(i) To find the modulus of \(u\), multiply numerator and denominator by the conjugate of the denominator: \((1 - 2i)\).

\(u = \frac{(6 - 3i)(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{6 - 12i - 3i + 6i^2}{1 - 4i^2} = \frac{6 - 12i - 3i - 6}{1 + 4} = \frac{-3i}{5}\)

The modulus of \(u\) is \(\left| \frac{-3i}{5} \right| = \frac{3}{5} \times 5 = 3\).

The argument of \(u\) is \(-\frac{\pi}{2}\) because it lies on the negative imaginary axis.

(ii) For \(\text{arg}(z - u) = \frac{1}{4}\pi\), the line from \(u\) makes an angle of \(\frac{1}{4}\pi\) with the real axis. The least possible value of \(|z|\) is the distance from the origin to this line, which is \(\frac{3}{2}\sqrt{2}\).

(iii) The locus \(|z - (1 + i)u| = 1\) is a circle with center \((1 + i)u\) and radius 1. The greatest possible value of \(|z|\) is the distance from the origin to the furthest point on this circle, which is \(3\sqrt{2} + 1\).