| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 24 |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Challenging +1.8 This is a substantial multi-part Further Maths question requiring de Moivre's theorem for multiple angle formulas, algebraic manipulation to extract a specific sine value, finding fifth roots in exponential form, and calculating a pentagon area using geometric reasoning. While the techniques are standard for Further Maths, the extended chain of reasoning and the final exact surd calculation elevate it above routine exercises. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \( | \omega | = 32\): B1 |
**(i)** $\sin 5\theta = \text{Im}(\cos\theta + i\sin\theta)^5 = \text{Im}(c+is)^5$: M1
$(c+is)^5 = c^5 + 5c^4is + 10c^3i^2s^2 + 10c^2i^3s^3 + 5ci^4s^4 + i^5s^5$: M1
Im part $= s(5c^4 - 10c^2s^2 + s^4)$: A1
$= s\left(5(1-s^2)^2 - 10(1-s^2)s^2 + s^4\right)$: M1
$= s(16s^4 - 20s^2 + 5)$, given answer legitimately: A1
$\sin 5\theta = 0 \Rightarrow 5\theta = 0, \pm\pi, \pm 2\pi$, etc $\Rightarrow \theta = 0, \pm\frac{\pi}{5}, \pm\frac{2\pi}{5}$, etc: M1
$s^2 = \frac{20 \pm \sqrt{80}}{32} = \frac{5\pm\sqrt{5}}{8}$: M1
Since $\frac{2\pi}{5}$ is acute and sine is an increasing function for acute angles,
$s = \sin\frac{2\pi}{5} = \sqrt{\frac{5+\sqrt{5}}{8}}$, with explanation (allow "largest positive root wanted"): A1 **[8]**
**(ii)** $|\omega| = 32$: B1
For use of $\tan^{-1}(\sqrt{3})$: M1
For $\arg\omega = \frac{2\pi}{3}$: A1 **[3]**
**(iii)(a)** $z^5 = \left(32, \frac{-10\pi}{3}\right),\ \left(32, \frac{-4\pi}{3}\right),\ \left(32, \frac{2\pi}{3}\right),\ \left(32, \frac{8\pi}{3}\right),\ \left(32, \frac{14\pi}{3}\right)$
For use of mods & args: M1; For considering at least two others $\pm 2n\pi$: M1
$\Rightarrow z = \left(2, \frac{-2\pi}{3}\right),\ \left(2, \frac{-4\pi}{15}\right),\ \left(2, \frac{2\pi}{15}\right),\ \left(2, \frac{8\pi}{15}\right),\ \left(2, \frac{14\pi}{15}\right)$
**ft** $\sqrt[5]{\text{mod}}$: B1; **ft** arg/5: M1; All correct: A1 **[5]**
**(b)** For 5 points on circle, centre $O$, radius 2, equally spread out: B1
Area $= 5 \times \frac{1}{2} \times 2 \times 2 \times \sin\frac{2\pi}{5}$: M1
$= 10\sqrt{\frac{5+\sqrt{5}}{8}}$ or exact equivalent: A1 **[3]**10
\begin{enumerate}[label=(\roman*)]
\item Use de Moivre's theorem to prove that $\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)$, where $s = \sin \theta$, and deduce that $\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }$.
The complex number $\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )$.
\item State the value of $| \omega |$ and find $\arg \omega$ as a rational multiple of $\pi$.
\item (a) Determine the five roots of the equation $z ^ { 5 } = \omega$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leq \pi$.\\
(b) These five roots are represented in the complex plane by the points $A , B , C , D$ and $E$. Show these points on an Argand diagram, and find the area of the pentagon $A B C D E$ in an exact surd form.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q10 [24]}}