style="font-family: arial;"> p + 2q = 2<br /></span><span style="font-family: arial;"> p = 2 - 2q</span><span style="font-family: arial;"> ---------------equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) First we will simplify the second equation 2x + 4y = 5xy:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2x + 4y = 5xy</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">(2x)/xy + (4y)/xy = 5</blockquote></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2/y + 4/x = 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">4/x + 2/y = 5, </span><span style="font-family: arial;">here, put 1/x = p and 1/y = q, we get,<br /></span><span style="font-family: arial;">4p + 2q = 5</span><span style="font-family: arial;"> ---------------equation 2</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">Substitute the value of </span><span style="font-family: arial;">p = 2 - 2q</span><span style="font-family: arial;"> from equation 1 in equation 2, we get,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4p + 2q = 5</span></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">4(</span><span style="font-family: arial;">2 - 2q)</span><span style="font-family: arial;"> + 2q = 5</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">8 - 8q</span><span style="font-family: arial;"> + 2q = 5<br /></span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">8 - 6q</span><span style="font-family: arial;"> = 5</span></span></blockquote></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> - </span><span style="font-family: arial;">6q = 5</span><span style="font-family: arial;"> - 8</span></span></div></blockquote></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> - 6q = - 3</span></div></blockquote></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span> q = (- 3)/(- 6)</span> </span></div></blockquote></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span><span style="font-family: arial;">q = 1/2</span></span><span style="font-family: arial;"> ---------------equation 3</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;">4) Put the value of </span></span><span style="font-family: arial;">q = 1/2</span><span style="font-size: medium;"><span style="font-family: arial;"> from equation 3 in equation 1, and we get</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = 2 - 2q</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = 2 - 2(1/2)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = 2 - 1</span></blockquote></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">p = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-size: medium;"><span style="font-family: arial;">5) The value of p = 1 and the value of q = 1/2.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">6) As, p = 1/</span><span style="font-family: arial;">x</span><span style="font-family: arial;"> and p = 1, we have </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 1.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As, q = 1/</span><span style="font-family: arial;">y</span><span style="font-family: arial;"> and q = 1/2, we have </span><span style="font-family: arial;">y = 2.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div></span></div></div><div><b><span style="font-family: arial; font-size: medium;">(vii) 10/(x + y) + 2/(x - y) = 4; 15/(x + y) - 5/(x - y) = - 2</span></b></div><div><span style="font-family: arial; font-size: medium;">1) Given equations are</span></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">10/(x + y) + 2/(x - y) = 4; 15/(x + y) - 5/(x - y) = - 2</blockquote></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) In the equation, 10/(x + y) + 2/(x - y) = 4, put 1/(x + y) = p, and 1/(x - y) = q, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">so we have,</span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">10/(x + y) + 2/(x - y) = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">10p + 2q = 4</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>5p + q = 2</span> </span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><span style="font-size: medium;"><span><span style="font-family: arial;">q = 2 - 5p</span></span><span style="font-family: arial;"> ---------------equation 1</span></span></div></div></div></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, 15/(x + y) - 5/(x - y) = - 2, put 1/(x + y) = p, and 1/(x - y) = q, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">so we have,</span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">15/(x + y) - 5/(x - y) = - 2</span><span><span style="font-family: arial;"> </span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span><span style="font-family: arial;">15p - 5q = - 2</span></span><span style="font-family: arial;"> ---------------equation 2</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of </span></span><span style="font-family: arial;">q = (2 - 5p)</span><span style="font-family: arial;"> from equation 1 in equation 2, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">15p - 5</span><span style="font-family: arial;">q</span><span style="font-family: arial;"> = - 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">15p - 5</span><span style="font-family: arial;">(2 - 5p)</span><span style="font-family: arial;"> = - 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">15p - 10 + 25p = - 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">15p + 25p = - 2 + 10</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">40p = 8</span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p = 8/40</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><span style="font-size: medium;"><span><span style="font-family: arial;">p = 1/5</span></span><span style="font-family: arial;"> ---------------equation 3</span></span></div></div></div></div></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) Put the value of </span><span style="font-family: arial;">p = 1/5</span><span style="font-family: arial;"> from equation 3 in equation 1, and we get</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">q = 2 - 5p</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">q = 2 - 5(1/5)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">q = 2 - 1</span><span style="font-family: arial;"><br />q = 1<span> </span><span>------------ equation 4</span></span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) The value of p = 1/5 and the value of q = 1.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As, p = 1/</span><span style="font-family: arial;">(x + y)</span><span style="font-family: arial;"> and p = 1/5, we have </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">(x + y)</span><span style="font-family: arial;"> = 5</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 5</span><span style="font-family: arial;"> </span></span></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">8) As, q = 1/</span><span style="font-family: arial;">(x - y)</span><span style="font-family: arial;"> and q = 1, we have </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(x - y)</span><span><span style="font-family: arial;"> = 1</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 6</span></span></div></div></div></blockquote><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;">9) Adding equation 5 and equation 6, we get,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x + y + x - y = 5 + 1</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2x = 6, so x = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 7</span></span></div></blockquote><span style="font-family: arial; font-size: medium;">10) put x = 3 from equation 7 in equation 5, and we get,</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(x + y)</span><span style="font-family: arial;"> = 5</span></span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(3 + y)</span><span style="font-family: arial;"> = 5</span></span></div></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = 5 - 3</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = 2<span> </span><span>------------ equation 8.</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">11) So, x</span><span style="font-family: arial;"> = 3, and y = 2.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(viii) 1/(3x + y) + 1/(3x - y) = 3/4; 1/[2(3x + y)] - 1/[2(3x - y)] = -1/8</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></div></div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">1/(3x + y) + 1/(3x - y) = 3/4; 1/[2(3x + y)] - 1/[2(3x - y)] = -1/8</span></span></blockquote></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) In the equation, 1/(3x + y) + 1/(3x - y) = 3/4, put 1/(3x + y) = p, and 1/(3x - y) = q, </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">so we have,</span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1/(3x + y) + 1/(3x - y) = 3/4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p + q = 3/4</span><span style="font-family: arial;"> ---------------equation 1</span></span></blockquote></div></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) In the equation, 1/[2(3x + y)] - 1/[2(3x - y)] = -1/8, put 1/(3x + y) = p, and </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span><span style="font-family: arial;">1/(3x - y) = q, </span></span><span style="font-family: arial;">so we have,</span></span></div></div></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1/[2(3x + y)] - 1/[2(3x - y)] = -1/8</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>p/2 - q/2 = -1/8</span> </span></div></div></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p - q = -1/4</span><span style="font-family: arial;"> ---------------equation 2</span></span></blockquote></div></div></div><div><div><div><div><span style="font-size: medium;"><span><span style="font-family: arial;"><span style="font-family: arial;">4) Adding</span></span></span><span><span style="font-family: arial;"> equation 1, and </span></span><span style="font-family: arial;">equation 2,</span><span style="font-family: arial;"> we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p + q = 3/4</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p - q = -1/4</span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span> -----------------------</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2p = 2/4</span> </span></div></div></div></blockquote><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = 2/8</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><span style="font-size: medium;"><span><span style="font-family: arial;">p = 1/4</span></span><span style="font-family: arial;"> ---------------equation 3</span></span></div></div></div></div></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) Put the value of </span><span style="font-family: arial;">p = 1/4</span><span style="font-family: arial;"> from equation 3 in equation 1, and we get</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p + q = 3/4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">1/4 + q = 3/4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">q = 3/4 - 1/4</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">q = (3 - 1)/4</span></div></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">q = 2/4</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">q = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></span></div></div></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) The value of p = 1/4 and the value of q = 1/2.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) As, p = 1/</span><span style="font-family: arial;">(3x + y)</span><span style="font-family: arial;"> and p = 1/4, we have </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span><span style="font-family: arial;">(3x + y)</span><span style="font-family: arial;"> = 4</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 5</span><span style="font-family: arial;"> </span></span></div></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">8) As, q = 1/</span><span style="font-family: arial;">(3x - y)</span><span style="font-family: arial;"> and q = 1/2, we have </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">(3x - y)</span><span><span style="font-family: arial;"> = 2</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 6</span></span></div></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">9) Adding equation 5 and equation 6, we get,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">3x + y = 4</span></div></div></div><div><div><span style="font-family: arial; font-size: medium;">3x - y = 2</span></div></div></div></div></blockquote><span style="font-family: arial; font-size: medium;"><span> -----------------------</span> <br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>6x = 6</span> </span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote></div><div><div><span style="font-size: medium;"><span><span style="font-family: arial;">x = 1</span></span><span style="font-family: arial;"> ---------------equation 7<br /></span></span></div></div></div></div></div></div></blockquote><div><span style="font-family: arial; font-size: medium;">10) put x = 1 from equation 7 in equation 5, and we get,</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(3x + y)</span><span style="font-family: arial;"> = 4</span></span></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">(3 + y)</span><span style="font-family: arial;"> = 4</span></span></div></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 4 - 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">y = 1<span> </span><span>------------ equation 8.</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">11) So, x</span><span style="font-family: arial;"> = 1, and y = 1.</span></span></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> </span></span></div></div><div style="text-align: left;"><div><span style="font-size: medium;"><b><span style="font-family: arial;">Q</span><span style="font-family: arial;">2. Formulate the following problems as a pair of equations, and hence find their solutions:</span></b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her </b></span><b>speed of rowing in still water and the speed of the current.</b></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 </b></span><b>women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to </b><b>finish the work, and also that taken by 1 man alone.</b></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 </b></span><b>hours if she travels 60 km by train and the remaining by bus. If she travels 100 km </b><b>by train and the remaining by bus, she takes 10 minutes longer. Find the speed of </b><b>the train and the bus separately.</b></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equations.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">equations to get the values of x and y. </span></blockquote><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><b>(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her </b></span><b>speed of rowing in still water and the speed of the current.</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div>1) Let Ritu's speed in still water be x km/h.</div><div>2) Let the speed of the current be y km/h.</div><div>3) Ritu's speed of rowing downstream (x + y) km/h.</div><div>4) Ritu's speed of rowing upstream (x - y) km/h.</div><div>5) According to the first condition, as Ritu can row downstream 20 km in 2 hrs, </div></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div>2(x + y) = 20</div></span></div></blockquote></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span>(x + y) = 10 </span><span>-------------------------- equation 1.</span></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">6) </span><span style="font-family: arial;">According to the first condition, as Ritu can row upstream 4 km in 2 hrs, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">2(x - y) = 4</span></div></blockquote></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span>(x - y) = 2</span><span>-------------------------- equation 2.</span></div></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">7) Adding equation 1 and equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">(x + y) + (x - y) = 10 + 2</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">2x = 12</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = 12/2</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = 6</span><span style="font-family: arial;"> -------------------------- equation 3.</span></div></blockquote><div><div><span style="font-family: arial;">8) Put the value of x = 6 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(x + y) = 10</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(6 + y) = 10</span></blockquote></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = 10 - 6</span></div></div></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">y = 4</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-family: arial;">9) <span style="font-family: arial;">Ritu's </span>speed of rowing in still water is 6 km/h and the speed of the current is 4</span></div></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">km/h.</span></div></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"><div><div><span style="font-family: arial;"> </span></div></div></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><b>(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 </b></span><b>women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to </b><b>finish the work, and also that taken by 1 man alone.</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial;"><div><span>1)</span><span> </span>Let the <span style="background-color: white; color: #333333;">number of days taken by one woman to finish the work = x</span>. So work done</div></span></div></div></div></div></div></span></div></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial;"><div style="text-align: left;">by a woman in 1 day will be 1/x</div></span></div></div></div></div></div></span></div></div></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div>2) Let the <span style="background-color: white; color: #333333;">number of days taken by one man to finish the work = y</span>. So work done</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">by a man in 1 day will be 1/y</span></span></span></div></blockquote><div>3) As 2 women and 5 men finish the work in 4 days, so we have,</div></span></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;">(2/x + 5/y) = 1/4, put 1/x = p and 1/y = q, we have,</div></span></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2p + 5q = 1/4</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4(2p + 5q) = 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial; font-size: medium;"><span>8p + 20q = 1</span><span> -------------------------- equation 1.</span> </span></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div>4) As 3 women and 6 men finish the work in 3 days, so we have,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;">(3/x + 6/y) = 1/3, </span></span></span><span style="font-family: arial;">put 1/x = p and 1/y = q, we have,</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3p + 6q = 1/3<br /></span><span style="font-family: arial;">3(p + 2q) = 1/3</span></blockquote></span></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><span style="font-family: arial;"><div style="text-align: left;">(p + 2q) = 1/9</div></span></span></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = 1/9 - 2q</span><span style="font-family: arial;"> -------------------------- equation 2.</span></blockquote></span></div></span></div></span></div></div></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">5) </span>Put the value of p = (1/9 - 2q) from equation 2 in equation 1, we get,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">8p + 20q = 1</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">8(1/9 - 2q) + 20q = 1</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">8/9 - 16q + 20q = 1</blockquote></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">8/9 + 4q = 1</div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">4q = 1 - 8/9</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4q = (9 - 8)/9</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;"> q = 1/36 -------------------------- equation 3.</div></span></span></div></blockquote></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">6) Put the value of q = 1/36 from equation 3 in equation 2, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">p = 1/9 - 2q</blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;">p = 1/9 - 2(1/36)</blockquote></div></span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial;"><div><div style="text-align: left;">p = 1/9 - 1/18</div></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p = 2/18 - 1/18<br /></span><span style="font-family: arial;">p = (2 - 1)/18<br /></span><span style="font-family: arial;">p = 1/18</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">7) The value of p = 1/18 and the value of q = 1/36.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) As, p = 1/</span><span style="font-family: arial;">x</span><span style="font-family: arial;"> and p = 1/18, we have </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 18.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>9) As, q = 1/</span><span>y</span><span> and q = 1/36, we have </span><span>y = 36.</span> </span></div><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">10) </span><span style="background-color: white; color: #333333;">Number of days taken by a woman to finish the work is 18 and by a man is 36</span>.</div></div></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b><span style="font-weight: 400;"><b>(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 </b></span><b>hours if she travels 60 km by train and the remaining by bus. If she travels 100 km </b><b>by train and the remaining by bus, she takes 10 minutes longer. Find the speed of </b><b>the train and the bus separately.</b></b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div><div><div><span style="font-family: arial;">1) Let the speed of the train be x km/h.</span></div><div><span style="font-family: arial;">2) Let the speed of the bus be y km/h.</span></div><div><span style="font-family: arial;">3) Total distance traveled by Roohi is 300 km.</span></div><div><span style="font-family: arial;">4) According to the first condition, she traveled 60 km by train and 240 km by bus in</span></div></div></div></div></b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4 hrs, so, we have,</span></div></div></div></div></b></span></div></div></div></blockquote><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">60/x + 240/y = 4</span><span style="font-family: arial;">, put 1/x = p and 1/y = q, we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">60p + 240q = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial;">4(15p + 60q) = 4</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"></blockquote><span style="font-family: arial;">15p + 60q = 1</span></blockquote></div></b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><b><div style="font-weight: 400;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">15(p + 4q) = 1</span></div></div></b></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p + 4q = 1/15<br /></span><span style="font-family: arial;">p = ((1/15) - 4q)</span><span style="font-family: arial;"> -------------------------- equation 1.</span></span></blockquote><div><div style="text-align: left;"><span style="font-family: arial;"><b><div style="font-weight: 400;"><div><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><span style="font-family: arial; font-size: medium;">5) According to the second condition, she traveled 100 km by train and 200 km by<br /></span></div></b></span></div></div></div></b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><div><div><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400; text-align: left;"><span style="font-family: arial;">bus in </span><span style="font-family: arial;">4 hrs and 10 minutes i.e. 4 + (10/60) = 4 + (1/6) = 25/6 hrs, so, we have,</span></div></b></span></div></div></div></div></b></span></div></div></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><div><div><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">100/x + 200/y = 25/6</span><span style="font-family: arial;">, put 1/x = p and 1/y = q, we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">100p + 200q = 25/6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">25(4p + 8q) = 25/6</span></blockquote></div></b></span></div></div></div></div></b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial;"><b><div style="font-weight: 400;"><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div style="text-align: left;"><span style="font-family: arial;">(4p + 8q) = 1/6</span><span style="font-family: arial;"> -------------------------- equation 2.</span></div></div></b></span></div></div></b></span></div></blockquote><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div><span style="font-family: arial;">6) Put the value of p = ((1/15) - 4q) from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(4p + 8q) = 1/6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(4</span><span style="font-family: arial;">((1/15) - 4q)</span><span style="font-family: arial;"> + 8q) = 1/6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">(4/15 - 16q + 8q) = 1/6</span></blockquote></div></b></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div style="text-align: left;"><span style="font-family: arial;">(4/15 - 8q) = 1/6</span> </div></div></b></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8q = 4/15 - 1/6</span> </blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8q = (8/30) - (5/30)</span></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8q = 3/30</span></blockquote></div></div></b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><div style="text-align: left;"><span style="font-size: medium;">8q = 1/10</span></div></div></b></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>q = 1/(10(8))</span> </span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">q = 1/80</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></div></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div><div><span style="font-family: arial;">7) Put the value of q = 1/80 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = ((1/15) - 4q)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = ((1/15) - 4(1/80))</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">p = ((1/15) - 1/20))</span></blockquote></div></b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div><span style="font-family: arial;"><b><div style="font-weight: 400;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">p = ((4/60) - 3/60))</span></div></div></b></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">p = (4 - 3)/60</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p = 1/60</span><span style="font-family: arial;"> </span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><div style="font-weight: 400;"><div><span><span style="font-family: arial;">8) The value of p = 1/60 and the value of q = 1/80.</span></span></div><div><span><span style="font-family: arial;">9) As, p = 1/</span><span style="font-family: arial;">x</span><span style="font-family: arial;"> and p = 1/60, we have </span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = 60.</span></span></div><div><span style="font-family: arial;">10) As, q = 1/</span><span style="font-family: arial;">y</span><span style="font-family: arial;"> and q = 1/80, we have </span><span style="font-family: arial;">y = 80.</span></div><div><div><span style="font-family: arial;">11) <span style="color: #333333;"><span style="background-color: white;">The speed of the train is 60 km/h. </span></span></span></div></div></div></b></span></div><div><span style="font-family: arial; font-size: medium;"><b><span style="font-weight: 400;">12) </span><span style="color: #333333; font-weight: 400;"><span style="background-color: white;">The speed of the bus is 80 km/h.</span></span></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/154-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.7</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></span></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-43115782969626182352023-06-08T11:14:00.001+05:302023-06-14T13:08:36.778+05:30152-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.5<div style="clear: both; color: #0400ff; text-align: left;"><div style="clear: both; color: black;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black;"><span style="font-family: arial; font-size: medium;">Exercise 3.5</span></div><div style="clear: both; color: black;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/151-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables-Ex-3.4</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.5</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>Q1. Which of the following pairs of linear equations has a unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using the cross-multiplication method.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(i) x – 3y – 3 = 0; 3x – 9y – 2 = 0, (ii) 2x + y = 5; 3x + 2y = 8</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 3x – 5y = 20; 6x – 10y = 40, (iv) x – 3y – 7 = 0, 3x – 3y – 15 = 0.</b></span></div></blockquote><div style="text-align: left;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div></div></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) </span><span style="font-family: arial;">For the equations, </span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">a) a</span><sub>1</sub><span style="font-family: arial;">x + b</span><sub>1</sub><span style="font-family: arial;">y + c</span><sub>1</sub><span style="font-family: arial;"> = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></div></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">b) a</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial; line-height: 19.26px;"> </span><span style="font-family: arial;">= 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">In cross multiplication </span><span style="font-family: arial;">method, if </span><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;">b</span><sub>2</sub><span style="font-family: arial;"> - a</span><sub>2</sub><span style="font-family: arial;">b</span><sub>1 </sub><b style="color: #333333;">≠</b><span style="font-family: arial;"> 0 </span><span style="font-family: arial;">we have,</span></div></span></div></div></blockquote></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">c) x = <a name="_Hlk123941678">[(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a>)/(a<sub>1</sub>b<sub>2</sub> - a<sub>2</sub>b<sub>1</sub>)] ------------ equation 3</span></div><div><span style="font-family: arial;">d) y = </span><a name="_Hlk123941678">[(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span style="font-family: arial;">)/(a</span><sub>1</sub><span style="font-family: arial;">b</span><sub>2</sub><span style="font-family: arial;"> - a</span><sub>2</sub><span style="font-family: arial;">b</span><sub>1</sub><span style="font-family: arial;">)]</span><span style="font-family: arial;"> ------------ equation 4</span></div></blockquote></span></div><div><span style="font-family: arial; font-size: medium;"><span>2) So, we can say that,<br /></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span>e) </span></span><span>If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub>,</sub></span><span>then </span><span>we get a unique solution.</span> </span></div></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>f) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there are infinitely many solutions.</span></span></div></div><div><div><span style="font-family: arial; font-size: medium;"><span>g) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there is no solution.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) From Equation 3 and Equation 4, we can say that,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>h) x/</span><a name="_Hlk123941678">(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a><span>) = y/</span><a name="_Hlk123941678">(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span>) = 1/</span><span>(a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><span>)</span><span> ------------ equation 5.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) Steps to solve these equations by cross multiplication method:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">i) Write the given equations in the form of equation 1 and equation 2.</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">ii) Write your equations using the formula given in the</span><span style="font-family: arial;"> equation 5.</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>iii) Simplify and get the values of x and y. (Here, we have </span><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1 </sub><b style="color: #333333;">≠</b><span> 0).</span></span></div></blockquote><div style="text-align: left;"><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><div><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(i) x – 3y – 3 = 0; 3x – 9y – 2 = 0</span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>1) </span><span>From equation </span><span>x – 3y – 3 = 0, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 1, b</span><sub>1</sub><span>= </span><span>–</span><span> 3, c</span><sub>1 </sub><span>= </span><span>–</span><span> 3.</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) From equation </span><span>3x – 9y – 2 = 0</span><span>, and equation </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= 3, b</span><sub>2</sub><span>= </span><span>–</span><span> 9, c</span><sub>2 </sub><span>= </span><span>–</span><span> 2.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = 1/3</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = (-3)/(-9)</span><span> = 1/3</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-3)/(-2)</span><span> = 3/2</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) From the above equations, we can say that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there is no solution.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 2x + y = 5; 3x + 2y = 8 </span></b></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) Our equations are </span><span style="font-family: arial;">2x + y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 5 = 0; 3x + 2y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 8 = 0.</span></span></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>From equation 2</span><span>x + y – 5 = 0, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 2, b</span><sub>1</sub><span>= 1, c</span><sub>1 </sub><span>= </span><span>– 5</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From equation </span><span>3x + 2y – 8 = 0</span><span>, and equation </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= 3, b</span><sub>2</sub><span>= 2, c</span><sub>2 </sub><span>= </span><span>– 8</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = 2/3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = 1/2</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-5)/(-8)</span><span> = 5/8</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) From the above equations, we can say that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub>,</sub></span><span>then </span><span>we get a unique solution</span><span>.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>6) We can find </span><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>as follows</span><span>,</span></span></div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><a name="_Hlk123941678"><b><span style="font-family: arial; font-size: medium;"><span>b</span><sub>1 </sub><span>c</span><sub><span><span>1 (1) (-5)</span></span></sub></span></b></a></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>2 </sub>c<sub>2</sub></a><sub><span><span> (2) (-8)</span></span></sub></span></b></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= (1)(-8) - (2)(-5)</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= -8 + 10</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= 2</span><span> </span><span>------------ equation 1</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>7) We can find </span><a name="_Hlk123941678"></a><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><sub> </sub><span>as follows</span><span>,</span></span></div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><span>c</span><sub>1 </sub><span>a</span><sub><span><span>1 (-5) (2)</span></span></sub></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>2 </sub>a<sub>2</sub></a><sub><span><span> (-8) (3)</span></span></sub></span></b></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= (-5)(3) - (-8)(2)</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= -15 + 16</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= 1</span><span> </span><span>------------ equation 2</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>8) We can find </span><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><sub> </sub><span>as follows</span><span>,</span></span></div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>b</span><sub><span><span>1 (2) (1)</span></span></sub></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">a<sub>2 </sub>b<sub>2</sub></a><a name="_Hlk123941678"><sub><span><span> (3) (2)</span></span></sub></a></span></b></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> (2)(2) - (3)(1)</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> 4 - 3</span></span></div></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> 1</span><span> </span><span>------------ equation 3</span></span></div></div></blockquote><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">9) By cross multiplication method, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x/</span><a name="_Hlk123941678">(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a><span>) = y/</span><a name="_Hlk123941678">(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span>) = 1/</span><span>(a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><span>)</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/</span><a name="_Hlk123941678">2</a><span> = y/</span><a name="_Hlk123941678">1</a><span> = 1/</span><span>1</span><span>, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/2</span><span> = 1, and y/</span><a name="_Hlk123941678">(1</a><span>) = 1, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x </span><span style="font-family: arial;">= 2 and y = 1</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">10) So, here x = 2 and y = 1. </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 3x – 5y = 20; 6x – 10y = 40</span></b></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) Our equations are </span><span style="font-family: arial;">3x - 5y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 20 = 0; 6x - 10y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 40 = 0.</span></span></div><div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>From equation </span><span>3x - 5y </span><span>–</span><span> 20 = 0</span><span>, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 3, b</span><sub>1</sub><span>= - 5, c</span><sub>1 </sub><span>= </span><span>– 20</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From equation </span><span>6x - 10y – 40 = 0</span><span>, and equation </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= 6, b</span><sub>2</sub><span>= - 10, c</span><sub>2 </sub><span>= </span><span>– 40</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = 3/6 = 1/2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = (-5)/(-10) = 5/10 = 1/2</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-20)/(-40)</span><span> = 1/2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) From the above equations, we can say that,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there are infinitely many solutions.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(iv) x – 3y – 7 = 0, 3x – 3y – 15 = 0.</b> </span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Our equations are </span><span style="font-family: arial;">x - 3y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 7 = 0; 3x - 3y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 15 = 0.</span></span></div><div><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>From equation </span><span>x - 3y </span><span>–</span><span> 7 = 0</span><span>, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 1, b</span><sub>1</sub><span>= - 3, c</span><sub>1 </sub><span>= </span><span>– 7</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From equation </span><span>3x - 3y </span><span>–</span><span> 15 = 0</span><span>, and equation </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= 3, b</span><sub>2</sub><span>= - 3, c</span><sub>2 </sub><span>= </span><span>– 15</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = 1/3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = (-3)/(-3) = 1</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-7)/(-15)</span><span> = 7/15</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">5) From the above equations, we can say that,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub>,</sub></span><span>then </span><span>we get a unique solution</span><span>. </span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>6) We can find </span><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>as follows</span><span>,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><a name="_Hlk123941678"><b><span style="font-family: arial; font-size: medium;"><span>b</span><sub>1<span> </span></sub><span>c</span><sub><span><span>1 (-3) (-7)</span></span></sub></span></b></a></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>2 </sub>c<sub>2</sub></a><a name="_Hlk123941678"><sub><span><span> (-3) (-15)</span></span></sub></a></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= (-3)(-15) - (-3)(-7)</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= 45 - 21</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= 24</span><span> </span><span>------------ equation 1</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>7) We can find </span><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><sub> </sub><span>as follows</span><span>,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><a name="_Hlk123941678"><b><span style="font-family: arial; font-size: medium;"><span>c</span><sub>1 </sub><span>a</span><sub><span><span>1 (-7) (1)</span></span></sub></span></b></a></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>2 </sub>a<sub>2</sub></a><a name="_Hlk123941678"><sub><span><span> (-15) (3)</span></span></sub></a></span></b></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= (-7)(3) - (-15)(1)</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= -21 + 15</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= -6</span><span> </span><span>------------ equation 2</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>8) We can find </span><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><sub> </sub><span>as follows</span><span>,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><a name="_Hlk123941678"><b><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>b</span><sub><span><span>1 (1) (-3)</span></span></sub></span></b></a></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">a<sub>2 </sub>b<sub>2</sub></a><a name="_Hlk123941678"><sub><span><span> (3) (-3)</span></span></sub></a></span></b></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> (1)(-3) - (3)(-3)</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> -3 + 9</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> 6</span><span> </span><span>------------ equation 3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">9) By cross multiplication method, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/</span><a name="_Hlk123941678">(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a><span>) = y/</span><a name="_Hlk123941678">(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span>) = 1/</span><span>(a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/</span><a name="_Hlk123941678">24</a><span> = y/</span><a name="_Hlk123941678">(-6)</a><span> = 1/</span><span>6</span><span>, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/24</span><span> = 1/6, and y/</span><a name="_Hlk123941678">(-6</a><span>) = 1/6, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x </span><span style="font-family: arial;">= 24/6 and y = 6/(-6)</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">10) So, here x = 4 and y = -1.</span><span style="font-family: arial;"> </span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. (i) For which values of a and b do the following pair of linear equations have an </b></span><b>infinite number of solutions?</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>2x + 3y = 7; <span> </span></b></span><b>(a – b) x + (a + b) y = 3a + b – 2</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) For which value of k will the following pair of linear equations have no solution?</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>3x + y = 1; <span> </span></b></span><b>(2k – 1) x + (k – 1) y = 2k + 1</b></span></div><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div></div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">1) </span><span style="font-family: arial;">For the equations, </span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">a) a</span><sub>1</sub><span style="font-family: arial;">x + b</span><sub>1</sub><span style="font-family: arial;">y + c</span><sub>1</sub><span style="font-family: arial;"> = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></div></span></div></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">b) a</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial; line-height: 19.26px;"> </span><span style="font-family: arial;">= 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></div></span></div></div></blockquote></span></div></div><div><div><div><span style="font-family: arial; font-size: medium;">2) So, we can say that,<br /></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c) </span><span>If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there are infinitely many solutions.</span></span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>d) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there is no solution.</span></span></div></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><span><b>(i) For which values of a and b do the following pair of linear equations have an </b></span><b>infinite number of solutions?</b></span></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>2x + 3y = 7; </b></span><b>(a – b) x + (a + b) y = 3a + b – 2.</b></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) Our equations </span><span>2x + 3y - 7 = 0 and </span><span>(a – b) x + (a + b) y - (3a + b – 2) = 0</span><b> </b><span>have</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">infinite number of solutions, so we have,</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) </span><span>From equation 2</span><span>x + 3y </span><span>–</span><span> 7 = 0</span><span>, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 2, b</span><sub>1</sub><span>= 3, c</span><sub>1 </sub><span>= </span><span>– 7</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">3) From the equation, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(a – b) x + (a + b) y - (3a + b – 2)</span><span> = 0</span><span>, and </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= (a - b), b</span><sub>2</sub><span>= (a + b), c</span><sub>2 </sub><span>= </span><span>–(3a + b - 2)</span><span>.</span></span></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">4) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = (2)/(a - b)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = (3)/(a + b)</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-7)/(-(3a + b -2))</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) As our equations have an </span><span style="font-family: arial;">infinite number of solutions, so we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2 ,</sub></span><span>So we have,</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(2)/(a - b) = </span><span style="font-family: arial;">(3)/(a + b) = </span><span style="font-family: arial;">(-7)/(-(3a + b - 2)) -------- equation 1</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) Equating the first two ratios of equation 1, we have</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(2)/(a - b) = </span><span style="font-family: arial;">(3)/(a + b)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(2)(a + b) = </span><span style="font-family: arial;">(3)(a - b)<br /></span><span style="font-family: arial;">(2)(a + b) = </span><span style="font-family: arial;">(3)(a - b)<br /></span><span style="font-family: arial;">2a + 2b = </span><span style="font-family: arial;">3a - 3b<br /></span><span style="font-family: arial;">2a - 3a = - </span><span style="font-family: arial;">3b - 2b<br /></span><span style="font-family: arial;">- a = - 5</span><span style="font-family: arial;">b</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a = 5</span><span style="font-family: arial;">b</span><span style="font-family: arial;"> -------- equation 2</span></span></blockquote><span style="font-family: arial; font-size: medium;"><span>7) Equating the first and third ratios of equation 1, we have</span><br /></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">(2)/(a - b) = </span><span style="font-family: arial;">(-7)/(-(3a + b - 2))</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(2)/(a - b) = </span><span style="font-family: arial;">7/(3a + b - 2)<br /></span><span style="font-family: arial;">(2)</span><span style="font-family: arial;">(3a + b - 2) </span><span style="font-family: arial;">= </span><span style="font-family: arial;">7</span><span style="font-family: arial;">(a - b)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">6a + 2b - 4 </span><span style="font-family: arial;">= </span><span style="font-family: arial;">7</span><span style="font-family: arial;">a - 7b<br /></span><span style="font-family: arial;">7b + 2b - 7a + 6a </span><span style="font-family: arial;">= 4</span><span style="font-family: arial;"><br /></span><span style="font-family: arial;">9b - a = 4</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------- equation 3</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">8) </span><span style="font-family: arial;">Put the value of </span><span style="font-family: arial;">a = 5</span><span style="font-family: arial;">b</span><span style="font-family: arial;"> from equation 2 in equation 3, we get,</span></span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9b - a = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9b - 5b = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4b = 4</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>b = 1</span><span>-------------------------- equation 4.</span> </span></blockquote></div></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial;">Put the value of </span><span style="font-family: arial;">b = 1</span><span style="font-family: arial;"> from equation 4 in equation 2, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a = 5</span><span style="font-family: arial;">b</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a = 5</span><span style="font-family: arial;">(1)</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a = 5 </span><span style="font-family: arial;">-------------------------- equation 5.</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>10) So, here a = 5, and b = 1.</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(ii) For which value of k will the following pair of linear equations have no solution?</b></span></div><div><span style="font-family: arial; font-size: medium;"><span><b>3x + y = 1; </b></span><b>(2k – 1) x + (k – 1) y = 2k + 1</b></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>1) Our equations </span><span>3x + y - 1 = 0 and </span><span>(2k – 1) x + (</span><span>k – 1</span><span>) y - (2k + 1) = 0</span><b> </b><span>have</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">no solutions, so we have,</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) </span><span>From equation </span><span>3x + y - 1 = 0</span><span>, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 3, b</span><sub>1</sub><span>= 1, c</span><sub>1 </sub><span>= </span><span>– 1</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">3) From the equation, </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>(2k – 1) x + (</span><span>k – 1</span><span>) y - (2k + 1) = 0</span><span>, and </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= (</span><span>2k – 1</span><span>), b</span><sub>2</sub><span>= (</span><span>k – 1</span><span>), c</span><sub>2 </sub><span>= </span><span>– (</span><span>2k + 1</span><span>)</span><span>.</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;">4) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = (3)/(</span><span>2k – 1</span><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = (1)/(</span><span>k – 1</span><span>)</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-1)/(-(</span><span>2k + 1</span><span>))</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) As our equations have </span><span style="font-family: arial;">no solutions, so we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2 ,</sub></span><span>So we have,</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(3)/(</span><span style="font-family: arial;">2k – 1</span><span style="font-family: arial;">)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">(1)/(</span><span style="font-family: arial;">k – 1</span><span style="font-family: arial;">)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">(-1)/(-(</span><span style="font-family: arial;">2k + 1</span><span style="font-family: arial;">))</span><span style="font-family: arial;"> -------- equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">6) Equating the first two ratios of equation 1, we have</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(3)/(</span><span style="font-family: arial;">2k – 1</span><span style="font-family: arial;">)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">(1)/(</span><span style="font-family: arial;">k – 1</span><span style="font-family: arial;">)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(3)(</span><span style="font-family: arial;">k – 1</span><span style="font-family: arial;">) = </span><span style="font-family: arial;">(1)(</span><span style="font-family: arial;">2k – 1</span><span style="font-family: arial;">)<br /></span><span style="font-family: arial;">3k - 3 = </span><span style="font-family: arial;">2k - 1<br /></span><span style="font-family: arial;">3k - 2k = - </span><span style="font-family: arial;">1 + 3<br /></span><span style="font-family: arial;">k = 2</span><span style="font-family: arial;"> -------- equation 2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">7) So, here k = 2.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-size: medium;"><span style="font-family: arial;">Q3. Solve the following pair of linear equations by the substitution and cross-multiplication </span><span style="font-family: arial;">methods :</span></span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span><b>8x + 5y = 9; <span> </span></b></span><b>3x + 2y = 4</b></span></div></div></blockquote><div style="text-align: left;"><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><span style="font-family: arial; font-size: medium;">Substitution method:</span></b></span></div></div></div><span style="font-family: arial; font-size: medium;"><span>1) Two equations in two variables are given.</span><br /><span>2) Simply get the value of x or y with the easy steps (say).</span><br /></span><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">3) Put the obtained value of x in the other equation and get the value of y.</span></div></span></div></div></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><b><span style="font-family: arial;">Cross-multiplication </span><span style="font-family: arial;">methods:</span></b></span></div><div><span style="font-family: arial;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">1) </span><span style="font-family: arial;">For the equations, </span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">a) a</span><sub>1</sub><span style="font-family: arial;">x + b</span><sub>1</sub><span style="font-family: arial;">y + c</span><sub>1</sub><span style="font-family: arial;"> = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></div></span></div></div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">b) a</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial;">y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="font-family: arial; line-height: 19.26px;"> </span><span style="font-family: arial;">= 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">In cross multiplication </span><span style="font-family: arial;">method, if </span><span style="font-family: arial;">a</span><sub>1</sub><span style="font-family: arial;">b</span><sub>2</sub><span style="font-family: arial;"> - a</span><sub>2</sub><span style="font-family: arial;">b</span><sub>1 </sub><b style="color: #333333;">≠</b><span style="font-family: arial;"> 0 </span><span style="font-family: arial;">we have,</span></div></span></div></blockquote></blockquote><div><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">c) x = <a name="_Hlk123941678">[(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a>)/(a<sub>1</sub>b<sub>2</sub> - a<sub>2</sub>b<sub>1</sub>)] ------------ equation 3</span></div><div><span style="font-family: arial;">d) y = </span><a name="_Hlk123941678">[(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span style="font-family: arial;">)/(a</span><sub>1</sub><span style="font-family: arial;">b</span><sub>2</sub><span style="font-family: arial;"> - a</span><sub>2</sub><span style="font-family: arial;">b</span><sub>1</sub><span style="font-family: arial;">)]</span><span style="font-family: arial;"> ------------ equation 4</span></div></blockquote></span></div></div></span></div></span></div><div><div><div><div><span style="font-family: arial; font-size: medium;">2) So, we can say that,<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>e) </span><span>If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub>,</sub></span><span>then </span><span>we get a unique solution.</span> </span></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>f) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there are infinitely many solutions.</span></span></div></div><div><div><span style="font-family: arial; font-size: medium;"><span>g) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>,</span><span><sub> </sub></span><span>then </span><span>there is no solution.</span></span></div></div></blockquote><div><span style="font-family: arial; font-size: medium;">3) From Equation 3 and Equation 4, we can say that,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>h) x/</span><a name="_Hlk123941678">(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a><span>) = y/</span><a name="_Hlk123941678">(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span>) = 1/</span><span>(a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><span>)</span><span> ------------ equation 5.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) Steps to solve these equations by cross multiplication method:</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">i) Write the given equations in the form of equation 1 and equation 2.</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">ii) Write your equations using the formula given in the</span><span style="font-family: arial;"> equation 5.</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>iii) Simplify and get the values of x and y. (Here, we have </span><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1 </sub><b style="color: #333333;">≠</b><span> 0).</span></span></blockquote></div></div></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><b>8x + 5y = 9; </b></span><b>3x + 2y = 4</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><span style="font-family: arial; font-size: medium;">a) Substitution method:</span></b></span></div><div>1) 8x + 5y = 9 ------------ equation 1</div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">2) 3x + 2y = 4 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 2, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x + 2y = 4</span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x = - 2y + 4</span></blockquote></div></div></div></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 2y + 4</span><span style="font-family: arial;">)/3 </span><span style="font-family: arial;">------------ equation 3</span></div></div></blockquote></div></div></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of </span></span><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 2y + 4</span><span style="font-family: arial;">)/3</span><span style="font-family: arial;"> from equation 3 in equation 1, we get</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8x + 5y = 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">8(</span><span style="font-family: arial;">- 2y + 4</span><span style="font-family: arial;">)/3 + 5y = 9<br /></span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;">[8(</span><span style="font-family: arial;">- 2y + 4</span><span style="font-family: arial;">) + 15y]/3 = 9</span></div></div></div></span></span></div></div></div></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">[-16y + 32</span><span style="font-family: arial;"> + 15y] = 9(3)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">- y + 32</span><span style="font-family: arial;"> = 27</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 32 - 27</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of </span><span style="font-family: arial;">y = 5</span><span style="font-family: arial;"> from equation 4 in equation 3, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 2y + 4</span><span style="font-family: arial;">)/3</span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 2(5) + 4</span><span style="font-family: arial;">)/3</span></div></div></div></span></span></div></div></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 10 + 4</span><span style="font-family: arial;">)/3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = (</span><span style="font-family: arial;">- 6</span><span style="font-family: arial;">)/3</span></div><span style="font-family: arial;">x = - 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 5</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;">6) </span><span style="font-family: arial;">So, x = - 2 and y = 5.</span></div><div><span style="font-family: arial;"><br /></span></div></div></div></span></span></div></div></div></span></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><b>8x + 5y = 9; </b></span><b>3x + 2y = 4</b></div><div><span style="font-family: arial; font-size: medium;"><b><span style="font-family: arial; font-size: medium;">b) Cross-multiplication </span><span style="font-family: arial;">methods:</span></b></span></div></span></div></div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;">7) Our equations are </span><span style="font-family: arial;">8x + 5y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 9 = 0; 3x + 2y </span><span style="font-family: arial;">–</span><span style="font-family: arial;"> 4 = 0.</span></span></div><div><div><span style="font-family: arial; font-size: medium;"><span>8) </span><span>From equation </span><span>8x + 5y </span><span>–</span><span> 9 = 0</span><span>, and equation </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>= 8, b</span><sub>1</sub><span>= 5, c</span><sub>1 </sub><span>= </span><span>– 9</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>9) From equation </span><span>3x + 2y </span><span>–</span><span> 4 = 0</span><span>, and equation </span><span>a</span><sub><span style="line-height: 16.05px;">2</span></sub><span>x + b</span><sub><span style="line-height: 16.05px;">2</span></sub><span>y + c</span><sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span><span>= 0</span><span>, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a</span><sub>2 </sub><span>= 3, b</span><sub>2</sub><span>= 2, c</span><sub>2 </sub><span>= </span><span>– 4</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">10) Here, we have </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a) (</span><span>a</span><sub>1</sub><span>/</span> <span>a</span><sub>2</sub><span>) = 8/3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) (</span><span>b</span><sub>1</sub><span>/</span> <span>b</span><sub>2</sub><span>) = 5/2</span></span></div><span style="font-family: arial; font-size: medium;"><span>c) (</span><span>c</span><sub>1</sub><span>/</span> <span>c</span><sub>2</sub><span>) = (-9)/(-4)</span><span> = 9/4</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">11) From the above equations, we can say that,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub>,</sub></span><span>then </span><span>we get a unique solution</span><span>.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>12) We can find </span><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>as follows</span><span>,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><span>b</span><sub>1 </sub><span>c</span><sub><span><span>1 (5) (-9)</span></span></sub></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>2 </sub>c<sub>2</sub></a><sub><span><span> (2) (-4)</span></span></sub></span></b></div></blockquote><div><span style="font-family: arial; font-size: medium;"> </span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= (5)(-4) - (2)(-9)</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= - 20 + 18</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1 </sub></a><span>= - 2</span><span> </span><span>------------ equation 6</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>13) We can find </span><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><sub> </sub><span>as follows</span><span>,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><span>c</span><sub>1 </sub><span>a</span><sub><span><span>1 (-9) (8)</span></span></sub></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>2 </sub>a<sub>2</sub></a><sub><span><span> (-4) (3)</span></span></sub></span></b></div></blockquote><div><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= (-9)(3) - (-4)(8)</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= -27 + 32</span></span></div><div><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><a name="_Hlk123941678"><sub> </sub></a><span>= 5</span><span> </span><span>------------ equation 7</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>14) We can find </span><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><sub> </sub><span>as follows</span><span>,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1 </sub><span>b</span><sub><span><span>1 (8) (5)</span></span></sub></span></b></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><b><span style="font-family: arial; font-size: medium;"><a name="_Hlk123941678">a<sub>2 </sub>b<sub>2</sub></a><a name="_Hlk123941678"><sub><span><span> (3) (2)</span></span></sub></a></span></b></div></blockquote><div><span style="font-family: arial; font-size: medium;"> </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> (8)(2) - (3)(5)</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> 16 - 15</span></span></div><div><span style="font-family: arial; font-size: medium;"><span>a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><a name="_Hlk123941678"><sub> </sub></a><span>=</span><span> 1</span><span> </span><span>------------ equation 8</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">15) By cross multiplication method, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/</span><a name="_Hlk123941678">(b<sub>1</sub>c<sub>2</sub> - b<sub>2</sub>c<sub>1</sub></a><span>) = y/</span><a name="_Hlk123941678">(c<sub>1</sub>a<sub>2</sub> - c<sub>2</sub>a<sub>1</sub></a><span>) = 1/</span><span>(a</span><sub>1</sub><span>b</span><sub>2</sub><span> - a</span><sub>2</sub><span>b</span><sub>1</sub><span>)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/(-</span><a name="_Hlk123941678">2)</a><span> = y/</span><a name="_Hlk123941678">5</a><span> = 1/</span><span>1</span><span>, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x/(-2)</span><span> = 1, and y/</span><a name="_Hlk123941678">(5</a><span>) = 1, so</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x </span><span style="font-family: arial;">= - 2 and y = 5</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">16) So, here x = - 2 and y = 5.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-size: medium;"><span style="font-family: arial;">Q4. Form the pair of linear equations in the following problems and find their solutions (if </span><span style="font-family: arial;">they exist) by any algebraic method :</span></span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><span><b>(i) A part of monthly hostel charges is fixed and the remaining depends on the </b></span><span><b>number of days one has taken food in the mess. When a student A takes food for </b></span><b>20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes </b><b><span>food for 26 days, pays </span><span>Rs</span><span> 1180 as hostel charges. Find the fixed charges and the </span></b><b>cost of food per day.</b></span></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-size: medium;"><b><span style="font-family: arial;">(ii) A fraction becomes 1/3 </span><span style="font-family: arial;">when 1 is subtracted from the numerator and it becomes </span></b><b><span style="font-family: arial;">1/4 </span><span style="font-family: arial;">when 8 is added to its denominator. Find the fraction.</span></b></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b><span>(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 </span><span>mark for each wrong answer. Had 4 marks been awarded for each correct answer </span></b><b>and 2 marks been deducted for each incorrect answer, then Yash would have </b><b>scored 50 marks. How many questions were there in the test?</b></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><b><span style="font-size: medium;"><span style="font-family: arial;">(iv) Places A and B are 100 km apart on a highway. One car starts from A and another </span><span style="font-family: arial;">from B at the same time. If the cars travel in the same direction at different speeds, </span><span style="font-family: arial;">they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What </span><span style="font-family: arial;">are the speeds of the two cars?</span></span></b></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><b><span style="font-size: medium;"><span style="font-family: arial;">(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by </span><span style="font-family: arial;">5 units and breadth is increased by 3 units. If we increase the length by 3 units and </span><span style="font-family: arial;">the breadth by 2 units, the area increases by 67 square units. Find the dimensions </span><span style="font-family: arial;">of the rectangle.</span></span></b></div></div></blockquote><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equations.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">equations to get the values of x and y. </span></blockquote><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><b>(i) A part of monthly hostel charges is fixed and the remaining depends on</b></span></span></div></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><b>the </b></span><span style="font-family: arial; font-size: medium;"><b>number of days one has taken food in the mess. When a student A takes food for </b></span><b>20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes </b><b><span style="font-family: arial; font-size: medium;">food for 26 days, pays </span><span style="font-family: arial;">Rs</span><span style="font-family: arial; font-size: medium;"> 1180 as hostel charges. Find the fixed charges and the </span></b><b>cost of food per day.</b></span></div></div></span></div></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;">1) Let the fixed charges be Rs x.</span></div><div><span style="font-family: arial; font-size: medium;">2) Let the charges for food per day be Rs y.</span></div><div><span style="font-family: arial; font-size: medium;">3) Student A paid Rs 1000 for 20 days, so we have,</span></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 20 y = 1000 </span><span style="font-family: arial;">------------ equation 1</span></blockquote><div><div><span style="font-family: arial; font-size: medium;">4) </span><span style="font-family: arial;">Student A paid Rs 1180 for 26 days, so we have,</span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + 26 y = 1180</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"></span></div></div><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">5) Subtract equation 1 from equation 2, and we get,</span></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x + 26 y = 1180</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 20 y = 1000</span></blockquote><div> <span style="font-family: arial;"> </span><span style="font-family: arial;">(-) (-) (-)</span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> </span></span><span style="font-family: arial;">----------------------------</span></div></div></span></span></div></span></div></span></div></div></span></div></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6 y = 180</span></blockquote></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"> y = 30</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></blockquote></span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 30 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 20 y = 1000</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 20 (30) = 1000</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x + 600 = 1000</span> </span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = 1000 - 600</span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 400</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, </span></span><span style="font-family: arial; font-size: medium;">the fixed charges are Rs 400 and the </span><span style="font-family: arial;">cost of food per day is </span><span style="font-family: arial;">Rs 30.</span></div><div><span style="font-family: arial;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(ii) A fraction becomes 1/3 </span><span style="font-family: arial;">when 1 is subtracted from the numerator and it</span></b></div></div></span></span></div></span></div></span></div></div></span></div></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><b><span style="font-family: arial;">becomes </span></b><b><span style="font-family: arial; font-size: medium;">1/4 </span><span style="font-family: arial;">when 8 is added to its denominator. Find the fraction.</span></b></div></div></span></span></div></span></div></span></div></div></span></div></span></div></span></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><b><span style="font-family: arial;"><br /></span></b></div><div><div><span style="font-family: arial; font-size: medium;">1) Let the numerator be x and the denominator be y.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 1)/y = 1/3 </span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3(x - 1) = y</span></blockquote></div></span></span></div></span></div></div></div></span></span></div></span></div></span></div></div></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">3x - 3 = y</span></div></div></span></span></div></span></div></span></span></div></span></div></span></div></div></span></div></span></div></span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - y = 3 ------------ equation 1</span></blockquote></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) </span><span style="font-family: arial;">According to the second condition, we have,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x)/(y + 8) = 1/4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x = y + 8</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 4x - 8</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div><div><span style="font-family: arial; font-size: medium;">3) </span><span style="font-family: arial;">Put the value of </span><span style="font-family: arial;">y = 4x - 8</span><span style="font-family: arial;"> from equation 2 in equation 1, we get,</span></div></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x - y = 3</span></blockquote></div></span></span></div></span></div></span></div></div></span></div></span></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3x - (4x - 8) = 3</span></div></div></span></span></div></span></div></span></div></div></span></div></span></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3x - 4x + 8 = 3<br /></span><span style="font-family: arial;">3x - 4x = 3 - 8<br /></span><span style="font-family: arial;">- x = - 5</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">x = 5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></span></span></div></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">4) Put the value of x = 5 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 4x - 8</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial; font-size: medium;">y = 4(5) - 8</span></span></blockquote></div></span></span></div></span></div></span></span></div></span></div></span></div></div></span></div></span></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">y = 20 - 8</span> </div></div></span></span></div></span></div></span></span></div></span></div></span></div></div></span></div></span></div></span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 12</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) So, here, </span><span style="font-family: arial;">the numerator is 5 and the denominator is 12</span><span style="font-family: arial;">. So the fraction is 5/12.</span></span></div></div></span></span></div></span></div></span></span></div></span></div></span></div></div></span></div></span></div></div><div style="text-align: left;"><br /></div></span></span></div><div style="text-align: left;"><div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) Yash scored 40 marks on a test, getting 3 marks for each right answer and</span></b></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><span>losing 1 </span><span>mark for each wrong answer. Had 4 marks been awarded for each correct answer </span></b><b>and 2 marks been deducted for each incorrect answer, then Yash would have </b><b>scored 50 marks. How many questions were there in the test?</b></span></div></div></div></blockquote><div style="text-align: left;"><div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div></div><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the right answers be x and the wrong answers be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) So the total questions will be (x + y).</span></div></div><div><span style="font-family: arial; font-size: medium;">3) According to the first condition, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3x - y = 40</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">4) According to the second condition, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">4x - 2y = 50</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2x - y = 25</span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 2x - 25 </span><span style="font-family: arial;">------------ equation 2</span></span></div></div></div></div></blockquote></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 2x - 25 from equation 2 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x - y = 40</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x - (2x - 25) = 40</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x - 2x + 25 = 40</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">3x - 2x = 40 - 25</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 15</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">6) Put the value of x = 15 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 2x - 25</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 2(15) - 25</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">y = 30 - 25</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 5</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial;">The right answers are 15 and the wrong answers are 5, so the total number of questions is 20</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;">(iv) Places A and B are 100 km apart on a highway. One car starts from A and</span></b></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;">another </span><span style="font-family: arial;">from B at the same time. If the cars travel in the same direction at different speeds, </span><span style="font-family: arial;">they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What </span><span style="font-family: arial;">are the speeds of the two cars?</span></b></span></span></div></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the speed of the car starting from place A be x km/h.</span></div><div><span style="font-family: arial; font-size: medium;">2) Let the speed of the car starting from place B be y km/h.</span></div><div><span style="font-family: arial; font-size: medium;">3) </span><span style="font-family: arial;">When a car travels in the same direction, and meets in 5 </span><span style="font-family: arial;">hrs, (speed = distance/time),</span></div></div></div></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div style="text-align: left;">a) Car started from place A covered 5x km.</div></div></div></span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">b) Car started from place B covered 5y km.</span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) According to the first condition, as the places are 100 km apart, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5x - 5y = 100</span></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x - y = 20</span></blockquote></div></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div style="text-align: left;">y = x - 20<span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></div></div></div></span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) When car travels towards each other, and meet in 1 hrs, (speed = distance/time),</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div>a) Car started from place A covered x km.</div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">b) Car started from place B covered y km.</span></blockquote><div style="text-align: left;"><span style="font-family: arial;">6) According to the second condition, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x + y = 100</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div></div><div><div><div><span style="font-family: arial; font-size: medium;">7) Put the value of y = x - 20 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + y = 100</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + (x - 20) = 100</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2x = 100 + 20</span></blockquote></div></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div style="text-align: left;"><span style="font-family: arial;">2x = 120</span> </div></div></div></span></span></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = 60</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 3.</span></div></div></blockquote></div><div><div><div><span style="font-family: arial; font-size: medium;">8) Put the value of x = 60 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = x - 20</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 60 - 20</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">y = 40</span><span style="font-family: arial;"> -------------------------- equation 4.</span></div></div></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">9) </span></span><span style="font-family: arial;">The speed of the car starting from place A is 60 km/h and </span><span style="font-family: arial;">the speed of the car</span></div></div></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div style="text-align: left;"><span style="font-family: arial;">starting from place B be 40 km/h.</span></div></div></div></span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;">(v) The area of a rectangle gets reduced by 9 square units if its length is</span></b></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b><span style="font-family: arial; font-size: medium;">reduced by </span><span style="font-family: arial;">5 units and breadth is increased by 3 units. If we increase the length by 3 units and </span><span style="font-family: arial;">the breadth by 2 units, the area increases by 67 square units. Find the dimensions </span><span style="font-family: arial;">of the rectangle.</span></b></span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><br /></span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div>1) Let the length be x units, and the breadth be y units.</div><div>2) So the area of the rectangle will be xy square units.</div><div>3) According to the first condition,</div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div>a) New area = xy - 9</div></span></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">b) New length = x - 5<br /></span><span style="font-family: arial;">c) New breadth = y + 3</span><div> </div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div>(x - 5)(y + 3) = xy - 9</div></span></div></span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">xy + 3x - 5y - 15 = xy - 9<br /></span><span style="font-family: arial;">3x - 5y - 15 = - 9</span><div><span style="font-family: arial;">3x - 5y = 15 - 9</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - 5y = 6</span></blockquote></div></span></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">5y = 3x - 6</div></div></span></div></span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = (3x - 6)/5 </span><span style="font-family: arial;">-------------------------- equation 1.</span></span></blockquote><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;">4) According to the second condition, </div></div></span></div></span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">a) New area = xy + 67</div></span></div></div></div></div></div></span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) New length = x + 3<br /></span><span style="font-family: arial;">c) New breadth = y + 2</span></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">(x + 3)(y + 2) = xy + 67</div></span></div></div></div></div></div></span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">xy + 2x + 3y + 6 = xy + 67<br /></span><span style="font-family: arial;">2x + 3y + 6 = 67<br /></span><span style="font-family: arial;">2x + 3y = 67 - 6</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2x + 3y = 61</span><span style="font-family: arial;"> -------------------------- equation 2.</span></span></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = (3x - 6)/5 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">2x + 3y = 61</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">2x + [3</span><span style="font-family: arial;">(3x - 6)/5]</span><span style="font-family: arial;"> = 61</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">[10x + 3</span><span style="font-family: arial;">(3x - 6)]/5</span><span style="font-family: arial;"> = 61</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">[10x + 3</span><span style="font-family: arial;">(3x - 6)]</span><span style="font-family: arial;"> = 61(5)</span></div></blockquote></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">[10x + 9x - 18] = 305</div></span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>19x - 18 = 305</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div style="text-align: left;">19x = 305 + 18</div></span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">19x = 323<br /></span><span style="font-family: arial;">x = 323/19</span></span><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 17</span><span style="font-family: arial;"> -------------------------- equation 3.</span></span></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;">6) Put the value of x = 17 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = (3x - 6)/5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = (3(17) - 6)/5</span></blockquote></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial;">y = (51 - 6)/5</span></div></div></span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial;">y = (45)/5</span></div></div></span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 9</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">7) The length of the rectangle will be 17 units and the breadth of the rectangle will</span></div></div></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">be 9 units.</span></div></div></span></span></div></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/153-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.6</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial;">ANIL SATPUTE</span></a></div></div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-50484330599494555032023-06-06T18:18:00.002+05:302023-06-14T13:10:34.622+05:30151-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.4<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 3.4</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/150-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables-Ex-3.3</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.4</span></h3></div><div><span style="font-family: arial; font-size: medium;"><span><b>Q1. Solve the following pair of linear equations by the elimination method and the substitution </b></span><b>method :</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 5 and 2x – 3y = 4 <span> </span>(ii) 3x + 4y = 10 and 2x – 2y = 2</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7 <span> </span>(iv) (x/2) + (2y/3) = - 1, x - (y/3) = 3</b></span></div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;">1) Two equations in two variables are given.</span></div><div><span style="font-family: arial; font-size: medium;">2) Make the coefficients of either x or y the same by multiplying by some non-zero</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">constant.</span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">3) Eliminate the variable by subtracting/adding one equation from/to the other.</span></div><div><span style="font-family: arial; font-size: medium;">4) This method is known as the elimination method.</span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 5 and 2x – 3y = 4</b></span></div><h3 style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>a) Elimination method</b></span></h3><div><span style="font-family: arial; font-size: medium;">1) Given equations are</span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">x + y = 5 ---------------equation 1</span></div></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x - 3y = 4 ---------------equation 2</span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">2) Here multiply equation 1 by 2 to get the coefficient of x same.</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2(x + y) = 2(5)</span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x + 2y = 10 ---------------equation 3</span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">3) Subtracting equation 2 from equation 3, we get,</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">2x + 2y = 10</span></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">2x - 3y = 4</span></div></span></div></blockquote><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span> </span>(-) (+) (-)</div><div style="text-align: left;">---------------------------</div></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">5y = 6</div></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 6/5 </span><span style="font-family: arial;">---------------equation 4</span></span></div></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Put the value of y =6/5 from equation 4 in equation 1, we get,</span></span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + (6/5) = 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 5 - (6/5)</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = (25 - 6)/5</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">x = (19)/5</span><span style="font-family: arial;"> </span></span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = 19/5 </span><span>-------------------------- equation 5.</span> </span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) So, x = 19/5 and y = 6/5.</span></div><h3 style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>b) Substitution </b></span><b>method</b></span></h3><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">6) x + y = 5 ------------ equation 6</span></div><div><span style="font-family: arial; font-size: medium;">7) 2x - 3y = 4 ------------ equation 7</span></div><div><span style="font-family: arial; font-size: medium;">8) Simplify equation 6, and we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 5</span></blockquote></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 5 - x </span><span style="font-family: arial;">------------ equation 8</span></span></div></div></blockquote></div></div></div></blockquote><div><div><div style="text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial;">Substitute the value of y = (5 - x) from equation 8 in equation 7, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x - 3y = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2x</span><span style="font-family: arial;"> - 3(5 - x) = 4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2x - 15</span><span style="font-family: arial;"> + 3x = 4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">5x - 15 = 4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">5x = 15 + 4</span></span></div><div><span style="font-family: arial; font-size: medium;">5x = 19</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">x = 19/5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 9</span></span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">10) Put the value of x = 19/5 from equation 9 in equation 8, and we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">y = 5 - x</span></div><span style="font-family: arial; font-size: medium;">y = 5 - (19/5)</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>y = (25</span><span> - 19)/5</span><span> </span> </span></div></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 6/5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 10</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">11) The value of x = 19/5 and the value of y = 6/5.</span></div></div><div><span style="font-family: arial; font-size: medium;"> </span></div></div></div></div></div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"></div></span><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><b>(ii) 3x + 4y = 10 and 2x – 2y = 2</b></span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">a) Elimination method</span></h3></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3x + 4y = 10 ---------------equation 1</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2x - 2y = 2 ---------------equation 2</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2) Here multiply equation 2 by 2 to get the coefficient of y same.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2(2x - 2y) = 2(2)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">4x - 4y = 4 ---------------equation 3</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3) Adding equation 1 and equation 3, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3x + 4y = 10</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">4x - 4y = 4</span></span></div></blockquote><span style="font-family: arial; font-size: medium;"><div><span> --</span>----------------------</div></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">7x = 14</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 14/7</span></blockquote></blockquote></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">x = 2<span style="font-family: arial; font-size: medium;"> </span>---------------equation 4</div></span></div></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><span style="font-family: arial;">4) </span><span style="font-family: arial;">Put the value of x = 2 from equation 4 in equation 2, we get,</span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2x - 2y = 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2(2) - 2y = 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2y = 4 - 2</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2y = 2</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">y = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 5.</span> </div></div></blockquote><div><div><span style="font-family: arial;">5) So, x = 2 and y = 1.</span></div></div></div></span></div><div><span style="font-family: arial;"><div><div><h3><span style="font-size: medium;"><span style="font-family: arial;"><b>b) Substitution </b></span><b>method</b></span></h3><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">6) 3x + 4y = 10 ------------ equation 6</span></div><div><span style="font-family: arial;">7) 2x - 2y = 2 ------------ equation 7</span></div><div><span style="font-family: arial;">8) Simplify equation 7, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">2x - 2y = 2</span></blockquote></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;"><span> </span>x - y = 1</span> </div></div></div></div></span></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = y + 1 </span><span style="font-family: arial;">------------ equation 8</span></div></div></blockquote></div></div></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial;">Substitute the value of x =(y + 1) from equation 8 in equation 6, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x + 4y = 10</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3(y + 1) + 4y = 10</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3y + 3 + 4y </span><span style="font-family: arial;">= 10</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7y + 3 = 10</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">7y = 10 - 3</span></span></div><div><span style="font-family: arial; font-size: medium;">7y = 7</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">y = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 9</span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">10) Put the value of y = 1 from equation 9 in equation 8, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = y + 1</span></div><span style="font-family: arial;">x = 1 + 1</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x = 2</span></span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 10</span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">11) </span><span style="font-family: arial;">So, x = 2 and y = 1.</span></div><div><span style="font-family: arial;"><br /></span></div><div><b>(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7</b></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">a) Elimination method</span></h3></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3x - 5y - 4 = 0</span></span></blockquote></span></div></span></div></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">3x - 5y = 4 ---------------equation 1</div></span></div></span></div></div></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">9x = 2y + 7</span></span></blockquote></span></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">9x - 2y = 7 ---------------equation 2</div></span></span></div></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2) Here multiply equation 1 by 3 to get the coefficient of x as same.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3(3x - 5y) = 3(4)</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9x - 15y = 12 ---------------equation 3</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3) Subtracting equation 2 from equation 3, we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">9x - 15y = 12</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">9x - 2y = 7</span></span></div></blockquote><div style="text-align: left;"> <span> </span>(-) (+) (-)</div><span style="font-family: arial; font-size: medium;"> ------------------------</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span> </span>-13 y = 5</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span> </span>y = - (5/13)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 4</span></blockquote></blockquote></span></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">4) </span><span style="font-family: arial;">Put the value of y = - (5/13) from equation 4 in equation 1, we get,</span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - 5y = 4</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - 5(-5/13) = 4</span></blockquote></div></div></div></span></div></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">3x + (25/13) = 4</span> </div></div></div></div></span></div></span></div></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x = 4 - (25/13)</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x = (52 - 25)/13</span></blockquote></div></div></span></div></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">3x = (27)/13</span> </div></div></div></span></div></span></div></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = (27)/(13 x 3)</span></div></div></blockquote></div></span></div></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">x = 9/13</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 5.</span> </div></div></span></div></span></div></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">5) So, x = 9/13 and y = - (5/13).</span></div></div></span></div><div><span style="font-family: arial;"><div><div><h3><span style="font-size: medium;"><span style="font-family: arial;"><b>b) Substitution </b></span><b>method</b></span></h3><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">6) 3x - 5y = 4 ------------ equation 6</span></div><div><span style="font-family: arial;">7) 9x - 2y = 7 ------------ equation 7</span></div><div><span style="font-family: arial;">8) Simplify equation 7, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - 5y = 4</span></blockquote></div></div></div></div></span></div></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">3x = 5y +4</span></blockquote></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = (5y + 4)/3 </span><span style="font-family: arial;">------------ equation 8</span></div></div></blockquote></div></div></blockquote><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">9) </span><span style="font-family: arial;">Substitute the value of x = </span></span><span style="font-family: arial;">(5y + 4)/3</span><span style="font-family: arial;"> from equation 8 in equation 7, we get</span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9x - 2y = 7</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9(5y + 4)/3 - 2y = 7<br /></span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">3(5y + 4) - 2y = 7</span> </div></div></div></div></span></span></div></div></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">15y + 12 - 2y </span><span style="font-family: arial;">= 7</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">13y + 12 = 7</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">13y = 7 - 12</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">13y = - 5</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">y = - (5/13)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 9</span></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">10) Put the value of y = </span><span style="font-family: arial;">- (5/13)</span><span style="font-family: arial;"> from equation 9 in equation 8, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = (5(-5/13) + 4)/3<br /></span></div></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">x = (- 25/13) + 4)/3</span></div></div></div></div></span></span></div></div></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = (- 25 + 52)/(13(3))</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = (27)/(13(3))</span></div><span style="font-family: arial; font-size: medium;"><span>x = (9)/(13)</span> </span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 9/13</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 10</span></blockquote></div><div><span style="font-family: arial; font-size: medium;">11) </span><span style="font-family: arial;">So, x = 9/13 and y = - (5/13).</span></div></div></div></span></span></div></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iv) (x/2) + (2y/3) = - 1, x - (y/3) = 3</b></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">a) Elimination method</span></h3></span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">1) Given equations are</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">(x/2) + (2y/3) = - 1 ---------------equation 1</span></blockquote></span></div></span></div></div></div></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face=""Arial",sans-serif" style="line-height: 107%; mso-ansi-language: EN-IN; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">x – (y/3) = 3 ---------------equation 2</span></blockquote></span></span></div></div></div></span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2) Here multiply equation 1 by 2 to get the coefficient of x as same.</span></span></div><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; margin: 0cm 0cm 0cm 36pt;"><span face=""Arial",sans-serif">2(x/2) + 2(2y/3) = 2(- 1)<o:p></o:p></span></p>
</span></span></div></div></div></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">x + (4y/3) = - 2 ---------------equation 3</span></span></span></blockquote></span></span></div></div></div></div></span></span></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">3) Subtracting equation 2 from equation 3, we get,</div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">x + 4y/3 = - 2</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">x - y/3 = 3</span></span></div></blockquote><div> (-) (+) (-)</div><span style="font-family: arial; font-size: medium;"> --------------------------</span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"> 5 y/3 = - 5</span></blockquote></span></span></div></div></div></div></span></span></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">y/3 = - 1</span></blockquote></span></span></div></div></div></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial;">y = - 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 4</span></blockquote></span></span></div></div></div></div></span></span></div></span></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">4) </span><span style="font-family: arial;">Put the value of y = - 3 from equation 4 in equation 1, we get,</span></div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">(x/2) + (2y/3) = - 1</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span face="Arial, sans-serif">(x/2) + (2(- 3)/3) = - 1</span></blockquote></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span face="Arial, sans-serif">(x/2) + (- 2) = - 1</span></div></div></span></span></div></div></div></span></span></blockquote></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span face="Arial, sans-serif">(x/2) = - 1 + 2</span></blockquote></div></div></div></span></span></div></div></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span face="Arial, sans-serif">(x/2) = 1</span></blockquote></div></div></div></span></span></div></div></div></span></span></div></span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial;">x = 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------------------- equation 5.</span> </blockquote></div></div></span></span></div></div></div></span></span></div></span></div></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;"></span><span style="font-family: arial;"></span><span style="font-family: arial;"><div><span style="font-family: arial;"><div><span style="font-family: arial;"></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;"><div><div><span style="font-family: arial;">5) So, x = 2 and y = - 3.</span></div></div></span></div><div><span style="font-family: arial;"><div><div><h3><span style="font-size: medium;"><span style="font-family: arial;"><b>b) Substitution </b></span><b>method</b></span></h3><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial;">6) 3x - 5y = 4 ------------ equation 6</span></div><div><span style="font-family: arial;">7) 9x - 2y = 7 ------------ equation 7</span></div><div><span style="font-family: arial;">8) Simplify equation 7, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x - 5y = 4</span></blockquote></div></div></div></div></span></div></span></div></div></div></div></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3x = 5y +4</span></blockquote></div></div></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = (5y + 4)/3 </span><span style="font-family: arial;">------------ equation 8</span></div></div></blockquote></div></div></blockquote><div><div><div><span><span style="font-family: arial;">9) </span><span style="font-family: arial;">Substitute the value of </span></span><span style="font-family: arial;">x = (5y + 4)/3</span><span><span style="font-family: arial;"> from equation 8 in equation 7, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9x - 2y = 7</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">9(5y + 4)/3 - 2y = 7</span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;">3(5y + 4) - 2y = 7</span> </div></div></div></span></span></div></div></div></span></span></blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">15y + 12 - 2y </span><span style="font-family: arial;">= 7</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">13y + 12 = 7</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span><span style="font-family: arial;">13y = 7 - 12</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">13y = - 5</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">y = - (5/13)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 9</span></div></div></blockquote><div><div><span style="font-family: arial;">10) Put the value of y = - (5/13) from equation 9 in equation 8, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = (5(-5/13) + 4)/3</span></blockquote></div></div></div></span></span></div></div></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;">x = (- 25/13) + 4)/3</span></div></div></div></span></span></div></div></div></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = (- 25 + 52)/(13(3))</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">x = (27)/(13(3))</span></div><span style="font-family: arial;">x = (9)/(13)</span> </blockquote><div><span style="font-family: arial;"></span><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><span style="font-family: arial;"><span style="font-family: arial;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 9/13</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 10</span></blockquote></div><div><span style="font-family: arial;">11) </span><span style="font-family: arial;">So, x = 9/13 and y = - (5/13).</span></div></div></div></span></span></div></div></div></span></span></div></span></div><div><span style="font-family: arial;"><br /></span></div><div><b><span style="font-family: arial;">Q</span>2. Form the pair of linear equations in the following problems, and find their solutions </b><b>(if they exist) by the elimination method :</b></div></span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><b>(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces </b><b>to 1. It becomes 1/2 </b><b>if we only add 1 to the denominator. What is the fraction?</b></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><b>(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as </b><b>old as Sonu. How old are Nuri and Sonu?</b></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><b>(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is </b><b>twice the number obtained by reversing the order of the digits. Find the number.</b></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><b>(iv) Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her </b><b>Rs</b><b> 50 and </b><b>Rs</b><b> 100 notes only. Meena got 25 notes in all. Find how many notes of </b><b>Rs</b><b> 50 and </b><b>Rs</b><b> 100 she received.</b></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><b>(v) A lending library has a fixed charge for the first three days and an additional charge </b><b>for each day thereafter. Saritha paid </b><b>Rs</b><b> 27 for a book kept for seven days, while Susy </b><b>paid </b><b>Rs</b><b> 21 for the book she kept for five days. Find the fixed charge and the charge </b><b>for each extra day.</b></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;">1) Two equations in two variables are given.</span></div><div><span style="font-family: arial; font-size: medium;">2) Make the coefficients of either x or y the same by multiplying by some non-zero</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">constant.</span></blockquote><div><div><span style="font-family: arial; font-size: medium;">3) Eliminate the variable by subtracting/adding one equation from/to the other.</span></div><div><span style="font-family: arial; font-size: medium;">4) This method is known as the elimination method.</span></div><div><span style="font-family: arial; font-size: medium;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><b>(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces </b><b>to 1. It becomes 1/2 </b><b>if we only add 1 to the denominator. What is the fraction?</b></span></div><div><span style="font-family: arial; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the numerator be x and the denominator be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the first condition, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 1)/(y - 1) = 1</span></blockquote></div></span></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">(x + 1) = (y - 1)</span> </div></div></span></div></span></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x - y = - 2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></blockquote></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">According to the second condition, we have,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x)/(y + 1) = 1/2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x = y + 1</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x - y = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">4) Subtract equation 1 from equation 2, and we get,</span></div></div></div></span></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;">2x - y = 1</span></div></div></span></span></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> x - y = - 2<br /></span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;"> (-) (+) (-)</span></div></span></span></div><span style="font-family: arial;">----------------------------</span></div></span></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;"> x <span> </span><span> </span>= 3</span></div></span></span></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">x = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></blockquote></span></span></div></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of x = 3 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x - y = - 2</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3 - y = - 2</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>y = 3 + 2</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 5</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, here, </span><span style="font-family: arial;">the numerator is 3 and the denominator is 5</span><span style="font-family: arial;">. So the fraction is 3/5.</span></span></div></div></span></span></div></div><div><b><br /></b></div><div><b>(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as </b><b>old as Sonu. How old are Nuri and Sonu?</b></div><div><b><br /></b></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let Nuri's present age be x and Sonu's present age be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) 5 years ago their ages were (x - 5) and (y - 5).</span></div><div><span style="font-family: arial; font-size: medium;">3) According to the first relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 5) = 3(y - 5)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 5) = 3y - 15</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;">x - 3y = - 15 + 5<br /><span style="font-family: arial; font-size: medium;">x - 3y = - 10 ---------------------- equation 1</span></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) 10 years later, their ages will be (x + 10) and (y + 10).</span> </span></div><div><span style="font-family: arial; font-size: medium;">5) According to the relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 10) = 2(y + 10)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 10) = 2y + 20</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;">x - 2y = 20 - 10<br /><span style="font-family: arial; font-size: medium;">x - 2y = 10 ---------------------- equation 2</span></blockquote><div><div><span style="font-family: arial; font-size: medium;">6) Subtract equation 1 from equation 2, and we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x - 2y = 10</span></blockquote></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;">x - 3y = -10</div></div></div></div></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span> </span>(-) (+) (+)</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>----------------------------</span> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = 20</span></div></div></div></div></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"></span></div></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">y = 20</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></div></span></span></div></div></span></div></blockquote></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;">7) Put the value of y = 20 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x - 2y = 10</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x - 2(20) = 10</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x = 10 + 40</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 50</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote></div></span></span></div></span></div></div><div><div><div><span style="font-size: medium;">8) </span><span style="font-family: arial;">So Nuri's present age is 50 years and Sonu's present age is 20 years.</span></div></div><div><span style="font-family: arial;"><br /></span></div><div><span style="font-family: arial;"><b>(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is </b><b>twice the number obtained by reversing the order of the digits. Find the number.</b></span></div><div><span style="font-family: arial;"><b><br /></b></span></div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the ones place digit be x and the tens place digit be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the first condition, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 9</span></blockquote></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x + y = 9 </span><span style="font-family: arial;">------------ equation 1</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) The original number: </span></span><span style="font-family: arial;">10 x + y</span></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial;">4) The number obtained by reversing the digits: 10 y + x</span></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) According to the second condition, we have,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9(10 x + y) = 2(10 y + x)</span></blockquote></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">90 x + 9 y = 20 y + 2 x</span> </div></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">90 x - 2 x + 9 y - 20 y = 0</span></blockquote></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">88 x - 11 y = 0</span></div></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><div style="text-align: left;"><span style="font-family: arial;">11(8 x - y) = 0</span></div></div></div></div></span></span></div></span></div></span></div></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">8 x - y = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div><div><span style="font-family: arial; font-size: medium;">4) Add equation 1 to equation 2, and we get,</span></div></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">8 x - y = 0</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> x + y = 9</span></span></blockquote><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> </span></span><span style="font-family: arial;">----------------------------</span></div></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">9x = 9</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></blockquote></span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of x = 1 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + y = 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">1 + y = 9</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>y = 9 - 1</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 8</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, the unit-placed digit</span><span style="font-family: arial;"> is 1 and the tens-placed digit is 8</span><span style="font-family: arial;">. So the number is 18.</span></span></div></div></span></span></div></span></div></span></div><div><span style="font-family: arial;"><br /></span></div></div></div></div><div><b>(iv) Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her </b><b>Rs</b><b> 50 and </b><b>Rs</b><b> 100 notes only. Meena got 25 notes in all. Find how many notes of </b><b>Rs</b><b> 50 and </b><b>Rs</b><b> 100 she received.</b></div><div><b><br /></b></div><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the number of Rs 50 be x and the number of Rs 100 be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the first condition, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 25</span></blockquote></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x + y = 25 </span><span style="font-family: arial;">------------ equation 1</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">3) According to the second condition, we have,</span></div></span></span></div></span></div></span></div></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">50 x + 100 y = 2000</span></div></span></span></div></span></div></span></div></div></span></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>50 (x + 2 y) = 50 (40)</span> </span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + 2 y = 40</span></blockquote></span></div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 2 y = 40 </span><span style="font-family: arial;">------------ equation 2</span></blockquote></div><div><span style="font-family: arial; font-size: medium;">4) Subtract equation 1 from equation 2, and we get,</span></div></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x + 2 y = 40</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + y = 25</span></blockquote><div style="text-align: left;"> <span style="font-family: arial;"> </span><span style="font-family: arial;">(-) (-) (-)</span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> </span></span><span style="font-family: arial;">----------------------------</span></div></div></span></span></div></span></div></span></div></div></span></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">y = 15</span></div></span></span></div></span></blockquote></span></div></div></span></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 15</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></blockquote></span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 15 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + y = 25</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 15 = 25</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x = 25 - 15</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 10</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, the </span><span style="font-family: arial;">number of Rs 50 notes is 10 and the number of Rs 100 notes is 15.</span></span></div></div></span></span></div></span></div></span></div></div></span></div></div><div><b><br /></b></div><div><b>(v) A lending library has a fixed charge for the first three days and an additional charge </b><b>for each day thereafter. Saritha paid </b><b>Rs</b><b> 27 for a book kept for seven days, while Susy </b><b>paid </b><b>Rs</b><b> 21 for a book she kept for five days. Find the fixed charge and the charge </b><b>for each extra day.</b></div><div><b><br /></b></div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the fixed charges for the first 3 days be Rs x.</span></div><div><span style="font-family: arial; font-size: medium;">2) Let the additional charges per extra day be Rs y.</span></div><div><span style="font-family: arial; font-size: medium;">3) Saritha paid Rs 27 for 7 days, so she paid Rs x for the first 3 days and the remaining for</span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4 days, so </span><span style="font-family: arial;">we have,</span></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + 4y = 27</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">x + 4y = 27</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">4) Susy paid Rs 21 for 5 days, so she paid Rs x for the first 3 days and the remaining for</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2 days, so we have,</span></blockquote></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + 2y = 21</span></div></div></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + 2y = 21</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></div></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">5) Subtract equation 2 from equation 1, and we get,</span></div></span></span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial;">x + 4 y = 27</span></div></span></span></div></span></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 2 y = 21</span></blockquote><div> <span style="font-family: arial;"> </span><span style="font-family: arial;">(-) (-) (-)</span></div><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"> </span></span><span style="font-family: arial;">----------------------------</span></div></div></span></span></div></span></div></span></div></div></span></div></div></span></div></div></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">2 y = 6</span></blockquote></div></div></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"> y = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">---------------equation 3</span></blockquote></span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 3 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 2 y = 21</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x + 2(3) = 21</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x + 6 = 21</span> </span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;">x = 21 - 6</span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 15</span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, </span><span style="font-family: arial;">the fixed charges for the first 3 days are Rs 15 and the additional charges per extra day are Rs 3.</span></span></div></div></span></span></div></span></div></span></div></div></span></div></span></div></div></div><div><span style="font-family: arial; font-size: medium;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/152-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.5</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE<br /></span></a></div></span></div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-11366741439990112582023-06-04T11:24:00.002+05:302023-06-14T13:11:02.643+05:30150-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.3<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 3.3</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/149-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.2</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.3</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>1. Solve the following pair of linear equations by the substitution method.</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 14, x – y = 4, <span> </span>(ii) s – t = 3, (s/3) + (t/2) = 6, <span> </span>(iii) 3x – y = 3, 9x – 3y = 9</b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(iv) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3, <span> </span>(v) <span style="white-space: pre-wrap;">√</span><span style="white-space: pre-wrap;">2 x + </span><span style="white-space: pre-wrap;">√3 y = 0, </span> <span style="white-space: pre-wrap;">√</span><span style="white-space: pre-wrap;">3 x - </span><span style="white-space: pre-wrap;">√8 y = 0</span></b></span></div><div><span style="font-family: arial; font-size: medium;"><b>(vi) (3x/2) - (5y/3) = - 2, (x/3) + (y/2) = 13/6</b></span></div><div><h3 style="font-size: medium;"><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><span style="font-family: arial; font-size: medium;">1) Take a simple equation.</span></div><div><span style="font-family: arial; font-size: medium;">2) Simply get the value of x or y with the easy steps (say).</span></div><div><span style="font-family: arial; font-size: medium;">3) Put the obtained value of x in the other equation and get the value of y.</span></div><div><span style="font-family: arial; font-size: medium;"><h3 style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 14, x – y = 4</b></span></div><div><span style="font-family: arial; font-size: medium;">1) x + y = 14 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) x - y = 4 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 2, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x - y = 4</span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = x - 4</span><span style="font-family: arial;"> ------------ equation 3</span></span></div></blockquote><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of y from equation 3 in equation 1, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 14</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + (x - 4) = 14</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x - 4 = 14</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x = 14 + 4</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x = 18</span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = 9 </span><span style="font-family: arial;">------------ equation 4</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5) Put the value of x from equation 4 in equation 3, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = x - 4</span></div><span style="font-size: medium;"><span style="font-family: arial;">y = 9 - 4</span><span style="font-family: arial;"> </span></span><div><span style="font-family: arial; font-size: medium;">y = 5</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) The value of x = 9 and the value of y = 5. </span></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b style="font-family: arial; font-size: large;">(ii) s – t = 3, (s/3) + (t/2) = 6</b></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) s - t = 3 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) (s/3) + (t/2) = 6 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 1, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">s - t = 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">s = t + 3</span><span style="font-family: arial;"> ------------ equation 3</span></span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of s from equation 3 in equation 2, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(s/3) + (t/2) = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">((t + 3)/3) + (t/2) = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">[(2t + 6) + (3t)]/6 = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(5t + 6)/6 = 6</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(5t + 6) = 36</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5t = 36</span><span style="font-family: arial;"> - 6</span></span></div></div><span style="font-family: arial; font-size: medium;"><span>5t = 30</span> <br /></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">t = 6 </span><span style="font-family: arial;">------------ equation 4</span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">5) Put the value of t from equation 4 in equation 3, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">s = t + 3</span></div><span style="font-family: arial; font-size: medium;"><span>s = 6 + 3</span><br /></span><div><span style="font-family: arial; font-size: medium;">s = 9</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">6) The value of s = 9 and the value of t = 6. </span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 3x – y = 3, 9x – 3y = 9</span></b></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) 3x - y = 3 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) 9x - 3y = 9 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 1, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3x - y</span><span style="font-family: arial;"> = 3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 3x + 3</span><span style="font-family: arial;"> ------------ equation 3</span></span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of y from equation 3 in equation 2, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9x - 3y = 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9x - 3(3x + 3) = 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9x - 9x + 9 = 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9 = 9</span></blockquote></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) This shows that the lines are </span><span style="font-family: arial;">coincident </span><span style="font-family: arial;">with infinitely many solutions.</span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(iv) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3</span></b></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) 0.2x + 0.3y = 1.3 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) 0.4x + 0.5y = 2.3 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Multiplying by 10 to equation 1 and equation 2, we get,</span></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">2x + 3y = 13 ------------ equation 3</span></div></span></div></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4x + 5y = 23 ------------ equation 4</span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"></span><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">4) Simplify equation 3, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x + 3y = 13</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x = 13 - 3y</span></blockquote></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x = (13 - 3y)/2</span><span style="font-family: arial;"> ------------ equation 5</span></span></div></div></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Substitute the value of x from equation 5 in equation 4, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x + 5y = 23</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">4[(</span><span style="font-family: arial;">13 - 3y)/2]</span><span style="font-family: arial;"> + 5y = 23</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2</span><span style="font-family: arial;">(</span><span style="font-family: arial;">13 - 3y)</span><span style="font-family: arial;"> + 5y = 23</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">26 - 6y</span><span style="font-family: arial;"> + 5y = 23</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>26 - y</span><span> = 23</span> </span></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 26 - 23</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 6</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">5) Put the value of y from equation 6 in equation 5, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x = (13 - 3y)/2</span></div><span style="font-family: arial; font-size: medium;"><span>x = (13 - 3(3))/2</span><br /></span><div><span style="font-family: arial; font-size: medium;">x = (13 - 9)/2</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = 4/2</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = 2 </span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">6) The value of x = 2 and the value of y = 3.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(v) <span style="white-space: pre-wrap;">√</span><span style="white-space: pre-wrap;">2 x + </span><span style="white-space: pre-wrap;">√3 y = 0, </span> <span style="white-space: pre-wrap;">√</span><span style="white-space: pre-wrap;">3 x - </span><span style="white-space: pre-wrap;">√8 y = 0</span></span></b></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) </span><span style="white-space: pre-wrap;"><b>√</b>2</span>x + <b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span>y = 0 ------------ equation 1</div><div><span style="font-family: arial; font-size: medium;">2) <span style="white-space: pre-wrap;"><b>√</b>3</span>x - <b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">8</span>y = 0 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 2, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>√</b>3</span><span>x - </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">8</span><span>y</span><span> = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>√</b>3</span><span>x = </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">8</span><span>y</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">(</span><span style="white-space: pre-wrap;">8/3)</span><span>y</span><span> ------------ equation 3</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of x from equation 3 in equation 1, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>√</b>2</span><span>x + </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>y = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>√</b>2</span><span>(8/3)y + </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3</span><span>y = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">[<b>√</b>2</span><span>(8/3) + </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3]</span><span>y = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">[<b>√</b>16</span><span>/3) + </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">3]</span><span>y = 0</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = 0</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">5) Put the value of y from equation 4 in equation 3, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>x = </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">(</span><span style="white-space: pre-wrap;">8/3)</span><span>y</span></span></div><span style="font-family: arial; font-size: medium;"><span>x = </span><b style="white-space: pre-wrap;">√</b><span style="white-space: pre-wrap;">(</span><span style="white-space: pre-wrap;">8/3)</span><span>x0</span><span> </span></span><div><span style="font-family: arial; font-size: medium;">x = 0</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">6) The value of x = 0 and the value of y = 0.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(vi) (3x/2) - (5y/3) = - 2, (x/3) + (y/2) = 13/6</span></b></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) (3x/2) - (5y/3) = - 2 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) (x/3) + (y/2) = 13/6 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">3) Simplify equation 1 and equation 2, and we get,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">9x - 10y = - 12 ------------ equation 3</span></span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;"><span style="font-family: arial; font-size: medium;">2x + 3y = 13 ------------ equation 4</span></span></span></blockquote></span></div><div><span style="font-family: arial; font-size: medium;">4) Simplify equation 3, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9x - 10y = - 12</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>9x = 10y - 12</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = (</span><span style="font-family: arial;">10y - 12)/9</span><span style="font-family: arial;"> ------------ equation 5</span></span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Substitute the value of x from equation 5 in equation 4, we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x + 3y = 13</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2[</span><span style="font-family: arial;">(</span><span style="font-family: arial;">10y - 12)/9]</span><span style="font-family: arial;"> + 3y = 13</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(20y - 24)/9</span><span style="font-family: arial;"> + 3y = 13</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">(20y - 24 + 27y)/9</span><span style="font-family: arial;"> = 13</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>(47y - 24)/9</span><span> = 13</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">47y - 24</span><span style="font-family: arial;"> = 13 x 9</span></span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">47y = 24</span><span style="font-family: arial;"> + 117</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">47y = 141</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>y </span><span>= 141/47</span> </span></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = 3 </span><span style="font-family: arial;">------------ equation 6</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;">6) Put the value of y from equation 6 in equation 4, we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2x + 3y = 13 </span></div><span style="font-family: arial; font-size: medium;"><span>2x + (3 x 3) = 13</span><br /></span><div><span style="font-family: arial; font-size: medium;">2x = 13 - 9 = 4</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = 4/2</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = 2</span> </span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">7) The value of x = 2 and the value of y = 3.</span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>Q</b><b>2. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which </b><b>y = mx + 3.</b></span></div><div><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;">1) 2x + 3y = 11 ------------ equation 1</span></div><div><span style="font-family: arial; font-size: medium;">2) 2x - 4y = - 24 ------------ equation 2</span></div><div><span style="font-family: arial; font-size: medium;">3) Simplify equation 2, we get</span></div></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x - 4y = - 24</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2x = 4y - 24</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = (4y - 24)/2</span> </span></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = </span><span style="font-family: arial;">2y - 12</span><span style="font-family: arial;"> ------------ equation 3</span></span></blockquote><span style="font-size: medium;"><span style="font-family: arial;">4) </span><span style="font-family: arial;">Substitute the value of x from equation 3 in equation 1, and we get</span></span><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2x + 3y = 11</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2(</span><span style="font-family: arial;">2y - 12)</span><span style="font-family: arial;"> + 3y = 11</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">4y - 24</span><span style="font-family: arial;"> + 3y = 11</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">7y - 24</span><span style="font-family: arial;"> = 11</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">7y = 24</span><span style="font-family: arial;"> + 11</span></span></div><div><span style="font-family: arial; font-size: medium;">7y = 35</span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = 35/7</span></div></div></div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 4</span></span></div></div></div></blockquote><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 5, from equation 4 in equation 3, and we get</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x = </span><span style="font-family: arial;">2y - 12</span></span></div><span style="font-size: medium;"><span style="font-family: arial;">x = (</span><span style="font-family: arial;">2 x 5) - 12</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = 10</span><span> - 12</span><span> </span> </span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x = - 2</span></div></blockquote><div><span style="font-family: arial; font-size: medium;">6) The value of x = - 2 and the value of y = 5.</span></div></div><div><span style="font-family: arial; font-size: medium;"><span>7)</span><b> </b><span>Put x = - 2 and y = 5 in the equation</span><b> </b><span>y = mx + 3, we get</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">y = mx + 3</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">5 = m(- 2) + 3</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">- 2m = 5 - 3</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">- 2m = 2</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">m = 2/(- 2)</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">m = - 1</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">Q3. Form the pair of linear equations for the following problems and find their solution by </span><span style="font-family: arial;">substitution method.</span></span></b></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(i) The difference between two numbers is 26 and one number is three times</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>the other. Find them.</b></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><b>(ii) The larger of two supplementary angles exceeds the smaller by 18</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>degrees. Find them.</b></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><b>(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball. </b></span></div></blockquote><div><b><span style="font-family: arial; font-size: medium;">(iv) The taxi charges in a city consist of a fixed charge together with the</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b><span>charge for the </span><span>distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a </span></b><b>journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the </b><b>charge per km? How much does a person have to pay for travelling a distance of </b><b>25 km?</b></span></div></blockquote><div><b><span style="font-size: medium;"><span style="font-family: arial;">(v) A fraction becomes 9/11</span><span style="font-family: arial;"> if 2 is added to both the numerator and the</span></span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">denominator. </span><span style="font-family: arial;">If, 3 is added to both the numerator and the denominator it becomes 5/6</span><span style="font-family: arial;">. Find the </span><span style="font-family: arial;">fraction.</span></span></b></div></blockquote><div><b><span style="font-family: arial; font-size: medium;">(vi) Five years hence, the age of Jacob will be three times that of his son. Five</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">years </span><span style="font-family: arial;">ago, Jacob’s age was seven times that of his son. What are their present ages?</span></span></b></div></blockquote><div style="text-align: left;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equations.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these</span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">equations to get the values of x and y. </span></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><b>(i) The difference between two numbers is 26 and one number is three times</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><b>the other. Find them.</b></span></blockquote><div style="text-align: left;">1) Let the greater number be x, and the smaller number be y.</div><div style="text-align: left;">2) According to the first condition, </div></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;">x - y = 26 -------------------------- equation 1.</div></span></div></div></div></div></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">3) According to the second condition, </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = 3y -------------------------- equation 2.</span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">4) Put the value of x from equation 2 in equation 1, we get,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x - y = 26</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3y - y = 26</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2y = 26</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">y = 26/2</span></div></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 13</span><span style="font-family: arial;"> -------------------------- equation 3.</span></span></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">5) Put the value of y from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 3y</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 3 x 13</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 39</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) The numbers are 13 and 39. </span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><span style="font-family: arial; font-size: medium;"><b>(ii) The larger of two supplementary angles exceeds the smaller by 18</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><b>degrees. Find them.</b></span></blockquote><div style="text-align: left;"><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div>1) Let the greater angle be x, and the smaller angle be y.</div><div>2) According to the first condition, </div></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x = y + 18 -------------------------- equation 1.</span></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">3) We know that the sum of the supplementary angles is 180 degrees, </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 180 -------------------------- equation 2.</span></blockquote><div><span style="font-family: arial; font-size: medium;">4) Put the value of x = y + 18 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(y + 18) + y = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">2y + 18 = 180</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2y = 180 - 18</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2y = 162</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = 81</span><sup><span style="font-family: arial;">0 </span></sup><span style="font-family: arial;">-------------------------- equation 3.</span></span></div></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 81 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = y + 18</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 81 + 18</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 99</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) The angles are </span><span style="font-family: arial;">81</span><sup><span style="font-family: arial;">0</span></sup><span style="font-family: arial;"> and 9</span><span style="font-family: arial;">9</span><sup><span style="font-family: arial;">0</span></sup><span style="font-family: arial;">. </span></span></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div><span style="font-family: arial; font-size: medium;"><b>(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><b>buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.</b></span></blockquote></div></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Let the cost of the bat be Rs x and the cost of the ball be Rs y.</span></div><div><span style="font-family: arial; font-size: medium;">2) As 7 bats and 6 balls cost Rs 3800, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">7x + 6y = 3800</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">7x = 3800 - 6y</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x = (3800 - 6y)/7</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></div></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">3) As 3 bats and 5 balls cost Rs 1750, we have,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3x + 5y = 1750</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">4) Put the value of x = (3800 - 6y)/7 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x + 5y = 1750</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3(</span><span style="font-family: arial;">3800 - 6y)/7</span><span style="font-family: arial;"> + 5y = 1750</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">[(11400 - 18y) + 35y]/7</span><span style="font-family: arial;"> = 1750</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">[(11400 - 18y) + 35y]</span><span style="font-family: arial;"> = 1750</span><span style="font-family: arial;"> x 7</span></span></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">11400 + 17y</span><span style="font-family: arial;"> = 1750</span><span style="font-family: arial;"> x 7</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">11400 + 17y</span><span style="font-family: arial;"> = 12250</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">17y = 12250 - 11400</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">17y = 850</span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 850/17</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 50 </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></div></div></blockquote><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 50 from equation 3 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x + 5y = 1750</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">3x + 5(50) = 1750</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3x + 250 = 1750</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3x = 1750 - 250</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3x = 1500</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>x = 1500/3</span> </span></div></div></blockquote><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 500</span><span style="font-family: arial;"> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial;">The cost of the bat is Rs 500 and the cost of the ball is Rs 50</span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(iv) The taxi charges in a city consist of a fixed charge together with the</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><b><span>charge for the </span><span>distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a </span></b><b>journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the </b><b>charge per km? How much does a person have to pay for travelling a distance of </b><b>25 km?</b></span></blockquote><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the fixed charges be Rs x and the charges per km be Rs y.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the first condition, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x + 10y = 105</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x = 105 - 10y</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">According to the second condition, we have,</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x + 15y = 155</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">4) Put the value of x = 105 - 10y from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + 15y = 155</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(105 - 10y) + 15y = 155</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">105 + 5y = 155</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">5y = 155 - 105</span></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">5y = 50</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">y = 10 </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of y = 10 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 105 - 10y</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 105 - 10(10)</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x = 105 - 100</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>x = 5</span> <span> -------------------------- equation 4.</span></span></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) </span><span style="font-family: arial;">The charges for 25 km travel will be x + 25y</span><span style="font-family: arial;">.</span></span></div></div><div><span style="font-family: arial; font-size: medium;">7) So the total charges for 25 km travel will be,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + 25y = 5 + 25(10)</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= 5 + 250</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= 255</span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">8) t</span><span style="font-family: arial;">he fixed charges will be Rs 5, the charges per km are Rs 10 and the travel charges for 25 km will be Rs 255.</span></span></div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div><b><span style="font-size: medium;"><span style="font-family: arial;">(v) A fraction becomes 9/11</span><span style="font-family: arial;"> if 2 is added to both the numerator and the</span></span></b></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">denominator. </span><span style="font-family: arial;">If, 3 is added to both the numerator and the denominator it becomes 5/6</span><span style="font-family: arial;">. Find the </span><span style="font-family: arial;">fraction.</span></span></b></div></div></blockquote><div style="text-align: left;"><div><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the numerator be x and the denominator be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) According to the first condition, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x + 2)/(y + 2) = 9/11</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">11x + 22 = 9y +18</span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">9y = 11x + 22 - 18</span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">9y = 11x + 4</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = (11x + 4)/9</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 1</span></span></div></div></div></div></blockquote><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) </span><span style="font-family: arial;">According to the second condition, we have,</span></span></div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x + 3)/(y + 3) = 5/6</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">6x + 18 = 5y +15</span></div></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6x - 5y = 15 - 18</span></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">6x - 5y = - 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></div></div><div><div><span style="font-family: arial; font-size: medium;">4) Put the value of y = (11x + 4)/9 from equation 1 in equation 2, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">6x - 5y = - 3</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">6x - 5</span><span style="font-family: arial;">(11x + 4)/9</span><span style="font-family: arial;"> = - 3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">[54x - 5</span><span style="font-family: arial;">(11x + 4)]/9</span><span style="font-family: arial;"> = - 3</span></span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">[54x - 5</span><span style="font-family: arial;">(11x + 4)]</span><span style="font-family: arial;"> = - 3 (9)</span></span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>[54x - 55</span><span>x - 20]</span><span> = - 27</span> </span></div></div></div></blockquote><div style="text-align: left;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">54x - 55</span><span style="font-family: arial;">x</span><span style="font-family: arial;"> = - 27</span><span style="font-family: arial;"> + 20</span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">- x = - 7 </span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">x = 7 </span><span style="font-family: arial;">-------------------------- equation 3.</span></span></blockquote><div><div><span style="font-family: arial; font-size: medium;">5) Put the value of x = 7 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = (11x + 4)/9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = (11(7) + 4)/9</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>y = (77 + 4)/9</span> </span></div></div></div></div></blockquote><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">y = 81/9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>y = 9</span> <span> -------------------------- equation 4.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">6) So, here, </span><span style="font-family: arial;">the numerator is 7 and the denominator is 9</span><span style="font-family: arial;">. So the fraction is 7/9.</span></span></div></div></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(vi) Five years hence, the age of Jacob will be three times that of his son.</span></b></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Five </span></b><b><span style="font-size: medium;"><span style="font-family: arial;">years </span><span style="font-family: arial;">ago, Jacob’s age was seven times that of his son. What are their present ages?</span></span></b></div></blockquote><div style="text-align: left;"><h2 style="clear: both;"><span style="font-family: arial; font-size: medium;"><div style="color: black; font-weight: 400;"><div style="text-align: left;"><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">1) Let Jacob's present age be x and his son's present age be y.</span></div><div><span style="font-family: arial; font-size: medium;">2) 5 years hence their ages will be (x + 5) and (y + 5).</span></div><div><span style="font-family: arial; font-size: medium;">3) According to the first relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x + 5) = 3(y + 5)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x + 5) = 3y + 15</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x = 3y + 15 - 5</span></div><span style="font-family: arial; font-size: medium;">x = 3y + 10 ---------------------- equation 1</span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) 5 years ago, their ages were (x - 5) and (y - 5).</span> </span></div><div><span style="font-family: arial; font-size: medium;">5) According to the relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 5) = 7(y - 5)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x - 5) = 7y - 35</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x = 7y - 35 + 5</span></div><span style="font-family: arial; font-size: medium;">x = 7y - 30 ---------------------- equation 2</span></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;">6) </span><span style="font-family: arial;">Put the value of x = 3y + 10 from equation 1 in equation 2, and we get,</span></div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 7y - 30</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3y + 10 = 7y - 30</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">3y - 7y = - 30 - 10</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;"><span> </span>- 4y = - 40</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial;"><span> </span> 4y = 40</span><span style="font-family: arial; font-size: medium;"> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">y = 10 </span><span style="font-family: arial;">-------------------------- equation 3.</span></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">7) Put the value of y = 10 from equation 3 in equation 1, we get,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 3y + 10</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 3(10) + 10</span></blockquote></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>x = 30 + 10</span> </span></div></div></blockquote><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial;">x = 40</span><span style="font-family: arial;"> </span><span style="font-family: arial;"> -------------------------- equation 4.</span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">8) </span></span><span style="font-family: arial;">So Jacob's present age is 40 years and his son's present age is 10 years.</span></div></div></div></div></div></div></span></h2></div><div style="text-align: left;"><div><div><h2 style="clear: both; color: #0400ff; font-family: arial;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/151-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.4</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-64385939082905965822023-05-25T10:35:00.002+05:302023-06-14T13:11:59.703+05:30149-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.2<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 3.2</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/148-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.1</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.2</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b><span>Q</span>1. Form the pair of linear equations in the following problems, and find their solutions </b><b>graphically.</b></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-size: medium;"><span style="font-family: arial;"><b>(i) 10 students of class X took part in a mathematics quiz. If the number of girls is 4 </b><b>more than the number of boys, find the number of boys and girls who took part in </b><b>the quiz.</b></span><span style="font-family: arial;"> </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><b>(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together </b><b>cost Rs 46. Find the cost of one pencil and that of one pen.</b> </span></div></blockquote><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be two variables.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equations.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these</span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">equations to get the values of x and y. </span></div></div></div></div></blockquote><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div><span style="font-family: arial; font-size: medium;"><span><b>(i) 10 students of class X took part in a mathematics quiz. If the number of girls is 4 </b></span><b>more than the number of boys, find the number of boys and girls who took part in </b><b>the quiz.</b></span></div><div><b><span style="font-family: arial; font-size: medium;"><br /></span></b></div><div><span style="font-family: arial; font-size: medium;">1) Let the number of girls be x and the number of boys be y.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) As </span><span style="font-family: arial;">10 students of class X took part in a mathematics quiz</span><span style="font-family: arial;">,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x + y = 10</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = 10 - x </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) As </span><span style="font-family: arial;">the number of girls is 4 </span><span style="font-family: arial;">more than the number of boys</span><span style="font-family: arial;">,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = y + 4</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> y = x - 4</span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote><div><div><div><span style="font-family: arial; font-size: medium;">4) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = 10 - x.</span></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbGI8krqsVuEFbaNkSJvvLjO3gi1Vhy8PKNzaNQMbndbzDt49Qu15JQovsw-6mZ73jM-selIhX_FEfx5YImsTB-Y6y_z7eU2_eATubZK5Ha5XOhLwCpZ7RPtqDeYaCcfMApJ4StTQPrKb00ydAOzw-t6jgpnY-Qdzbe9sEW7oAxIR1dChG-BqsnLvM/s305/1-1-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="83" data-original-width="305" height="83" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbGI8krqsVuEFbaNkSJvvLjO3gi1Vhy8PKNzaNQMbndbzDt49Qu15JQovsw-6mZ73jM-selIhX_FEfx5YImsTB-Y6y_z7eU2_eATubZK5Ha5XOhLwCpZ7RPtqDeYaCcfMApJ4StTQPrKb00ydAOzw-t6jgpnY-Qdzbe9sEW7oAxIR1dChG-BqsnLvM/s1600/1-1-1.png" width="305" /></span></a></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = x - 4.</span></div></div></div></div></blockquote><div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe_DFY2-EBmqru0_CH3uC4Cilq0QMeg90G_N7zIyiY6ErB8fTLTygdUi8PKbIpEn62epFwRMRcRktkhOmtfezRshkUJS-06qGrid30eUHJuDtZwNaYNjWVBKs0nHs21pAWPDUYvN5tN6BTsgQ8iHo_61frPdSFLrY_clKugrYuC5dfiJfhNj4JdAwQ/s305/1-1-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="85" data-original-width="305" height="85" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe_DFY2-EBmqru0_CH3uC4Cilq0QMeg90G_N7zIyiY6ErB8fTLTygdUi8PKbIpEn62epFwRMRcRktkhOmtfezRshkUJS-06qGrid30eUHJuDtZwNaYNjWVBKs0nHs21pAWPDUYvN5tN6BTsgQ8iHo_61frPdSFLrY_clKugrYuC5dfiJfhNj4JdAwQ/s1600/1-1-2.png" width="305" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">5) The graphical representation will be as follows.</span></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWsDGRMk0wb7oTgOg4wmAUx6n5bSHLKZz6guMuVwdEztz-KdNzK0Zq93JS9M-PtTdC2zL9cpnSiQhioHZh2uoTHaTAGYnJgQBpR4YpUWGokzJ8E6mdviLtgGOUpLGxoem9IF5d7R0S59hdMfWCBNo_ZLp2N_kuddgn1S7ag8TRE2tGC2-e8B2zljdU/s1213/1-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="899" data-original-width="1213" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWsDGRMk0wb7oTgOg4wmAUx6n5bSHLKZz6guMuVwdEztz-KdNzK0Zq93JS9M-PtTdC2zL9cpnSiQhioHZh2uoTHaTAGYnJgQBpR4YpUWGokzJ8E6mdviLtgGOUpLGxoem9IF5d7R0S59hdMfWCBNo_ZLp2N_kuddgn1S7ag8TRE2tGC2-e8B2zljdU/s320/1-1.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>6) Here the lines intersect at (7,3), so the number of girls is 7 and the number of boys is 3.</span><br /><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together </b></span><b>cost Rs 46. Find the cost of one pencil and that of one pen.</b> </span></div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">1) Let the cost of the pencil be Rs x and the cost of the pen be Rs y.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) As 5 </span><span style="font-family: arial;">pencil</span><span style="font-family: arial;">s and 7 pens cost Rs 50, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">5 x + 7 y = 50</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">7 y = (50 - 5 x)</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"> y = (50 - 5 x)/7 </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) As </span><span style="font-family: arial;">7 </span><span style="font-family: arial;">pencil</span><span style="font-family: arial;">s and 5 pens</span><span style="font-family: arial;"> cost Rs 46, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7 x + 5 y = 46</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;">5 y = (46 - 7 x)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> y = (46 - 7 x)/5 </span><span style="font-family: arial;"> </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></blockquote><div><div><div><span style="font-family: arial; font-size: medium;">4) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) We will take 2 points for y = (50 - 5 x)/7.</span> </span></div></blockquote></div></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q2. On comparing the ratios a<sub>1</sub>/a<sub>2, </sub>b<sub>1</sub>/b<sub>2, and </sub>c<sub>1</sub>/c<sub>2</sub>, find out whether the lines representing the </b></span><b>following pairs of linear equations intersect at a point, are parallel or coincident:</b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><b>(i) 5x – 4y + 8 = 0, 7x + 6y – 9 = 0 </b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0</b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 6x – 3y + 10 = 0, 2x – y + 9 = 0</b></span></div></div></div></blockquote><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub>
=
0 and a<sub><span style="line-height: 107%;">2</span></sub>x + b<sub><span style="line-height: 107%;">2</span></sub>y + c<sub><span style="line-height: 107%;">2</span></sub><span style="line-height: 107%;"> </span>= 0<br /></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2 </sub></span><span>then the lines are </span><span>coincident.<br /></span></span></div></div></div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>b) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face=""Arial",sans-serif" style="line-height: 107%; mso-ansi-language: EN-IN; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2 </sub></span><span>then the lines are </span><span>parallel.<br /></span></span></div></div></div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>c) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span><sub> </sub></span><span>then the lines are </span><span>intersecting</span><span>.</span></span></div></div></div></blockquote><div><div><p class="MsoNormal"><span style="font-family: arial; font-size: medium;"><o:p></o:p></span></p><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-family: arial; font-size: medium;"><b>(i) 5x – 4y + 8 = 0, 7x + 6y – 9 = 0 </b></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">7, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 4</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6, </span><span style="font-family: arial;">c<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">8</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 9.</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5/7 -------------1</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 4/6</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">8/(- 9)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) From 1, 2, and 3, we can say that a<sub>1</sub>/a<sub>2 </sub><span style="line-height: 19.9733px;">≠ </span>b<sub>1</sub>/b<sub>2</sub>, so the lines are intersecting.</span></div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0</span></b></div><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">9</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">18, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6, </span><span style="font-family: arial;">c<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">12</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">24.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">9/18 = 1/2 -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3/6 </span><span style="font-family: arial;">= 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">12/24 </span><span style="font-family: arial;">= 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) From 1, 2, and 3, we can say that </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>, so the lines are </span><span>coincident.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) 6x – 3y + 10 = 0, 2x – y + 9 = 0</span></b></div><div style="text-align: left;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 3</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 1, </span><span style="font-family: arial;">c<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">10</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">9.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6/2 = 3 -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 3/(- 1) = 3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">10/9</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>2) From 1, 2, and 3, we can say that a<sub>1</sub>/a<sub>2 </sub><span style="line-height: 19.9733px;">= </span>b<sub>1</sub>/b<sub>2</sub> </span><span>≠ </span><span>c</span><sub>1</sub><span>/c</span><sub>2</sub><span>, so the lines are </span><span>parallel</span><span>.</span></span></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q3. On comparing the ratios a<sub>1</sub>/a<sub>2, </sub>b<sub>1</sub>/b<sub>2, and </sub>c<sub>1</sub>/c<sub>2</sub>, find out whether the following pair of linear </b></span><b>equations are consistent, or inconsistent.</b></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><b>(i) 3x + 2y = 5 ; 2x – 3y = 7 </b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(ii) 2x – 3y = 8 ; 4x – 6y = 9</b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(iii) (3/2)x + (5/3)y =7; 9x – 10y = 14 </b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(iv) 5x – 3y = 11 ; – 10x + 6y = –22</b></span></div></div></div><div><div><div><span style="font-family: arial; font-size: medium;"><b>(v) (4/3)x + 2y = 8; 2x + 3y = 12</b></span></div></div></div></blockquote><div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>coincident </span><span>with </span><span>infinitely </span><span>many solutions</span></span></div></div></blockquote></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></blockquote></div></div></div></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span>b) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>parallel with no </span><span>solutions, so they</span></span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></blockquote></div></div></div></div></blockquote><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2,</sub></span><span><sub> </sub></span><span>the lines are </span><span>intersecting with unique solution, so </span><span>they</span></span></div></blockquote></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.<br /></span></span></div></blockquote></div></div></div></div></blockquote><div><div><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div><b><span style="font-family: arial; font-size: medium;">(i) 3x + 2y = 5 ; 2x – 3y = 7</span></b></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">3x + 2y - 5 = 0; 2x – 3y - 7 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 3, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">7.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3/2 -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2/(- 3)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 5)/(- 7)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) From 1, 2, 3, we can say that (a<sub>1</sub>/a<sub>2 </sub><span style="line-height: 19.9733px;">≠ </span>b<sub>1</sub>/b<sub>2</sub></span><span style="font-family: arial;">)</span><span style="font-family: arial;">, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so the lines are intersecting with unique solution</span><span style="font-family: arial;">, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> </span></div><div><div><div><b><span style="font-family: arial; font-size: medium;">(ii) 2x – 3y = 8 ; 4x – 6y = 9</span></b></div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">2x - 3y - 8 = 0; 4x – 6y - 9 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">4, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 3</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">– 6</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">8</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">9.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2/4 = 1/2 -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 3)</span><span style="font-family: arial;">/(- 6) = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 8)/(- 9) = 8/9</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">parallel with no </span><span style="font-family: arial;">solutions, so they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iii) (3/2)x + (5/3)y =7; 9x – 10y = 14</span></b></div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">(3/2)</span><span style="font-family: arial;">x + </span><span style="font-family: arial;">(5/3)</span><span style="font-family: arial;">y - 7 = 0; 9x – 10y - 14 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(3/2)</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">9, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(5/3)</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">– 10</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">7</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">14.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(3/2)/(9/1) = 1/6</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(5/3)</span><span style="font-family: arial;">/(- 10/1) = (- 1/6) </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 7)/(- 14) = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) From 1, 2, 3, we can say that (a<sub>1</sub>/a<sub>2 </sub><span style="line-height: 19.9733px;">≠ </span>b<sub>1</sub>/b<sub>2</sub></span><span style="font-family: arial;">)</span><span style="font-family: arial;">, </span></span></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>so the lines are intersecting with unique solution</span><span>, so </span><span>they </span><span>are </span><span>consistent</span><span>.</span> </span></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(iv) 5x – 3y = 11 ; – 10x + 6y = –22</span></b></div><div style="text-align: left;"><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">5x - 3y - 11 = 0; </span><span style="font-family: arial;">– 10</span><span style="font-family: arial;">x + 6y + 22 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 10, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 3</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">11</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= 22</span><span style="font-family: arial;">.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5/(- 10) = (- 1/2) -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 3)</span><span style="font-family: arial;">/(6) = </span><span style="font-family: arial;">(- 1/2)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 11)/(22) = </span><span style="font-family: arial;">(- 1/2)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">coincident </span><span style="font-family: arial;">with infinitly many solutions, </span><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></blockquote></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(v) (4/3)x + 2y = 8; 2x + 3y = 12</span></b></div><div style="text-align: left;"><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">(4/3)</span><span style="font-family: arial;">x + </span><span style="font-family: arial;">2</span><span style="font-family: arial;">y - 8 = 0; 2x + 3y - 12 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(4/3)</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">8</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">12.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(4/3)/(2/1) = 2/3</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">/3 </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 8)/(- 12) = 2/3</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">coincident </span><span style="font-family: arial;">with infinitly many solutions, </span><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q4. Which of the following pairs of linear equations are consistent/inconsistent? If </b></span><b>consistent, obtain the solution graphically:</b></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 5, 2x + 2y = 10</b></span></div></div></div></div></div></div><div><div><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><b>(ii) x – y = 8, 3x – 3y = 16</b></span></div></div></div></div></div></div><div><div><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><b>(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0</b></span></div></div></div></div></div></div><div><div><div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><b>(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0</b></span></div></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>coincident </span><span>with infinitly many solutions</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></blockquote></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>parallel with no </span><span>solutions, so they</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></blockquote></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2,</sub></span><span><sub> </sub></span><span>the lines are </span><span>intersecting with unique solution, so </span><span>they</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.<br /></span></span></div></blockquote></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><b>(i) x + y = 5, 2x + 2y = 10</b></span></div></div></div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">x + </span><span style="font-family: arial;">y - 5 = 0; 2x + 2y - 10 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">10.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1/2</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">/2 </span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 5)/(- 10) = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">coincident </span><span style="font-family: arial;">with infinitly many solutions, </span><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) We will simplify our equations </span><span style="font-family: arial;">x + y = 5 and </span><span style="font-family: arial;">2x + 2y = 10,</span></span></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x + y = 5</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = (5 - x)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------4</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2x + 2y = 10</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>x + y = 5</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = (5 - x)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Now, we will represent these equations graphically.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (5 - x).</span></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdY_SVMtBKllV7yYlpacQjrLwvrDYqfz2dGg7GDH8tB8ctWW73fQDpkbiCVjQaLwpKOX2A9mPsraJY_bxfJ9OksWIaY_YDUlGbputoCYbG7FhSfL_H_ZrCRTrdd3ITGQVgjq3nat2TExDB5oiAxx-_L5NVt7rHdut7maOwJJtUUkIjWlZt8W__uCLC/s303/4-1-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="79" data-original-width="303" height="79" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdY_SVMtBKllV7yYlpacQjrLwvrDYqfz2dGg7GDH8tB8ctWW73fQDpkbiCVjQaLwpKOX2A9mPsraJY_bxfJ9OksWIaY_YDUlGbputoCYbG7FhSfL_H_ZrCRTrdd3ITGQVgjq3nat2TExDB5oiAxx-_L5NVt7rHdut7maOwJJtUUkIjWlZt8W__uCLC/s1600/4-1-1.png" width="303" /></span></a></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (5 - x).</span></div></div></div></div></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUQ_5cXTkZadqvrgtpCXJhqpeVxtDlX1AzEwSOC7lXGTZmwVPmrU17pMytctmx2LjXlBtiZhdlqnTaDWtUR0HfnR89UA_QP54XadjtEnlhOAjyLx1BrYUPDg3wa0r3qxlbNtGE771kNaVHBueVJTt_hTCYZBmjOFvOFZiSmdmHPjXNMm0K3nuHTJuA/s301/4-1-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="81" data-original-width="301" height="81" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUQ_5cXTkZadqvrgtpCXJhqpeVxtDlX1AzEwSOC7lXGTZmwVPmrU17pMytctmx2LjXlBtiZhdlqnTaDWtUR0HfnR89UA_QP54XadjtEnlhOAjyLx1BrYUPDg3wa0r3qxlbNtGE771kNaVHBueVJTt_hTCYZBmjOFvOFZiSmdmHPjXNMm0K3nuHTJuA/s1600/4-1-2.png" width="301" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>6) The graphical representation will be as follows.</span> </span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipbL6GQRqon1m5rMoZL2Tq6RI7VD1TRuf2PFPg3PEBviEFf5yC7WXB25amMTxvIkWJsY5VUMyUsIz0RTlLVL2czRcf-6ojFa3J7AwbyUraOEuXU5ZRfVDbqttaw9zdgdkkk-C4AtAu_qVa2k2wceJAHeXy8n9o6KULOMLDYN3ixXR7nsBlKreoaatr/s1219/4-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="895" data-original-width="1219" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipbL6GQRqon1m5rMoZL2Tq6RI7VD1TRuf2PFPg3PEBviEFf5yC7WXB25amMTxvIkWJsY5VUMyUsIz0RTlLVL2czRcf-6ojFa3J7AwbyUraOEuXU5ZRfVDbqttaw9zdgdkkk-C4AtAu_qVa2k2wceJAHeXy8n9o6KULOMLDYN3ixXR7nsBlKreoaatr/w343-h252/4-1.png" width="343" /></span></a></div><span style="font-size: medium;"><span style="font-family: arial;">7) T</span><span style="font-family: arial;">he lines are </span><span style="font-family: arial;">coincident </span><span style="font-family: arial;">with infinitely many solutions, </span><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div><div><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">(ii) x – y = 8, 3x – 3y = 16</span></b></div><div style="text-align: left;"><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">x - </span><span style="font-family: arial;">y - 8 = 0; 3x - 3y - 16 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- </span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">8</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">16.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1/3</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 1)/(- 3)</span><span style="font-family: arial;"> = 1/3</span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 8)/(- 16) = 1/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2</sub></span><span><sub> </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div style="text-align: left;"><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">parallel with no </span><span style="font-family: arial;">solutions, so they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div style="text-align: left;"><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) We will simplify our equations </span><span style="font-family: arial;">x - y = 8 and </span><span style="font-family: arial;">3x - 3y = 16,</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x - y = 8</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = (x - 8)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">3x - 3y = 16</span></div></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>3(x - y) = 16</span> </span></div></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"> <span>x - y = 16/3</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = x - (16/3)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Now, we will represent these equations graphically.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) We will take 3 points for y = (x - 8).</span> </span></div></blockquote></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNJKXaA0o0N6SaZGZK6MeTAXeyDYb6WmfCcuQOUz_g6V9nIYrPChME1-Mp82S_lqf0bwDBRy90VlnjE3LBr7eCinalaqTxKyWBub50C1Jou2Iarr7BYor9jErsUnQjuJezTNf3oMJXgPd97_zLCp2bogq9RnATLHHi4u0tIRZ_OBNVoJiA3U9qgMRn/s285/4-2-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="75" data-original-width="285" height="75" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNJKXaA0o0N6SaZGZK6MeTAXeyDYb6WmfCcuQOUz_g6V9nIYrPChME1-Mp82S_lqf0bwDBRy90VlnjE3LBr7eCinalaqTxKyWBub50C1Jou2Iarr7BYor9jErsUnQjuJezTNf3oMJXgPd97_zLCp2bogq9RnATLHHi4u0tIRZ_OBNVoJiA3U9qgMRn/s1600/4-2-1.png" width="285" /></span></a></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = x - (16/3).</span></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwo2dXvq33y9_5YSzjdROZjAjObZo-HwM-Ri8y4Gy7jhZ0ub6UW9OJCJy-bTD5eEfQLZHvM7AwaVJkX7OoKFu96UiWETcOD_RNALa-Dy5hgphtkMyuZEAfayVwT7QfGmxX8IWGLE7ryKZq8VAO-b802sdcymdUa648qy8PCGyJEuFDe6vq7IHR4i-7/s363/4-2-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="81" data-original-width="363" height="77" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwo2dXvq33y9_5YSzjdROZjAjObZo-HwM-Ri8y4Gy7jhZ0ub6UW9OJCJy-bTD5eEfQLZHvM7AwaVJkX7OoKFu96UiWETcOD_RNALa-Dy5hgphtkMyuZEAfayVwT7QfGmxX8IWGLE7ryKZq8VAO-b802sdcymdUa648qy8PCGyJEuFDe6vq7IHR4i-7/w348-h77/4-2-2.png" width="348" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>6) The graphical representation will be as follows.</span> </span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL1zKZJwxLIz3BD-wHXLCpNlDQgKjacKSWMcRyLuf3cOxlVzbLUDhmJs4wyFtMVWhEyTTmV4c-aeQxuHIhRriM4hjhZb63z3xhcN1NaAQdk45rkvuIl6b3YCck5cYRZfy2mwc-FeTID_HxIFn_GFsCu7tKWuus45O8YIlfzVgwgwCxXgoY34R5Z_B7/s1209/4-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="899" data-original-width="1209" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL1zKZJwxLIz3BD-wHXLCpNlDQgKjacKSWMcRyLuf3cOxlVzbLUDhmJs4wyFtMVWhEyTTmV4c-aeQxuHIhRriM4hjhZb63z3xhcN1NaAQdk45rkvuIl6b3YCck5cYRZfy2mwc-FeTID_HxIFn_GFsCu7tKWuus45O8YIlfzVgwgwCxXgoY34R5Z_B7/s320/4-2.png" width="320" /></span></a></div><span style="font-size: medium;"><span style="font-family: arial;">7) The lines are </span><span style="font-family: arial;">parallel with no </span><span style="font-family: arial;">solutions, so they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0</span></b></div><div><div><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, 2</span><span style="font-family: arial;">x + </span><span style="font-family: arial;">y - 6 = 0; 4x - 2y - 4 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">4, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">6</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">4.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2/4 = 1/2</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1/(- 2)</span><span style="font-family: arial;"> = -(1/2)</span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 6)/(- 4) = 3/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div><div><div><div><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">intersecting with unique solution, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></div></div></div></div></div></div></div></blockquote><div><div><div><div style="text-align: left;"><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) We will simplify our equations </span><span style="font-family: arial;">2</span><span style="font-family: arial;">x + </span><span style="font-family: arial;">y - 6 = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">4x - 2y - 4 = 0</span><span style="font-family: arial;">,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2x + y = 6</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = (6 - 2x)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">4x - 2y = 4</span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">2(2x - y) = 4</span></div></div></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(2x - y) = 2</span></div></div></div></div></div></div></blockquote><div><div><div style="text-align: left;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"> <span>y = 2x - 2</span><span> </span><span>-------------5</span></span></div></div></blockquote><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Now, we will represent these equations graphically.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) We will take 3 points for y = (6 - 2x).</span> </span></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwkCC4XMLWt52PD9o9i7EMKHz_VJpZV7IkPZA68ptXmv9d0mtrGwXP3uflv5vQEksITyagq11Cnrb_eBznq1Qacgjnh1GUcOVyfFipiTrlFKOVam1LTFZtcUhXeEOnoQB554E7z6p2SkMLFwTfDCvMFAPdDCgGmmLuL2ZRLlnBsjXM-hPzgMAET8Np/s259/4-3-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="79" data-original-width="259" height="79" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwkCC4XMLWt52PD9o9i7EMKHz_VJpZV7IkPZA68ptXmv9d0mtrGwXP3uflv5vQEksITyagq11Cnrb_eBznq1Qacgjnh1GUcOVyfFipiTrlFKOVam1LTFZtcUhXeEOnoQB554E7z6p2SkMLFwTfDCvMFAPdDCgGmmLuL2ZRLlnBsjXM-hPzgMAET8Np/s1600/4-3-1.png" width="259" /></span></a></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (2x - 2).</span></div></blockquote><div style="clear: left; display: inline; margin-bottom: 1em; margin-left: 1em; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUvtVrfZmlIAJl0eKJ4YPFiRT6LghXJP8fRTJNlvyybIG7GjdsbITdUP3gOW2fQb2yfhcsmGp_DxQNZjEUWPFpODzf7GNTfOoXODD_Dmn4DytfPlvaTDjyjZK_KaSLS238wInTMNnlnyEQAXVgtvYqNcl9Wm0tA3DeQBpvFxgub_mUl8R3-8Udd4LH/s255/4-3-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="81" data-original-width="255" height="81" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUvtVrfZmlIAJl0eKJ4YPFiRT6LghXJP8fRTJNlvyybIG7GjdsbITdUP3gOW2fQb2yfhcsmGp_DxQNZjEUWPFpODzf7GNTfOoXODD_Dmn4DytfPlvaTDjyjZK_KaSLS238wInTMNnlnyEQAXVgtvYqNcl9Wm0tA3DeQBpvFxgub_mUl8R3-8Udd4LH/s1600/4-3-2.png" width="255" /></span></a></div></div></div></div></div></div><div><span style="font-family: arial; font-size: medium;">6) The graphical representation will be as follows.</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbEa62ff2Zq08m8WQtvxO4ejdsm0ld9Ye4GrFMo0117I3d6dtI14c1EifEqh0vsC1A3oVnclzoMslq7mgcKG049DLMOAYBWnQ1M0dPSl4UIhY91uix4f2b-Jy2i4rIPpG0_DGlvBM0rdSkMUiTMD3Y5xwZ7J9Yq6JBuHUS9pkXeEBdAKmitASf6R2x/s1219/4-3.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="891" data-original-width="1219" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbEa62ff2Zq08m8WQtvxO4ejdsm0ld9Ye4GrFMo0117I3d6dtI14c1EifEqh0vsC1A3oVnclzoMslq7mgcKG049DLMOAYBWnQ1M0dPSl4UIhY91uix4f2b-Jy2i4rIPpG0_DGlvBM0rdSkMUiTMD3Y5xwZ7J9Yq6JBuHUS9pkXeEBdAKmitASf6R2x/s320/4-3.png" width="320" /></span></a></div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">7) The lines are </span><span style="font-family: arial;">intersecting with a unique solution, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">8) The solution of these equations is (2, 2)</span></div></div></div><div><span style="font-family: arial; font-size: medium;"><br /></span></div><div><b><span style="font-family: arial; font-size: medium;">(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0</span></b></div><div><div><div><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, 2</span><span style="font-family: arial;">x - 2</span><span style="font-family: arial;">y - 2 = 0; 4x - 4y - 5 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">4, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">- 2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">4</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= -</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">5.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2/4 = 1/2</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 2)</span><span style="font-family: arial;">/(- 4)</span><span style="font-family: arial;"> = </span><span style="font-family: arial;">1/2</span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 2)/(- 5) = 2/5</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2</sub></span><span><sub> </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span>)</span><span>, </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">parallel with no </span><span style="font-family: arial;">solutions, so they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></blockquote></div></div></div></div></div></div></div><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) We will simplify our equations </span><span style="font-family: arial;">2</span><span style="font-family: arial;">x - 2</span><span style="font-family: arial;">y - 2</span><span style="font-family: arial;"> = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">4x - 4y - 5</span><span style="font-family: arial;"> = 0</span><span style="font-family: arial;">,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2</span><span style="font-family: arial;">x - 2</span><span style="font-family: arial;">y </span><span style="font-family: arial;">= 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = (x - 1)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">4x - 4y = 5</span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">4(x - y) = 5</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">(x - y) = 5/4</span></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"> <span>y = x - (5/4)</span><span> </span><span>-------------5</span></span></div></div></blockquote><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Now, we will represent these equations graphically.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (x - 1).</span></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTwE6pG6h2ReW0ms_9vA6t6EFKEzchKOG6Dz8cTEc0njsYkQoyHoHRVoKzrt0uMQ_vHphuBCnHTiCrv5XBhm4L5kI709CdsZ_9JObWn8EwE2nzHf5pjIISK7hwZk8s7rS1nDr_uFIIprRfto3DVwF458IaJIvqfkRWaBxrglaSG0broG92RooHHyp8/s255/4-4-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="75" data-original-width="255" height="75" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTwE6pG6h2ReW0ms_9vA6t6EFKEzchKOG6Dz8cTEc0njsYkQoyHoHRVoKzrt0uMQ_vHphuBCnHTiCrv5XBhm4L5kI709CdsZ_9JObWn8EwE2nzHf5pjIISK7hwZk8s7rS1nDr_uFIIprRfto3DVwF458IaJIvqfkRWaBxrglaSG0broG92RooHHyp8/s1600/4-4-1.png" width="255" /></span></a></div></div></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (x - (5/4).<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-x9cYuONgWeSGMB4IsDE80buTdleQMudxQrnKDIR79p6KssXKf57qxTCvzXCWwCPq8Cn1FuCLZNfdWdpR0Enb7Kv-F7oEjgfV_z9CQATZQZIV_5BUliwNEFZdCaKBlhpWz6soZWBT38PCUbRsAu38tMnPGmyeUfNrok7CNIXgwHovGuTvHA7Le5AU/s383/4-4-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="79" data-original-width="383" height="69" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-x9cYuONgWeSGMB4IsDE80buTdleQMudxQrnKDIR79p6KssXKf57qxTCvzXCWwCPq8Cn1FuCLZNfdWdpR0Enb7Kv-F7oEjgfV_z9CQATZQZIV_5BUliwNEFZdCaKBlhpWz6soZWBT38PCUbRsAu38tMnPGmyeUfNrok7CNIXgwHovGuTvHA7Le5AU/w333-h69/4-4-2.png" width="333" /></a></div></span></div></div></div></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><div class="separator" style="clear: both; text-align: left;">6) The graphical representation will be as follows.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRNUyVs7mRTQ74gluTtg3caKcOULUG7qHCOCt5lfvADDSm3w3kSpVPw4czsUgmpezp0TEpx7c1Uc4EL-tdxouNuymDKSSG0gnjYcIkkTdIQTKg2OQlYpA4xnV4tTwx55SsNNQ8pGmyjCrQViOrai4SKulV-GG5NkuVAl357X-U1sNFvOS65LJnM4Zy/s1220/4-4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="893" data-original-width="1220" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRNUyVs7mRTQ74gluTtg3caKcOULUG7qHCOCt5lfvADDSm3w3kSpVPw4czsUgmpezp0TEpx7c1Uc4EL-tdxouNuymDKSSG0gnjYcIkkTdIQTKg2OQlYpA4xnV4tTwx55SsNNQ8pGmyjCrQViOrai4SKulV-GG5NkuVAl357X-U1sNFvOS65LJnM4Zy/s320/4-4.png" width="320" /></a></div></span><div><span style="font-size: medium;"><span style="font-family: arial;">7) The lines are </span><span style="font-family: arial;">parallel with no </span><span style="font-family: arial;">solutions, so they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><b><span style="font-family: arial; font-size: medium;">Q5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.</span></b></div><div class="separator" style="clear: both;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be the length and width of the <span style="font-size: medium;"><span style="font-family: arial; font-size: medium;">rectangular garden</span></span>.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equation.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these equations to get the values of x and y. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial;">Let the </span><span style="font-family: arial;">length </span><span style="font-family: arial;">be x and the </span><span style="font-family: arial;">width</span><span style="font-family: arial;"> be y.</span></span></div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">2) As the </span></span><span style="font-family: arial; font-size: medium;">length is 4 m more than its width</span><span style="font-size: medium;"><span style="font-family: arial;">, we have,</span></span></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x = y + 4</span></div></div></div></div></div></div></div></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">y = x - 4 </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) We know that the perimeter of a rectangle is 2(x + y), </span></span></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">so half of perimeter is (x + y).</span></span></div></div></div></div></div></div></div></div></div></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) Therefore, as half of the perimeter is 36 m, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x + y = 36</span></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>y = 36 - x </span><span>------------ equation 2</span></span></div></div></div></blockquote><div><div><div><span style="font-family: arial; font-size: medium;">5) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (x - 4).</span></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial; font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6FXrQsvO2KUYWi2lGzbF7hWutCCNm_fc2sDC_K46gAdddipURbMIP0nJYm5yMf2g-izRNlxN2owOdusYXGLFYfuhTa8zYhrUQPDzEYIWBg5fyAkGurjIZtzp5J_IZ9IwDRdvK3f-0ku-nGLn1g-qb4ChrIsN2tXkZsP6VFQcnAIskoE3rLy-suWkI/s294/5-1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="69" data-original-width="294" height="69" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6FXrQsvO2KUYWi2lGzbF7hWutCCNm_fc2sDC_K46gAdddipURbMIP0nJYm5yMf2g-izRNlxN2owOdusYXGLFYfuhTa8zYhrUQPDzEYIWBg5fyAkGurjIZtzp5J_IZ9IwDRdvK3f-0ku-nGLn1g-qb4ChrIsN2tXkZsP6VFQcnAIskoE3rLy-suWkI/s1600/5-1.png" width="294" /></a><span style="text-align: left;"> </span></span></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (36 - x).</span></div></div></div></div></div></blockquote><div style="margin-left: 1em; margin-right: 1em; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjicGKOfCHBme40AdLziUF4ZYO7vrY_RimmLInr9KOMNbvL1mMGifaSTDQtVJ1fm2fsY2FrwqMRvFaii3RUaZGohexl6V9br__6xzul5MkRCxk7cqmSN5QxWeF2bCf-F33corat0rVNUpd4ArHUUVQluozYHsy4k1H6JcrfgfZUU3JT2HgBc2n14FG/s306/5-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="72" data-original-width="306" height="72" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjicGKOfCHBme40AdLziUF4ZYO7vrY_RimmLInr9KOMNbvL1mMGifaSTDQtVJ1fm2fsY2FrwqMRvFaii3RUaZGohexl6V9br__6xzul5MkRCxk7cqmSN5QxWeF2bCf-F33corat0rVNUpd4ArHUUVQluozYHsy4k1H6JcrfgfZUU3JT2HgBc2n14FG/s1600/5-2.png" width="306" /></span></a></div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">6) The graphical representation will be as follows.</span></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgya0T9GcNFDJ6zdW71dFd_eb9sqYRO_hReqq54mx7Iz6mDsr1wk3SvWb8WOz0vxxQ1dFYoDprnUZgY39KeWOw-EpKHFAF-HSXAtk5OHD471_o8XMI3ogTbdGS2HxclfVLP-6sn7ty8nQEK1tT28PK3X6Q0oEtJIrbF9pjEdvXs2oVPYglRjiM8_MCV/s1227/5.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="899" data-original-width="1227" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgya0T9GcNFDJ6zdW71dFd_eb9sqYRO_hReqq54mx7Iz6mDsr1wk3SvWb8WOz0vxxQ1dFYoDprnUZgY39KeWOw-EpKHFAF-HSXAtk5OHD471_o8XMI3ogTbdGS2HxclfVLP-6sn7ty8nQEK1tT28PK3X6Q0oEtJIrbF9pjEdvXs2oVPYglRjiM8_MCV/s320/5.png" width="320" /></span></a></div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">7) The lines are </span><span style="font-family: arial;">intersecting with a unique solution, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) Here, the dimensions </span><span style="font-family: arial;">of the rectangular garden</span><span style="font-family: arial;"> are as follows.</span></span></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) length is 20 m, </span></div></div></div></div></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span style="font-family: arial;">width is 16 m.</span></span></div></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div><span style="font-family: arial; font-size: medium;"><br /></span></div></div></div></div></div></div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q6. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables </b></span><b>such that the geometrical representation of the pair so formed is:</b></span></div><div class="separator" style="clear: both;"><span style="font-family: arial; font-size: medium;"><span><b>(i) intersecting lines (ii) parallel lines </b></span><b>(iii) coincident lines</b></span></div><div class="separator" style="clear: both;"><div><div><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>coincident </span><span>with infinitly many solutions</span></span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">so they are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></blockquote></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>parallel with no </span><span>solutions, so they</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></blockquote></div></blockquote><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2,</sub></span><span><sub> </sub></span><span>the lines are </span><span>intersecting with unique solution, so </span><span>they</span></span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.<br /></span></span></div></blockquote></div></blockquote><div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;"><b>(i) Intersecting lines</b></span></div></div></div></div></div></div></div><div><div style="text-align: left;"><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0,<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2,</sub></span><span><sub> </sub></span><span>the lines will be </span><span>intersecting with unique solution</span><span>.</span></span></div></blockquote></div></div></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2) Let our equation be </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0</span><span>, and the given equation is 2x + 3y – 8 = 0.</span></span></div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both;"><span style="font-family: arial; font-size: medium;"><span>3) So, here</span><span> </span><span>a<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>,</span><span><sub> </sub>a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>2, </span><span>b<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>,</span><span><sub> </sub>b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>3</span><span>, </span><span>c<sub>1</sub></span><span> </span><span>= </span><span>c</span><sub>1</sub><span>,</span><span><sub> </sub>c<sub>2 </sub></span><span>= </span><span>-</span><span><sub> </sub></span><span>8.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>/2</span><span> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b<sub>1</sub>/b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>/3</span><span>-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c<sub>1</sub>/c<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>c</span><sub>1</sub><span>/(- 8)</span><span> </span><span>-------------3</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) As our lines are intersecting, we have </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2,</sub></span></span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>so, we have </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">≠ </span><span>b</span><sub>1</sub><span>/3. Let us take any values for </span><span>a</span><sub>1, </sub><span>b</span><sub>1 </sub><span>and </span><span>c</span><sub>1 </sub><span>which satisfies, </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">≠ </span><span>b</span><sub>1</sub><span>/3</span><span> </span></span></div></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>let </span><span>a</span><sub>1 </sub><span>= 3, </span><span>b</span><sub>1</sub><span> = (-2), and </span><span>c</span><sub>1</sub><span>= 14.</span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) So, our equation will be 3x - 2y + 14 = 0, the given equation is </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;">.</span></span></div><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">6) We will simplify our equations </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">3x - 2y + 14 = 0</span><span style="font-family: arial;">,</span></span></div><div style="text-align: left;"><div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2</span><span style="font-family: arial;">x + 3</span><span style="font-family: arial;">y </span><span style="font-family: arial;">= 8</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, 3y = (8 - 2x)</span><span style="font-family: arial;"> </span></span></blockquote></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div><div><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>y = (8 - 2x)/3 </span><span>-------------4</span></span></div></div></div></div></div></div></div></div></div></div></blockquote></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">3x - 2y = -14</span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">2y = (3x + 14)</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = </span><span style="font-family: arial;">(3x + 14)/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></div></blockquote></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">7) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (8 - 2x)/3.</span></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s322/6-a-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="74" data-original-width="322" height="74" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s320/6-a-1.png" width="320" /></span></a></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (3x + 14)/2.</span></div></div></div></div></div></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyKAu6j5AmfHNFZJW4khB5QfamDfTF2OWHOSnboIr2lk4oOeQ6Ys-Kw1QXXongtlP5N5J45IjsUv1cxQVK5DCkvHUMZD-Ud_JKM9hW9JuhXWjMusdEpHT0JRaNnU7bPOISMbxoVA0DQssF-FzfiQsB_6AL5Jy_MJa11EjXNugHALmz4dWWDGS0eiHp/s322/6-a-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="70" data-original-width="322" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyKAu6j5AmfHNFZJW4khB5QfamDfTF2OWHOSnboIr2lk4oOeQ6Ys-Kw1QXXongtlP5N5J45IjsUv1cxQVK5DCkvHUMZD-Ud_JKM9hW9JuhXWjMusdEpHT0JRaNnU7bPOISMbxoVA0DQssF-FzfiQsB_6AL5Jy_MJa11EjXNugHALmz4dWWDGS0eiHp/s320/6-a-2.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><span>8) The graphical representation will be as follows.</span> </span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyPWg10bEFh09GHUU6J1NWTIrhjHU0KOL-_NcsbwxFnm6FaKfok50pzzsbw9SIePtCnS7JTKKKbuI0ycaKMd74MiSEYb8lhnnbWIUwkvDprwn0pNgyF3AaeSqLLRqztH0ILukouPolEAe62whsYz7tT97jpP-_1W2mRd6iBG96IJN6FATTKeQU8wZd/s1227/6-a.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="897" data-original-width="1227" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyPWg10bEFh09GHUU6J1NWTIrhjHU0KOL-_NcsbwxFnm6FaKfok50pzzsbw9SIePtCnS7JTKKKbuI0ycaKMd74MiSEYb8lhnnbWIUwkvDprwn0pNgyF3AaeSqLLRqztH0ILukouPolEAe62whsYz7tT97jpP-_1W2mRd6iBG96IJN6FATTKeQU8wZd/s320/6-a.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>9) Here the point of intersection is (-2, 4).</span><br /></span><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><b>(ii) parallel lines</b><br /></span><div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>parallel with no </span><span>solutions, so they</span></span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">are </span><span style="font-family: arial;">in</span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></blockquote></div></div></div></div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;"><span>2) Let our equation be </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0</span><span>, and the given equation is 2x + 3y – 8 = 0.</span></span></div><div><div class="separator" style="clear: both;"><span style="font-family: arial; font-size: medium;"><span>3) So, here</span><span> </span><span>a<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>,</span><span><sub> </sub>a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>2, </span><span>b<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>,</span><span><sub> </sub>b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>3</span><span>, </span><span>c<sub>1</sub></span><span> </span><span>= </span><span>c</span><sub>1</sub><span>,</span><span><sub> </sub>c<sub>2 </sub></span><span>= </span><span>-</span><span><sub> </sub></span><span>8.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>/2</span><span> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b<sub>1</sub>/b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>/3</span><span>-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c<sub>1</sub>/c<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>c</span><sub>1</sub><span>/(- 8)</span><span> </span><span>-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) As our lines are parallel, we have </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span><sub>,</sub></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>so, we have </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">= </span><span>b</span><sub>1</sub><span>/3. Let us take any values for </span><span>a</span><sub>1, </sub><span>b</span><sub>1 </sub><span>and </span><span>c</span><sub>1 </sub><span>which satisfies, </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">= </span><span>b</span><sub>1</sub><span>/3</span><span> </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>let </span><span>a</span><sub>1 </sub><span>= 4, </span><span>b</span><sub>1</sub><span> = 6, and </span><span>c</span><sub>1</sub><span>= 9.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) So, our equation will be 4x + 6y + 9 = 0, the given equation is </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;">.</span></span></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) We will simplify our equations </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">4x + 6y + 10 = 0</span><span style="font-family: arial;">,</span></span></div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2</span><span style="font-family: arial;">x + 3</span><span style="font-family: arial;">y </span><span style="font-family: arial;">= 8</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, 3y = (8 - 2x)</span><span style="font-family: arial;"> </span></span></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><span style="font-family: arial; font-size: medium;"> <span>y = (8 - 2x)/3 </span><span>-------------4</span></span></div></div></div></div></blockquote></blockquote><div><div><div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x + 6y = 10</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">6y = (10 - 4x)</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = </span><span style="font-family: arial;">(10 - 4x)</span><span style="font-family: arial;">/6</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></div></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">7) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (8 - 2x)/3.</span></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s322/6-a-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="74" data-original-width="322" height="74" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s320/6-a-1.png" width="320" /></span></a></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 3 points for y = (10 - 4x)/6.</span></div></div></div></div></div></div></div></blockquote><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbQaqqW_zJ140Bqr7IuWQoLCAFLJcNq3GXT3nGSLkgn8a_zoBvrAHhg0TH84BCmlU-yyID0mDFtQE-DmvoPjpbnswaqIeDdK6zvo2lH3jUR43iK_11qcrBLFcEPIYkhYpLYsrnSjfesFaiHQz_qotwatvLtlN5gVHUPPcihqWbszRJqLV2opOyySD/s322/6-b-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="70" data-original-width="322" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbQaqqW_zJ140Bqr7IuWQoLCAFLJcNq3GXT3nGSLkgn8a_zoBvrAHhg0TH84BCmlU-yyID0mDFtQE-DmvoPjpbnswaqIeDdK6zvo2lH3jUR43iK_11qcrBLFcEPIYkhYpLYsrnSjfesFaiHQz_qotwatvLtlN5gVHUPPcihqWbszRJqLV2opOyySD/s320/6-b-2.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>8) The graphical representation will be as follows.</span> </span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDq-0eB64AiMq7MCoLt_W5RHPb_3wgabxdT0d2xTJ8tLIYVJOCwd54U6Sw-yxEVe1XxjOcMhbVMFQOet_DlPIYHCaNpSWD8S419d536UFeqGcpgkR29N3K5HlNShrBG7JwcY_w-EumLNO2GfwrZRlHDXPohI7QFF38K1WaLLm0ME7OcpYtZrYfikw3/s1215/6-b.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="901" data-original-width="1215" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDq-0eB64AiMq7MCoLt_W5RHPb_3wgabxdT0d2xTJ8tLIYVJOCwd54U6Sw-yxEVe1XxjOcMhbVMFQOet_DlPIYHCaNpSWD8S419d536UFeqGcpgkR29N3K5HlNShrBG7JwcY_w-EumLNO2GfwrZRlHDXPohI7QFF38K1WaLLm0ME7OcpYtZrYfikw3/s320/6-b.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>9) Here the lines are parallel.</span><br /><br /><b>(iii) coincident lines</b><br /></span><div><div><div><div><span style="font-family: arial; font-size: medium;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) If </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2, </sub></span><span>the lines are </span><span>coincident </span><span>with infinitly many solutions</span><span>.</span></span></div></div></div></div></div></blockquote></div></div><div><div><span style="font-family: arial; font-size: medium;"><span>2) Let our equation be </span><span>a</span><sub>1</sub><span>x + b</span><sub>1</sub><span>y + c</span><sub>1</sub><span> = 0</span><span>, and the given equation is 2x + 3y – 8 = 0.</span></span></div><div><div class="separator" style="clear: both;"><span style="font-family: arial; font-size: medium;"><span>3) So, here</span><span> </span><span>a<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>,</span><span><sub> </sub>a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>2, </span><span>b<sub>1 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>,</span><span><sub> </sub>b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>3</span><span>, </span><span>c<sub>1</sub></span><span> </span><span>= </span><span>c</span><sub>1</sub><span>,</span><span><sub> </sub>c<sub>2 </sub></span><span>= </span><span>-</span><span><sub> </sub></span><span>8.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a<sub>1</sub>/a<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>a</span><sub>1</sub><span>/2</span><span> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>b<sub>1</sub>/b<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>b</span><sub>1</sub><span>/3</span><span>-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>c<sub>1</sub>/c<sub>2 </sub></span><span>=</span><span><sub> </sub></span><span>c</span><sub>1</sub><span>/(- 8)</span><span> </span><span>-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>4) As our lines are parallel, we have </span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif">=</span><span><sub> </sub>b<sub>1</sub>/b<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">=</span><span><sub> </sub></span><span>c<sub>1</sub>/c<sub>2</sub></span><span><sub>,</sub></span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>so, we have </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">= </span><span>b</span><sub>1</sub><span>/3. Let us take any values for </span><span>a</span><sub>1, </sub><span>b</span><sub>1 </sub><span>and </span><span>c</span><sub>1 </sub><span>which satisfies, </span><span>a</span><sub>1</sub><span>/2 </span><span face="Arial, sans-serif">= </span><span>b</span><sub>1</sub><span>/3</span><span> </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span>let </span><span>a</span><sub>1 </sub><span>= 4, </span><span>b</span><sub>1</sub><span> = 6, and </span><span>c</span><sub>1</sub><span>= -16.</span></span></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">5) So, our equation will be 4x + 6y - 16 = 0, the given equation is </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;">.</span></span></div><div><div><span style="font-size: medium;"><span style="font-family: arial;">6) We will simplify our equations </span><span style="font-family: arial;">2x + 3y – 8 = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">4x + 6y - 16 = 0</span><span style="font-family: arial;">,</span></span></div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">2</span><span style="font-family: arial;">x + 3</span><span style="font-family: arial;">y </span><span style="font-family: arial;">= 8</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, 3y = (8 - 2x)</span><span style="font-family: arial;"> </span></span></blockquote></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><span style="font-family: arial; font-size: medium;"> <span>y = (8 - 2x)/3 </span><span>-------------4</span></span></div></div></div></div></blockquote></blockquote><div><div><div><div><div><div><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">4x + 6y = 16</span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">6y = (16 - 4x)</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = </span><span style="font-family: arial;">(16 - 4x)</span><span style="font-family: arial;">/6</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></div></blockquote></div><div><div><span style="font-family: arial; font-size: medium;">7) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">a) We will take 3 points for y = (8 - 2x)/3.</span></blockquote><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s322/6-a-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="74" data-original-width="322" height="74" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW4N8XOGVuj1srkIVkoc1CL49rKegNdQKBCsIcyqshzzEtHCcHF5wC41KbgrkOH3f3boMQdGlvdLIaDTr3lpDROX6c2QIYW8VMJBS7B-1mvLPBuiLVkZGxiPANIe7B1hiwM4Vo6QFkzh4jSHrmCkQVzm5I8GHxnVxOdaxdqeIOegsXp0cvcAEMcOaD/s320/6-a-1.png" width="320" /></span></a></div></div></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div style="text-align: left;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>b) We will take 3 points for y = (16 - 4x)/6.</span></span></div></div></div></div></div></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR1p5tcpf3KGjuwGbv24_3jaLv3vijuwlPSA7drb98yPKCZtjoPNyvROr1uKRxKpNIwKFO-A1GwzAabi9nm-y3IrIagwx-WVGYyKtKty4warWhAIKgtJBF0wxMbf_Ov5aYsNmGBPOUeezzBEm29okKwEVK9Nxxm-7Sls2CLLJj1p0p9T85x6pGUVR3/s322/6-c-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="68" data-original-width="322" height="68" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR1p5tcpf3KGjuwGbv24_3jaLv3vijuwlPSA7drb98yPKCZtjoPNyvROr1uKRxKpNIwKFO-A1GwzAabi9nm-y3IrIagwx-WVGYyKtKty4warWhAIKgtJBF0wxMbf_Ov5aYsNmGBPOUeezzBEm29okKwEVK9Nxxm-7Sls2CLLJj1p0p9T85x6pGUVR3/s320/6-c-2.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>8) The graphical representation will be as follows.</span> </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8U568KXJjb1LygAT4Ior7m5ncReYi9jDqX1yLQdDMbuImh0qq4RhqXrEBDnvfTNJg7gs-Oexm1D0JRBTASZp7Ix77b-JWw9AJs4vQzTYBAoCD_o3X9bIpAZdjtuA7-2MQCVF8jEdtUsV06Etd9UmhdCMc1K3hb4NoGYu22UlgpkfAzSIFqTQlxT-m/s1223/6-c.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="899" data-original-width="1223" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8U568KXJjb1LygAT4Ior7m5ncReYi9jDqX1yLQdDMbuImh0qq4RhqXrEBDnvfTNJg7gs-Oexm1D0JRBTASZp7Ix77b-JWw9AJs4vQzTYBAoCD_o3X9bIpAZdjtuA7-2MQCVF8jEdtUsV06Etd9UmhdCMc1K3hb4NoGYu22UlgpkfAzSIFqTQlxT-m/s320/6-c.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">9) Here the lines are </span><span style="font-family: arial;">coincident</span><span style="font-family: arial;">.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;"><span><b>Q7. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the </b></span><b>coordinates of the vertices of the triangle formed by these lines and the x-axis, and </b><b>shade the triangular region.</b></span></div><div class="separator" style="clear: both; text-align: left;"><div><div><div><div><div><div><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><div><h3><span style="font-family: arial;">Explanation:</span></h3></div><div><div><div><span style="font-family: arial;"><span>1) For the equations, </span>a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub><span style="line-height: 16.05px;">2</span></sub>x + b<sub><span style="line-height: 16.05px;">2</span></sub>y + c<sub><span style="line-height: 16.05px;">2</span></sub><span style="line-height: 19.26px;"> </span>= 0<br /></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">a) </span><span style="font-family: arial;">If </span><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span style="font-family: arial;">b<sub>1</sub>/b<sub>2,</sub></span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">intersecting with unique solution, so </span><span style="font-family: arial;">they</span></div></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.<br /></span></div></blockquote></div></blockquote><div><div><h3><span style="font-family: arial;">Solution:</span></h3></div></div></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Write our equations as, </span><span style="font-family: arial;">x - </span><span style="font-family: arial;">y + 1 = 0; 3x + 2y - 12 = 0.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span style="font-family: arial;">a<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">3, </span><span style="font-family: arial;">b<sub>1 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">-</span><span style="font-family: arial;">1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">2</span><span style="font-family: arial;">, </span><span style="font-family: arial;">c<sub>1</sub></span><span style="font-family: arial;"> </span><span style="font-family: arial;">= 1</span><span style="font-family: arial;">,</span><span style="font-family: arial;"><sub> </sub>c<sub>2 </sub></span><span style="font-family: arial;">= </span><span style="font-family: arial;">-</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">12.</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a<sub>1</sub>/a<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1/3</span><span style="font-family: arial;"> -------------1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">b<sub>1</sub>/b<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">(- 1)</span><span style="font-family: arial;">/2</span><span style="font-family: arial;"> = -(1/2)</span><span style="font-family: arial;">-------------2 </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">c<sub>1</sub>/c<sub>2 </sub></span><span style="font-family: arial;">=</span><span style="font-family: arial;"><sub> </sub></span><span style="font-family: arial;">1/(- 12) = </span><span style="font-family: arial;">-(1/12)</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------3</span></span></div></blockquote><div><span style="font-family: arial; font-size: medium;"><span>3) From 1, 2, 3, we can say that (</span><span>a<sub>1</sub>/a<sub>2 </sub></span><span face="Arial, sans-serif" style="line-height: 19.9733px;">≠ </span><span>b<sub>1</sub>/b<sub>2</sub></span><span>)</span><span>, </span></span></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">the lines are </span><span style="font-family: arial;">intersecting with unique solution, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></div></div></blockquote><div><div><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">4) We will simplify our equations </span><span style="font-family: arial;">x - </span><span style="font-family: arial;">y + 1 = 0</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">3x + 2y - 12 = 0</span><span style="font-family: arial;">,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">x - </span><span style="font-family: arial;">y + 1 = 0</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">so, y = x + 1</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------4</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">3x + 2y</span><span style="font-family: arial;"> = 12</span></span></blockquote></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;">2y = 12 - 3x</span></div></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">y = (</span><span style="font-family: arial;">12 - 3x)/2</span><span style="font-family: arial;"> </span><span style="font-family: arial;">-------------5</span></span></div></blockquote><div><div><div><div><div><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">Now, we will represent these equations graphically.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;">a) We will take 3 points for y = </span><span style="font-family: arial;">x + 1</span><span style="font-family: arial;">.</span></span></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial; font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkNB8K2a8gmDsltMUnE49NxO8YQQhVl5fQSVCQG8A7rsSZsZG4rIXaehohY8CUk88iLn4x0VZyrE3lZ1wgdZUmrmiSXc9pHuYlHXwkDASwtKGQrjS3Kaobz1Zv3rDvUZbHf6Nn4WT97D4BrenMlF1tcS9pOYy3Y5Z8TFr96SL7-_pKtEHimyC5--4l/s294/7-1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="70" data-original-width="294" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkNB8K2a8gmDsltMUnE49NxO8YQQhVl5fQSVCQG8A7rsSZsZG4rIXaehohY8CUk88iLn4x0VZyrE3lZ1wgdZUmrmiSXc9pHuYlHXwkDASwtKGQrjS3Kaobz1Zv3rDvUZbHf6Nn4WT97D4BrenMlF1tcS9pOYy3Y5Z8TFr96SL7-_pKtEHimyC5--4l/s1600/7-1.png" width="294" /></a><span style="text-align: left;"> </span></span></div></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div><div><div><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b) We will take 3 points for y = </span><span style="font-family: arial;">(</span><span style="font-family: arial;">12 - 3x)/2</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></div></div></div></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYcvbo-XKxk276fNElcPNDuNL2NUM9F5O8hvqCL5HSSq8-SaLxII6JGbiaw9jpeb9a7RJMZkjgjFiFjCqKlDdyan5FYxZ3jq8uR5eGJn3xdq46qK-83K8Cm9zFqnyGQ5YlIL2f-9EO4esAE-wsiY94oDFYsknNBXucT3MrRQABklq0yc_h0KXTmkZ9/s293/7-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="70" data-original-width="293" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYcvbo-XKxk276fNElcPNDuNL2NUM9F5O8hvqCL5HSSq8-SaLxII6JGbiaw9jpeb9a7RJMZkjgjFiFjCqKlDdyan5FYxZ3jq8uR5eGJn3xdq46qK-83K8Cm9zFqnyGQ5YlIL2f-9EO4esAE-wsiY94oDFYsknNBXucT3MrRQABklq0yc_h0KXTmkZ9/s1600/7-2.png" width="293" /></span></a></div><span style="font-family: arial; font-size: medium;">6) The graphical representation will be as follows.</span></div><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitL0BnPoE_KC0FHvmclj44D9ynlQ33WzksWYm9M9IoXXCGiiqjwyYxMm_KsQ1bvaDXTCgC0YgJdj6YnBPw-qGbzUSc3IgStgaJhE2NOV39dqqxu74L4Y9mgu3QCzMxkqgH90pFp-np6-vjZrl7Djlk1InZZwZt5II-mNxI05ipFqCrVOHHufPhmCMH/s1225/7.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="899" data-original-width="1225" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitL0BnPoE_KC0FHvmclj44D9ynlQ33WzksWYm9M9IoXXCGiiqjwyYxMm_KsQ1bvaDXTCgC0YgJdj6YnBPw-qGbzUSc3IgStgaJhE2NOV39dqqxu74L4Y9mgu3QCzMxkqgH90pFp-np6-vjZrl7Djlk1InZZwZt5II-mNxI05ipFqCrVOHHufPhmCMH/s320/7.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div><div class="separator" style="clear: both;"><div class="separator" style="clear: both;"><div><div><span style="font-size: medium;"><span style="font-family: arial;">7) The lines are </span><span style="font-family: arial;">intersecting with unique solutions, so </span><span style="font-family: arial;">they </span><span style="font-family: arial;">are </span><span style="font-family: arial;">consistent</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">8) Here, </span><span style="font-family: arial;">the </span><span style="font-family: arial;">coordinates of the vertices of the triangle formed by these lines and the</span></span></div></div></div></div></div></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div><div class="separator" style="clear: both;"><div class="separator" style="clear: both;"><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">x-axis, and </span><span style="font-family: arial;">shade the triangular region are A(-1, 0), B (2, 3), and C(4, 0)</span><span style="font-family: arial;">.</span></span></div></div></div></div></div></div></div></div></div></blockquote><div><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/06/150-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.3</a></span></h2><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com04Q6FHQPM+PR-45.413157600000012 129.7846107-73.723391436178858 94.6283607 -17.102923763821167 164.9408607tag:blogger.com,1999:blog-2945240619290990604.post-34789833562643663772023-05-22T15:31:00.002+05:302023-06-14T13:12:30.550+05:30148-NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.1<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 3.1</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 3 Pair of Linear Equations in Two Variables</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/147-ncert-10-2-polynomials-ex-24.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-2-Polynomials - Ex-2.4</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 3.1</span></h3></div><div style="text-align: left;"><b><span style="font-family: arial; font-size: medium;">Q1. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically. </span></b></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) In such cases, always consider their present age.</span></div><div><span style="font-family: arial; font-size: medium;">2) In age-related problems, ago means subtracting those years from both ages.</span></div><div><span style="font-family: arial; font-size: medium;">3) In the case of hence or later any such words will tell us to add those ages in both. </span></div></div><div style="text-align: left;"><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) Let Aftab's present age be x and his daughter's age be y.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) 7 years ago their ages were (x - 7) and (y - 7).</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) According to the first relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x - 7) = 7(y - 7)</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x - 7) = 7y - 49</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">x-7+49 = 7y</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7y = x+42</span></div><span style="font-family: arial; font-size: medium;">y = (x+42)/7 ---------------------- equation 1</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>4) 3 years from now, their ages will be (x + 3) and (y + 3).</span> </span></div><div><span style="font-family: arial; font-size: medium;">5) According to the relation given in the problem,</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 3) = 3(y + 3)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">(x + 3) = 3y + 9</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">x+3-9 = 3y</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">3y = x-6</span></div><span style="font-family: arial; font-size: medium;">y = (x-6)/3 ---------------------- equation 2</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">a) We will take 2 points for y = (x + 42)/7.</span></div></blockquote><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiEX7Sm7LI8RGj0SPIaB0YKpTT1hAN1l5GE7fFAouuQwkr1hI2xfZVPSpeif35EEXjH54xVTdYs8Yu9EHy33H0RdfTacuhi4XU8Hpsh5p8ZGf-2vmBVEZ3COlT6sdTg0rMtjcy29bHD7OtFkleYJe5j9dtu-DZA0WGI3rH7IYRhg1qzK4GwLatQeGm/s423/0-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="81" data-original-width="423" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiEX7Sm7LI8RGj0SPIaB0YKpTT1hAN1l5GE7fFAouuQwkr1hI2xfZVPSpeif35EEXjH54xVTdYs8Yu9EHy33H0RdfTacuhi4XU8Hpsh5p8ZGf-2vmBVEZ3COlT6sdTg0rMtjcy29bHD7OtFkleYJe5j9dtu-DZA0WGI3rH7IYRhg1qzK4GwLatQeGm/w315-h60/0-1.png" width="315" /></span></a></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 2 points for y = (x - 6)/3.</span></div></blockquote><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLf76q-BxQjxp996gD6m9xfSlTNfpRbDsvV77H-9MAmCKjB3XIJtpH4SMtx2BJT1IcpCZcI9iZmXJDFkizPEETdyvDkOETf4RP7mvepJFuREq1x6ElCwe6EeqCqiHR4vdi3t26XTEgnQihvOCm16yXHjkEtW0JltZG6f2yRWVO0VmEkneUQY0-4kHe/s425/0-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="82" data-original-width="425" height="62" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLf76q-BxQjxp996gD6m9xfSlTNfpRbDsvV77H-9MAmCKjB3XIJtpH4SMtx2BJT1IcpCZcI9iZmXJDFkizPEETdyvDkOETf4RP7mvepJFuREq1x6ElCwe6EeqCqiHR4vdi3t26XTEgnQihvOCm16yXHjkEtW0JltZG6f2yRWVO0VmEkneUQY0-4kHe/s320/0-2.png" width="320" /></span></a></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">7) The graphical representation will be as follows.</span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPE4fOP6NpyiAZAhVj3RziuZuJFp-qM7ZUhZc3lRVRR4_mplwpJfxxTcI-5jk1X5BHNpQ-sQtcAUT_vZJ97Y5871nZ5bpq4DyJrHw3ogcFRHAO7xXNpn9myJVel6VIhwRF9Q11gk_szbiUNiwF3eemmGjOBGIos4LnDSf3gOoTESnZtXngIMZvwkz1/s1225/1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="787" data-original-width="1225" height="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPE4fOP6NpyiAZAhVj3RziuZuJFp-qM7ZUhZc3lRVRR4_mplwpJfxxTcI-5jk1X5BHNpQ-sQtcAUT_vZJ97Y5871nZ5bpq4DyJrHw3ogcFRHAO7xXNpn9myJVel6VIhwRF9Q11gk_szbiUNiwF3eemmGjOBGIos4LnDSf3gOoTESnZtXngIMZvwkz1/w344-h221/1.png" width="344" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>8) So their present ages are 42 years and 12 years.</span><br /><span><br /><div><b>Q2. The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 3 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically.</b></div></span></span><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Let x and y be the costs of the bat and balls.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equation.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these equations to get the values of x and y. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">1) Let the cost of the bat be Rs x and the cost of the ball be Rs y.</span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) As 3 bats and 6 balls cost Rs 3900, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3 x + 6 y = 3900</span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3 (x + 2 y) = 3900</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">(x + 2 y) = 1300 </span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2 y = (1300 - x)</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> y = (1300 - x)/2 </span><span style="font-family: arial;">------------ equation 1</span></span></div></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;">3) As 1 bat and 3 balls cost Rs 1300, we have,</span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">x + 3 y = 1300</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-family: arial; font-size: medium;"><span> </span>3 y = (1300 - x)</span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"> y = (1300 - x)/3 </span><span style="font-family: arial;">------------ equation 2</span></span></blockquote></blockquote><div><div style="text-align: left;"><div><span style="font-family: arial; font-size: medium;">4) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><span>a) We will take 2 points for y = (1300 - x)/2.</span> </span></div></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH1VTigE9kepRh7jUlntE9oID_vF_KnzMTs7GdO5S9pRqd5gC9mcQPg6xUspU_G-NbyPSgC2BhzJ0OX0jRs6gTYMvlUarh62BnfwJL7nfUt6mNesJM8Ren0n1ZlMXG2IVgwRnesSwyCmGrcGGTwlhhbeVBRfCAM9DhVkdzwR5MLagFuXAuDK7PKcGa/s347/2-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="85" data-original-width="347" height="72" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH1VTigE9kepRh7jUlntE9oID_vF_KnzMTs7GdO5S9pRqd5gC9mcQPg6xUspU_G-NbyPSgC2BhzJ0OX0jRs6gTYMvlUarh62BnfwJL7nfUt6mNesJM8Ren0n1ZlMXG2IVgwRnesSwyCmGrcGGTwlhhbeVBRfCAM9DhVkdzwR5MLagFuXAuDK7PKcGa/w295-h72/2-1.png" width="295" /></span></a></div></div><div style="text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">b) We will take 2 points for y = (1300 - x)/3.</span></div></blockquote></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr7PyOreB_emW1wgXnpSHnP4XyxZXHCOHHzGRhHiziCODOBZhuEGj7Nz4N42KnJi9y4BYOMh8OA1DjFMwuc825W2kaO6GblZofKBBrzHtCsa-6doDdCEAy3hOoTVVN85Rh1g4S002JQO3hCMyaIrtIwKVRr7Gjy0EXJVfEsdcIdDBPVLBMk0X18xk5/s345/2-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="83" data-original-width="345" height="69" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr7PyOreB_emW1wgXnpSHnP4XyxZXHCOHHzGRhHiziCODOBZhuEGj7Nz4N42KnJi9y4BYOMh8OA1DjFMwuc825W2kaO6GblZofKBBrzHtCsa-6doDdCEAy3hOoTVVN85Rh1g4S002JQO3hCMyaIrtIwKVRr7Gjy0EXJVfEsdcIdDBPVLBMk0X18xk5/w288-h69/2-2.png" width="288" /></span></a></div><span style="font-family: arial; font-size: medium;">5) The graphical representation will be as follows.<br /></span><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6l8wLEv8x506bAupzvt_RvQxRTDTBYrhAuCdAun_uAHaOBEARchmD5RZbgT1__mfykHMHZSz_VpQFtcfBV3MhMEh0rbcMwMrPm9XicthJuymbWxJmGerMqC3MDyPjKyttW1sQCeaT76m8r0dEF18PGtXFIZdy0rjMog7eAopbTAhuay58n2ZKiY2o/s1203/2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="890" data-original-width="1203" height="244" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6l8wLEv8x506bAupzvt_RvQxRTDTBYrhAuCdAun_uAHaOBEARchmD5RZbgT1__mfykHMHZSz_VpQFtcfBV3MhMEh0rbcMwMrPm9XicthJuymbWxJmGerMqC3MDyPjKyttW1sQCeaT76m8r0dEF18PGtXFIZdy0rjMog7eAopbTAhuay58n2ZKiY2o/w329-h244/2.png" width="329" /></span></a></div><span style="font-family: arial; font-size: medium;"><span>6) The point of intersection is (1300,0).</span><br /></span><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">Q3. The cost of 2 kg of apples and 1kg of grapes in a day was found to be Rs 160. After a </span><span style="font-family: arial;">month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation </span><span style="font-family: arial;">algebraically and geometrically.</span></span></b></div><div style="text-align: left;"><div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here, let x and y be the costs of apples and grapes.</span></div><div><span style="font-family: arial; font-size: medium;">2) Apply the given conditions and frame the equation.</span></div><div><span style="font-family: arial; font-size: medium;">3) We will get two equations from the above two conditions, then solve these equations to get the values of x and y. </span></div></div><div><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;">1) Let the cost of 1 kg of apple be Rs x and the cost of 1 kg of grapes be Rs y.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) As </span><span style="font-family: arial;">2 kg of apple</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">1 kg of grapes</span><span style="font-family: arial;"> cost Rs 160, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">2 x + y = 160</span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;"> y = 160 - 2x </span><span style="font-family: arial;">------------ equation 1</span></span></blockquote></blockquote><div><div><span style="font-size: medium;"><span style="font-family: arial;">3) As </span><span style="font-family: arial;">4 kg of apple</span><span style="font-family: arial;"> and </span><span style="font-family: arial;">2 kg of grapes</span><span style="font-family: arial;"> cost Rs 300, we have,</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">4 x + 2 y = 300</span></div></blockquote></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2 x + y = 300/2</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>2 x + y = 150</span> </span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"> <span>y = 150 - 2x </span><span>------------ equation 2</span></span></div></div></div></blockquote></blockquote><div><div style="text-align: left;"><div><div><div><span style="font-family: arial; font-size: medium;">4) Now, we will represent these equations graphically.</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;">a) We will take 2 points for y = (160 - 2x).</span></div></blockquote></div></div></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIYojyRLwTNnWtwygaPIfqo8mXRGTZK9yq27HzutajH5xT8gHZK7y5JpishqzvrQ7QjA4mbtaBpD5IlibhILV88vyqneygMxuAUpCpO-HZJbKhS629-OVsHtW82tiKryWNhkPeKnfoPulxhJI6AqOgMJMOY5sjA9L678mvd5kWOz9Pjz3_m9X5jef/s343/3-1.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="83" data-original-width="343" height="77" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIYojyRLwTNnWtwygaPIfqo8mXRGTZK9yq27HzutajH5xT8gHZK7y5JpishqzvrQ7QjA4mbtaBpD5IlibhILV88vyqneygMxuAUpCpO-HZJbKhS629-OVsHtW82tiKryWNhkPeKnfoPulxhJI6AqOgMJMOY5sjA9L678mvd5kWOz9Pjz3_m9X5jef/s320/3-1.png" width="320" /></span></a></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">b) We will take 2 points for y = (150 - 2x).</span></div></div></blockquote><div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitgLEsotw8RP08vxXapN-8AXZh1416vM5tlES8v4txTojRPzLLPcz1Cq0xh6u36lWsQqKAuJ1OI-9WngEzE4Zytg7c8FiQzk3bS-Xuq3vw5qebCuCPYSCg5LneOoGIeIvYkVBcKlDW47zLt9UmY1bMPqFJ4kdXErIaDpAjy9rFhzukCkuntc3d_Zhk/s343/3-2.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="81" data-original-width="343" height="76" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitgLEsotw8RP08vxXapN-8AXZh1416vM5tlES8v4txTojRPzLLPcz1Cq0xh6u36lWsQqKAuJ1OI-9WngEzE4Zytg7c8FiQzk3bS-Xuq3vw5qebCuCPYSCg5LneOoGIeIvYkVBcKlDW47zLt9UmY1bMPqFJ4kdXErIaDpAjy9rFhzukCkuntc3d_Zhk/s320/3-2.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">5) The graphical representation will be as follows.</span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigcKbMdNM93WhjB8b-BonuX1afp7i2XbUz84AFk8PnV5ntNeeRndNFYNfvymVfG5H3hBJ5CYzrvIYZl2neVJZjEjm-_NCZRhOx29OMPrGxXtgHcR-1T5R5IPnPVpltSP5aDf74gV2lFc8AWVnbrw4BnMj24-OuPyRkDeuSScb7y9n40VK-hhihaxLT/s1219/3.png" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial; font-size: medium;"><img border="0" data-original-height="901" data-original-width="1219" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigcKbMdNM93WhjB8b-BonuX1afp7i2XbUz84AFk8PnV5ntNeeRndNFYNfvymVfG5H3hBJ5CYzrvIYZl2neVJZjEjm-_NCZRhOx29OMPrGxXtgHcR-1T5R5IPnPVpltSP5aDf74gV2lFc8AWVnbrw4BnMj24-OuPyRkDeuSScb7y9n40VK-hhihaxLT/s320/3.png" width="320" /></span></a></div><span style="font-family: arial; font-size: medium;">6) Here the lines are parallel, so there will not be any point common.</span><div style="text-align: left;"><div><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/149-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.2</a></span></h2></div><div><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE</span></a></div></div></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0tag:blogger.com,1999:blog-2945240619290990604.post-59809491606011382212023-05-08T11:12:00.005+05:302023-06-14T13:13:24.752+05:30147-NCERT-10-2-Polynomials - Ex-2.4<h2 style="clear: both; color: #0400ff;"><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">NCERT</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">10th Mathematics</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Exercise 2.4</span></div><div style="clear: both; color: black; font-size: medium; font-weight: 400;"><span style="font-family: arial; font-size: medium;">Topic: 2 Polynomials</span></div></h2><h2 style="clear: both; color: #0400ff;"><span style="font-family: arial; font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/146-ncert-10-2-polynomials-ex-23.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span> ⇨ NCERT-10-2-Polynomials - Ex-2.3</a></span></h2><div></div><div><h3><span style="font-family: arial; font-size: medium;">EXERCISE 2.4</span></h3><div><span style="font-family: arial; font-size: medium;"><b><div>1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:</div><div>(i) <span face="Arial, sans-serif" style="font-size: 13.5pt; line-height: 107%;">2x<sup>3</sup> + x<sup>2</sup> – 5x + 2</span>; <span> </span>1/2, 1, -2</div><div>(ii) <span face="Arial, sans-serif" style="font-size: 13.5pt; line-height: 107%;">x<sup>3</sup> – 4x<sup>2</sup> + 5x – 2</span>; <span> </span>2, 1, 1</div></b></span></div><div><span style="font-family: arial; font-size: medium;"><div style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: arial; font-size: medium;"><h3 style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></span></div><div style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: arial; font-size: medium;"></span></div></span></div><div><span style="font-family: arial; font-size: medium;"><span>1) </span><span>We know that if </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰, </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃</span><span>, and </span><span style="white-space: pre-wrap;">𝜸</span><span> are the zeroes of the cubic polynomial</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">ax</span><sup>3</sup><span face="Arial, sans-serif"> + bx</span><sup>2</sup><span face="Arial, sans-serif">
+ cx + d, </span><span>then </span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-family: arial; font-size: medium;"><div style="text-align: left;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰 + </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="white-space: pre-wrap;">𝜸 = - b/a = [- (</span><span><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup>2</sup><span style="white-space: pre-wrap;">)/(</span><span><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup>3</sup><span style="white-space: pre-wrap;">)]</span></div></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰</span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="text-align: center; white-space: pre-wrap;">𝛃</span><span style="white-space: pre-wrap;">𝜸 </span><span style="text-align: center; white-space: pre-wrap;">+ </span><span style="white-space: pre-wrap;">𝜸</span><span style="text-align: center; white-space: pre-wrap;">𝝰 </span><span style="white-space: pre-wrap;">= c/a </span><span style="white-space: pre-wrap;">= [(</span><span><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><span style="white-space: pre-wrap;">)/(</span><span><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup>3</sup><span style="white-space: pre-wrap;">)]<br /></span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰 x </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="white-space: pre-wrap;">𝜸 = - d/a </span><span style="white-space: pre-wrap;">= [- (constant</span><span style="white-space: pre-wrap;">)/(</span><span><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup>3</sup><span style="white-space: pre-wrap;">)]</span></span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">2) If <span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰, </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃</span>, and <span style="white-space: pre-wrap;">𝜸</span> are the zeroes of the cubic polynomial</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">p(x) = ax<sup>3</sup> + bx<sup>2</sup> + cx + d, then </span><span style="font-family: arial;">p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰) = p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃) = p(</span><span style="font-family: arial; white-space: pre-wrap;">𝜸) = 0.</span></span></div></blockquote><h3><span style="font-family: arial; font-size: medium;"><span>Solution:</span></span></h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700;">(i) </span><span face="Arial, sans-serif" style="font-weight: 700; line-height: 19.26px;">2x<sup>3</sup> + x<sup>2</sup> – 5x + 2</span><span style="font-family: arial; font-weight: 700;">; </span><span style="font-family: arial; font-weight: 700;"> </span><span style="font-family: arial; font-weight: 700;">1/2, 1, -2</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) Let </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="font-family: arial; text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">𝝰 = 1/2, </span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 = 1</span><span style="font-family: arial;">, and </span><span style="font-family: arial; white-space: pre-wrap;">𝜸 = -2</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span style="white-space: pre-wrap;">2) Let p(x) = </span></span><span face="Arial, sans-serif">2x</span><sup>3</sup><span face="Arial, sans-serif"> + x</span><sup>2</sup><span face="Arial, sans-serif"> – 5x + 2</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="text-align: center; white-space: pre-wrap;">1/2) = </span><span face="Arial, sans-serif">2(1/2)</span><sup>3</sup><span face="Arial, sans-serif"> + (1/2)</span><sup>2</sup><span face="Arial, sans-serif"> – 5(1/2) + 2</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">2(1/8)</span><span face="Arial, sans-serif"> + (1/4)</span><span face="Arial, sans-serif"> – 5(1/2) + 2</span> </span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(1/4)</span><span face="Arial, sans-serif"> + (1/4)</span><span face="Arial, sans-serif"> – (5/2) + 2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= [</span><span face="Arial, sans-serif">(1 + 1 - 10 + 8)/4]</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= [</span><span face="Arial, sans-serif">(0)/4]</span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that 1/2 is the zeroes of p(x).</span> </span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="text-align: center; white-space: pre-wrap;">1) = </span><span face="Arial, sans-serif">2(1)</span><sup>3</sup><span face="Arial, sans-serif"> + (1)</span><sup>2</sup><span face="Arial, sans-serif"> – 5(1) + 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">2(1)</span><span face="Arial, sans-serif"> + (1)</span><span face="Arial, sans-serif"> – 5(1) + 2</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(2)</span><span face="Arial, sans-serif"> + (1)</span><span face="Arial, sans-serif"> – (5) + 2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= [</span><span face="Arial, sans-serif">(2 + 1 - 5 + 2)]</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that 1 is the zeroes of p(x).</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="text-align: center; white-space: pre-wrap;">-2) = </span><span face="Arial, sans-serif">2(-2)</span><sup>3</sup><span face="Arial, sans-serif"> + (</span><span face="Arial, sans-serif">-2</span><span face="Arial, sans-serif">)</span><sup>2</sup><span face="Arial, sans-serif"> – 5(</span><span face="Arial, sans-serif">-2</span><span face="Arial, sans-serif">) + 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">2(-8)</span><span face="Arial, sans-serif"> + (4)</span><span face="Arial, sans-serif"> – 5(-2) + 2</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(-16)</span><span face="Arial, sans-serif"> + (4)</span><span face="Arial, sans-serif"> – (-10) + 2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= [</span><span face="Arial, sans-serif">(-16 + 4 + 10 + 2)]</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;"><span style="text-align: center; white-space: pre-wrap;">= [</span><span face="Arial, sans-serif">(-16 + 16)]</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that -2 is the zeroes of p(x).</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">3) So 1/2, 1, -2 are the zeroes of the cubic polynomial <span style="white-space: pre-wrap;">p(x) = </span>2x<sup>3</sup> + x<sup>2</sup> – 5x + 2.</span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">4) Here </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="font-family: arial; text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">𝝰 = 1/2, </span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 = 1</span><span style="font-family: arial;">, and </span><span style="font-family: arial; white-space: pre-wrap;">𝜸 = -2 and a = 2, b = 1, c = - 5, and d = 2.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) Now we will verify the relation between zeroes and </span><span style="font-family: arial;">the coefficients.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>a) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰 + </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="white-space: pre-wrap;">𝜸 = - b/a</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰 + </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="white-space: pre-wrap;">𝜸</span><span> </span></span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2 + </span></span></span><span style="text-align: center; white-space: pre-wrap;">1 - </span><span style="white-space: pre-wrap;">2</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2 -------------------- equation 1</span></span></span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">- b/a</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>- 1/2</span></span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">1/2 -------------------- equation 2</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">From equation 1 and equation 2, LHS = RHS.</span></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= c/a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= [(</span><span style="font-family: arial;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2) x (1)] + [(</span></span></span><span style="text-align: center; white-space: pre-wrap;">1) x (- </span><span style="white-space: pre-wrap;">2)] + [</span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">(- </span><span style="font-family: arial; white-space: pre-wrap;">2) x (1/2)]</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= [(</span><span style="font-family: arial;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2)] + [(</span></span></span><span style="text-align: center; white-space: pre-wrap;">- </span><span style="white-space: pre-wrap;">2)] + [</span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">(- </span><span style="font-family: arial; white-space: pre-wrap;">1)]</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2) - 3</span></span></span></span> </span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">5/2 -------------------- equation 3</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">c/a</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">- 5)/2</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">5/2 -------------------- equation 4</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">From equation 3 and equation 4, LHS = RHS.</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - d/a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>1/2) x (</span></span></span><span style="text-align: center; white-space: pre-wrap;">1) x (- </span><span style="white-space: pre-wrap;">2)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">1 -------------------- equation 5</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">- d/a</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">- 2/2</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= - </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">1 -------------------- equation 6</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">From equation 5 and equation 6, LHS = RHS.</span> </span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) So, the relation between zeroes and coefficient is verified.</span></div><div style="text-align: left;"><span style="font-size: medium;"> </span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial; font-weight: 700;">(ii) </span><span face="Arial, sans-serif" style="font-weight: 700; line-height: 19.26px;">x<sup>3</sup> – 4x<sup>2</sup> + 5x – 2</span><span style="font-family: arial; font-weight: 700;">; </span><span style="font-family: arial; font-weight: 700;"> </span><span style="font-family: arial; font-weight: 700;">2, 1, 1</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) Let </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="font-family: arial; text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">𝝰 = 2, </span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 = 1</span><span style="font-family: arial;">, and </span><span style="font-family: arial; white-space: pre-wrap;">𝜸 = 1</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">2) Let p(x) = </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4x</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">+ 5x – 2</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">2) = </span><span face="Arial, sans-serif">(2)</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(2)</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + 5(2) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(8)</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(4)</span><span face="Arial, sans-serif"> + 5(2) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="text-align: center; white-space: pre-wrap;">= </span></span><span face="Arial, sans-serif">8</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 16</span><span face="Arial, sans-serif"> + 10 </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-align: center; white-space: pre-wrap;">= 18 - 18</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that 2 is the zeroes of p(x).</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">1) = </span><span face="Arial, sans-serif">(1)</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(1)</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + 5(1) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(1)</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(1)</span><span face="Arial, sans-serif"> + 5(1) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="text-align: center; white-space: pre-wrap;">= </span></span><span face="Arial, sans-serif">1</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif"> + 5 </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">6</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 6</span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that 1 is the zeroes of p(x).</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span face="Arial, sans-serif">p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">1) = </span><span face="Arial, sans-serif">(1)</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(1)</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + 5(1) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">(1)</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif">(1)</span><span face="Arial, sans-serif"> + 5(1) </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span> </span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span style="font-family: arial;"><span style="text-align: center; white-space: pre-wrap;">= </span></span><span face="Arial, sans-serif">1</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4</span><span face="Arial, sans-serif"> + 5 </span><span face="Arial, sans-serif">–</span><span face="Arial, sans-serif"> 2</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial; text-align: center; white-space: pre-wrap;">= </span><span face="Arial, sans-serif">6</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 6</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium; text-align: center; white-space: pre-wrap;">= 0 ---------------- this shows that 1 is the zeroes of p(x).</span></blockquote></blockquote><div><span style="font-size: medium;"><span style="font-family: arial;">3) So 2, 1, 1 are the zeroes of the cubic polynomial <span style="white-space: pre-wrap;">p(x) = </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4x</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">+ 5x – 2</span><span style="font-family: arial;">.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">4) Here </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="font-family: arial; text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">𝝰 = 2, </span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 = 1</span><span style="font-family: arial;">, and </span><span style="font-family: arial; white-space: pre-wrap;">𝜸 = 1 and a = 1, b = - 4, c = 5, and d = - 2.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">5) Now we will verify the relation between zeroes and </span><span style="font-family: arial;">the coefficients.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">a) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - b/a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>2 + </span></span></span><span style="text-align: center; white-space: pre-wrap;">1 + </span><span style="white-space: pre-wrap;">1</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 4</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 1</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">- (- 4)/1</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 4</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 2</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">From equation 1 and equation 2, LHS = RHS.</span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">b) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= c/a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= [(</span><span style="font-family: arial;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>2) x (1)] + [(</span></span></span><span style="text-align: center; white-space: pre-wrap;">1) x (1</span><span style="white-space: pre-wrap;">)] + [</span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">(1</span><span style="font-family: arial; white-space: pre-wrap;">) x (2)]</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= [(</span><span style="font-family: arial;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>2)] + [(1</span></span></span><span style="white-space: pre-wrap;">)] + [</span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">(2</span><span style="font-family: arial; white-space: pre-wrap;">)]</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">= 2 + 1 + 2</span> </span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 5</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 3</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">c/a</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">5)/1</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 5</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 4</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">From equation 3 and equation 4, LHS = RHS.</span> </span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">c) </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - d/a</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">LHS = </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= (</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>2) x (</span></span></span><span style="text-align: center; white-space: pre-wrap;">1) x (1</span><span style="white-space: pre-wrap;">)</span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 2</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 5</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">RHS = </span><span style="font-family: arial; white-space: pre-wrap;">- d/a</span><span style="font-family: arial;"> </span></span></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= </span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">- (- 2)/1</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">= 2</span><span style="font-family: arial; font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"> -------------------- equation 6</span></span></span></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; font-size: medium;">From equation 5 and equation 6, LHS = RHS.</span></blockquote></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">6) So, the relation between zeroes and coefficient is verified.</span></div><div style="text-align: left;"><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><span style="font-family: arial;"><b>Q2. Find a cubic polynomial with the sum, the sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.</b></span> </span></div><div><div><span style="font-family: arial; font-size: medium;"><div style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;"><h3 style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></span></div><div style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;"></span></div></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial;">We know that if </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰, </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: arial;">, and </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial;"> are the zeroes of the cubic polynomial </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span face="Arial, sans-serif">ax</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> + bx</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + cx + d, </span><span style="font-family: arial;">then </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - b/a = [- (</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">2</sup><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= c/a </span><span style="font-family: Arial; white-space: pre-wrap;">= [(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]<br /></span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - d/a </span><span style="font-family: Arial; white-space: pre-wrap;">= [- (constant</span><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) If <span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰, </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃</span>, and <span style="white-space: pre-wrap;">𝜸</span> are the zeroes of the cubic polynomial</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">p(x) = ax<sup>3</sup> + bx<sup>2</sup> + cx + d, then </span><span style="font-family: arial;">p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰) = p(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃) = p(</span><span style="font-family: arial; white-space: pre-wrap;">𝜸) = 0.</span></span></blockquote><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) Let </span><span face="Arial, sans-serif">ax</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> + bx</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + cx + d be our </span><span style="font-family: arial;">cubic polynomial.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) Here </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = 2, </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= - 7, </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - 14.</span></span></div><div><div><span style="font-family: arial; font-size: medium;">3) We know that </span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - b/a</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="text-align: left;"><span style="font-family: Arial;"><span style="font-size: medium; white-space: pre-wrap;">2/1 = - b/a ------------- therefore a = 1, and b = -2.</span></span></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= c/a</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: Arial; font-size: medium; white-space: pre-wrap;">-7/1 = c/a ------------- therefore a = 1, and c = - 7.</span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"> </span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - d/a</span></span></div><div style="text-align: left;"><span style="font-family: Arial; font-size: medium; white-space: pre-wrap;">-14/1 = -d/a ------------- therefore a = 1, and c = 14.</span></div></blockquote><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">4) our cubic polynomial will be x<sup>3</sup> - 2x<sup>2</sup> - 7x + 14. </span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><b>Q3. If the zeroes of the polynomial </b></span><span style="font-family: arial;"><b>x<sup>3</sup> – 3x<sup>2</sup> + x + 1</b></span><span style="font-family: arial;"><b> are a – b, a, a + b, find a and b.</b></span></span></div><div style="text-align: left;"><div><div><span style="font-family: arial; font-size: medium;"><div style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;"><h3 style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;">Explanation:</span></h3></span></div><div style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;"></span></div></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial;">We know that if </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰, </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: arial;">, and </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial;"> are the zeroes of the cubic polynomial </span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span face="Arial, sans-serif">px</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> + qx</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + rx + s, </span><span style="font-family: arial;">then </span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - q/p = [- (</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">2</sup><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰</span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃</span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 </span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">+ </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰 </span><span style="font-family: Arial; white-space: pre-wrap;">= r/p </span><span style="font-family: Arial; white-space: pre-wrap;">= [(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]<br /></span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - s/p </span><span style="font-family: Arial; white-space: pre-wrap;">= [- (constant</span><span style="font-family: Arial; white-space: pre-wrap;">)/(</span><span style="font-family: Arial;"><span style="white-space: pre-wrap;">coeficient of </span></span><span face="Arial, sans-serif">x</span><sup style="font-family: Arial, sans-serif;">3</sup><span style="font-family: Arial; white-space: pre-wrap;">)]</span></span></blockquote><div><span style="font-family: arial; font-size: medium;">2) If <span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span>𝝰, </span></span></span><span style="text-align: center; white-space: pre-wrap;">𝛃</span>, and <span style="white-space: pre-wrap;">𝜸</span> are the zeroes of the cubic polynomial</span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span style="font-family: arial;">f(x) = </span><span face="Arial, sans-serif">px</span><sup style="font-family: Arial, sans-serif;">3</sup><span face="Arial, sans-serif"> + qx</span><sup style="font-family: Arial, sans-serif;">2</sup><span face="Arial, sans-serif"> + rx + s</span><span style="font-family: arial;">, then </span><span style="font-family: arial;">f(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝝰) = f(</span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃) = f(</span><span style="font-family: arial; white-space: pre-wrap;">𝜸) = 0.</span></span></blockquote><h3><span style="font-family: arial; font-size: medium;">Solution:</span></h3><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1) (</span><span style="font-family: arial;">a – b), (a), (a + b) are zeroes of </span><span style="font-family: arial;">polynomial </span><span style="font-family: arial;">x<sup>3</sup> – 3x<sup>2</sup> + x + 1.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">2) So, </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 = ( a - b), </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 = a, and </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = ( a + b), p = 1, q = - 3, r = 1, s = 1.</span></span></div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: Arial; white-space: pre-wrap;">3) As </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 + </span></span></span><span style="font-family: arial; text-align: center; white-space: pre-wrap;">𝛃 + </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - q/p</span></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><div style="text-align: left;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">(</span></span></span><span style="font-family: arial;">a – b) + (a) + (a + b)</span><span style="font-family: Arial; white-space: pre-wrap;"> = - (- 3)/1</span></span></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">(3</span></span></span><span style="font-family: arial;">a)</span><span style="font-family: Arial; white-space: pre-wrap;"> = 3</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;">So, a = 1 --------------- equation 1.</span></div></blockquote><div><span style="font-size: medium;"><span style="font-family: Arial; white-space: pre-wrap;">4) As </span><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">𝝰 x </span></span></span><span style="font-family: Arial; text-align: center; white-space: pre-wrap;">𝛃 x </span><span style="font-family: Arial; white-space: pre-wrap;">𝜸 = - s/p</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">(</span></span></span><span style="font-family: arial;">a – b) x (a) x (a + b)</span><span style="font-family: Arial; white-space: pre-wrap;"> = - (1)/1</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">(</span></span></span><span style="font-family: arial;">a</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;"> – b</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;">)</span><span style="font-family: Arial; white-space: pre-wrap;"> x a = - 1 </span><span style="font-family: arial;">--------------- equation 2.</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">From </span><span style="font-family: arial;">equation 1, put a = 1 in </span><span style="font-family: arial;">equation 2, we get,</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: medium;"><span id="docs-internal-guid-986db003-7fff-1477-7737-9301d16941bb" style="text-align: center;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: arial;">(</span></span></span><span style="font-family: arial;">1</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;"> – b</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;">)</span><span style="font-family: Arial; white-space: pre-wrap;"> x 1 = - 1</span></span></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">1</span><span style="font-family: arial;"> – b</span><sup style="font-family: arial;">2</sup><span style="font-family: Arial; white-space: pre-wrap;"> = - 1</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">b</span><sup style="font-family: arial;">2</sup><span style="font-family: Arial; white-space: pre-wrap;"> = 2</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">So, b = </span><span style="font-family: arial;">±</span><span style="font-family: arial;"><span style="white-space: pre-wrap;"> </span><span style="white-space: pre-wrap;">√2.</span></span></span></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">So, a = 1 and </span><span style="font-family: arial;">b = </span><span style="font-family: arial;"> </span><span style="font-family: arial;">±</span><span style="font-family: arial; white-space: pre-wrap;"> </span><span style="font-family: arial; white-space: pre-wrap;">√2</span><span style="font-family: arial;">.</span></span></div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><br /></span></div><div style="text-align: left;"><b><span style="font-size: medium;"><span style="font-family: arial;">Q</span><span style="font-family: arial;">4. If two zeroes of the polynomial </span><span style="line-height: 107%;"><span style="font-family: arial;">x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> +
138x – 35</span></span><span style="font-family: arial;"> are 2 ± </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3, find other zeroes.</span></span></b></div><h3><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;">1) Let p(x) = </span><span face="Arial, sans-serif"><span style="font-family: "Times New Roman";"><span style="line-height: 17.12px;"><span style="font-family: arial;">x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35</span></span></span> and </span><span style="font-family: arial;"><span style="white-space: pre-wrap;"><span style="font-family: "Times New Roman"; white-space: normal;"><span style="font-family: arial;">2 - </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3</span></span> and </span><span style="font-family: "Times New Roman";"><span style="font-family: arial;">2 + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3</span></span><span style="white-space: pre-wrap;"> are zeroes of p(x).</span></span></span></div><div style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;">2) So, [x - </span></span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 - </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">] x [x + </span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 + </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">] is a factor of </span>x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35<span face="Arial, sans-serif">.</span></span></div><div style="font-family: arial;"><span face="Arial, sans-serif" style="font-size: medium;">3) Now we will find other factors of p(x).</span></div><div style="font-family: arial;"><h3 style="font-family: "Times New Roman";"><span style="font-family: arial; font-size: medium;">Solution:</span></h3></div><div style="font-family: arial;"><span style="font-size: medium;"><span style="font-family: arial;">1) </span><span style="font-family: arial;"><span style="white-space: pre-wrap;">Here </span></span><span style="font-family: arial;"><span style="white-space: pre-wrap;">[x - </span></span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 - </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">] x [x + </span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 + </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">]</span><span style="white-space: pre-wrap;"> is a factor of </span>x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35<span face="Arial, sans-serif">.</span></span></div><div style="font-family: arial;"><span style="font-size: medium;"><span style="white-space: pre-wrap;">2) </span><span style="font-family: arial;"><span style="white-space: pre-wrap;">[x - </span></span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 - </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">] x [x + </span><span style="white-space: pre-wrap;">(</span><span style="font-family: arial;">2 + </span><span style="white-space: pre-wrap;">√</span>3)<span style="white-space: pre-wrap;">]</span><span style="white-space: pre-wrap;"> </span><span style="white-space: pre-wrap;"> = </span><span style="font-family: arial;"><span style="white-space: pre-wrap;">[(x - </span></span><span style="font-family: arial;">2) - </span><span style="white-space: pre-wrap;">√</span>3<span style="white-space: pre-wrap;">] x [(x + </span><span style="font-family: arial;">2) + </span><span style="white-space: pre-wrap;">√</span>3<span style="white-space: pre-wrap;">]</span><span face="Arial, sans-serif">.</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-family: arial; text-align: left;"><span style="font-size: medium;"><span face="Arial, sans-serif"><span style="white-space: pre-wrap;"> = </span><span style="font-family: arial;"><span style="white-space: pre-wrap;">(x - </span></span><span style="font-family: arial;">2)</span></span><sup>2</sup><span style="font-family: arial;"> - (</span><span style="white-space: pre-wrap;">√</span>3<span face="Arial, sans-serif"><span style="font-family: arial;">)</span></span><sup>2</sup></span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;"> = </span><span style="font-family: arial;"><span style="white-space: pre-wrap;">(</span></span></span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;"> - 4x + 4)</span><span style="font-family: arial;"> - (<span style="white-space: pre-wrap;">3</span></span><span style="font-family: arial;"><span style="font-family: arial;">)</span></span></span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"><span style="white-space: pre-wrap;"> = </span></span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;"> - 4x + 1</span> </span></div></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><span style="font-size: medium;"><span style="font-family: arial;">3) So, </span><span style="font-family: arial; white-space: pre-wrap;">(</span><span style="font-family: arial;">x</span><sup style="font-family: arial;">2</sup><span style="font-family: arial;"> - 4x + 1</span><span style="font-family: arial;">) divides the polynomial </span><span style="font-family: arial;"><span>x</span><sup>4</sup><span> – 6x</span><sup>3</sup><span> – 26x</span><sup>2</sup><span> + 138x – 35</span>.</span></span><div><span style="font-family: arial;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuMbvO0ADF781BuPmUQLKdK2CWK-GB8ox1Lk4qxUfQMzzCE5qyax0Dr_Fpgqv69XTolVdzcv2pElHkxBSjfH5BhdMFla0DmBaXgY2VueEYpfjbxgGrQIq0vYRNeNMlT5NmRlwq7znn_t2hJ60BhoTi4H5ZscqDhRQLBQiOyGuLPqObZxhFAD7pxCv2/s543/4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="303" data-original-width="543" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuMbvO0ADF781BuPmUQLKdK2CWK-GB8ox1Lk4qxUfQMzzCE5qyax0Dr_Fpgqv69XTolVdzcv2pElHkxBSjfH5BhdMFla0DmBaXgY2VueEYpfjbxgGrQIq0vYRNeNMlT5NmRlwq7znn_t2hJ60BhoTi4H5ZscqDhRQLBQiOyGuLPqObZxhFAD7pxCv2/w351-h196/4.png" width="351" /></a></div></span><div><span><div><span style="font-size: medium;"><div><span style="font-family: arial;">4) Here <span style="white-space: pre-wrap;">(</span><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 4x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">1) is one factor of </span><span style="font-family: arial;">x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35 and the other one</span></div></span></div></span></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><span><div><span style="font-size: medium;"><div style="text-align: left;"><span style="font-family: arial;">is (</span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 2x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">35). </span></div></span></div></span></div></div></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) Now we will find all factors of </span><span style="font-family: arial;">(</span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 2x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">35).</span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> </span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 2x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">35 = </span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 7x + 5x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">35</span></span></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= (</span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><sup>2</sup><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 7x) + (5x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">35)</span></span></div></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">= x(</span><span style="font-family: arial;"><span face="Arial, sans-serif">x</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 7) + 5(x </span></span><span style="font-family: arial;">– </span><span style="font-family: arial;">7)</span></span></div></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>= (</span><span><span face="Arial, sans-serif">x</span><span face="Arial, sans-serif"> </span><span face="Arial, sans-serif">– 7) x (x </span></span><span>+ </span><span>5)</span> </span></div></blockquote></blockquote></blockquote></blockquote><div><div><span style="font-family: arial; font-size: medium;"><span>6) Here x = (<span><span><span style="font-family: arial; white-space: normal;">2 - </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3), x =<span style="white-space: pre-wrap;">(</span></span></span></span><span><span style="font-family: arial;">2 + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3)</span></span>, x = - 5, and x = 7 are the other zeroes of </span><span face="Arial, sans-serif">x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35.</span></span></div><div><span style="font-size: medium;"><span style="font-family: arial;">7) All zeroes of </span><span style="font-family: arial;">x<sup>4</sup> – 6x<sup>3</sup> – 26x<sup>2</sup> + 138x – 35 are </span><span style="font-family: arial; white-space: pre-wrap;"><span style="white-space: normal;">(</span><span style="white-space: normal;"><span><span style="font-family: arial;">2 - </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3), <span style="white-space: pre-wrap;">(</span></span></span></span><span style="white-space: normal;"><span style="font-family: arial;">2 + </span><span style="font-family: arial; white-space: pre-wrap;">√</span><span style="font-family: arial;">3)</span></span><span style="white-space: normal;">, x = - 5, and x = 7</span>.</span></span></div><div><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><br /></span></span></div><div><b><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;">Q5. If the polynomial </span></span><span style="font-family: arial; font-size: medium;">x<sup>4</sup> – 6x<sup>3</sup> + 16x<sup>2</sup>
– 25x + 10 </span></b><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>is divided by another polynomial </b></span></span><span style="line-height: 107%;"><span style="font-family: arial; font-size: medium;"><b>x<sup>2</sup> – 2x + k</b></span></span><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"><b>, the remainder comes out to be x + a, find k and a.</b></span></span></div><div><div><div style="line-height: normal; margin-bottom: 0cm;"><h3 style="font-size: medium;"><span style="font-family: arial; font-size: medium;">Explanation:</span></h3><div><span style="font-family: arial; font-size: medium;">1) We know that Dividend = Divisor × Quotient + Remainder</span></div><div><span style="font-family: arial;"><span style="font-size: medium;">2) </span><span style="font-size: medium;">If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find</span></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="line-height: normal; margin-bottom: 0cm;"><div><span style="font-family: arial; font-size: medium;">polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x).</span></div></div></div></blockquote><div><div style="line-height: normal; margin-bottom: 0cm;"><h3 style="text-align: left;"><span style="font-family: arial; font-size: large;">Solution:</span></h3><div><span style="font-family: arial; font-size: medium;">1) Here p(x) = </span><span style="font-family: arial; font-size: medium;">x<sup>4</sup> – 6x<sup>3</sup> + 16x<sup>2</sup> – 25x + 10 and g(x) = </span><span style="font-family: arial; font-size: medium;">x<sup>2</sup> – 2x + k.</span></div><div><span style="font-size: medium;"><span style="font-family: arial;">2) Now we will find the </span><span style="font-family: arial;">quotient and remainder using long division.</span></span></div><div><span style="font-family: arial; font-size: medium;">3) Here remainder r(x) is given as r(x) = (x + a) ---------------------equation 1</span></div></div></div></div><div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7gEwssA_eetLQ4GJRNw91_SbfPj4lC6ZrzHGYZIvAu8nWTgYo3dgipfMnsYSUZmYKiLVtH6VAMlU8CNnBQb1O8qkKQmPOQ5xtgMSD28L_cIQ9X4-uEv-RPIE7nF6iZKDX2d9eDGLPsnOSpGlNASr9QFCjQzIA92eYift9-RBh2GEx5je6iyEjJ3WI/s588/5.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="285" data-original-width="588" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7gEwssA_eetLQ4GJRNw91_SbfPj4lC6ZrzHGYZIvAu8nWTgYo3dgipfMnsYSUZmYKiLVtH6VAMlU8CNnBQb1O8qkKQmPOQ5xtgMSD28L_cIQ9X4-uEv-RPIE7nF6iZKDX2d9eDGLPsnOSpGlNASr9QFCjQzIA92eYift9-RBh2GEx5je6iyEjJ3WI/w370-h179/5.png" width="370" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial; font-size: medium;">4) From equation 1, we have r(x) = (x + a),</span></div></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span>so, </span><span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">(-9+2k)x + (10-8k</span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">+</span></span><span>k</span><sup>2</sup><span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">)</span> = (x + a) ------------- equation 2</span></span></div></div></div></blockquote></blockquote><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;">5) </span><span style="font-family: arial;">From equation 2, we have,</span><span style="font-family: arial;"> </span></span></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-size: medium;"><span style="font-family: arial;"> </span><span style="font-family: arial; white-space: pre-wrap;">(-9 + 2k) = 1</span></span></div></div></div></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div style="text-align: left;"><span style="font-family: arial; font-size: medium; white-space: pre-wrap;">2k = 10</span></div></div></div></blockquote></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span style="white-space: pre-wrap;"> k = 5</span> <span>------------- equation 3</span></span></div></blockquote></blockquote></blockquote></blockquote><div><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;"><div><div><span style="font-family: arial; font-size: medium;">6) From equation 2, we have,</span></div></div></span></div></div></span></div></div></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div><div><div><span style="font-family: arial;"><div><div><span style="font-family: arial; font-size: medium;"><div><div style="text-align: left;"><span style="font-family: arial;"><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">10 - 8k </span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">+ </span></span><span style="font-family: arial;">k</span><sup>2 </sup><span style="font-family: arial;">= a from equation 3, put k = 5, we get,</span></div></div></span></div></div></span></div></div></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">10 - 8(5) </span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">+ (5)</span></span><sup>2 </sup><span>= a</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">10 - 40 </span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">+ 25</span></span><sup> </sup><span>= a</span></span></div></blockquote></blockquote></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: left;"><span style="font-family: arial; font-size: medium;"><span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">- 30 </span><span style="font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline; white-space: pre-wrap;">+ 25</span></span><sup> </sup><span>= a</span> </span></div></blockquote></blockquote></blockquote><div><span style="font-family: arial; font-size: medium;"><div><span style="font-family: arial; font-size: medium;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-family: arial; white-space: pre-wrap;"> a = - 5</span></blockquote></blockquote></blockquote></blockquote><div style="text-align: left;">7) So, k = 5 and a = - 5.</div><h3 style="text-align: left;"><span style="font-size: medium;"><a href="https://anil7pute.blogspot.com/2023/05/148-ncert-10-3-pair-of-linear-equations.html" rel="nofollow" target="_blank"><span style="color: #0400ff;">Click here for</span><span style="color: #0400ff;"> ⇨ NCERT-10-3-Pair of Linear Equations in Two Variables - Ex-3.1</span></a></span></h3></span></div><div><div style="font-family: "Times New Roman"; font-size: medium;"><a href="https://plus.google.com/107775571667386395180?rel=author" style="color: #9f220d; line-height: 19.404px; text-align: justify; text-transform: uppercase;"><span style="font-family: arial; font-size: medium;">ANIL SATPUTE<br /></span></a></div></div></span></div>Anil7putehttp://www.blogger.com/profile/18393693958131871439noreply@blogger.com0