| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2021 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of unity applications |
| Difficulty | Standard +0.8 This question combines roots of unity with geometric reasoning (part i) and loci intersection with area calculation (part ii). Part (i) requires converting to modulus-argument form, understanding rotation by 2π/5, and applying this four times. Part (ii) involves visualizing the intersection of a disc with an angular sector, then computing the area using sector formulas and potentially trigonometry. While the individual techniques are standard Further Maths content, the multi-step nature, need for geometric insight, and exact area calculation with non-standard angles elevate this above routine exercises. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \( | z | = \sqrt{6^2+6^2} = 6\sqrt{2}\) or \(\sqrt{72}\); \(\arg z = \tan^{-1}\left(\frac{6}{6}\right) = \frac{\pi}{4}\) |
| Adding multiples of \(\frac{2\pi}{5}\): \(z = 6\sqrt{2}e^{\left(\frac{\pi}{4}+\frac{2\pi k}{5}\right)i}\) | M1 | Correct method for all other 4 vertices; adding/subtracting multiples of \(\frac{2\pi}{5}\) |
| \(z = re^{\left(\theta+\frac{2\pi}{5}\right)i},\, re^{\left(\theta+\frac{4\pi}{5}\right)i},\, re^{\left(\theta+\frac{6\pi}{5}\right)i},\, re^{\left(\theta+\frac{8\pi}{5}\right)i}\) | A1ft | All 4 vertices following through on modulus and argument |
| \(z = 6\sqrt{2}e^{\frac{13\pi}{20}i},\, 6\sqrt{2}e^{\frac{21\pi}{20}i},\, 6\sqrt{2}e^{\frac{29\pi}{20}i},\, 6\sqrt{2}e^{\frac{37\pi}{20}i}\) | A1 | All 4 vertices correct in required form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Circle centre \((0,2)\) and radius \(2\), with vertex on origin | B1 | Correct description |
| Fully correct region shaded | B1 | Fully correct shaded region shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{area} = \frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} 4\sin\theta^2\, d\theta\) | M1 | Writes required area using polar coordinates |
| Uses \(\sin^2\theta = \frac{1}{2} - \frac{1}{2}\cos 2\theta\); integrates to form \(A\theta + B\sin 2\theta\) | M1 | Correct identity and integration form |
| \(\left[4\left(\frac{\pi}{3}\right) - 2\sin\left(\frac{2\pi}{3}\right)\right] - \left[4\left(\frac{\pi}{4}\right) - 2\sin\left(\frac{2\pi}{4}\right)\right]\) | M1 | Uses limits \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), subtracts correct way round |
| \(\text{Area} = \frac{\pi}{3} - \sqrt{3} + 2\) | A1 | Correct exact area |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $|z| = \sqrt{6^2+6^2} = 6\sqrt{2}$ or $\sqrt{72}$; $\arg z = \tan^{-1}\left(\frac{6}{6}\right) = \frac{\pi}{4}$ | M1, A1 | M1: finds modulus and argument; A1: both correct |
| Adding multiples of $\frac{2\pi}{5}$: $z = 6\sqrt{2}e^{\left(\frac{\pi}{4}+\frac{2\pi k}{5}\right)i}$ | M1 | Correct method for all other 4 vertices; adding/subtracting multiples of $\frac{2\pi}{5}$ |
| $z = re^{\left(\theta+\frac{2\pi}{5}\right)i},\, re^{\left(\theta+\frac{4\pi}{5}\right)i},\, re^{\left(\theta+\frac{6\pi}{5}\right)i},\, re^{\left(\theta+\frac{8\pi}{5}\right)i}$ | A1ft | All 4 vertices following through on modulus and argument |
| $z = 6\sqrt{2}e^{\frac{13\pi}{20}i},\, 6\sqrt{2}e^{\frac{21\pi}{20}i},\, 6\sqrt{2}e^{\frac{29\pi}{20}i},\, 6\sqrt{2}e^{\frac{37\pi}{20}i}$ | A1 | All 4 vertices correct in required form |
## Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle centre $(0,2)$ and radius $2$, with vertex on origin | B1 | Correct description |
| Fully correct region shaded | B1 | Fully correct shaded region shown |
## Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{area} = \frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} 4\sin\theta^2\, d\theta$ | M1 | Writes required area using polar coordinates |
| Uses $\sin^2\theta = \frac{1}{2} - \frac{1}{2}\cos 2\theta$; integrates to form $A\theta + B\sin 2\theta$ | M1 | Correct identity and integration form |
| $\left[4\left(\frac{\pi}{3}\right) - 2\sin\left(\frac{2\pi}{3}\right)\right] - \left[4\left(\frac{\pi}{4}\right) - 2\sin\left(\frac{2\pi}{4}\right)\right]$ | M1 | Uses limits $\frac{\pi}{4}$ and $\frac{\pi}{3}$, subtracts correct way round |
| $\text{Area} = \frac{\pi}{3} - \sqrt{3} + 2$ | A1 | Correct exact area |\begin{enumerate}
\item (i) The point $P$ is one vertex of a regular pentagon in an Argand diagram.
\end{enumerate}
The centre of the pentagon is at the origin.\\
Given that $P$ represents the complex number $6 + 6 \mathrm { i }$, determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$\\
(ii) (a) On a single Argand diagram, shade the region, $R$, that satisfies both
$$| z - 2 i | \leqslant 2 \quad \text { and } \quad \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi$$
(b) Determine the exact area of $R$, giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel CP2 2021 Q8 [11]}}