Inductors & DC Transients (Mixed) question from JEE Main 2013

Answer: (c)

1. With S1 closed and S2 open, C charges through R: $q = CV(1 - e^{-t/\tau})$.
2. (a) is wrong: battery work $= CV^2$, heat in R $= \frac{1}{2}CV^2$, so work is twice the heat.
3. (b) is wrong: at $t = \tau$, $q = CV(1 - e^{-1}) \approx 0.63\,CV$.
4. (c) is right and (d) is wrong: at $t = 2\tau$, $q = CV(1 - e^{-2})$.
5. So the answer is (c).