| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This is a standard multi-part complex numbers question requiring routine techniques: rationalizing a complex fraction, converting to exponential form, and finding square roots using De Moivre's theorem. All steps are algorithmic with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply numerator and denominator by \(1-2i\), or equivalent | M1 | At least one multiplication completed |
| Obtain correct numerator \((1-2a)\sqrt{2} - (2+a)\sqrt{2}i\) | A1 | OE |
| Obtain final answer \(\frac{1-2a}{5}\sqrt{2} - \frac{2+a}{5}\sqrt{2}i\) | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply \(x+iy\) by \(1+2i\) and compare real and imaginary parts | M1 | |
| Obtain \(x - 2y = \sqrt{2}\) and \(2x + y = a\sqrt{2}\) | A1 | |
| Obtain final answer \(\frac{1-2a}{5}\sqrt{2} - \frac{2+a}{5}\sqrt{2}i\) | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(r = 2\) | B1 FT | |
| Obtain \(\theta = -\frac{3}{4}\pi\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to find \(r\) or \(\theta\) | M1 | |
| State answer \(\sqrt{2}e^{-\frac{3}{8}\pi i}\) | A1 FT | |
| State answer \(\sqrt{2}e^{\frac{5}{8}\pi i}\) | A1 FT |
## Question 7:
### Part 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply numerator **and** denominator by $1-2i$, or equivalent | M1 | At least one multiplication completed |
| Obtain correct numerator $(1-2a)\sqrt{2} - (2+a)\sqrt{2}i$ | A1 | OE |
| Obtain final answer $\frac{1-2a}{5}\sqrt{2} - \frac{2+a}{5}\sqrt{2}i$ | A1 | OE |
**Alternative method for 7(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply $x+iy$ by $1+2i$ and compare real and imaginary parts | M1 | |
| Obtain $x - 2y = \sqrt{2}$ and $2x + y = a\sqrt{2}$ | A1 | |
| Obtain final answer $\frac{1-2a}{5}\sqrt{2} - \frac{2+a}{5}\sqrt{2}i$ | A1 | OE |
### Part 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $r = 2$ | B1 FT | |
| Obtain $\theta = -\frac{3}{4}\pi$ | B1 | |
### Part 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to find $r$ or $\theta$ | M1 | |
| State answer $\sqrt{2}e^{-\frac{3}{8}\pi i}$ | A1 FT | |
| State answer $\sqrt{2}e^{\frac{5}{8}\pi i}$ | A1 FT | |
---7 The complex number $u$ is defined by $u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }$, where $a$ is a positive integer.
\begin{enumerate}[label=(\alph*)]
\item Express $u$ in terms of $a$, in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.\\
It is now given that $a = 3$.
\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 Using your answer to part (b), find the two square roots of $u$. Give your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$, giving the exact values of $r$ and $\theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q7 [8]}}