| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Challenging +1.2 Part (a) is a standard de Moivre's theorem application requiring binomial expansion and separation of real/imaginary parts - a routine FP2 technique. Part (b) requires recognizing the connection between the derived identity and the given equation (substituting x = cos θ and solving cos 5θ = -1), which involves some insight but follows a familiar pattern for this topic. The multi-step nature and need to connect parts elevates it slightly above average difficulty. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown4.02q De Moivre's theorem: multiple angle formulae |
\begin{enumerate}[label=(\alph*)]
\item Use de Moivre's theorem to show that
$$\cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.$$ [6]
\item Hence find $3$ distinct solutions of the equation $16x^5 - 20x^3 + 5x + 1 = 0$, giving your answers to $3$ decimal places where appropriate. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q23 [10]}}