| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Solve equations using trigonometric identities |
| Difficulty | Standard +0.8 This is a multi-part FP2 question requiring de Moivre's theorem, binomial expansion of (z + 1/z)^n, and solving a trigonometric equation. Part (a) is routine, but part (b) requires careful algebraic manipulation with complex exponentials and the binomial theorem, and part (c) involves non-trivial equation solving. The 11-mark total and extended reasoning place it moderately above average difficulty for A-level, though it follows a standard FP2 template. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
The complex number $z = e^{i\theta}$, where $\theta$ is real.
\begin{enumerate}[label=(\alph*)]
\item Use de Moivre's theorem to show that
$$z^n + \frac{1}{z^n} = 2\cos n\theta$$
where $n$ is a positive integer. [2]
\item Show that
$$\cos^n \theta = \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta)$$ [5]
\item Hence find all the solutions of
$$\cos 5\theta + 5\cos 3\theta + 12\cos \theta = 0$$
in the interval $0 \leq \theta < 2\pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q6 [11]}}