| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | September |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question. Part (a) requires recognizing that arg(z+i)=π/6 represents a half-line from -i at angle π/6, which is standard bookwork. Part (b) involves finding the point on this ray closest to the origin using perpendicular distance or basic geometry—a routine optimization requiring no novel insight, just careful application of standard techniques. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}[label=(\alph*)]
\item On the Argand diagram below, sketch the locus, $L$, of points satisfying the equation
$$\arg(z + i) = \frac{\pi}{6}$$ [2 marks]
\includegraphics{figure_5}
\item $z_1$ is a point on $L$ such that $|z|$ is a minimum.
Find the exact value of $z_1$ in the form $a + bi$ [4 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q5 [6]}}