Real Numbers, Class 10, Mathematics - Solutions of NCERT (CBSE) Math Exercise 1.3

Class 10, NCERT (CBSE) solution of Mathematics

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Q 1: Prove that √5 is irrational.

Solution:

Let us assume √5 is a rational number.

Then, we can find two positive integers a, b (b ≠ 0) such that

Suppose, a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime.

or, a = √5b

or, a² = 5b²

Therefore, a² is divisible by 5 and so, it can be said that a is divisible by 5.

Now suppose,

a = 5k, where k is an integer.

From the above expression, b² is divisible by 5 and hence, b is divisible by 5.

This implies that a and b have at least 5 as a common factor.

And this contradicts the fact that a and b are co-prime.

Hence, √5 is an irrational number.

Q2: Prove that 3 + 2√5 is irrational.

Solution:

Let us assume on the contrary that 3 + 2√5 is rational.

Therefore, there exist co-prime positive integers say a, b (b ≠ 0) such that

This contradicts the fact that √5 is irrational. So, our assumption that 3 + 2√5 is rational is false.

Therefore, 3 + 2√5 is an irrational number.

Q3: Prove that the following are irrationals:

Solution:

(i)

Therefore, we can have two integers a, b (b ≠ 0) such that

(ii) Let us assume on the contrary that 7√5 is rational.

Then, we can find two integers a, b (b ≠ 0) such that for some integers a and b

Therefore, √5 must be rational.

This contradicts to the fact that √5 is irrational.

Or, our assumption that 7√5 is rational is false. Hence, 7√5 is irrational.

(iii)

Let 6 + √2 be rational.

Now, let there be two integers a, b (b ≠ 0) such that