| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.3 This is a straightforward multi-part complex numbers question requiring standard operations (subtraction, division), basic argument calculations using tan^(-1), and recognizing geometric relationships. While it involves multiple steps, each part uses routine techniques 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 interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that \(u - 2w = -7 - i\) | B1 | |
| EITHER: Multiply numerator and denominator of \(\frac{u}{w}\) by \(3 - 4i\), or equivalent | (M1) | |
| Simplify the numerator to \(25 + 25i\) or denominator to \(25\) | A1 | |
| Obtain final answer \(1 + i\) | A1 | |
| OR: Obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | (M1) | |
| Obtain \(x = 1\) or \(y = 1\) | A1 | |
| Obtain final answer \(1 + i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find the argument of \(\frac{u}{w}\) | M1 | |
| Obtain the given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that \(OB\) and \(CA\) are parallel | B1 | |
| State that \(CA = 2OB\), or equivalent | B1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that $u - 2w = -7 - i$ | B1 | |
| **EITHER:** Multiply numerator and denominator of $\frac{u}{w}$ by $3 - 4i$, or equivalent | (M1) | |
| Simplify the numerator to $25 + 25i$ or denominator to $25$ | A1 | |
| Obtain final answer $1 + i$ | A1 | |
| **OR:** Obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | (M1) | |
| Obtain $x = 1$ or $y = 1$ | A1 | |
| Obtain final answer $1 + i$ | A1 | |
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Find the argument of $\frac{u}{w}$ | M1 | |
| Obtain the given answer | A1 | |
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that $OB$ and $CA$ are parallel | B1 | |
| State that $CA = 2OB$, or equivalent | B1 | |7 Throughout this question the use of a calculator is not permitted.\\
The complex numbers $u$ and $w$ are defined by $u = - 1 + 7 \mathrm { i }$ and $w = 3 + 4 \mathrm { i }$.\\
(i) Showing all your working, find in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, the complex numbers $u - 2 w$ and $\frac { u } { w }$.\\
In an Argand diagram with origin $O$, the points $A , B$ and $C$ represent the complex numbers $u , w$ and $u - 2 w$ respectively.\\
(ii) Prove that angle $A O B = \frac { 1 } { 4 } \pi$.\\
(iii) State fully the geometrical relation between the line segments $O B$ and $C A$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q7 [8]}}