| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Integration using De Moivre identities |
| Difficulty | Standard +0.8 This is a standard FP3 question on De Moivre's theorem and multiple angle formulas. Part (i) requires systematic binomial expansion and collecting terms, part (ii) uses a trigonometric identity substitution, and part (iii) applies straightforward integration. While algebraically intensive and requiring careful bookkeeping, it follows a well-established template that FP3 students practice extensively. The techniques are routine for this module, though the algebra is more involved than typical A-level questions. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08d Evaluate definite integrals: between limits4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
\begin{enumerate}[label=(\roman*)]
\item By expressing $\sin \theta$ in terms of $e^{i\theta}$ and $e^{-i\theta}$, show that
$$\sin^6 \theta \equiv \frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10).$$ [5]
\item Replace $\theta$ by $\left(\frac{1}{2}\pi - \theta\right)$ in the identity in part (i) to obtain a similar identity for $\cos^6 \theta$. [3]
\item Hence find the exact value of $\int_0^{2\pi} \left(\sin^6 \theta - \cos^6 \theta\right) d\theta$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q8 [12]}}