(i) The modulus of \(z\) is given by \(|z| = \sqrt{(\sqrt{2})^2 + (-\sqrt{6})^2} = \sqrt{2 + 6} = \sqrt{8} = 2\sqrt{2}\).

The argument of \(z\) is \(\arctan\left(\frac{-\sqrt{6}}{\sqrt{2}}\right) = \arctan(-\sqrt{3}) = -\frac{1}{3}\pi\).

(ii) (a) The complex conjugate \(z^* = \sqrt{2} + \sqrt{6}i\). Thus, \(z + 2z^* = (\sqrt{2} - \sqrt{6}i) + 2(\sqrt{2} + \sqrt{6}i) = 3\sqrt{2} + \sqrt{6}i\).

(b) \(\frac{z^*}{iz} = \frac{\sqrt{2} + \sqrt{6}i}{i(\sqrt{2} - \sqrt{6}i)} = \frac{\sqrt{2} + \sqrt{6}i}{i\sqrt{2} + 6}\).

Multiply numerator and denominator by the conjugate of the denominator: \(\frac{(\sqrt{2} + \sqrt{6}i)(-i\sqrt{2} + 6)}{(i\sqrt{2} + 6)(-i\sqrt{2} + 6)} = \frac{4\sqrt{3} + 4i}{8} = \frac{1}{2}\sqrt{3} + \frac{1}{2}i\).

(iii) On the Argand diagram, point \(A\) represents \(z^* = \sqrt{2} + \sqrt{6}i\) and point \(B\) represents \(iz = i(\sqrt{2} - \sqrt{6}i) = \sqrt{6} + i\sqrt{2}\).

The argument of \(\frac{z^*}{iz}\) is \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), which corresponds to an angle of \(\frac{1}{6}\pi\).