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]