# UP Board Solutions for Class 10 Maths Chapter 1 Exercise 1.3 Real Numbers

## EXERCISE 1.3

Sol. Let us assume, to the contrary, that is rational.
Now, let , where a and b are coprime and Squaring on both sides, we get => … (1)

This shows that is divisible by 5.
It follows that a is divisible by 5                          … (2)
=>   a = 5m for some integer m.
Substituting a = 5m in (1), we get or => is divisible by 5
and hence b is divisible by 5                               … (3)
From (2) and (3), we can conclude that 5 is a common factor of both a and b.
But this contradicts our supposition that a and b are coprime.
Hence, is irrational.

## 2. Prove that 3 + is irrational.

Sol. Let us assume, to the contrary, that 3 + is a rational number.
Now, let 3 + , where a and b are coprime and  a and b are integers. is a rational number

=> is a rational number.
But is an irrational number.
This shows that our assumption is incorrect.
So, 3 + 2 is an irrational number.

### Sol. (i) Let us assume, to the contrary, that is rational.

That is, we can find co-prime integers p and q such that

Since p and q are integers,  is rational, and so is rational.

But this contradicts the fact that is irrational.
So, we conclude that  is irrational.

### (ii) Let us assume, to the contrary, that is rational.

That is, we can find co-prime integers p and q such that Since p and q are integers, is rational and so is But this contradicts the fact that is irrational.
So, we conclude that is irrational.

### (iii) Let us assume, to the contrary, that is rational.

That is, we can find integers p and q such that

Since p and q are integers, we get 6  –  = is rational, and so is rational.
But this contradicts the fact that is irrational.
So, we conclude that  6+ is irrational.