Spherical Surfaces & Thin Lenses question from JEE Main 2025 (28 Jan, Shift 1)

Answer: (d)

1. The last ray from O that can reach E leaves the liquid at the rim B and must just graze along the surface towards E.
2. Ray OB from the bottom to the rim makes $45^\circ$ with the vertical normal at B (since $OC = CB = R$).
3. Grazing emergence means $45^\circ$ is the critical angle: $\sin 45^\circ = \frac{1}{\mu}$.
4. $\mu_{min} = \sqrt{2}$. So the answer is (d).