1. Interpret the system

A plano-convex lens is silvered on its curved surface. So light:

• first refracts through the lens,
• then reflects from the silvered curved surface,
• then again refracts through the lens while coming back.

Thus the lens is effectively traversed twice.

We need the object position such that the final image is real and has the same size as the object.

2. Focal length of the plano-convex lens

Using lens maker formula:

For a plano-convex lens:

• one surface is plane, so $R_1=\infty$
• curved surface has radius $R=30\,\text{cm}$
• refractive index $\mu=1.5$

Hence,

So,

3. Equivalent effect of silvered lens

When a lens is silvered at one surface, the light passes through the lens twice. Therefore the optical power doubles.

If lens power is

then equivalent power is

So equivalent focal length is

Thus the combination behaves like a concave mirror of focal length $30\,\text{cm}$.

4. Condition for real image of same size

For a mirror, a real image of the same size as the object is formed when the object is at the center of curvature:

Here,

Therefore the object should be placed at

from the lens.

5. Check with options

• A: $60\,\text{cm}$ ✅
• B: $30\,\text{cm}$
• C: $20\,\text{cm}$
• D: $80\,\text{cm}$

So the correct option is A.

6. Comparison with stored answer

Stored correct answer is C ($20\,\text{cm}$), but our derivation gives A ($60\,\text{cm}$).

The stored answer appears incorrect.