| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | January |
| Marks | 12 |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question. Part (i) is routine conversion to modulus-argument form (|w|=4, arg(w)=π/3). Part (ii) requires sketching a sector region and finding the maximum distance from w to points in that region—a standard geometric optimization that requires identifying the furthest vertex. While it involves multiple steps and geometric visualization, it uses only standard techniques without requiring novel insight. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
You are given the complex number $w = 2 + 2\sqrt{3}i$.
\begin{enumerate}[label=(\roman*)]
\item Express $w$ in modulus-argument form. [3]
\item Indicate on an Argand diagram the set of points, $z$, which satisfy both of the following inequalities.
$$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$
Mark $w$ on your Argand diagram and find the greatest value of $|z - w|$. [9]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q6 [12]}}