| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus-argument form conversions |
| Difficulty | Standard +0.3 This is a straightforward modulus-argument question requiring standard conversions between forms. Part (i) involves converting from mod-arg to Cartesian form (routine), part (ii) is basic sketching, and part (iii) requires recognizing that multiplying by 2 scales the modulus and multiplying by w rotates by π/3, then performing algebraic manipulation. All techniques are standard P3 material with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain answer \(w = \frac{1}{2} + \frac{\sqrt{3}}{2}\text{i}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show point representing \(u\) | B1 | |
| Show point representing \(v\) in relatively correct position | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Explain why the moduli are equal | B1 | |
| Explain why the arguments are equal | B1 | |
| Use \(\text{i}^2 = -1\) and obtain \(2uw\) in the given form | M1 | |
| Obtain answer \(1 - 2\sqrt{3} + (2 + \sqrt{3})\text{i}\) | A1 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain answer $w = \frac{1}{2} + \frac{\sqrt{3}}{2}\text{i}$ | B1 | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show point representing $u$ | B1 | |
| Show point representing $v$ in relatively correct position | B1 | |
## Question 6(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Explain why the moduli are equal | B1 | |
| Explain why the arguments are equal | B1 | |
| Use $\text{i}^2 = -1$ and obtain $2uw$ in the given form | M1 | |
| Obtain answer $1 - 2\sqrt{3} + (2 + \sqrt{3})\text{i}$ | A1 | |(i) Express $w$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.\\
The complex number $1 + 2 \mathrm { i }$ is denoted by $u$. The complex number $v$ is such that $| v | = 2 | u |$ and $\arg v = \arg u + \frac { 1 } { 3 } \pi$.\\
(ii) Sketch an Argand diagram showing the points representing $u$ and $v$.\\
(iii) Explain why $v$ can be expressed as $2 u w$. Hence find $v$, giving your answer in the form $a + \mathrm { i } b$, where $a$ and $b$ are real and exact.\\
\hfill \mbox{\textit{CAIE P3 2019 Q6 [7]}}