a
(a) From the following figure

\(r + i = 90^\circ ==> i = 90^\circ -r\)

For ray not to emerge from curved surface \(i > C\)

\( ==> sin\, i > sin \,C ==> sin (90° -r) > sin C\)

\(==> cos\, r > sin\, C \)

==> \(\sqrt {1 - {{\sin }^2}r} > \frac{1}{n}\)            \(\left\{ {\,\sin C = \frac{1}{n}} \right\}\)

==> \(1 - \frac{{{{\sin }^2}\alpha }}{{{n^2}}} > \frac{1}{{{n^2}}}\)==> \(1 > \frac{1}{{{n^2}}}(1 + {\sin ^2}\alpha )\)

==> \({n^2} > 1 + {\sin ^2}\alpha \) ==> \(n > \sqrt 2 \) {\(sin i → 1\)}

==> Least value \( = \sqrt 2 \)