| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus modulus/argument |
| Difficulty | Moderate -0.5 This is a straightforward complex numbers question requiring division (multiplying by conjugate), finding argument, and basic Argand diagram geometry. All techniques are standard P3 procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| EITHER Multiply numerator and denominator of \(\frac{u}{v}\) by \(4-2i\), or equivalent | (M1) | |
| Simplify the numerator to \(10+10i\) or denominator to 20 | A1) | |
| OR Obtain two equations in \(x\) and \(y\), and solve for \(x\) or for \(y\) | (M1 | |
| Obtain \(x = \frac{1}{2}\) or \(y = \frac{1}{2}\), or equivalent | A1) | |
| Obtain final answer \(\frac{1}{2} + \frac{1}{2}i\) | A1 | Unsupported answer receives 0 marks |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State argument is \(\frac{1}{4}\pi\) (or 0.785 radians or 45°) | B1FT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that \(OC\) and \(BA\) are equal (in length) | B1 | |
| State that \(OC\) and \(BA\) are parallel or have the same direction | B1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| EITHER Use angle \(AOB = \arg u - \arg v = \arg\left(\frac{u}{v}\right)\) | (M1 | |
| Obtain given answer (or 45°) | A1) | AG |
| OR Obtain \(\tan AOB\) from gradients of \(OA\) and \(OB\) and \(\tan(A\pm B)\) formula | (M1 | |
| Obtain given answer (or 45°) | A1) | AG |
| Total | 2 |
# Question 6:
## Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| **EITHER** Multiply numerator and denominator of $\frac{u}{v}$ by $4-2i$, or equivalent | (M1) | |
| Simplify the numerator to $10+10i$ or denominator to 20 | A1) | |
| **OR** Obtain two equations in $x$ and $y$, and solve for $x$ or for $y$ | (M1 | |
| Obtain $x = \frac{1}{2}$ or $y = \frac{1}{2}$, or equivalent | A1) | |
| Obtain final answer $\frac{1}{2} + \frac{1}{2}i$ | A1 | Unsupported answer receives 0 marks |
| **Total** | **3** | |
## Part 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State argument is $\frac{1}{4}\pi$ (or 0.785 radians or 45°) | B1FT | |
## Part 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that $OC$ and $BA$ are equal (in length) | B1 | |
| State that $OC$ and $BA$ are parallel or have the same direction | B1 | |
| **Total** | **2** | |
## Part 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| **EITHER** Use angle $AOB = \arg u - \arg v = \arg\left(\frac{u}{v}\right)$ | (M1 | |
| Obtain given answer (or 45°) | A1) | AG |
| **OR** Obtain $\tan AOB$ from gradients of $OA$ and $OB$ and $\tan(A\pm B)$ formula | (M1 | |
| Obtain given answer (or 45°) | A1) | AG |
| **Total** | **2** | |
---6 The complex numbers $1 + 3 \mathrm { i }$ and $4 + 2 \mathrm { i }$ are denoted by $u$ and $v$ respectively.\\
(a) Find $\frac { u } { v }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(b) State the argument of $\frac { u } { v }$.\\
In an Argand diagram, with origin $O$, the points $A , B$ and $C$ represent the complex numbers $u , v$ and $u - v$ respectively.\\
(c) State fully the geometrical relationship between $O C$ and $B A$.\\
(d) Show that angle $A O B = \frac { 1 } { 4 } \pi$ radians.\\
\hfill \mbox{\textit{CAIE P3 2020 Q6 [8]}}