(i) Multiply the numerator and denominator of \(u = \frac{1 + 2i}{1 - 3i}\) by the conjugate of the denominator, \(1 + 3i\):

\(u = \frac{(1 + 2i)(1 + 3i)}{(1 - 3i)(1 + 3i)}\).

Calculate the denominator: \((1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 + 9 = 10\).

Calculate the numerator: \((1 + 2i)(1 + 3i) = 1 + 3i + 2i + 6i^2 = 1 + 5i - 6 = -5 + 5i\).

Thus, \(u = \frac{-5 + 5i}{10} = -\frac{1}{2} + \frac{1}{2}i\).

(ii) On an Argand diagram, plot the points: A at \(-\frac{1}{2} + \frac{1}{2}i\), B at 1 + 2i, and C at 1 - 3i.

(iii) The argument of 1 + 2i is \(\arctan 2\) and the argument of 1 - 3i is \(\arctan 3\).

The argument of \(u\) is \(\arg(1 + 2i) - \arg(1 - 3i) = \arctan 2 - \arctan 3\).

Given \(\arg u = \frac{3}{4} \pi\), we have \(\arctan 2 + \arctan 3 = \frac{3}{4} \pi\).