A particle $P$, of mass $m$, is able to move in a vertical circle on the smooth inner surface of a sphere with centre $O$ and radius $a$. Points $A$ and $B$ are on the inner surface of the sphere and $A O B$ is a horizontal diameter. Initially, $P$ is projected vertically downwards with speed $\sqrt { } \left( \frac { 21 } { 2 } a g \right)$ from $A$ and begins to move in a vertical circle. At the lowest point of its path, vertically below $O$, the particle $P$ collides with a stationary particle $Q$, of mass $4 m$, and rebounds. The speed acquired by $Q$, as a result of the collision, is just sufficient for it to reach the point $B$.\\
(i) Find the speed of $P$ and the speed of $Q$ immediately after their collision.\\

In its subsequent motion, $P$ loses contact with the inner surface of the sphere at the point $D$, where the angle between $O D$ and the upward vertical through $O$ is $\theta$.\\
(ii) Find $\cos \theta$.\\

\hfill \mbox{\textit{CAIE FP2 2019 Q11 EITHER}}