Throughout this question the use of a calculator is not permitted. The complex number (√3) + i is denoted by u.

1. (i)Express u in the form r e^(iθ), where r > 0 and −π < θ ≤ π, giving the exact values of r and θ. Hence or otherwise state the exact values of the modulus and argument of u⁴.[5]
2. (ii)Verify that u is a root of the equation z³ – 8z + 8√3 = 0 and state the other complex root of this equation.[3]
3. (iii)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z - u| ≤ 2 and Im z ≥ 2, where Im z denotes the imaginary part of z.[5]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.