| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Argand diagram sketching and regions |
| Difficulty | Standard +0.8 This question requires converting to exponential form, applying De Moivre's theorem, and sketching an Argand diagram region defined by argument and real part constraints. Parts (a) and (b) are standard, but part (c) requires geometric visualization of the intersection of a wedge region and a half-plane, then finding the maximum modulus through coordinate geometry or optimization—this elevates it above routine exercises. |
| Spec | 4.02d Exponential form: re^(i*theta)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 | Marks | Guidance |
| State or imply \(r = 2\) | B1 | |
| State or imply \(\theta = \frac{5}{6}\pi\) | B1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use a correct method for finding the modulus or argument of \(u^6\) | M1 | |
| Show correctly that \(u^6\) is real and has value \(-64\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show half lines from the point representing \(-\sqrt{3}+\mathrm{i}\) | B1 | |
| Show correct half lines | B1 | |
| Show the line \(x = 2\) in the first quadrant | B1 | |
| Shade the correct region | B1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a correct method to find the greatest value of \( | z | \) |
| Obtain answer \(5.14\) | A1 | |
| Total | 2 |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $r = 2$ | B1 | |
| State or imply $\theta = \frac{5}{6}\pi$ | B1 | |
| **Total** | **2** | |
## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use a correct method for finding the modulus or argument of $u^6$ | M1 | |
| Show correctly that $u^6$ is real and has value $-64$ | A1 | |
| **Total** | **2** | |
## Question 11(c)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show half lines from the point representing $-\sqrt{3}+\mathrm{i}$ | B1 | |
| Show correct half lines | B1 | |
| Show the line $x = 2$ in the first quadrant | B1 | |
| Shade the correct region | B1 | |
| **Total** | **4** | |
## Question 11(c)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a correct method to find the greatest value of $|z|$ | M1 | |
| Obtain answer $5.14$ | A1 | |
| **Total** | **2** | |11 The complex number $- \sqrt { 3 } + \mathrm { i }$ is denoted by $u$.
\begin{enumerate}[label=(\alph*)]
\item Express $u$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$, giving the exact values of $r$ and $\theta$.
\item Hence show that $u ^ { 6 }$ is real and state its value.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $0 \leqslant \arg ( z - u ) \leqslant \frac { 1 } { 4 } \pi$ and $\operatorname { Re } z \leqslant 2$.
\item Find the greatest value of $| z |$ for points in the shaded region. Give your answer correct to 3 significant figures.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q11 [10]}}