$(a)\;60^{\circ} \\ (b)\;45^{\circ} \\ (c)\;no\;emergence\;takes\;place \\ (d)\;the \;light\; retraces \; its\;path $

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$i_1$= angle of incidence $=30^{\circ}$

$\large\frac{\sin i_1}{\sin r_1}=\frac{3}{2}$

$\large\frac{\Large\frac{1}{2}}{\sin r_1}=\frac{3}{2}$

$\sin r_1 =\large\frac{3}{2}$

Since $r_1 +r_2 =90^{\circ}$

$r_2 =90^{\circ}-r_1$

$\therefore \cos r_2 =\large\frac{1}{3}$

$\sin r_2 =\sqrt {1- \cos ^2 r_2}$

$\sin r_2=\sqrt {1- \bigg(\frac{1}{3}\bigg)^2}$

$\qquad= \large\frac{2 \sqrt 2}{3}$-----(1)

Also,

$\sin c= \large\frac{1}{\mu}$

$\sin c =\large\frac{2}{3}$----(2)

From (1) and (2) we infer that

$r_2 >c$

$\therefore $ No refraction takes place at 2nd face total internal reflection takes place.

$\therefore $ no emergent ray exists.

Hence c is the correct answer.

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