\((a) The modulus r of u = -\sqrt{3} + i is calculated as:\)

\(r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2.\)

\(The argument \theta is given by:\)

\(\theta = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) = \frac{5}{6}\pi.\)

\(Thus, u = 2e^{i\frac{5}{6}\pi}.\)

(b) To find u^6, use De Moivre's Theorem:

\(u^6 = (2e^{i\frac{5}{6}\pi})^6 = 2^6 e^{i(6 \times \frac{5}{6}\pi)} = 64 e^{i5\pi} = 64(-1) = -64.\)

Thus, u^6 is real and equals -64.

(c)(i) On the Argand diagram, draw half-lines from -\sqrt{3} + i at angles 0 and \frac{1}{4}\pi. Also, draw the vertical line x = 2. Shade the region bounded by these lines.

(c)(ii) The greatest value of |z| occurs at the intersection of the line x = 2 and the boundary of the shaded region. Calculate |z| at this point to find the maximum value:

\(The maximum distance from the origin to the line x = 2 is \sqrt{(2 - (-\sqrt{3}))^2 + 1^2} = 5.14.\)