8 The diagram shows a sketch of a curve.\\
\includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726}

The polar equation of the curve is

$$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$

The point $P$ is the point of the curve at which $\theta = \frac { \pi } { 3 }$.

The perpendicular from $P$ to the initial line meets the initial line at the point $N$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the exact value of $r$ when $\theta = \frac { \pi } { 3 }$.
\item Show that the polar equation of the line $P N$ is $r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta$.
\item Find the area of triangle $O N P$ in the form $\frac { k \sqrt { 3 } } { 128 }$, where $k$ is an integer.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Using the substitution $u = \sin \theta$, or otherwise, find $\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta$, where $n \geqslant 2$.
\item Find the area of the shaded region bounded by the line $O P$ and the arc $O P$ of the curve. Give your answer in the form $a \pi + b \sqrt { 3 } + c$, where $a , b$ and $c$ are constants.\\
(8 marks)
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA FP3 2013 Q8 [16]}}