a) EITHER: Square $x + iy$ and equate real and imaginary parts to 1 and $ - 2\sqrt 6 $ respectively

Obtain ${x^2} - {y^2} = 1$ and $2xy =  - 2\sqrt 6 $

Eliminate one variable and find an equation in the other

Obtain ${x^4} - {x^2} - 6 = 0$ or ${y^4} + {y^2} - 6 = 0$, or 3-term equivalent

Obtain answers $ \pm (\sqrt 3  - i\sqrt 2 )$

OR: Denoting $1 - 2\sqrt {6i} $ by $Rcis\theta $, state, or imply, square roots are $ \pm \sqrt R cis(\frac{1}{2}\theta )$ and find values of $R$ and either $\cos \theta $ or $\sin \theta $ or $\tan \theta $

Obtain $ \pm \sqrt 5 (\cos \frac{1}{2}\theta  + i\,\sin \frac{1}{2}\theta )$, and $\cos \theta  = \frac{1}{5}$ or $\sin \theta  =  - \frac{{2\sqrt 6 }}{5}$ or $\tan \theta  =  - 2\sqrt 6 $

Use correct method to find an exact value of $\cos \frac{1}{2}\theta $ or $\sin \frac{1}{2}\theta $

Obtain $\cos \frac{1}{2}\theta  =  \pm \sqrt {\frac{3}{5}} $ and $\sin \frac{1}{2}\theta  =  \pm \sqrt {\frac{2}{5}} $, or equivalent

Obtain answers $ \pm \left( {\sqrt 3  - i\sqrt 2 } \right)$, or equivalent

[Condone omission of ± except in the final answers.]

b) Show point representing 3i on a sketch of an Argand diagram

Show a circle with centre at the point representing 3i and radius 2

Shade the interior of the circle

Carry out a complete method for finding the greatest value of arg $z$

Obtain answer ${131.8^ \circ }$ or $2.30$ (or $2.3$) radians