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,

5b^{2} = a^{2} …………. (i)

∴ 5 divides a^{2}.

5 divides a.

∴ We can write a = 5c.

Substituting the value of ‘a’ in eqn. (i),

5b^{2} = (5c)^{2} = 25c^{2}

b^{2} = 5c^{2}

It means 5 divides b^{2}.

∴ 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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