Answer only one of the following two alternatives.

\textbf{EITHER}

The points $A$ and $B$ are a distance 1.2 m apart on a smooth horizontal surface. A particle $P$ of mass $\frac{2}{3}$ kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point $A$. A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to $P$ and the other end attached to $B$.

\begin{enumerate}[label=(\roman*)]
\item Show that when $P$ is in equilibrium $AP = 0.75$ m.
[3]
\end{enumerate}

The particle $P$ is displaced by 0.05 m from the equilibrium position towards $A$ and then released from rest.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show that $P$ performs simple harmonic motion and state the period of the motion.
[6]

\item Find the speed of $P$ when it passes through the equilibrium position.
[2]

\item Find the speed of $P$ when its acceleration is equal to half of its maximum value.
[3]
\end{enumerate}

\textbf{OR}

The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Number of repairs in a week & 0 & 1 & 2 & 3 & 4 & 5 & $\geqslant 6$ \\
\hline
Number of weeks & 6 & 15 & 9 & 6 & 3 & 1 & 0 \\
\hline
\end{tabular}

\begin{enumerate}[label=(\roman*)]
\item Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data.
[3]
\end{enumerate}

Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Number of repairs in a week & 0 & 1 & 2 & 3 & 4 & 5 & $\geqslant 6$ \\
\hline
Expected frequency & 8.076 & 12.921 & 10.337 & 5.513 & 2.205 & $a$ & $b$ \\
\hline
\end{tabular}

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show that $a = 0.706$ and find the value of the constant $b$.
[3]

\item Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level.
[8]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2019 Q11 [28]}}