5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-10_483_528_260_772}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)
- Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
- Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
- Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\)
A copy of Figure 1, labelled Diagram 1, is given on page 12.
- Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality
$$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
\includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
\section*{Diagram 1}