Explanation

(c)
Given, $x = \frac{1}{2}\left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$
$\Rightarrow x^{2} = \frac{1}{4}\left( 7 + \frac{1}{7} + 2 \right) = \frac{16}{7}$
Now, $\frac{\sqrt{x^{2} - 1}}{x - \sqrt{x^{2} - 1}} = \frac{\sqrt{x^{2} - 1}}{x - \sqrt{x^{2} - 1}} \times \frac{\left( x + \sqrt{x^{2} - 1} \right)}{\left( x + \sqrt{x^{2} - 1} \right)}$
$= \frac{x\sqrt{x^{2} - 1} + x^{2} - 1}{1}$
$= \frac{1}{2}\left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)\sqrt{\frac{16}{7} - 1} + \frac{16}{7} - 1$
$= \frac{1}{2}\left( \sqrt{7} + \frac{1}{\sqrt{7}} \right) \times \frac{3}{\sqrt{7}} + \frac{9}{7}$
$= \frac{1}{2}\left( 3 + \frac{3}{7} \right) + \frac{9}{7}$
$= 3$