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Home » NCERT Solutions » Class 10 » Maths » Chapter 1 Real Numbers » Exercise 1.3

Exercise 1.3

Last Updated on April 5, 2023 By Mrs Shilpi Nagpal

NCERT Solutions for Class 10 Maths
Chapter 1 Real Numbers Ex 1.3

Page 14 

1. Prove that √5 is irrational.

Answer :

Let’s assume, that √5 is rational number.

i.e. √5 = x/y (where, x and y are co-primes)

y√5= x

Squaring both the sides, we get,

(y√5)2 = x2

⇒ 5y2 = x2……………………………….. (1)

Thus, x2 is divisible by 5, so x is also divisible by 5.

Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get,

5y2 = (5k)2

⇒ y2 = 5k2

is divisible by 5 it means y is divisible by 5.

Therefore, x and y are co-primes. Since, our assumption about  is rational is incorrect.

Hence, √5 is irrational number.

2. Prove that 3 2 5 + is irrational.

Answer :

Let’s assume that 3 + 2√5 is a rational number.

So we can write this number as

3 + 2√5 = a/b

Here a and b are two co prime number and b is not equal to 0

Subtract 3 both sides we get

2√5 = a/b – 3

2√5 = (a-3b)/b

Now divide by 2, we get

√5 = (a-3b)/2b

Here a and b are integer so (a-3b)/2b is a rational number so √5 should be a rational number But √5 is a irrational number so it contradicts.

Hence, 3 + 2√5 is a irrational number.

3. Prove that the following are irrationals : (i) 1/√2 (ii) 7√ 5 (iii) 6 +√2

Answer :

(i) 1/√2

Let us assume 1/√2 is rational.

Then we can find co-prime x and y (y ≠ 0) such that 1/√2 = x/y

Rearranging, we get,

√2 = y/x

Since, x and y are integers, thus, √2 is a rational number, which contradicts the fact that √2 is irrational.

Hence, we can conclude that 1/√2 is irrational.

(ii) 7√5

Let us assume 7√5 is a rational number.

Then we can find co-prime a and b (b ≠ 0) such that 7√5 = x/y

Rearranging, we get,

√5 = x/7y

Since, x and y are integers, thus, √5 is a rational number, which contradicts the fact that √5 is irrational.

Hence, we can conclude that 7√5 is irrational.

(iii) 6 +√2

Let us assume 6 +√2 is a rational number.

Then we can find co-primes x and y (y ≠ 0) such that 6 +√2 = x/y⋅

Rearranging, we get,

√2 = (x/y) – 6

Since, x and y are integers, thus (x/y) – 6 is a rational number and therefore, √2 is rational. This contradicts the fact that √2 is an irrational number.

Hence, we can conclude that 6 +√2 is irrational.

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Filed Under: Chapter 1 Real Numbers, Class 10, Maths, NCERT Solutions

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