| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| 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 sketching loci (circle and ray), identifying their intersection region, and calculating area using sector and triangle geometry. While the individual components are standard Core Pure 2 content, the area calculation requires careful geometric decomposition and exact form manipulation, making it moderately challenging but still within typical A-level scope. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Circle drawn with centre on the real axis | M1 | Circle drawn with centre on real axis; real axis acts as line of symmetry. |
| Circle in correct position with imaginary axis as tangent | A1 | Centre need not be labelled. |
| Half line starting at origin, in first quadrant only | M1 | Must be in first quadrant. Do not award if line continues into third quadrant. |
| Fully correct diagram: circle in correct position, half line intersecting circle at origin and in first quadrant, with \(x\)-coordinate of intersection in first quadrant to left of centre | A1 | All three conditions required for fully correct diagram. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Shading of region inside circle, above real axis and below the half line | B1ft | Shades region in their circle above real axis and below half line. Their line must intersect the circle for this mark. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((x-4)^2+y^2=16,\ y=\sqrt{3}x \Rightarrow x^2-8x+16+3x^2=16 \Rightarrow x=...\), giving \(x=2,\ y=2\sqrt{3}\) | M1 A1 | Correct strategy for both coordinates of intersection. Award for substituting \(y=kx\) into \((x-4)^2+y^2=16\), proceeding to find \(x \neq 0\). Correct coordinates: \((2, 2\sqrt{3})\) or \(2+2\sqrt{3}i\). |
| \(\frac{1}{2}\pi\times4^2 - \left(\frac{1}{2}\times4^2\times\frac{\pi}{3}-\frac{1}{2}\times4^2\times\sin\frac{\pi}{3}\right)\) or \(\frac{1}{2}\times4^2\times\frac{2\pi}{3}+\frac{1}{2}\times4\times2\sqrt{3}\) or \(\frac{1}{2}\times4^2\times\frac{2\pi}{3}+\frac{1}{2}\times4\times4\times\frac{\sqrt{3}}{2}\) | dM1 | Fully correct strategy for area, consistent with half line at angle \(\frac{\pi}{3}\) to real axis. Can subtract segment from semicircle or add sector to triangle. |
| \(= \frac{16}{3}\pi + 4\sqrt{3}\) | A1 | Correct answer in required form. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((x-4)^2+y^2=16 \Rightarrow r^2=8r\cos\theta \Rightarrow r=8\cos\theta\); Area \(= \int\frac{1}{2}r^2\,d\theta = \int\frac{1}{2}\cdot64\cos^2\theta\,d\theta = \int(16+16\cos2\theta)\,d\theta\) | M1 A1 | |
| \(= \left[16\theta+8\sin2\theta\right]_0^{\frac{\pi}{3}} = ...\) | dM1 | |
| \(= \frac{16}{3}\pi+4\sqrt{3}\) | A1 |
## Question 5(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Circle drawn with centre on the real axis | M1 | Circle drawn with centre on real axis; real axis acts as line of symmetry. |
| Circle in correct position with imaginary axis as tangent | A1 | Centre need not be labelled. |
| Half line starting at origin, in first quadrant only | M1 | Must be in first quadrant. Do not award if line continues into third quadrant. |
| Fully correct diagram: circle in correct position, half line intersecting circle at origin and in first quadrant, with $x$-coordinate of intersection in first quadrant to left of centre | A1 | All three conditions required for fully correct diagram. |
---
## Question 5(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Shading of region inside circle, above real axis and below the half line | B1ft | Shades region in their circle above real axis and below half line. Their line must intersect the circle for this mark. |
---
## Question 5(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(x-4)^2+y^2=16,\ y=\sqrt{3}x \Rightarrow x^2-8x+16+3x^2=16 \Rightarrow x=...$, giving $x=2,\ y=2\sqrt{3}$ | M1 A1 | Correct strategy for both coordinates of intersection. Award for substituting $y=kx$ into $(x-4)^2+y^2=16$, proceeding to find $x \neq 0$. Correct coordinates: $(2, 2\sqrt{3})$ or $2+2\sqrt{3}i$. |
| $\frac{1}{2}\pi\times4^2 - \left(\frac{1}{2}\times4^2\times\frac{\pi}{3}-\frac{1}{2}\times4^2\times\sin\frac{\pi}{3}\right)$ or $\frac{1}{2}\times4^2\times\frac{2\pi}{3}+\frac{1}{2}\times4\times2\sqrt{3}$ or $\frac{1}{2}\times4^2\times\frac{2\pi}{3}+\frac{1}{2}\times4\times4\times\frac{\sqrt{3}}{2}$ | dM1 | Fully correct strategy for area, consistent with half line at angle $\frac{\pi}{3}$ to real axis. Can subtract segment from semicircle or add sector to triangle. |
| $= \frac{16}{3}\pi + 4\sqrt{3}$ | A1 | Correct answer in required form. |
**Alternative 3 (Polar coordinates):**
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(x-4)^2+y^2=16 \Rightarrow r^2=8r\cos\theta \Rightarrow r=8\cos\theta$; Area $= \int\frac{1}{2}r^2\,d\theta = \int\frac{1}{2}\cdot64\cos^2\theta\,d\theta = \int(16+16\cos2\theta)\,d\theta$ | M1 A1 | |
| $= \left[16\theta+8\sin2\theta\right]_0^{\frac{\pi}{3}} = ...$ | dM1 | |
| $= \frac{16}{3}\pi+4\sqrt{3}$ | A1 | |\begin{enumerate}
\item The locus $C$ is given by
\end{enumerate}
$$| z - 4 | = 4$$
The locus $D$ is given by
$$\arg z = \frac { \pi } { 3 }$$
(a) Sketch, on the same Argand diagram, the locus $C$ and the locus $D$
The set of points $A$ is defined by
$$A = \{ z \in \mathbb { C } : | z - 4 | \leqslant 4 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\}$$
(b) Show, by shading on your Argand diagram, the set of points $A$\\
(c) Find the area of the region defined by $A$, giving your answer in the form $p \pi + q \sqrt { 3 }$ where $p$ and $q$ are constants to be determined.
\hfill \mbox{\textit{Edexcel CP2 2024 Q5 [9]}}