8
\begin{enumerate}[label=(\alph*)]
\item Express in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$ :
\begin{enumerate}[label=(\roman*)]
\item $\quad 4 ( 1 + i \sqrt { 3 } )$;
\item $4 ( 1 - i \sqrt { 3 } )$.
\end{enumerate}\item The complex number $z$ satisfies the equation

$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$

Show that $z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }$.
\item \begin{enumerate}[label=(\roman*)]
\item Solve the equation

$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$

giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\item Illustrate the roots on an Argand diagram.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Explain why the sum of the roots of the equation

$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$

is zero.
\item Deduce that $\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA FP2 2011 Q8 [17]}}