A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time $t$ seconds is denoted by $x$ cm.

Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which $x = 0$.

\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\Alph*)]
\item Write down a differential equation to model this motion. [3]

\item Give the general solution of the differential equation in part (i) (A). [1]
\end{enumerate}
\end{enumerate}

Subsequent observations indicate that the object's motion would be better modelled by the differential equation
$$\frac{d^2x}{dt^2} + 2k \frac{dx}{dt} + (k^2 + 9)x = 0 \qquad (*)$$
where $k$ is a positive constant.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\Alph*)]
\item Obtain the general solution of (*). [3]

\item State two ways in which the motion given by this model differs from that in part (i). [2]
\end{enumerate}
\end{enumerate}

The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the value of $k$. [3]
\end{enumerate}

At the start of the object's motion, $x = 0$ and the velocity is 12 cm s$^{-1}$ in the positive $x$ direction.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find an equation for $x$ as a function of $t$. [4]

\item Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm. [2]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core  Q16 [18]}}