A plano-convex lens of refractive index 1.5 has the radius of curvature of its curved surface equal to 30 cm. Its plane face is silvered. A small…

The silvered plane face has zero power, so the system acts as a single mirror whose power is the sum of the two passes through the lens: $P=2P_{lens}+P_{plane\ mirror}=2\dfrac{(\mu-1)}{R}+0=2\times\dfrac{0.5}{30}=\dfrac1{30}\ \mathrm{cm^{-1}}$. Thus the equivalent concave mirror has $f=30\ \mathrm{cm}$. Using the mirror formula with $u=-45\ \mathrm{cm}$: $\dfrac1v=\dfrac1f-\dfrac1{|u|}=\dfrac1{30}-\dfrac1{45}=\dfrac1{90}$, so $v=90\ \mathrm{cm}$, positive in the sense of a real image on the same side as the object. Forgetting the double pass uses $f=60\ \mathrm{cm}$ and gives $180\ \mathrm{cm}$; treating it as a convex mirror gives $18\ \mathrm{cm}$. Option (A) is correct.