| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | January |
| Marks | 10 |
| Topic | Complex Numbers Argand & Loci |
| Type | Area calculations in complex plane |
| Difficulty | Challenging +1.2 Part (a) requires reading an Argand diagram to identify a center and two radii defining an annulus—straightforward visual interpretation. Part (b) involves finding the intersection with a half-plane defined by |z - i| ≤ |z - 3i| (which simplifies to Im(z) ≤ 2), then computing the area of the resulting region. This requires understanding loci in the complex plane and calculating areas of circular segments, which is moderately challenging but follows standard Further Maths techniques without requiring novel insight. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
\includegraphics{figure_11}
Figure 1 shows an Argand diagram.
The set $P$ of points that lie within the shaded region including its boundaries, is defined by
$$P = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\}$$
where $a$, $b$, $c$ and $d$ are integers.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $a$, $b$, $c$ and $d$. [3]
\end{enumerate}
The set $Q$ is defined by
$$Q = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\} \cap \{z \in \mathbb{C} : |z - i| \leq |z - 3i|\}$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the exact area of the region defined by $Q$, giving your answer in simplest form. [7]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q11 [10]}}