\includegraphics{figure_3}

In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations
$$x = 10 \cos \theta + 5 \cos 2\theta, \quad y = 10 \sin \theta + 5 \sin 2\theta, \quad (0 \leqslant \theta < 2\pi),$$
where $x$ and $y$ are in metres.

\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{\text{d}y}{\text{d}x} = -\frac{\cos \theta + \cos 2\theta}{\sin \theta + \sin 2\theta}$.

Verify that $\frac{\text{d}y}{\text{d}x} = 0$ when $\theta = \frac{1}{3}\pi$. Hence find the exact coordinates of the highest point A on the path of C. [6]

\item Express $x^2 + y^2$ in terms of $\theta$. Hence show that
$$x^2 + y^2 = 125 + 100 \cos \theta.$$ [4]

\item Using this result, or otherwise, find the greatest and least distances of C from O. [2]
\end{enumerate}

You are given that, at the point B on the path vertically above O,
$$2 \cos^2 \theta + 2 \cos \theta - 1 = 0.$$

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q4 [16]}}