| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Standard +0.8 This is a standard FP3 de Moivre's theorem question requiring binomial expansion of (cos θ + i sin θ)^6, equating real parts, and expressing in terms of cos θ only. Part (ii) requires recognizing the substitution x = cos θ and solving for the largest root. While it involves multiple steps and Further Maths content, it follows a well-established template that FP3 students practice extensively, making it moderately above average difficulty but not exceptional. |
| Spec | 4.02q De Moivre's theorem: multiple angle formulae |
\begin{enumerate}[label=(\roman*)]
\item Use de Moivre's theorem to prove that
$$\cos 6\theta = 32\cos^6 \theta - 48\cos^4 \theta + 18\cos^2 \theta - 1.$$ [4]
\item Hence find the largest positive root of the equation
$$64x^6 - 96x^4 + 36x^2 - 3 = 0,$$
giving your answer in trigonometrical form. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q5 [8]}}