Escape Velocity question from MHT CET 2024 (10 May, Shift 1)

Answer: (b)

1. With $M = \frac{4}{3}\pi R^3\rho$: $v_e = \sqrt{\frac{2GM}{R}} = R\sqrt{\frac{8\pi G\rho}{3}} \propto R\sqrt{\rho}$.
2. Planet: $R' = 4R$, $\rho' = 4\rho \Rightarrow v_e' = 4\times 2\,v_e = 8v_e$.
3. $v_{earth} : v_{planet} = 1 : 8$.
4. So the answer is (b) $1 : 8$.