NCERT Solutions for class 10 maths chapter 1

Real Numbers

NCERT Solutions for class 10 maths chapter 1 Exercise 1.3

Question 1

Prove that √5 is irrational.

Answer 1

Let us assume, that √5 is rational number.
√5 = .. where, a and b are co-primes
b√5= a
On squaring both the sides,
(b√5)² = a²
b² =
a² is divisible by 5,
'a' is also divisible by 5.
Let 'a' = 5c,
5b² = (5c)²
b² = 5c²
c² =
b² is divisible by 5
'b' is divisible by 5.
this implies that a and b have 5 as common factor. And this is a contradiction to the fact that a and b are co-prime . So our assumption about is rational is incorrect.
Hence, √5 is irrational number.

Question 2

Prove that 3 + 2√5 + is irrational.

Answer 2

Let us assume 3 + 2√5 is rational.
Then we can find co-prime a and b (b ≠ 0) such that 3 + 2√5 =
2√5= -3
2√5= ( -3)
Since a and b are integers ( -3) will also be rational number Therefore, √5 is also a rational number. But this contradicts the fact that √5 is irrational.
So, we conclude that 3 + 2√5 is irrational.

Question 3

Prove that the following are irrationals:
(i) (ii) 7√5
(iii) 6 + √2

Answer 3

(i) 1/√2
Let us assume is rational.
Then we can find co-prime a and b (b ≠ 0) such that =
√2 =
Since a and b are integers will also be rational number Therefore, √2 is also a rational number. But this contradicts the fact that √2 is irrational.
So, we conclude that √2 is irrational.

(ii) 7√5
Let us assume 7√5 is rational.
Then we can find co-prime a and b (b ≠ 0) such that 7√5 =
√5 =
Since a and b are integers will also be rational number Therefore, √5 is also a rational number. But this contradicts the fact that √5 is irrational.
So, we conclude that 7√5 is irrational.

(iii) 6 +√2
Then we can find co-prime a and b (b ≠ 0) such that 6 +√2 =
√2 = - 6
Since a and b are integers - 6 will also be rational number Therefore, √2 is also a rational number. But this contradicts the fact that √2 is irrational.
So, we conclude that 6 +√2 is irrational.