(i) Substitute \(x = -2 + i\) into the equation \(x^3 - 11x - k = 0\). Calculate \((-2 + i)^3\) and use \(i^2 = -1\) to simplify. Solve for \(k\) to obtain \(k = 20\).

(ii) The other complex root is the conjugate of \(-2 + i\), which is \(-2 - i\).

(iii) The modulus of \(u = -2 + i\) is \(\sqrt{(-2)^2 + 1^2} = \sqrt{5}\). The argument is \(\arctan\left(\frac{1}{-2}\right)\), which is approximately \(153.4^\circ\) or \(2.68\) radians.

(iv) On the Argand diagram, plot the point \(-2 + i\). Draw a vertical line through \(z = 1\) and half-lines from \(u\) with gradients 0 and 1. Shade the region satisfying the inequalities \(|z| < |z - 2|\) and \(0 < \arg(z - u) < \frac{1}{4}\pi\).