10
\begin{enumerate}[label=(\alph*)]
\item Show on an Argand diagram the points representing the three cube roots of unity.
\item \begin{enumerate}[label=(\roman*)]
\item Find the exact roots of the equation $z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }$, expressing them in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta < \pi$.
\item The points representing the cube roots of unity form a triangle $\Delta _ { 1 }$. The points representing the roots of the equation $z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }$ form a triangle $\Delta _ { 2 }$.

State a sequence of two transformations that maps $\Delta _ { 1 }$ onto $\Delta _ { 2 }$.
\item The three roots in part (b)(i) are $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$.

By simplifying $z _ { 1 } + z _ { 2 } + z _ { 3 }$, verify that the sum of these roots is zero.
\item Hence show that $\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q10 [13]}}