| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| 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 (circle and half-line from a point), finding their tangency condition, then converting to modulus-argument form. Part (a) is routine, but parts (b)(i)-(ii) require insight that uniqueness implies tangency, followed by coordinate geometry and trigonometric manipulation—more demanding than standard A-level complex number exercises but not requiring exceptional creativity. |
| Spec | 4.02d Exponential form: re^(i*theta)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 mark]
\includegraphics{figure_8}
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item There is a unique complex number $w$ that satisfies both
$$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$
where $\alpha$ is a constant such that $0 < \alpha < \pi$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item[\textbf{(b) (i)}] Find the value of $\alpha$.
[2 marks]
\item[\textbf{(b) (ii)}] Express $w$ in the form $r(\cos \theta + i \sin \theta)$.
[4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q8 [7]}}