In the figure shown a point object O is placed in air on the principal axis. The radius of curvature of the spherical surface is 60cm. $I_{f}$ is the final image formed after all the refractions and reflections:

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If $d_{1} = 240 c m$ , then the ' $I_{f}$ ' cannot be formed on ‘O’

If $d_{1} = 240 c m$ , then the ' $I_{f}$ ' is formed on ‘O’ only if $d_{2} = 360 c m$

If $d_{1} = 120 c m$ , then the ' $I_{f}$ ' is formed on $' O'$ for any value of $d_{2}$

If $d_{1} = 240 c m$ , then the ' $I_{f}$ ' is formed on ‘O’ for all value of $d_{2}$

answer is A, B.

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Detailed Solution

If $d_{1} = 120$ cm then by refraction at curved surface, by refraction formula, we have
$\frac{3}{2 v} - \frac{1}{- 120} = \frac{(\frac{3}{2} - 1)}{60} \Rightarrow v = \infty$

Thus parallel light incident normally on mirror and hence image will form on object itself, hence option (A) is correct.
If $d_{1} = 240$ cm then using refraction formula we get $v = 360 c m$ so if $d_{2} = 360 c m$ , then plane mirror form image at same place and again final image form on object itself hence option (B) is also correct.