| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Topic | Complex Numbers Arithmetic |
| Type | Roots of unity and special equations |
| Difficulty | Easy -1.2 This is a straightforward application of factorising a difference of cubes and solving a quadratic equation. Students need only factor (z-1)(z²+z+1)=0 and apply the quadratic formula to z²+z+1=0, requiring minimal problem-solving beyond routine algebraic manipulation. |
| Spec | 4.02r nth roots: of complex numbers |
**Question 3: Complex roots of $z^3 - 1 = 0$**
$z = 1$ is a root implies $(z-1)$ is a factor **B1** (Must be stated or used)
$\frac{z^3 - 1}{z - 1} = z^2 + z + 1$ **M1** (Must reach a 3 term quadratic. Accept alternative methods e.g. coefficient matching)
$z = \frac{-1 \pm \sqrt{-3}}{2}$ **M1** (Allow one error of substitution in correct quadratic formula if stated. If formula not stated and error present M0)
$z = \frac{-1 \pm \text{i}\sqrt{3}}{2}$ **A1**
**Total: 4 marks**3 Given that $z = 1$ is the real root of the equation $z ^ { 3 } - 1 = 0$, find the two complex roots.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2018 Q3 [4]}}