| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | nth roots with preliminary simplification |
| Difficulty | Standard +0.8 This question requires finding fifth roots of unity (standard recall), converting a complex number to polar form (r=32, θ=2π/3), then applying De Moivre's theorem to find all five roots. While the technique is standard for Further Maths, it involves multiple steps including polar conversion, angle arithmetic with fractions of π, and systematic enumeration of all roots—more demanding than routine C3/C4 questions but still a textbook application of a core FP1 technique. |
| Spec | 4.02r nth roots: of complex numbers4.02s Roots of unity: geometric applications |
Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation
$$z^5 = -16 + (16\sqrt{3})i,$$
giving each root in the form $re^{i\theta}$. [4]
\hfill \mbox{\textit{CAIE FP1 2005 Q1 [4]}}