# 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.