| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| 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 question on complex number basics and Argand diagram geometry. Parts (i)-(iii) involve routine calculations (subtraction, division, finding argument) and recognizing that u-v represents vector BA. Part (iv) requires finding arguments of u and v then subtracting, which is standard technique. All steps are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(u – v = –3 + i\) | B1 | |
| EITHER: Carry out multiplication of numerator and denominator of \(u/v\) by \(4 – 2i\), or equivalent | M1 | |
| Obtain answer \(\frac{1}{2} + \frac{1}{2}i\), or any equivalent | A1 | |
| OR: Obtain two equations in \(x\) and \(y\), and solve for \(x\) or for \(y\) | M1 | |
| Obtain answer \(\frac{1}{2} + \frac{1}{2}i\), or any equivalent | A1 | Total: 3 marks |
| (ii) State argument is \(\frac{1}{4}\pi\) (or 0.785 radians or \(45°\)) | A1√ | Total: 1 mark |
| (iii) 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 marks |
**(i)** State $u – v = –3 + i$ | B1 | |
**EITHER:** Carry out multiplication of numerator and denominator of $u/v$ by $4 – 2i$, or equivalent | M1 | |
Obtain answer $\frac{1}{2} + \frac{1}{2}i$, or any equivalent | A1 | |
**OR:** Obtain two equations in $x$ and $y$, and solve for $x$ or for $y$ | M1 | |
Obtain answer $\frac{1}{2} + \frac{1}{2}i$, or any equivalent | A1 | **Total: 3 marks** |
**(ii)** State argument is $\frac{1}{4}\pi$ (or 0.785 radians or $45°$) | A1√ | **Total: 1 mark** |
**(iii)** 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 marks** |6 The complex numbers $1 + 3 \mathrm { i }$ and $4 + 2 \mathrm { i }$ are denoted by $u$ and $v$ respectively.\\
(i) Find, in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, the complex numbers $u - v$ and $\frac { u } { v }$.\\
(ii) State the argument of $\frac { u } { v }$.
In an Argand diagram, with origin $O$, the points $A , B$ and $C$ represent the numbers $u , v$ and $u - v$ respectively.\\
(iii) State fully the geometrical relationship between $O C$ and $B A$.\\
(iv) Prove that angle $A O B = \frac { 1 } { 4 } \pi$ radians.
\hfill \mbox{\textit{CAIE P3 2004 Q6 [8]}}