| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Topic | Complex numbers 2 |
| Type | Integration using De Moivre identities |
| Difficulty | Challenging +1.8 This Further Maths question requires multiple sophisticated techniques: expressing sin θ using exponentials, binomial expansion of (e^(iθ) - e^(-iθ))^6, extracting real parts, then using the identity to evaluate a specific value involving nested radicals. Part (b) adds Maclaurin series and root-finding for π approximation. While the steps are guided, executing the algebra correctly across multiple stages and connecting the parts requires strong technical facility beyond standard A-level. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
4 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item By writing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ show that
$$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
\item Hence show that $\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$.\\
(a) Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.\\
(b) By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q4 [8]}}