The displacement of an object attached to a spring and executing simple harmonic motion is given by $\mathrm{x}=2\times {10}^{-2}\cos \mathrm{\pi t}$ meter. The time in second at which the maximum speed first occurs is

$$\begin{gathered} \mathrm{x}=\left(2\times {10}^{-2}\right)\cos \mathrm{\pi t} \\ \mathrm{Here} \mathrm{a}=2\times {10}^{-2}\mathrm{m}=2\mathrm{cm} \\ \mathrm{At} \mathrm{t}=0,\mathrm{x}=2\mathrm{cm} \end{gathered}$$

i.e., the object is at positive extreme, so to acquire maximum speed (i.e., to reach mean position) it takes $\frac{1}{4}$th of time period.

 $$\begin{gathered} \therefore \mathrm{Required} \mathrm{time} =\frac{\mathrm{T}}{4} \\ \mathrm{Where} \mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}=\mathrm{\pi } \\ \Rightarrow \mathrm{T}=2\mathrm{s} \end{gathered}$$

$\mathrm{So}, \mathrm{required} \mathrm{time} =\frac{\mathrm{T}}{4}=\frac{2}{4}=0.5\mathrm{s}$