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

The argument of \(u\) is \(\arctan\left(\frac{2}{2}\right) = \frac{\pi}{4}\) or 45°.

(ii) On the Argand diagram, plot the points 1, i, and u. The perpendicular bisector of the line joining 1 and i is the line \(x = y\). The circle centered at \(u = 2 + 2i\) with radius 1 is drawn. The region satisfying \(|z - 1| \leq |z - i|\) is below the line \(x = y\), and the region satisfying \(|z - u| \leq 1\) is inside the circle. Shade the intersection of these regions.

(iii) The point in the shaded region for which \(\arg z\) is least lies on the tangent from the origin to the circle centered at \(u\). The distance from the origin to this tangent point is \(\sqrt{7}\).