we have, $x=A \cos \omega t .$ At $t=0, x=A$

When $t=\tau, x=A-a$ and

$\text { when } t=2 \tau, x=A-3 a$

$\Rightarrow A-a=A \times \cos \times \omega \tau$ and              $...(i)$

$A-3 a=A \times \cos \times 2 \omega \tau$                    $...(ii)$

As, $\cos 2 \omega \tau=2 \cos ^{2} \omega \tau-1$

$\Rightarrow \quad \frac{A-3 a}{A}=2\left(\frac{A-a}{A}\right)^{2}-1$

$\therefore \quad \frac{A-3 a}{A}=\frac{2 A^{2}+2 a^{2}-4 A a-A^{2}}{A^{2}}$

$\therefore \quad A^{2}-3 a A=A^{2}+2 a^{2}-4 A a$

$\therefore \quad a^{2}=2 a A \Rightarrow A=2 a$

$\text { Now, } A-a=A \times \cos \times \omega \tau \ldots \ldots[\text { From }(1)]$

$\Rightarrow \quad \cos x \omega \tau=\frac{1}{2}$

$\therefore \quad \frac{2 \pi}{T} \tau=\frac{\pi}{3} \Rightarrow T=6 \tau$