Answer only one of the following two alternatives.

**EITHER**

\includegraphics{figure_11a}

A uniform disc, of mass $4m$ and radius $a$, and a uniform ring, of mass $m$ and radius $2a$, each have centre $O$. A wheel is made by fixing three uniform rods, $OA$, $OB$ and $OC$, each of mass $m$ and length $2a$, to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through $A$, perpendicular to the plane of the wheel, is $42ma^2$. [5]

The axis through $A$ is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with $O$ above the level of $A$ and $AO$ making an angle of $30°$ with the horizontal. Find the angular speed of the wheel when $AO$ is horizontal. [3]

When $AO$ is horizontal the disc becomes detached from the wheel. Find the angle that $AO$ makes with the horizontal when the wheel first comes to instantaneous rest. [6]

**OR**

The continuous random variable $T$ has probability density function given by
$$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

\begin{enumerate}[label=(\roman*)]
\item Find the distribution function of $T$, and find also P$(T > 5)$. [3]
\item Consecutive independent observations of $T$ are made until the first observation that exceeds $5$ is obtained. The random variable $N$ is the total number of observations that have been made up to and including the observation exceeding $5$. Find P$(N > E(N))$. [3]
\item Find the probability density function of $Y$, where $Y = \frac{1}{T-1}$. [8]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2010 Q11 [28]}}