| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Integration using De Moivre identities |
| Difficulty | Challenging +1.2 This is a standard FP2 De Moivre question with three routine parts: (a) proving a standard identity using z = e^(iθ), (b) expressing cos^4θ in terms of multiple angles using binomial expansion and part (a), and (c) applying a volume of revolution formula. While it requires knowledge of Further Maths content and multiple techniques, each step follows a well-established procedure with no novel insight required. The multi-part structure and integration of complex numbers with calculus places it above average difficulty, but it remains a textbook-style exercise. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae4.08d Volumes of revolution: about x and y axes |
\begin{enumerate}
\item (a) Given that $z = e ^ { i \theta }$, show that
\end{enumerate}
$$z ^ { p } + \frac { 1 } { z ^ { p } } = 2 \cos p \theta$$
where $p$ is a positive integer.\\
(b) Given that
$$\cos ^ { 4 } \theta = A \cos 4 \theta + B \cos 2 \theta + C$$
find the values of the constants $A , B$ and $C$.
The region $R$ bounded by the curve with equation $y = \cos ^ { 2 } x , - \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }$, and the $x$-axis is rotated through $2 \pi$ about the $x$-axis.\\
(c) Find the volume of the solid generated.
\hfill \mbox{\textit{Edexcel FP2 Q8 [14]}}