Work-Energy Theorem question from JEE Main 2005 (AIEEE)

Answer: (c)

1. Constant resistance $F$: work done $Fx$ equals KE lost.
2. Speed $v \to \frac{v}{2}$: KE falls from $K$ to $\frac{K}{4}$, so $\frac{3K}{4}$ is lost in $3\text{ cm}$.
3. Remaining $\frac{K}{4}$ is one-third of $\frac{3K}{4}$, so it needs $\frac{3}{3} = 1\text{ cm}$ more.
4. So the answer is (c) $1.0\text{ cm}$.