Exercise 1.3

Question 1. Prove that √5 is irrational number

Solution : let, that √5 is rational number.

√5 = a/b

b√5 = a

squaring both sides.

(b√5)² = a²

5b² = a² ..............(1)

Thus, a² is divisible by 5, so a is also divisible by 5.

Let us say, a=5c ,for some value of C and substituting the value of a in equation (1)

5b²= (5c)²

5b²=25c²

b²=5c²

b² is divisible by 5 it means b is divisible by 5.

Therefore, a and b are co-Primes.

Since. Our assumption about is rational is incorrect.

Hence, √5 is irrational number.

Question 2. Prove that 3+2√5 is irrational.

Solution : Let 3+2√5 is rational

3+2√5 = a/b

Rearranging,

2√5 =a/b -3

√5=1/2 (a/b-3)

Since, a ane b are integers

Thus, 1/2(a/b-3)is a rational number. But this contradiction is wrong.

So, we conclude that 3+2√5 is irrational.

Question 3. Prove that the following are irrational.

(1) 1/√2

Solution : Let 1/√2 Is rational

1/√2 = a/b

√2 =b/a

Since, a and b are integers.

Thus, √2 is a rational number.

Which contradicts the fact that √2 is irrational.

Hence, we can conclude that 1/√2 is irrational

(2) 7/√5

Solution : Let 7/√5 is a rational number

7√5 = a/b

√5 = a/7b

Since,a and b are integers.

Thus, √5 is a rational number.

Which contradicts the fact that √5 is irrational.

Hence, we can conclude that 7√5 is irrational.

(3) 6+√2

Solution: Let 6+√2 is a rational number

6+√2 = a/b

√2=(a/b)-6

Since, a and b are integers.

Thus, (a/b)-6 √2 is rational.

This contradicts the fact that √2 is an irrational number.

Hence, we can conclude that 6+√2 is irrational

We hope CBSE/MP Board Solution of Class 10th Chapter 1 "Real Numbers" Exercise 1.3 will help you.

Written by - Abhishek Dohare