Question

Prove that the following is irrational : $6+\sqrt{2}$ [3 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Proof : 1 Mark

Lets assume that $(6+\sqrt{2})$ is rational.

It means that we have co-prime integers a and b $(b\ne 0)$ such that $\frac{a}{b}=6+\sqrt{2}$

$\Rightarrow \frac{a}{b}-6=\sqrt{2}$

$\Rightarrow \frac{(a-6b)}{b}=\sqrt{2}$ ...(1)

a and b are integers. It means L.H.S of (1) is rational.

But we know that $\sqrt{2}$ is irrational.

$\Rightarrow L.H.S\ne R.H.S$

This contradiction arises because our assumption is wrong.

$\therefore (6+\sqrt{2})$ cannot be rational.

Hence, $(6+\sqrt{2})$ is irrational.