Real Numbers Exercise 1.3

NCERT solutions for Class 10 Maths Real Numbers 

NCERT Solutions for Class 10 Maths Real Numbers

1. Prove that is irrational.

Ans. Let us prove irrational by contradiction.

Let us suppose that is rational. It means that we have co-prime integers a and b(b ≠ 0) such that

⇒ b=a

Squaring both sides, we get

… (1)

It means that 5 is factor of

Hence, 5 is also factor ofa by Theorem. … (2)

If, 5 is factor of a, it means that we can write a = 5c for some integer c.

Substituting value of ain (1),

It means that 5 is factor of .

Hence, 5 is also factor of b by Theorem. … (3)

From (2) and (3), we can say that 5 is factor of both a and b.

But, a and b are co-prime.

Therefore, our assumption was wrong. cannot be rational. Hence, it is irrational.

NCERT Solutions class-10 Maths Exercise 1.3

2. Prove that (3 + 2) is irrational.

Ans. We will prove this by contradiction.

Let us suppose that (3+2) is rational.

It means that we have co-prime integers a and b(b ≠ 0) such that

⇒

⇒

⇒ … (1)

a and b are integers.

It means L.H.S of (1) is rational but we know that is irrational. It is not possible. Therefore, our supposition is wrong. (3+2) cannot be rational.

Hence, (3+2) is irrational.

NCERT Solutions class-10 Maths Exercise 1.3

3. Prove that the following are irrationals.

(i)

(ii)

(iii)

Ans. (i) We can prove irrational by contradiction.

Let us suppose that is rational.

It means we have some co-prime integers a and b (b ≠ 0) such that

=ab

⇒ … (1)

R.H.S of (1) is rational but we know that is irrational.

It is not possible which means our supposition is wrong.

Therefore, cannot be rational.

Hence, it is irrational.

(ii) We can prove irrational by contradiction.

Let us suppose that is rational.

It means we have some co-prime integers a and b (b ≠ 0) such that

⇒ … (1)

R.H.S of (1) is rational but we know that is irrational.

It is not possible which means our supposition is wrong.

Therefore, cannot be rational.

Hence, it is irrational.

(iii) We will prove irrational by contradiction.

Let us suppose that () is rational.

It means that we have co-prime integers a and b(b ≠ 0) such that

⇒

⇒ …(1)

a and b are integers.

It means L.H.S of (1) is rational but we know that is irrational. It is not possible.

Therefore, our supposition is wrong. () cannot be rational.

Hence, () is irrational.

NCERT Solutions class-10 Maths Exercise 1.3