| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Topic | Complex numbers 2 |
| Type | Solve equations using trigonometric identities |
| Difficulty | Challenging +1.2 This is a standard Further Maths question on De Moivre's theorem and multiple angle identities. Part (i) is immediate from De Moivre's theorem (1 mark routine verification). Part (ii) requires expanding (z + 1/z)^5 using binomial theorem and matching terms—a well-practiced technique worth 4 marks. Part (iii) involves algebraic manipulation to factor the equation and solve, requiring careful work but following standard methods. While this is harder than typical A-level questions due to the Further Maths content, it's a textbook application of De Moivre's theorem without requiring novel insight. |
| Spec | 1.05o Trigonometric equations: solve in given intervals4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
\begin{enumerate}[label=(\roman*)]
\item Given the complex number $z = \cos \theta + \text{i} \sin \theta$, show that $z^n + \frac{1}{z^n} = 2 \cos n\theta$. [1]
\item Deduce the identity $16 \cos^5 \theta \equiv \cos 5\theta + 5 \cos 3\theta + 10 \cos \theta$. [4]
\item For $0 < \theta < 2\pi$, solve the equation $\cos 5\theta + 5 \cos 3\theta + 9 \cos \theta = 0$. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2015 Q6 [9]}}