$\theta =\frac{\mathrm{\pi }}{2}$.

$\theta =\mathrm{\pi }$.

$\theta =\frac{3\mathrm{\pi }}{2}$.

Find this value of $\theta$.

For this value of $\theta$, plot the approximate position of $z_{\theta }$ on the Argand diagram.

          A1

Note: Award A1 for correct modulus and A1 for correct argument for part (a)(i), and A1 for other two points correct.
The points may not be labelled, and they may be shown by line segments.

[1 mark]

          A1

Note: Award A1 for correct modulus and A1 for correct argument for part (a)(i), and A1 for other two points correct.
The points may not be labelled, and they may be shown by line segments.

[1 mark]

          A1

Note: Award A1 for correct modulus and A1 for correct argument for part (a)(i), and A1 for other two points correct.
The points may not be labelled, and they may be shown by line segments.

[1 mark]

 $\frac{1}{2}\theta =4$         (M1)

$\Rightarrow \theta =8$         A1

[2 marks]

$z_8$ is shown in the diagram above         A1A1

Note: Award A1 for a point plotted on the circle and A1 for a point plotted in the second quadrant.

[2 marks]

This question was challenging to many candidates, and some left the answer blank. Those who attempted it often failed to gain any marks. It would have helped examiners credit responses if points that were plotted on the Argand diagram were labelled. Certainly, there was some confusion caused by the appearance of θ both in the modulus and argument of the complex numbers in Euler form. Better use of technology to help visualize the complex numbers by simply getting decimal approximations of values in terms of π or by converting from Euler to Cartesian form would have helped in this question.