| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.8 This is a straightforward complex numbers question requiring only basic techniques: recognizing conjugate roots, plotting points on an Argand diagram, and computing z₁² by direct multiplication. All parts are routine recall and calculation with no problem-solving or insight required, making it easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
**Part (i)**
- State $3 - \mathrm{i}$: B1 **[1]**
**Part (ii)**
- Show $3 + \mathrm{i}$ on an Argand diagram: B1
- Show $3 - \mathrm{i}$: B1$\checkmark$ **[2]**
**Part (iii)**
- Show $9 + 6\mathrm{i} - 1$: B1
- $= 8 + 6\mathrm{i}$: B1 **[2]**
**[Total: 5]**6 The roots of the equation $z ^ { 2 } - 6 z + 10 = 0$ are $z _ { 1 }$ and $z _ { 2 }$, where $z _ { 1 } = 3 + \mathrm { i }$.\\
(i) Write down the value of $z _ { 2 }$.\\
(ii) Show $z _ { 1 }$ and $z _ { 2 }$ on an Argand diagram.\\
(iii) Show that $z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q6 [5]}}