| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question testing De Moivre's theorem proof by induction (5 marks) and a complex number calculation (2 marks). The induction proof is a standard bookwork result requiring careful algebraic manipulation with trigonometric identities, while part (b) appears routine. The induction component elevates this above average A-level difficulty, but it's a well-known proof that students explicitly prepare for in FP2. |
| Spec | 4.01a Mathematical induction: construct proofs4.02q De Moivre's theorem: multiple angle formulae |
\begin{enumerate}[label=(\alph*)]
\item Given that
$$z = r(\cos n\theta + i \sin n\theta), \quad r \in \mathbf{R}$$
prove, by induction, that $z^n = r^n(\cos n\theta + i \sin n\theta)$, $n \in \mathbf{Z}^+$. [5]
\item Find the exact value of $w^2$, giving your answer in the form $a + ib$, where $a, b \in \mathbf{R}$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q4 [7]}}