| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Marks | 11 |
| 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 trigonometric identity, deducing an exact value, finding fifth roots in polar form, and calculating a pentagon area. While each component uses standard Further Maths techniques, the combination of proof, algebraic manipulation to extract sin(2π/5), and geometric calculation with exact surds makes this significantly harder than average A-level questions but not exceptionally difficult for Further Maths students. |
| Spec | 1.05m Geometric proofs: of trig sum and double angle formulae4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
**(i)** $\frac{\mathrm{d}y}{\mathrm{d}x}+y=3xy^4$ is a Bernoulli (differential) equation
$u=\frac{1}{y^3} \Rightarrow \frac{\mathrm{d}u}{\mathrm{d}x}=-\frac{3}{y^4}\times\frac{\mathrm{d}y}{\mathrm{d}x}$ **B1**
Then $\frac{\mathrm{d}y}{\mathrm{d}x}+y=3xy^4$ becomes $-\frac{3}{y^4}\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{3}{y^3}=-9x \Rightarrow \frac{\mathrm{d}u}{\mathrm{d}x}-3u=-9x$ **AG** **M1 A1**
**(ii) Method 1:**
IF is $\mathrm{e}^{-3x}$ **M1 A1**
$\Rightarrow u\mathrm{e}^{-3x}=\int -9x\mathrm{e}^{-3x}\,\mathrm{d}x$ **M1**
$=3x\mathrm{e}^{-3x}-\int 3\mathrm{e}^{-3x}\,\mathrm{d}x$ — Use of "parts" **M1**
$=(3x+1)\mathrm{e}^{-3x}+C$ **A1**
General solution: $u=3x+1+C\mathrm{e}^{3x}$ **ft** **B1**
$\Rightarrow y^{-3}=\frac{1}{3x+1+C\mathrm{e}^{3x}}$ **ft** **B1**
Using $x=0$, $y=\frac{1}{2}$ to find $C$: $C=7$ or $y^3=\frac{1}{3x+1+7\mathrm{e}^{3x}}$ **M1 A1**
**Method 2:**
Auxiliary equation $m-3=0 \Rightarrow u_C=A\mathrm{e}^{3x}$ is the complementary function **M1 A1**
For particular integral try $u_P=ax+b$, $u_P'=a$ **M1**
Substituting $u_P=ax+b$ and $u_P'=a$ into the d.e. and comparing terms **M1**
$a-3ax-3b=-9x \Rightarrow a=3,\,b=1$ i.e. $u_P=3x+1$ **A1**
General solution: $u=3x+1+A\mathrm{e}^{3x}$ **ft** **B1**
$\Rightarrow y^{-3}=\frac{1}{3x+1+A\mathrm{e}^{3x}}$ **ft** **B1**
Using $x=0$, $y=\frac{1}{2}$ to find $A$: $A=7$ or $y^3=\frac{1}{3x+1+7\mathrm{e}^{3x}}$ **M1 A1**
**Total: 11 marks**11
\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 ( $\mathrm { r } , \theta$ ), where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\
(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 2016 Q11 [11]}}