(i) The complex number \(u = 1 + i \sqrt{3}\) can be expressed in polar form as \(r(\cos \theta + i \sin \theta)\).

Calculate the modulus: \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2\).

Calculate the argument: \(\theta = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}\).

Thus, \(u = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})\).

For \(u^2\):

Modulus: \(2^2 = 4\).

Argument: \(2 \times \frac{\pi}{3} = \frac{2\pi}{3}\).

For \(u^3\):

Modulus: \(2^3 = 8\).

Argument: \(3 \times \frac{\pi}{3} = \pi\).

(ii) Substitute \(u = 1 + i \sqrt{3}\) into the equation \(z^2 - 2z + 4 = 0\):

\((1 + i \sqrt{3})^2 - 2(1 + i \sqrt{3}) + 4 = 0\).

Calculate \((1 + i \sqrt{3})^2 = 1 + 2i \sqrt{3} - 3 = -2 + 2i \sqrt{3}\).

\(-2 + 2i \sqrt{3} - 2 - 2i \sqrt{3} + 4 = 0\).

Thus, \(u\) is a root. The other root is \(1 - i \sqrt{3}\).

(iii) On the Argand diagram, plot \(i\) at \((0, 1)\) and \(u\) at \((1, \sqrt{3})\).

Draw a circle centered at \(i\) with radius 1.

Draw a line from the origin with angle \(\frac{\pi}{3}\) (the argument of \(u\)).

Shade the region inside the circle and above the line.