| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 Part (a) is a standard Further Maths complex numbers question requiring knowledge that arg(z+i)=π/6 represents a half-line from -i. Part (b) requires finding the point on this ray closest to the origin using geometry (perpendicular from origin to the ray) or basic optimization, which is routine for FM students. The question involves standard techniques with no novel insight required, making it slightly easier than average. |
| 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]
\end{enumerate}
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item $z_1$ is a point on $L$ such that $|z_1|$ 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 2023 Q5 [6]}}