| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Moderate -0.3 This is a straightforward application of exponential form operations for complex numbers. Part (a) requires basic manipulation (squaring, division) using index laws with exponential form, which is routine. Part (b) asks for the smallest n where w^n is a positive real number, requiring recognition that the argument must be a multiple of 2π—a standard exercise in complex number periodicity. Both parts are mechanical with no problem-solving insight needed, making this slightly easier than average. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(u^2 = 4e^{\frac{1}{2}\pi i}\) | B1 | |
| Obtain answer \(v = \frac{4}{3}e^{\frac{1}{6}\pi i}\) | B1+B1 | For the modulus and the argument |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(n = 6\) | B1 | |
| Total | 1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $u^2 = 4e^{\frac{1}{2}\pi i}$ | B1 | |
| Obtain answer $v = \frac{4}{3}e^{\frac{1}{6}\pi i}$ | B1+B1 | For the modulus and the argument |
| **Total** | **3** | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $n = 6$ | B1 | |
| **Total** | **1** | |
---5 The complex numbers $u$ and $w$ are defined by $u = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }$ and $w = 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { u ^ { 2 } } { w }$, giving your answer in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Give the exact values of $r$ and $\theta$.
\item State the least positive integer $n$ such that both $\operatorname { Im } w ^ { n } = 0$ and $\operatorname { Re } w ^ { n } > 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [4]}}