(i) The modulus of \(z = \sqrt{3} + i\) is given by \(|z| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2\).

The argument of \(z\) is \(\arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\) or 30°.

(ii) (a) The complex conjugate \(z^* = \sqrt{3} - i\). Therefore, \(2z + z^* = 2(\sqrt{3} + i) + (\sqrt{3} - i) = 3\sqrt{3} + i\).

(b) \(\frac{iz^*}{z} = \frac{i(\sqrt{3} - i)}{\sqrt{3} + i}\).

Multiply numerator and denominator by \(\sqrt{3} - i\):

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

(iii) On the Argand diagram, point A represents \(z = \sqrt{3} + i\) and point B represents \(iz^* = i(\sqrt{3} - i) = 1 + i\sqrt{3}\).

The angle \(AOB\) is given by \(\arg(iz^*) - \arg(z) = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}\).