| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Convert to exponential/polar form |
| Difficulty | Moderate -0.8 This is a straightforward conversion question requiring standard techniques: finding modulus and argument for part (a), then applying division rules for complex numbers in polar form for part (b). The numbers are chosen to give exact values (multiples of π/6), making this easier than average with no problem-solving or novel insight required. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(r = 2\) | B1 | |
| State or imply \(\theta = -\frac{2}{3}\pi\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(r = \frac{5}{2}\) | B1FT | FT \(\frac{5}{their\ 2}\) |
| State or imply \(\theta = \frac{5}{6}\pi\) | B1FT | FT \(\frac{1}{6}\pi - their\ {-\frac{2}{3}\pi}\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $r = 2$ | B1 | |
| State or imply $\theta = -\frac{2}{3}\pi$ | B1 | |
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $r = \frac{5}{2}$ | B1FT | FT $\frac{5}{their\ 2}$ |
| State or imply $\theta = \frac{5}{6}\pi$ | B1FT | FT $\frac{1}{6}\pi - their\ {-\frac{2}{3}\pi}$ |
---4 The complex number $u$ is given by $u = - 1 - \mathrm { i } \sqrt { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Express $u$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Give the exact values of $r$ and $\theta$.\\
The complex number $v$ is given by $v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)$.
\item Express the complex number $\frac { \mathrm { v } } { \mathrm { u } }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q4 [4]}}