(a)(i) To express \(u = \frac{3i}{a + 2i}\) in Cartesian form, multiply the numerator and denominator by the conjugate of the denominator: \(a - 2i\).

\(u = \frac{3i(a - 2i)}{(a + 2i)(a - 2i)} = \frac{3ai - 6i^2}{a^2 + 4} = \frac{3ai + 6}{a^2 + 4}\)

Thus, \(u = \frac{6}{a^2 + 4} + \frac{3ai}{a^2 + 4}\).

(a)(ii) Given \(\arg u^* = \frac{1}{3} \pi\), we need to find \(a\).

\(\arg u = \arg \left( \frac{6}{a^2 + 4} + \frac{3ai}{a^2 + 4} \right) = \arctan \left( \frac{3a}{6} \right) = \frac{1}{3} \pi\).

\(\frac{3a}{6} = \sqrt{3} \Rightarrow a = -2\sqrt{3}\).

(b)(i) The region is defined by the perpendicular bisector of points representing \(2i\) and \(1 + i\), and a circle with radius 2 centered at \(2 + i\).

(b)(ii) The least value of \(\arg z\) is at the critical point \(2 + 3i\), which is 56.3° or 0.983 radians.