| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | September |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question requiring conversion between forms and checking inequalities. Part (i) reads coordinates from a diagram, part (ii) converts modulus-argument to Cartesian form using standard formulas, and part (iii) checks two given complex numbers against the region's constraints. All techniques are routine for FM students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by
$$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$
\includegraphics{figure_9}
\begin{enumerate}[label=(\roman*)]
\item Find, in modulus-argument form, the complex number represented by the point P. [2]
\item Find, in the form $a + ib$, where $a$ and $b$ are exact real numbers, the complex number represented by the point Q. [3]
\item In this question you must show detailed reasoning.
Determine whether the points representing the complex numbers
\begin{itemize}
\item $3 + 5i$
\item $5.5(\cos 0.8 + i\sin 0.8)$
\end{itemize}
lie within this region. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q12 [9]}}