Friction on an Incline question from NEET 1988 (AIPMT)

Answer: (b)

1. At $45^\circ$: smooth $a_0 = \dfrac{g}{\sqrt{2}}$, rough $a = \dfrac{g}{\sqrt{2}}(1-\mu)$.
2. Same distance: $t \propto \dfrac{1}{\sqrt{a}}$, so $t_r = 2t_s \Rightarrow 1 - \mu = \dfrac{1}{4}$.
3. So the answer is (b) $\mu = 0.75$.