$\Rightarrow y=A(\cos 2 \omega t+1)$

$\Rightarrow y-A=A \cos 2 \omega t$

$\Rightarrow A_{1}=A ; \omega_{1}=2 \omega$

$y=A(\sin \omega t+\sqrt{3} \cos \omega t)$

$\Rightarrow y=2 A\left(\frac{1}{2} \sin \omega t+\frac{\sqrt{3}}{2} \cos \omega t\right)$

$\Rightarrow y=2 A \sin \left(\omega t+\frac{\pi}{3}\right)$

$\Rightarrow A_{2}=2 A ; \omega_{2}=\omega$

Ratio of maximum speeds $=\frac{A_{1} \omega_{1}}{A_{2} \omega_{2}}=\frac{A(2 \omega)}{(2 A) \omega}=\frac{1}{1}$