| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2019 |
| Session | Specimen |
| Marks | 8 |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Challenging +1.8 This is a substantial multi-part question requiring de Moivre's theorem for deriving a multiple angle formula, algebraic manipulation to extract a specific trigonometric value, finding fifth roots in polar form, and calculating a pentagon area using geometry. While each component uses standard Further Maths techniques, the combination of proof, root extraction, and geometric application with exact surd answers requires sustained reasoning across multiple domains, placing it well above average difficulty. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| \( | \omega | = 32\) [B1] |
$\sin 5\theta = \text{Im}(\cos\theta + \text{i}\sin\theta)^5 = \text{Im}(c + \text{i}s)^5$ [M1]
$(c + \text{i}s)^5 = c^5 + 5c^4\text{i}s + 10c^3\text{i}^2s^2 + 10c^2\text{i}^3s^3 + 5c\text{i}^4s^4 + \text{i}^5s^5$ [M1]
Im part $= s(5c^4 - 10c^2s^2 + s^4)$ [A1]
$= s(5(1-s^2)^2 - 10(1-s^2)s^2 + s^4)$ [M1]
$= s(16s^4 - 20s^2 + 5)$ legitimately **AG** [A1]
$\sin 5\theta = 0 \Rightarrow 5\theta = 0, \pm\pi, \pm 2\pi$, etc. $\Rightarrow \theta = 0, \pm\dfrac{\pi}{5}, \pm\dfrac{2\pi}{5}$, etc. [M1]
$s^2 = \dfrac{20 \pm \sqrt{80}}{32} = \dfrac{5 \pm \sqrt{5}}{8}$ [M1]
Since $\dfrac{2\pi}{5}$ is acute and sine is an increasing function for acute angles,
$s = \sin\dfrac{2\pi}{5} = \sqrt{\dfrac{5+\sqrt{5}}{8}}$ with explanation (allow "largest positive root wanted") [A1]
**Total: 8 marks**
**(b)**
$|\omega| = 32$ [B1]
for use of $\tan^{-1}(\sqrt{3})$ [M1]
for $\arg\omega = \dfrac{2\pi}{3}$ [A1]
**Total: 3 marks**
**(c)(i)**
$z^5 = \left(32, \dfrac{-10\pi}{3}\right), \left(32, \dfrac{-4\pi}{3}\right), \left(32, \dfrac{2\pi}{3}\right), \left(32, \dfrac{8\pi}{3}\right), \left(32, \dfrac{14\pi}{3}\right)$ for use of modulus & argument [M1]
for considering at least two others $\pm 2n\pi$ [M1]
$\Rightarrow z = \left(2, \dfrac{-2\pi}{3}\right), \left(2, \dfrac{-4\pi}{15}\right), \left(2, \dfrac{2\pi}{15}\right), \left(2, \dfrac{8\pi}{15}\right), \left(2, \dfrac{14\pi}{15}\right)$ ft $\sqrt[5]{\text{mod}}$ [B1, B1ft]
their arg/5 [M1]
all correct [A1]
**Total: 5 marks**
**(c)(ii)**
5 points on circle, centre $O$, radius 2, equally spread out [B1]
Area $= 5 \times \dfrac{1}{2} \times 2 \times 2 \times \sin\dfrac{2\pi}{5}$ [M1]
$= 10\sqrt{\dfrac{5+\sqrt{5}}{8}}$ or exact equivalent [A1]
**Total: 3 marks**11
\begin{enumerate}[label=(\alph*)]
\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 \begin{enumerate}[label=(\roman*)]
\item Determine the five roots of the equation $z ^ { 5 } = \omega$, giving your answers in the form ( $\mathrm { r } , \theta$ ), where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\item 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.[3]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2019 Q11 [8]}}