What is the ratio of the angular velocity of the hour hand of a watch to the angular velocity of Earth's rotation about its axis?

The correct option is A $2:1$
Time taken by hour hand for one rotation is $12\text{hrs}$
So, ${\omega }_H$= $\frac{2\pi }{12}rad/hr$........(i)
Time taken by Earth for one rotation is $24\text{hrs}$
So, ${\omega }_E$ =$\frac{2\pi }{24}rad/hr$

Hence, required ratio is
$\frac{{\omega }_H}{{\omega }_E}=\frac{\frac{2\pi }{12}}{\frac{2\pi }{24}}=\frac{24}{12}=2:1$