Solve the equation ${z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}$ giving your answers in the form $z = r(\cos \theta  + {\text{i}}\sin \theta )$ and in the form $z = a + b{\text{i}}$ where $a,{\text{ }}b \in \mathbb{R}$.

Consider the complex numbers ${z_1} = 1 + {\text{i}}$ and ${z_2} = 2\left( {\cos \left( {\frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\frac{\pi }{6}} \right)} \right)$.

(i)     Write ${z_1}$ in the form $r(\cos \theta  + {\text{i}}\sin \theta )$.

(ii)     Calculate ${z_1}{z_2}$ and write in the form $z = a + b{\text{i}}$ where $a,{\text{ }}b \in \mathbb{R}$.

(iii)     Hence find the value of $\tan \frac{{5\pi }}{{12}}$ in the form $c + d\sqrt 3$, where $c,{\text{ }}d \in \mathbb{Z}$.

(iv)     Find the smallest value $p > 0$ such that ${({z_2})^p}$ is a positive real number.

Note:     Accept answers and working in degrees, throughout.

${z^3} = 8\left( {\cos \left( {\frac{\pi }{2} + 2\pi k} \right) + {\text{i}}\sin \left( {\frac{\pi }{2} + 2\pi k} \right)} \right)$     (A1)

attempt the use of De Moivre’s Theorem in reverse     M1

$z = 2\left( {\cos \left( {\frac{\pi }{6}} \right) + {\text{i}}\sin \left( {\frac{\pi }{6}} \right)} \right);{\text{ }}2\left( {\cos \left( {\frac{{5\pi }}{6}} \right) + {\text{i}}\sin \left( {\frac{{5\pi }}{6}} \right)} \right);$

$2\left( {\cos \left( {\frac{{9\pi }}{6}} \right) + {\text{i}}\sin \left( {\frac{{9\pi }}{6}} \right)} \right)$     A2

Note:     Accept cis form.

$z =  \pm \sqrt 3  + {\text{i}},{\text{ }} - 2{\text{i}}$     A2

Note:     Award A1 for two correct solutions in each of the two lines above.

[6 marks]

Note:     Accept answers and working in degrees, throughout.

(i)     ${z_1} = \sqrt 2 \left( {\cos \left( {\frac{\pi }{4}} \right) + {\text{i}}\sin \left( {\frac{\pi }{4}} \right)} \right)$     A1A1

(ii)     $\left( {{z_2} = \left( {\sqrt 3  + {\text{i}}} \right)} \right)$

${z_1}{z_2} = (1 + {\text{i}})\left( {\sqrt 3  + {\text{i}}} \right)$     M1

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

(iii)     ${z_1}{z_2} = 2\sqrt 2 \left( {\cos \left( {\frac{\pi }{6} + \frac{\pi }{4}} \right) + {\text{i}}\sin \left( {\frac{\pi }{6} + \frac{\pi }{4}} \right)} \right)$     M1A1

Note:     Interpret “hence” as “hence or otherwise”.

$\tan \frac{{5\pi }}{{12}} = \frac{{\sqrt 3  + 1}}{{\sqrt 3  - 1}}$     A1

$= 2 + \sqrt 3$     M1A1

Note:     Award final M1 for an attempt to rationalise the fraction.

(iv)     ${z_2}^p = {2^p}\left( {{\text{cis}}\left( {\frac{{p\pi }}{6}} \right)} \right)$     (M1)

${z_2}^p$ is a positive real number when $p = 12$     A1

Note:     Accept a solution based on part (a).

[11 marks]

Total [17 marks]