189-NCERT New Syllabus Grade 10 Real Numbers Ex-1.2

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NCERT New Syllabus Mathematics

Class: 10

Exercise 1.2

Topic: Real Numbers

Understanding Real Numbers: Key to Mathematical Mastery.

Welcome to the path of discovering one of the most fundamental topics in mathematics: real numbers. This chapter from the NCERT class 10 syllabus offers the framework for comprehending advanced mathematical concepts by concentrating on properties, theorems, and real-number operations. In this blog, we'll look at exercise 1.2, breaking down each difficulty carefully and addressing it. Whether you're studying for an exam or creating a solid foundation, this post will easily guide you through each answer. Let's discover the power of real numbers together!

EXERCISE 1.2

Q1. Prove that √5 is irrational.

Explanation:

1) We can prove this using indirect proof. It is also known as proof by contradiction.

Solution:

1) Let us assume, for the sake of contradiction, that √5 is a rational number.

2) Therefore, we can express √5 = p/q where p and q are coprime integers 

(having no common factor other than 1) and q ≠ 0.

(√5)² = (p/q)²

5 = (p)²/(q)²

p² = 5q²  ---------- equation 1

3) Equation (1) shows that 5 divides p², meaning that 5 must also divide p (since

if a prime divides the square of a number, it must divide the number itself).

4) Let $p=5r$ for some integer r. Substituting this value into equation (1):

(5r)² = 5(q)²

25r² = 5q²  ---------- equation 2

5) Dividing equation (2) by 5, we get:

q² = 5r²

6) Thus, we have shown that 5 divides both $p$ and $q$, which contradicts our initial

assumption that $p$ and $q$ are coprime.

7) Therefore, our assumption that √5 is a rational number must be incorrect.

8) Hence, √5 is an irrational number.

Q2. Prove that 3 + 2√5 is irrational.

Explanation:

1) We can prove this using indirect proof. It is also known as proof by contradiction.

Solution:

1) Let us assume, for contradiction, that 3 + 2√5 is a rational number.

2) Therefore, we can express 3 + 2√5 = p/q where p and q are coprime integers

(with no common factor other than 1) and q ≠ 0.

3 + 2√5 = p/q 

2√5 = (p/q) – 3

2√5 = [(p – 3q)/q]

√5 = (p – 3q)/2q

3) Since p and q are co-primes, (p – 3q)/2q is rational, so √5 is also rational.

However, this contradicts the well-known fact that √5 is irrational.

4) Therefore, our assumption was wrong, and 3 + 2√5 is an irrational number.

Q3. Prove that the following are irrationals :

(i) 1/√2    (ii) 7√5    (iii) 6 + √2

Explanation:

1) We can prove this using indirect proof. It is also known as proof by contradiction.

Solution:

(i) 1/√2

1) Assume, for contradiction, that 1/√2 is a rational number.

2) Thus, 1/√2 = p/q where p and q are co-primes and q ≠ 0. (i.e., no common

factors other than 1).

1/√2 = p/q

√2 = q/p

3) Since p and q are co-primes, q/p is a rational number, which implies √2 is 

rational. But this contradicts the known fact that which contradicts the fact that √2 is irrational.

4) Therefore, 1/√2 is an irrational number.

(ii) 7√5

1) Let us assume, for contradiction, that 7√5 is a rational number.

2) So, 7√5 = p/q where p and q are co-primes and q ≠ 0. (i.e., no common

factors other than 1).

7√5 = p/q

√5 = p/7q

3) Since p and q are co-primes, p/7q is a rational number, which implies √5 is 

rational. However, this contradicts the fact that √5 is irrational.

4) Thus, 7√5 is an irrational number.

(iii) 6 + √2

1) Assume that 6 + √2 is a rational number.

2) Therefore, 6 + √2 = p/q where p and q are co-primes integers and q ≠ 0. That is, there is no common factor other than 1.

6 + √2 = p/q 

√2 = (p/q) – 6

√2 = (p – 6q)/q

3) Since p and q are co-primes, (p – 6q)/q is rational, which implies √2 is rational. But this contradicts the fact that √2 is irrational.

4) Therefore, 6 + √2 is irrational number.

Conclusion: Unveiling the Power of Real Numbers

In this chapter on Real Numbers, we've explored the foundational blocks of mathematics, laying the groundwork for a deeper understanding of algebra and beyond. From the beauty of irrational numbers to the precision of the Euclidean algorithm, real numbers are at the core of every mathematical operation. As you continue to delve into the world of numbers, remember that these principles stretch far beyond the classroom, influencing technology, science, and everyday calculations. Keep exploring, keep calculating, and unlock the endless possibilities of real numbers!

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NCERT New Syllabus Class 10 - Polynomials Exercise 2.1

Anil Satpute