| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | Specimen |
| Marks | 8 |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus other arithmetic operations |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic complex number operations: conjugate arithmetic, squaring a complex number, and division by a complex number. All three parts are routine textbook exercises requiring standard techniques (conjugate multiplication for division) with no problem-solving insight needed. Easier than average A-level content. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
**(i)**
- $z^* = 5 + 3\text{i}$ seen or implied [B1]
- $25 - 21\text{i}$ obtained [B1]
**(ii)**
- Correct $z - \text{i}$ or expansion of $(z - \text{i})^2$ seen [B1]
- $9 - 40\text{i}$: Real part correct [B1 ft]
- Imaginary part correct [B1 ft]
**(iii)**
- Multiply by conjugate [M1]
- $\frac{25}{34} + \frac{15}{34}\text{i}$ o.e. Numerators correct [A1]
- Denominator correct [A1]
**Total: 8 marks**6 The complex number $5 - 3 \mathrm { i }$ is denoted by $z$. Giving your answers in the form $x + \mathrm { i } y$, and showing clearly how you obtain them, find\\
(i) $\quad 6 z - z ^ { * }$,\\
(ii) $\quad ( z - \mathrm { i } ) ^ { 2 }$,\\
(iii) $\frac { 5 } { z }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q6 [8]}}