| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Area calculations in complex plane |
| Difficulty | Standard +0.8 This question requires interpreting an Argand diagram to identify an annulus center and radii (part a), then finding the intersection with a perpendicular bisector locus to calculate the area of a semi-annulus (part b). While the concepts are standard A-level Further Maths, the multi-step nature combining loci interpretation, geometric reasoning about the perpendicular bisector |z-i|=|z-3i| (which gives y=2), and area calculation of the resulting semi-annular region makes this moderately challenging but still within typical Further Maths scope. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(a = 1,\ d = 2\) | B1 | 1.1b |
| \(b = 2\) | B1 | 1.1b |
| \(c = -1\) | B1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \( | z - \mathrm{i} | = |
| Area between circles \(= \pi \times 2^2 - \pi \times 1^2\) | M1 | 1.1a – Selects correct procedure: large circle area minus small circle area |
| Angle subtended at centre \(= 2\times\cos^{-1}\!\left(\frac{1}{2}\right)\); alternatively \((x+2)^2+(y-1)^2=4,\ y=2 \Rightarrow x=\ldots\) or \(x=\sqrt{2^2-1^2}\); leading to angle \(= 2\times\tan^{-1}\!\left(\frac{\sqrt{3}}{1}\right)\) | M1 | 3.1a – Correct method to find angle at centre (or half angle) |
| Segment area \(= \frac{1}{2}\times\frac{2\pi}{3}\times 2^2 - \frac{1}{2}\times 2^2\times\sin\!\left(\frac{2\pi}{3}\right) = \left\{\frac{4}{3}\pi - \sqrt{3}\right\}\) | M1, A1 | 2.1, 1.1b – Correct method for minor segment area |
| Area of \(Q\): \(\pi\times 2^2 - \pi\times 1^2 - \left(\frac{1}{2}\times\frac{2\pi}{3}\times 2^2 - \frac{1}{2}\times 2^2\times\sin\!\left(\frac{2\pi}{3}\right)\right)\) | M1 | 3.1a – Fully correct strategy: subtracting segment from annulus |
| \(= \frac{5\pi}{3} + \sqrt{3}\) | A1 | 1.1b – Correct exact answer |
# Question 5:
## Part (a):
| Working | Mark | Notes |
|---------|------|-------|
| $a = 1,\ d = 2$ | B1 | 1.1b |
| $b = 2$ | B1 | 1.1b |
| $c = -1$ | B1 | 1.1b |
## Part (b):
| Working | Mark | Notes |
|---------|------|-------|
| $|z - \mathrm{i}| = |z - 3\mathrm{i}| \Rightarrow y = 2$ | B1 | 2.2a – Deduces perpendicular bisector with equation $y=2$ |
| Area between circles $= \pi \times 2^2 - \pi \times 1^2$ | M1 | 1.1a – Selects correct procedure: large circle area minus small circle area |
| Angle subtended at centre $= 2\times\cos^{-1}\!\left(\frac{1}{2}\right)$; alternatively $(x+2)^2+(y-1)^2=4,\ y=2 \Rightarrow x=\ldots$ or $x=\sqrt{2^2-1^2}$; leading to angle $= 2\times\tan^{-1}\!\left(\frac{\sqrt{3}}{1}\right)$ | M1 | 3.1a – Correct method to find angle at centre (or half angle) |
| Segment area $= \frac{1}{2}\times\frac{2\pi}{3}\times 2^2 - \frac{1}{2}\times 2^2\times\sin\!\left(\frac{2\pi}{3}\right) = \left\{\frac{4}{3}\pi - \sqrt{3}\right\}$ | M1, A1 | 2.1, 1.1b – Correct method for minor segment area |
| Area of $Q$: $\pi\times 2^2 - \pi\times 1^2 - \left(\frac{1}{2}\times\frac{2\pi}{3}\times 2^2 - \frac{1}{2}\times 2^2\times\sin\!\left(\frac{2\pi}{3}\right)\right)$ | M1 | 3.1a – Fully correct strategy: subtracting segment from annulus |
| $= \frac{5\pi}{3} + \sqrt{3}$ | A1 | 1.1b – Correct exact answer |
**Total: 10 marks**
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-12_584_830_246_639}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows an Argand diagram.\\
The set $P$, of points that lie within the shaded region including its boundaries, is defined by
$$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$
where $a$, $b$, $c$ and $d$ are integers.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $a , b , c$ and $d$.
The set $Q$ is defined by
$$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
\item Determine the exact area of the region defined by $Q$, giving your answer in simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2021 Q5 [10]}}