Given is a thin convex lens of glass (refractive index $\mu$) and each side having radius of curvature $R$. One side is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that the image gets formed on the object itself?
Answer: (D) $R/(2\mu-1)$
The silvered lens acts as a mirror. Light passes through the lens, reflects from the silvered face and passes through the lens again, so the powers add:
$$P = 2P_L + P_M$$
Lens: $P_L = (\mu-1)\left(\dfrac{1}{R}+\dfrac{1}{R}\right) = \dfrac{2(\mu-1)}{R}$. Concave silvered surface: $P_M = \dfrac{2}{R}$.
$$P = \frac{4(\mu-1)}{R} + \frac{2}{R} = \frac{2(2\mu-1)}{R} \Rightarrow f_{eq} = \frac{R}{2(2\mu-1)}$$
The image coincides with the object when the object is at the centre of curvature of this equivalent mirror, i.e. at $2f_{eq}$:
$$u = \frac{R}{2\mu-1}$$
Solution by Sreeraj P, M.Sc Physics