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Question

A thin equiconvex lens of refractive index 3/2 is placed on a horizontal plane mirror as shown in figure. The space between the lens and the mirror is filled with a liquid of refractive index 4/3. It is found what when a point object is placed 15 cm above the lens on its principal axis, the object coincides with its own image.
The radius of curvature of the convex surface is
160935_145d70588eb14a4ba58eb70c8169a8da.png

A
10cm
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B
15cm
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C
20cm
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D
25cm
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Solution

The correct option is C 10cm

The setup would be detailed as shown above. A thin layer of air would exist between mirror and liquid. Water could be treated as a plano-concave lens of refractive index 4/3.
From the principle of reversibility of light we can say that if the final image is to be formed on the object itself, the image after refraction from liquid to air would be formed on the mirror itself. If the mirror and that image have a finite distance, the ray cannot be traced back to the same point after reflection from the mirror.
Consider refraction from first surface,

32v1115=321R

Refraction from second surface,

43v232v1=4332R

Refraction from liquid to air,

1v343v2=143

Hence, 1v3=115+23R

Since v3=0

Hence R=10cm

410092_160935_ans_ad8b6ebece4444b9b636675b3457ef1b.png

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