$x_{1}=x_{2}$
$\Rightarrow-A \sin \omega t=A \cos \omega t$
$\Rightarrow \tan \omega t=-1$
$\Rightarrow \omega t=-\frac{\pi}{4}, \frac{3 \pi}{4}$
$\omega t=-\frac{\pi}{4}$ will give the negative value of time which is not possible.
so
$\omega t=\frac{3 \pi}{4}$
$\Rightarrow \frac{2 \pi}{T} t=\frac{3 \pi}{4}$
$\Rightarrow t=\frac{3 T}{8}$
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$(A)$ If $V_2=2 V_1$ and $T_2=3 T_1$, then the energy stored in the spring is $\frac{1}{4} P_1 V_1$
$(B)$ If $V_2=2 V_1$ and $T_2=3 T_1$, then the change in internal energy is $3 P_1 V_1$
$(C)$ If $V_2=3 V_1$ and $T_2=4 T_1$, then the work done by the gas is $\frac{7}{3} P_1 V_1$
$(D)$ If $V_2=3 V_1$ and $T_2=4 T_1$, then the heat supplied to the gas is $\frac{17}{6} P_1 V_1$