| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2020 |
| 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 to derive a trigonometric identity, solve it to find an exact value, find fifth roots in polar form, and calculate a pentagon area. While the techniques are standard for Further Maths (de Moivre's theorem, polar form, roots of unity), the combination of algebraic manipulation, exact value extraction, and geometric application across multiple parts requires sustained reasoning and careful execution beyond typical questions. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b 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 |
|---|---|---|
| \( | \omega | = 32\) [B1] |
$\sin 5\theta = \text{Im}(\cos 5\theta + \text{i}\sin 5\theta) = \text{Im}(c+\text{i}s)^5$ [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.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2020 Q11 [8]}}