NIFT UG 2017 Question Paper with solutions

Let $\sqrt{7}$ is not rational number
$\Rightarrow \sqrt{7}$ is rational number
$\Rightarrow \sqrt{7}=\frac{\mathit{p}}{\mathit{q}}$
$\frac{{\mathit{p}}^2}{{\mathit{q}}^2}=7$
$\Rightarrow {\mathit{p}}^2=7{\mathit{q}}^2$
⇒ 7 is a factor of ${\mathit{p}}^2$
⇒ 7 is a factor of p
Let p = 7m for some natural number m
Then $\mathit{p}=7\mathit{m}$
${\mathit{p}}^2=7^2{\mathit{m}}^2$
$\Rightarrow 7{\mathit{q}}^2=7^2{\mathit{m}}^2$
$\Rightarrow {\mathit{q}}^2=7{\mathit{m}}^2;\left[\because {\mathit{p}}^2=7{\mathit{q}}^2\right]$
⇒ 7 is a factor of ${\mathit{q}}^2$
⇒ 7 is a factor of q
But, 7 is a factor of p and 7 is a factor of q means that 7 is a factor of both p and q. This contradicts the assumption that p and q have no common factor. Hence, our supposition is wrong. Hence, $\sqrt{7}$ can not be a rational number.