Explanation

(a) From the following figure
$r + i = 90{^\circ} \Rightarrow i = 90{^\circ} - r$
For ray not to emerge from curved surface $i > C$
$\Rightarrow \sin i > \sin C \Rightarrow \sin{(90{^\circ} - r)} > \sin C \Rightarrow \cos r > \sin C$
$\Rightarrow \sqrt{1 - \sin^{2}r} > \frac{1}{n}\ \ \ \ \ \ \ \ \ \ \ \left\lbrack \because\sin C = \frac{1}{n} \right\rbrack$
$\Rightarrow 1 - \frac{\sin^{2}\alpha}{n^{2}} > \frac{1}{n^{2}} \Rightarrow 1 > \frac{1}{n^{2}}(1 + \sin^{2}\alpha)$
$\Rightarrow n^{2} > 1 + \sin^{2}\alpha \Rightarrow n > \sqrt{2}\ \ \ \lbrack\sin{i \rightarrow 1}\rbrack$
$\Rightarrow$ Least value $= \sqrt{2}$