| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 This is a Further Maths complex numbers question requiring geometric understanding of loci in the Argand diagram. Part (a) is routine (circle sketching), but part (b) requires finding the unique intersection point where a circle meets a half-line from a different center, then converting to polar form. This involves non-trivial geometric reasoning and coordinate work, placing it moderately above average difficulty. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the Argand diagram below, the locus of points satisfying the equation
$|z - 3| = 2$
[1]
\end{enumerate}
\includegraphics{figure_9}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item There is a unique complex number $w$ that satisfies both
$|w - 3| = 2$ and $\arg(w + 1) = \alpha$
where $\alpha$ is a constant such that $0 < \alpha < \pi$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\alpha$.
[2]
\item Express $w$ in the form $r(\cos \theta + i \sin \theta)$.
Give each of $r$ and $\theta$ to two significant figures.
[4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q9 [7]}}