A smooth sphere with centre $O$ and of radius $a$ is fixed to a horizontal plane. A particle $P$ of mass $m$ is projected horizontally from the highest point of the sphere with speed $u$, so that it begins to move along the surface of the sphere. The particle $P$ loses contact with the sphere at the point $Q$ on the sphere, where $OQ$ makes an angle $\theta$ with the upward vertical through $O$.

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\item Show that $\cos\theta = \frac{u^2 + 2ag}{3ag}$. [4]
\end{enumerate}

It is given that $\cos\theta = \frac{5}{9}$.

\begin{enumerate}[label=(\alph*)]
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\item Find, in terms of $a$ and $g$, an expression for the vertical component of the velocity of $P$ just before it hits the horizontal plane to which the sphere is fixed. [3]
\item Find an expression for the time taken by $P$ to fall from $Q$ to the plane. Give your answer in the form $k\sqrt{\frac{a}{g}}$, stating the value of $k$ correct to 3 significant figures. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q7 [9]}}