Escape Velocity question from MHT CET 2024 (11 May, Shift 2)

Answer: (a)

1. Energy conservation: $\frac{1}{2}m n^2\frac{2GM}{R} = GMm\left(\frac{1}{R} - \frac{1}{R+h}\right)$.
2. $\frac{1}{R+h} = \frac{1-n^2}{R} \Rightarrow R+h = \frac{R}{1-n^2}$.
3. $h = \frac{n^2R}{1-n^2}$.
4. So the answer is (a) $\frac{n^{2}R}{(1-n^{2})}$.