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You are here: Home / NCERT Solutions / Class 10 / Maths / Chapter 1 Real Numbers / Exercise 1.3

Exercise 1.3

Last Updated on April 1, 2021 By Mrs Shilpi Nagpal Leave a Comment

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.

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

About Mrs Shilpi Nagpal

Author of this website, Mrs Shilpi Nagpal is MSc (Hons, Chemistry) and BSc (Hons, Chemistry) from Delhi University, B.Ed (I. P. University) and has many years of experience in teaching. She has started this educational website with the mindset of spreading Free Education to everyone.

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