b
\(\quad x=a \sin (\omega t+\pi / 6)\)

\(\frac{d x}{d t}=a \omega \cos (\omega t+\pi / 6)\)

Max. velocity \(=a \omega\)

\(\therefore \quad \frac{a \omega}{2}=a \omega \cos (\omega t+\pi / 6)\)

\(\therefore \quad \cos (\omega t+\pi / 6)=\frac{1}{2}\)

\(\Rightarrow 60^{\circ}\) or \(\frac{2 \pi}{6}\) radian \(=\frac{2 \pi}{T} \cdot t+\pi / 6\)

\(\Rightarrow \frac{2 \pi}{T} \cdot t=\frac{2 \pi}{6}-\frac{\pi}{6}=+\frac{\pi}{6}\)

\(\therefore \quad t=+\frac{\pi}{6} \times \frac{T}{2 \pi}=\left|+\frac{T}{12}\right|\)