c
\(Mg \sin \theta R =\left( mk ^{2}+ mR ^{2}\right) \alpha\)

\(\alpha=\frac{ Rg \sin \theta}{ k ^{2}+ R ^{2}} \Rightarrow a =\frac{ g \sin \theta}{1+\frac{ k ^{2}}{ R ^{2}}}\)

\(t =\sqrt{\frac{2 s }{ a }}=\sqrt{\frac{2 s }{ g \sin \theta}\left(1+\frac{ k ^{2}}{ R ^{2}}\right)}\)

for least time, \(k\) should be least \(\&\) we know \(k\) is least for solid sphere.