**NCERT solutions for class 10 Maths**

**Chapter 1**

**Real Numbers**

**Exercise 1.3**

**Ex 1.3 Question 1.**

**Prove that √5 is irrational.**

**Solution:**

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.**

**Solution:**

**Ex 1.3 Question 3.**

**Prove that the following are irrational.**

**Solution:**