In this post, we will share NCERT Class 10th Maths Book Solutions Chapter 1 Real Numbers Ex 1.3. These solutions are based on new NCERT Syllabus.

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

Question 1.

Prove that \(\sqrt{5}\) is irrational.

Solution:

Let us assume, to the contrary, that \(\sqrt{5}\) is rational.

∴ \(\sqrt{5}=\frac{a}{b}\)

∴ b × \(\sqrt{5}\) = 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.

∴ \(\sqrt{5}\) is an irrational number.

Question 2.

Prove that 3 +2\(\sqrt{5}\) is irrational.

Solution:

Let 3 + 2\(\sqrt{5}\) is rational.

⇒ From (1), \(\sqrt{5}\) is rational

But this contradicts the fact that \(\sqrt{5}\) is irrational.

∴ Our supposition is wrong.

Hence, 3 + 2\(\sqrt{5}\) is irrational.

Question 3.

Prove that the following are irrationals.

(i) \(\frac{1}{\sqrt{2}}\)

(ii) 7\(\sqrt{5}\)

(iii) 6 + \(\sqrt{2}\)

Solution:

(i) We have

From (1), \(\sqrt{2}\) is rational number which contradicts the fact that \(\sqrt{2}\) is irrational.

∴ Our assumption is wrong.

Thus, \(\frac{1}{\sqrt{2}}\) is irrational.

(ii) Let \(7 \sqrt{5}\) is rational.

∴ We can find two co-prime integers a and b such that \(7 \sqrt{5}=\frac{a}{b}\), where b ≠ 0

This contradicts the fact that \(\sqrt{5}\) is irrational.

∴ Out assumption is wrong.

Thus, we conclude that 7\(\sqrt{5}\) is irrational.

(iii) Let 6 + \(\sqrt{2}\) is rational.

∴ We can find two co-prime integers a and b

= Rational [ ∵ a and b are integers]

From (1), \(\sqrt{2}\) is a rational number,

which contradicts the fact that \(\sqrt{2}\) is an irrational number.

∴ Our supposition is wrong.

⇒ 6 + \(\sqrt{2}\) is an irrational number.

In this post, we will share NCERT Class 10th Maths Book Solutions Chapter 1 Real Numbers Ex 1.3. These solutions are based on new NCERT Syllabus.

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

Question 1.

Prove that \(\sqrt{5}\) is irrational.

Solution:

Let us assume, to the contrary, that \(\sqrt{5}\) is rational.

∴ \(\sqrt{5}=\frac{a}{b}\)

∴ b × \(\sqrt{5}\) = 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.

∴ \(\sqrt{5}\) is an irrational number.

Question 2.

Prove that 3 +2\(\sqrt{5}\) is irrational.

Solution:

Let 3 + 2\(\sqrt{5}\) is rational.

⇒ From (1), \(\sqrt{5}\) is rational

But this contradicts the fact that \(\sqrt{5}\) is irrational.

∴ Our supposition is wrong.

Hence, 3 + 2\(\sqrt{5}\) is irrational.

Question 3.

Prove that the following are irrationals.

(i) \(\frac{1}{\sqrt{2}}\)

(ii) 7\(\sqrt{5}\)

(iii) 6 + \(\sqrt{2}\)

Solution:

(i) We have

From (1), \(\sqrt{2}\) is rational number which contradicts the fact that \(\sqrt{2}\) is irrational.

∴ Our assumption is wrong.

Thus, \(\frac{1}{\sqrt{2}}\) is irrational.

(ii) Let \(7 \sqrt{5}\) is rational.

∴ We can find two co-prime integers a and b such that \(7 \sqrt{5}=\frac{a}{b}\), where b ≠ 0

This contradicts the fact that \(\sqrt{5}\) is irrational.

∴ Out assumption is wrong.

Thus, we conclude that 7\(\sqrt{5}\) is irrational.

(iii) Let 6 + \(\sqrt{2}\) is rational.

∴ We can find two co-prime integers a and b

= Rational [ ∵ a and b are integers]

From (1), \(\sqrt{2}\) is a rational number,

which contradicts the fact that \(\sqrt{2}\) is an irrational number.

∴ Our supposition is wrong.

⇒ 6 + \(\sqrt{2}\) is an irrational number.