NCERT solutions for class 10 Maths
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.