# NCERT Class 10 Solutions: Real Numbers (Chapter 1) Exercise 1.3 –Part 1

Q-1 Prove that is irrational.

Solution:

Let us assume, to the contrary, that is rational.

That is, we can find integers a and b such that

Suppose a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are co-prime.

Therefore

Squaring on both sides (Equation 1)

The above implies that is divisible by 5 and also a is divisible by 5.

Therefore we can write that for some integer f.

Substituting in equation

Or

Is divisible by 5 which mean b is also divisible by 5.

Therefore a and b have 5 as a common factor.

This contradicts the fact that a and b are co-prime.

We arrived at the contradictory statement as above since our assumption is not correct.

Hence, we can conclude that is irrational.

Q-2 Prove that is irrational.

Solution:

If possible let be a rational number.

Squaring both side

(Equation 1)

Since 'a' is a rational number that expression is also rational number.

Is a rational number

This is contradiction. Hence, is irrational.

Hence proved

Q-3 Prove that the following are irrationals:

Solution:

1. Let be a rational number.

2a is rational number since product of two rational numbers is a rational number.

Which will imply that is a rational number. But it is a contradiction since is an irrational number.

Therefore our assumption is wrong

Therefore is irrational

2. Let be a rational number.

Here, is a rational number

Since product of two rational number is a rational number.

The above will imply that is a rational number

But Is an irrational number

Therefore we can conclude that is an irrational number

3. If possible let be a rational number

Squaring

(Equation 1)

Since a is a rational number the expression is also rational number

is a rational number