Class 10, NCERT (CBSE) solution of Mathematics
Chapter 1, Real Numbers
NCERT solution for Math Textbook Exercise 1.3
Q 1: Prove that √5 is irrational.
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.
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.
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.
(ii) Let us assume on the contrary that 7√5 is rational.
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.
Let 6 + √2 be rational.