(a) To express \(u = \frac{3 - 5i}{1 + 4i}\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator:

\(u = \frac{(3 - 5i)(1 - 4i)}{(1 + 4i)(1 - 4i)}\)

Calculate the denominator: \((1 + 4i)(1 - 4i) = 1 - 16i^2 = 1 + 16 = 17\).

Calculate the numerator: \((3 - 5i)(1 - 4i) = 3 - 12i - 5i + 20i^2 = 3 - 17i - 20 = -17 - 17i\).

Thus, \(u = \frac{-17 - 17i}{17} = -1 - i\).

(b)(i) On the Argand diagram, plot the point \(2 + i\) and draw a circle with center \(2 + i\) and radius 1. The inequality \(|z - 2 - i| \leq 1\) represents this circle.

The inequality \(|z - i| \leq |z - 2|\) represents the region on or closer to the line perpendicular bisector of the segment joining \(i\) and \(2\). Shade the region inside the circle and on the correct side of the bisector.

(b)(ii) The maximum value of \(\arg z\) is the angle between the tangents from the origin to the circle. This angle is \(0.927\) radians (or \(53.1^\circ\)).