Answer only one of the following two alternatives.

EITHER

A particle $P$ of mass $m$ is free to move on the smooth inner surface of a fixed hollow sphere of radius $a$. The centre of the sphere is $O$ and the point $C$ is on the inner surface of the sphere, vertically below $O$. The points $A$ and $B$ on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter $AOB$ makes an acute angle $\alpha$ with the vertical, where $\cos \alpha = \frac{4}{5}$, with $A$ below the horizontal level of $B$. The particle is projected from $A$ with speed $u$, and moves along the inner surface of the sphere towards $C$. The normal reaction forces on the particle at $A$ and $C$ are in the ratio $8 : 9$.

\begin{enumerate}[label=(\roman*)]
\item Show that $u^2 = 4ag$. [6]

\item Determine whether $P$ reaches $B$ without losing contact with the inner surface of the sphere. [6]
\end{enumerate}

OR

A machine is used to produce metal rods. When the machine is working efficiently, the lengths, $x$ cm, of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm. The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.

\begin{tabular}{|c|c|c|c|c|}
\hline
Interval & $146 \leqslant x < 147$ & $147 \leqslant x < 148$ & $148 \leqslant x < 149$ & $149 \leqslant x < 150$ \\
\hline
Observed frequency & 1 & 2 & 23 & 52 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|c|}
\hline
$150 \leqslant x < 151$ & $151 \leqslant x < 152$ & $152 \leqslant x < 153$ & $153 \leqslant x < 154$ \\
\hline
69 & 36 & 15 & 2 \\
\hline
\end{tabular}

As a first check, the sample is used to calculate an estimate for the mean.

\begin{enumerate}[label=(\roman*)]
\item Show that an estimate for the mean from this sample is close to 150 cm. [2]
\end{enumerate}

As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm. The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.

\begin{tabular}{|c|c|c|c|c|}
\hline
Interval & $x < 147$ & $147 \leqslant x < 148$ & $148 \leqslant x < 149$ & $149 \leqslant x < 150$ \\
\hline
Observed frequency & 1 & 2 & 23 & 52 \\
\hline
Expected frequency & 1.24 & 8.32 & 30.94 & 59.50 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|c|}
\hline
$150 \leqslant x < 151$ & $151 \leqslant x < 152$ & $152 \leqslant x < 153$ & $153 \leqslant x$ \\
\hline
69 & 36 & 15 & 2 \\
\hline
59.50 & 30.94 & 8.32 & 1.24 \\
\hline
\end{tabular}

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show how the expected frequency for $151 \leqslant x < 152$ is obtained. [3]

\item Test, at the 5\% significance level, the goodness of fit of the normal distribution to the results. [7]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2018 Q11 [24]}}