# MP Board Class 10th Maths Chapter 1 Real Numbers Ex 1.3 Solutions

In this article, we will share MP Board Class 10th Maths Book Solutions Chapter 1 Real Numbers Ex 1.3 Pdf, These solutions are solved subject experts.

## MP Board Class 10th Maths Solutions Chapter 1 Real Numbers Ex 1.3

Question 1. Prove that 5–√ is irrational.

Solution:

Let us assume, to the contrary, that 5–√ is rational.
∴ 5–√=ab
∴ b × 5–√ = a
By Squaring on both sides,
5b2 = a2 …………. (i)
∴ 5 divides a2.
5 divides a.
∴ We can write a = 5c.
Substituting the value of ‘a’ in eqn. (i),
5b2 = (5c)2 = 25c2
b2 = 5c2
It means 5 divides b2.
∴ 5 divides b.
∴ ‘a’ and ‘b’ have at least 5 as a common factor.
But this contradicts the fact that a’ and ‘b’ are prime numbers.
∴ 5–√ is an irrational number.

MP Board Class 10th Maths Solutions

Question 2. Prove that 3 +25–√ is irrational.

Solution:

Let 3 + 25–√ is rational.

⇒ From (1), 5–√ is rational
But this contradicts the fact that 5–√ is irrational.
∴ Our supposition is wrong.
Hence, 3 + 25–√ is irrational.

MP Board Class 10th Maths Solutions

Question 3.
Prove that the following are irrationals.
(i) 12√
(ii) 75–√
(iii) 6 + 2–√
Solution:
(i) We have

From (1), 2–√ is rational number which contradicts the fact that 2–√ is irrational.
∴ Our assumption is wrong.
Thus, 12√ is irrational.

### (ii) Let 75–√ is rational.

∴ We can find two co-prime integers a and b such that 75–√=ab, where b ≠ 0

This contradicts the fact that 5–√ is irrational.
∴ Out assumption is wrong.
Thus, we conclude that 75–√ is irrational.

(iii) Let 6 + 2–√ is rational.
∴ We can find two co-prime integers a and b

= Rational [ ∵ a and b are integers]
From (1), 2–√ is a rational number,
which contradicts the fact that 2–√ is an irrational number.
∴ Our supposition is wrong.
⇒ 6 + 2–√ is an irrational number.

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