Consider $z = r(\cos \theta  + {\text{i}}\sin \theta ),{\text{ }}z \in \mathbb{C}$.

Use mathematical induction to prove that ${z^n} = {r^n}(\cos n\theta  + {\text{i}}\sin n\theta ),{\text{ }}n \in {\mathbb{Z}^ + }$.

Given $u = 1 + \sqrt 3 {\text{i}}$ and $v = 1 - {\text{i}}$,

(i)     express $u$ and $v$ in modulus-argument form;

(ii)     hence find ${u^3}{v^4}$.

Plot point A and point B on the Argand diagram.

Find the area of triangle O${\text{A}}'$${\text{B}}'$.

Given that $u$ and $v$ are roots of the equation ${z^4} + b{z^3} + c{z^2} + dz + e = 0$, where $b,{\text{ }}c,{\text{ }}d,{\text{ }}e \in \mathbb{R}$,

find the values of $b,{\text{ }}c,{\text{ }}d$ and $e$.

let ${\text{P}}(n)$ be the proposition ${z^n} = {r^n}(\cos n\theta  + {\rm{i}}\sin n\theta ),n \in {¢^ + }$

let $n = 1 \Rightarrow$

${\text{LHS}} = r(\cos \theta  + {\text{i}}\sin \theta )$

${\text{RHS}} = r(\cos \theta  + {\text{i}}\sin \theta ),{\text{ }}\therefore {\text{P}}(1)$ is true     R1

assume true for $n = k \Rightarrow {r^k}{(\cos \theta  + {\text{i}}\sin \theta )^k} = {r^k}\left( {\cos (k\theta ) + {\text{i}}\sin (k\theta )} \right)$     M1

Note:     Only award the M1 if truth is assumed.

now show $n = k$ true implies $n = k + 1$ also true

${r^{k + 1}}{(\cos \theta  + {\text{i}}\sin \theta )^{k + 1}} = {r^{k + 1}}{(\cos \theta  + {\text{i}}\sin \theta )^k}(\cos \theta  + {\text{i}}\sin \theta )$     M1

$= {r^{k + 1}}\left( {\cos (k\theta ) + {\text{i}}\sin (k\theta )} \right)(\cos \theta  + {\text{i}}\sin \theta )$

$= {r^{k + 1}}\left( {\cos (k\theta )\cos \theta  - \sin (k\theta )\sin \theta  + {\text{i}}\left( {\sin (k\theta )\cos \theta  + \cos (k\theta )\sin \theta } \right)} \right)$     A1

$= {r^{k + 1}}\left( {\cos (k\theta  + \theta ) + {\text{i}}\sin (k\theta  + \theta )} \right)$     A1

$= {r^{k + 1}}\left( {\cos (k + 1)\theta  + {\text{i}}\sin (k + 1)\theta } \right) \Rightarrow n = k + 1$ is true     A1

${\text{P}}(k)$ true implies ${\text{P}}(k + 1)$ true and ${\text{P}}(1)$ is true, therefore by mathematical induction statement is true for $n \geqslant 1$     R1

Note:     Only award the final R1 if the first 4 marks have been awarded.

[7 marks]

(i)     $u = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)$     A1

$v = \sqrt 2 {\text{cis}}\left( { - \frac{\pi }{4}} \right)$     A1

Notes:     Accept 3 sf answers only. Accept equivalent forms.

     Accept $2{e^{\frac{\pi }{3}i}}$ and $\sqrt 2 {e^{ - \frac{\pi }{4}i}}$.

(ii)     ${u^3} = {2^3}{\text{cis}}(\pi ) =  - 8$

${v^4} = 4{\text{cis}}( - \pi ) =  - 4$     (M1)

${u^3}{v^4} = 32$     A1

Notes:     Award (M1) for an attempt to find ${u^3}$ and ${v^4}$.

     Accept equivalent forms.

[4 marks]

     A1

Note:     Award A1 if A or ${\text{1 + }}\sqrt 3 i$ and B or $1 - i$ are in their correct quadrants, are aligned vertically and it is clear that $\left| u \right| > \left| v \right|$.

[1 mark]

Area $= \frac{1}{2} \times 2 \times \sqrt 2  \times \sin \left( {\frac{{5\pi }}{{12}}} \right)$     M1A1

$= 1.37{\text{ }}\left( { = \frac{{\sqrt 2 }}{4}\left( {\sqrt 6  + \sqrt 2 } \right)} \right)$     A1

Notes:     Award M1A0A0 for using $\frac{{7\pi }}{{12}}$.

[3 marks]

$(z - 1 + {\text{i}})(z - 1 - {\text{i}}) = {z^2} - 2z + 2$     M1A1

Note:     Award M1 for recognition that a complex conjugate is also a root.

$\left( {z - 1 - \sqrt 3 {\text{i}}} \right)\left( {z - 1 + \sqrt 3 {\text{i}}} \right) = {z^2} - 2z + 4$     A1

$\left( {{z^2} - 2z + 2} \right)\left( {{z^2} - 2z + 4} \right) = {z^4} - 4{z^3} + 10{z^2} - 12z + 8$     M1A1

Note:     Award M1 for an attempt to expand two quadratics.

[5 marks]