| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Geometric properties in Argand diagram |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard complex number operations (multiplication, polar form conversion) and geometric interpretation. Parts (a)-(c) are routine A-level exercises requiring direct application of formulas. Part (d) requires recognizing that arg(zω) = arg(z) + arg(ω) and using the tangent of the resulting angle, which is a standard technique but adds slight problem-solving element. Overall slightly easier than average due to the guided structure and straightforward calculations. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=14.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(z\omega = \left(-3+3\sqrt{3}\right) + \left(3+3\sqrt{3}\right)\text{i}\) | B1 | Or exact equivalent with real and imaginary parts collected. Need brackets around the coefficient of i. Allow for \(a =, b =\) stated correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \( | z | = \sqrt{2}\) |
| Obtain \(\arg z = -\frac{\pi}{4}\) final answer | B1 | |
| Obtain \( | \omega | = 6\) |
| Obtain \(\arg \omega = \frac{2\pi}{3}\) final answer | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show \( | OA | = |
| \(\angle AOB = \arg\omega - \arg z\omega = -\arg z = \frac{\pi}{4}\), hence third angle is a right angle | B1 | Note: question does not require the diagram. If they use \(\frac{5\pi}{12}\) they need to demonstrate where it comes from. Complex number equivalent to \(AB\) is \(3\sqrt{3}+3\text{i}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\arg z\omega = \arg z + \arg\omega \left(= \frac{2\pi}{3} - \frac{\pi}{4} = \frac{5\pi}{12}\right)\) | M1 | For showing correct use of their angles from part (b). Must demonstrate where \(\frac{5\pi}{12}\) comes from |
| \(\arg z\omega = \tan^{-1}\dfrac{3+3\sqrt{3}}{-3+3\sqrt{3}}\) | M1 | Correct method for their \(z\omega\) from part (a). Must link to point \(B\) on diagram or to \(\arg z\omega\). Need to see \(\tan^{-1}\frac{3+3\sqrt{3}}{-3+3\sqrt{3}}\) or \(\tan\theta = \frac{3+3\sqrt{3}}{-3+3\sqrt{3}}\) and not just \(\tan^{-1}\frac{1+\sqrt{3}}{-1+\sqrt{3}}\) |
| \(\Rightarrow \tan\!\left(\frac{5}{12}\pi\right) = \dfrac{\sqrt{3}+1}{\sqrt{3}-1}\) | A1 | Obtain given answer from full and correct working |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $z\omega = \left(-3+3\sqrt{3}\right) + \left(3+3\sqrt{3}\right)\text{i}$ | B1 | Or exact equivalent with real and imaginary parts collected. Need brackets around the coefficient of i. Allow for $a =, b =$ stated correctly |
---
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $|z| = \sqrt{2}$ | B1 | |
| Obtain $\arg z = -\frac{\pi}{4}$ final answer | B1 | |
| Obtain $|\omega| = 6$ | B1 | |
| Obtain $\arg \omega = \frac{2\pi}{3}$ final answer | B1 | |
---
## Question 9(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show $|OA| = |AB| = 6$, hence isosceles | B1 | One mark for 'isosceles' and one mark for 'right angle'. Alternatives e.g. use of Pythagoras (ratio of lengths is $1:1:\sqrt{2}$), expressing each number in "vector" form and using scalar product, or explaining the effect of multiplying by $1-\text{i}$ |
| $\angle AOB = \arg\omega - \arg z\omega = -\arg z = \frac{\pi}{4}$, hence third angle is a right angle | B1 | Note: question does not require the diagram. If they use $\frac{5\pi}{12}$ they need to demonstrate where it comes from. Complex number equivalent to $AB$ is $3\sqrt{3}+3\text{i}$ |
---
## Question 9(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\arg z\omega = \arg z + \arg\omega \left(= \frac{2\pi}{3} - \frac{\pi}{4} = \frac{5\pi}{12}\right)$ | M1 | For showing correct use of their angles from part (b). Must demonstrate where $\frac{5\pi}{12}$ comes from |
| $\arg z\omega = \tan^{-1}\dfrac{3+3\sqrt{3}}{-3+3\sqrt{3}}$ | M1 | Correct method for their $z\omega$ from part (a). Must link to point $B$ on diagram or to $\arg z\omega$. Need to see $\tan^{-1}\frac{3+3\sqrt{3}}{-3+3\sqrt{3}}$ or $\tan\theta = \frac{3+3\sqrt{3}}{-3+3\sqrt{3}}$ and not just $\tan^{-1}\frac{1+\sqrt{3}}{-1+\sqrt{3}}$ |
| $\Rightarrow \tan\!\left(\frac{5}{12}\pi\right) = \dfrac{\sqrt{3}+1}{\sqrt{3}-1}$ | A1 | Obtain given answer from full and correct working |
---9 The complex numbers $z$ and $\omega$ are defined by $z = 1 - i$ and $\omega = - 3 + 3 \sqrt { 3 } i$.
\begin{enumerate}[label=(\alph*)]
\item Express $z \omega$ in the form $\mathrm { a } + \mathrm { bi }$, where $a$ and $b$ are real and in exact surd form.
\item Express $z$ and $\omega$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Give the exact values of $r$ and $\theta$ in each case.
\item On an Argand diagram, the points representing $\omega$ and $z \omega$ are $A$ and $B$ respectively.
Prove that $O A B$ is an isosceles right-angled triangle, where $O$ is the origin.
\item Using your answers to part (b), prove that $\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q9 [10]}}