Answer only one of the following two alternatives.

\textbf{EITHER}

\includegraphics{figure_11a}

The diagram shows a uniform thin rod $AB$ of length $3a$ and mass $8m$. The end $A$ is rigidly attached to the surface of a sphere with centre $O$ and radius $a$. The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass $m$ and radius $a$ surrounded by a thin uniform spherical shell of mass $m$ and also of radius $a$. The horizontal axis $l$ is perpendicular to the rod and passes through the point $C$ on the rod where $AC = a$.

\begin{enumerate}[label=(\roman*)]
\item Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis $l$ is $\frac{289}{15}ma^2$. [6]
\end{enumerate}

The object is free to rotate about the axis $l$. The object is held so that $CA$ makes an angle $\alpha$ with the downward vertical and is released from rest.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Given that $\cos \alpha = \frac{1}{6}$, find the greatest speed achieved by the centre of the sphere in the subsequent motion. [6]
\end{enumerate}

\textbf{OR}

The times taken to run $200$ metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by $x$ and the time taken, in seconds, at the end of the year is denoted by $y$. For a random sample of $8$ members, the results are shown in the following table.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Member & A & B & C & D & E & F & G & H \\
\hline
$x$ & 24.2 & 23.8 & 22.8 & 25.1 & 24.5 & 24.0 & 23.8 & 22.8 \\
\hline
$y$ & 23.9 & 23.6 & 22.8 & 24.5 & 24.2 & 23.5 & 23.6 & 22.7 \\
\hline
\end{tabular}

$[\Sigma x = 191, \quad \Sigma x^2 = 4564.46, \quad \Sigma y = 188.8, \quad \Sigma y^2 = 4458.4, \quad \Sigma xy = 4510.99.]$

\begin{enumerate}[label=(\roman*)]
\item Find, showing all necessary working, the equation of the regression line of $y$ on $x$. [4]
\end{enumerate}

The athletics coach believes that, on average, the time taken by an athlete to run $200$ metres decreases between the beginning and the end of the year by more than $0.2$ seconds.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the $10\%$ significance level. [8]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2017 Q11 [24]}}