| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This is a Further Maths complex numbers question requiring visualization of loci (circle and argument rays) and finding maximum argument on a circle. Part (i) involves standard intersection of regions but with careful attention to inequalities. Part (ii) requires geometric insight that maximum argument occurs at a tangent point, then trigonometry to calculate it. More demanding than typical A-level complex loci questions due to the intersection and optimization, but uses well-established techniques without requiring novel problem-solving approaches. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}[label=(\roman*)]
\item Shade on an Argand diagram the set of points
$$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]
\end{enumerate}
The complex number $w$ satisfies $|w - 4i| = 3$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the maximum value of $\arg w$ in the interval $(-\pi, \pi]$. Give your answer in radians correct to 2 decimal places. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q8 [7]}}