JEE Advanced 2016 · Paper 1, Physics question 9

Lens. A real image twice the size gives $m=-2$, so $v=60$ cm with $u=-30$ cm:
$$\frac1f=\frac1{60}+\frac1{30}\implies f=20\text{ cm}.$$
(D) is true.

Reflection from the convex face. The curved face acts as a convex mirror and forms a virtual, erect image 10 cm behind it:
$$\frac1v+\frac1u=\frac2R\implies\frac1{10}-\frac1{30}=\frac2R\implies R=30\text{ cm}.$$
So (B) is false, and (C) is false because the image is virtual, not real.

Refractive index. $\dfrac1f=\dfrac{n-1}R\implies n-1=\dfrac{30}{20}$, so $n=2.5$. (A) is true.

(A), (D)