NCERT solutions for class 10 Maths

Chapter 1

Real Numbers

Exercise 1.3

Ex 1.3 Question 1.

Prove that √5 is irrational.
Let √5 is a rational number.

Therefore, we can find two integers a, b (b ≠ 0) such that √5 = 𝑎/𝑏

Let 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.

𝑎 = √5𝑏

⇒ 𝑎² = 5𝑏²

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

Let a = 5k, where k is an integer

(5𝑘)² = 5𝑏² ⇒ 5𝑘² = 𝑏²

This means that b² is divisible by 5 and hence, b is divisible by 5.

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

And this is a contradiction to the fact that a and b are co-prime. Hence, √5 cannot be expressed as 𝑝/𝑞 or it can be said that √5 is irrational.

Ex 1.3 Question 2.

Show that 3 + √5 is irrational.

Ex 1.3 Question 3.

Prove that the following are irrational.



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