| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 4 |
| Topic | Complex Numbers Arithmetic |
| Type | Quadratic equations involving z² and z* |
| Difficulty | Standard +0.8 This is a Further Maths complex number equation involving the conjugate z*, requiring substitution of z = a + bi, expanding to separate real and imaginary parts, then solving a system of simultaneous equations. It's more sophisticated than standard A-level complex number questions but follows a methodical approach once the technique is recognized. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
The complex number $z$ satisfies the equation $z^2 - 4iz^* + 11 = 0$.
Given that $\text{Re}(z) > 0$, find $z$ in the form $a + bi$, where $a$ and $b$ are real numbers. [4]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q1 [4]}}