| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 question testing well-rehearsed techniques: proof by induction (routine for this theorem), binomial expansion to derive multiple angle formulas, and algebraic manipulation using special angle properties. While it requires multiple steps and FP2 content is inherently harder than single maths, these are textbook exercises without novel insight required. |
| Spec | 4.01a Mathematical induction: construct proofs4.02q De Moivre's theorem: multiple angle formulae |
De Moivre's theorem states that $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ for $n \in \mathbb{R}$
\begin{enumerate}[label=(\alph*)]
\item Use induction to prove de Moivre's theorem for $n \in \mathbb{Z}^+$. (5)
\item Show that $\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$ (5)
\item Hence show that $2\cos\frac{\pi}{10}$ is a root of the equation
$$x^4 - 5x^2 + 5 = 0$$ (3)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2008 Q11}}