| Exam Board | Edexcel |
|---|
| Module | F1 (Further Pure Mathematics 1) |
|---|
| Year | 2023 |
|---|
| Session | June |
|---|
| Marks | 10 |
|---|
| Paper | Download PDF ↗ |
|---|
| Topic | Complex Numbers Arithmetic |
|---|
| Type | Real and imaginary part expressions |
|---|
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic complex number operations: modulus calculation, multiplication/division with rationalization, and solving simultaneous equations. While it's Further Maths content (inherently harder), the techniques are routine and mechanical with no novel insight required. Part (a) is simple arithmetic, (b) is standard rationalization, (c) involves equating real/imaginary parts, and (d) is a calculator arctangent. Slightly above average difficulty due to being Further Maths and multi-part, but well below typical Further Maths challenging questions. |
|---|
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
|---|
- In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$
- Determine the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\)
Given that \(w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\)
- determine \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y\), where \(x , y \in \mathbb { R }\)
Given also that \(w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }\)
- determine the value of \(a\) and the value of \(b\)
- determine arg \(w\), giving your answer in radians to 4 significant figures.