(i) To express \(u\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator:

\(u = \frac{4i}{1 - \sqrt{3}i} \times \frac{1 + \sqrt{3}i}{1 + \sqrt{3}i} = \frac{4i(1 + \sqrt{3}i)}{(1)^2 - (\sqrt{3}i)^2}\)

\(= \frac{4i + 4\sqrt{3}i^2}{1 + 3} = \frac{4i - 4\sqrt{3}}{4} = -\sqrt{3} + i\)

(ii) The modulus of \(u = -\sqrt{3} + i\) is \(\sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2\).

The argument of \(u\) is \(\arctan\left(\frac{1}{-\sqrt{3}}\right) = \frac{5\pi}{6}\) (or 150°).

(iii) On the Argand diagram, draw a circle centered at the origin with radius 2. Plot \(u = -\sqrt{3} + i\) in the correct position. Draw the perpendicular bisector of the line joining \(u\) and the origin. Shade the region satisfying \(|z| < 2\) and \(|z - u| < |z|\).