The time period of a mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be -

For a spring, the spring constant is inversely proportional to its length: $kl$ = constant.
Each quarter has length $\dfrac{l}{4}$, so its spring constant is $k' = 4k$.
Time period: $T = 2\pi\sqrt{\dfrac{m}{k}}$
$T' = 2\pi\sqrt{\dfrac{m}{4k}} = \dfrac{1}{2} \times 2\pi\sqrt{\dfrac{m}{k}} = \dfrac{T}{2}$
So the new time period is $\dfrac{T}{2}$.