In an experiment a convex lens of focal length $15$ cm is placed coaxially on an optical bench in front of a convex mirror at a distance of $5$ cm from it. It is found that an object and its image coincide, if the object is placed at a distance of $20$ cm from the lens. The focal length of the convex mirror is:
Answer: (D) $27.5$ cm
The image coincides with the object only if the rays leaving the lens strike the mirror normally and retrace their path. So the lens must form its image at the centre of curvature of the convex mirror (behind it).
Lens: $u = -20$ cm, $f = 15$ cm:
$$\frac1v = \frac1{15} - \frac1{20} = \frac1{60} \;\Rightarrow\; v = 60\ \text{cm}$$
This point is $60 - 5 = 55$ cm behind the mirror, so $R = 55$ cm and
$$f = \frac R2 = 27.5\ \text{cm}$$
Solution by Sreeraj P, M.Sc Physics