NCERT Class 10 Solutions: Real Numbers (Chapter 1) Exercise 1.3 –Part 1
Q1 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 coprime.
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 coprime.
We arrived at the contradictory statement as above since our assumption is not correct.
Hence, we can conclude that is irrational.
Q2 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
Q3 Prove that the following are irrationals:
Solution:

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

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
This contradicts our assumption.
Therefore we can conclude that is an irrational number

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
This is a contradiction
Hence is irrational