(i) The modulus of \(u = 2 + 2i\) is given by \(|u| = \sqrt{2^2 + 2^2} = \sqrt{8}\).

The argument of \(u\) is \(\arctan\left(\frac{2}{2}\right) = \frac{\pi}{4}\) or 45°.

(ii) On the Argand diagram, plot the points 1, i, and u. The perpendicular bisector of the line joining 1 and i is the line \(x = y\). The circle centered at \(u = 2 + 2i\) with radius 1 is drawn. The region satisfying \(|z - 1| \leq |z - i|\) is below the line \(x = y\), and the region satisfying \(|z - u| \leq 1\) is inside the circle. Shade the intersection of these regions.

(iii) The point in the shaded region for which \(\arg z\) is least lies on the tangent from the origin to the circle centered at \(u\). The distance from the origin to this tangent point is \(\sqrt{7}\).

(i) (a) To find \(u + v\), add the real and imaginary parts of \(u = -2 + i\) and \(v = 3 + i\):

\((-2 + 3) + (1 + 1)i = 1 + 2i\).

(b) To find \(\frac{u}{v}\), multiply numerator and denominator by the conjugate of \(v\):

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

Calculate the numerator: \((-2)(3) + (-2)(-i) + i(3) + i(-i) = -6 + 2i + 3i - 1 = -7 + 5i\).

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

Thus, \(\frac{u}{v} = \frac{-7 + 5i}{10} = -\frac{7}{10} + \frac{5}{10}i = -\frac{1}{2} + \frac{1}{2}i\).

(ii) The argument of \(\frac{u}{v}\) is \(\frac{3}{4}\pi\) as given.

(iii) Use the fact that angle \(AOB = \text{arg}(u) - \text{arg}(v) = \text{arg}\left(\frac{u}{v}\right) = \frac{3}{4}\pi\).

(iv) Since \(u + v\) is the vector sum of \(u\) and \(v\), \(OA = BC\) and \(OA\) is parallel to \(BC\).

(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 \(0 < \arg(z - u) < \frac{1}{4}\pi\).

(i) To solve the equation \(z^2 + (2\sqrt{3})iz - 4 = 0\), we use the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 2\sqrt{3}i\), and \(c = -4\).

Calculate the discriminant: \(b^2 - 4ac = (2\sqrt{3}i)^2 - 4 \times 1 \times (-4) = -12 - (-16) = 4\).

Thus, \(z = \frac{-2\sqrt{3}i \pm \sqrt{4}}{2} = \frac{-2\sqrt{3}i \pm 2}{2}\).

This gives the roots \(z = 1 - \sqrt{3}i\) and \(z = -1 - \sqrt{3}i\).

(ii) On the Argand diagram, plot the points \((1, -\sqrt{3})\) and \((-1, -\sqrt{3})\).

(iii) The modulus of each root is \(\sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2\).

The argument of \(1 - \sqrt{3}i\) is \(\arctan\left(\frac{-\sqrt{3}}{1}\right) = -60^\circ\).

The argument of \(-1 - \sqrt{3}i\) is \(\arctan\left(\frac{-\sqrt{3}}{-1}\right) = -120^\circ\).

(iv) The points \((0,0)\), \((1, -\sqrt{3})\), and \((-1, -\sqrt{3})\) form an equilateral triangle because the distances between each pair of points are equal, each being 2.

(i) The modulus of w is calculated as \(\sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1\).

The argument of w is \(\arctan\left(\frac{\sqrt{3}/2}{-1/2}\right) = \arctan(-\sqrt{3}) = \frac{2}{3}\pi\) or 120°.

(ii) The modulus of wz is \(R \times 1 = R\).

The argument of wz is \(\theta + \frac{2}{3}\pi\).

The modulus of \(\frac{z}{w}\) is \(\frac{R}{1} = R\).

The argument of \(\frac{z}{w}\) is \(\theta - \frac{2}{3}\pi\).

(iii) In an Argand diagram, the points z, wz, and \(\frac{z}{w}\) are equidistant from the origin and subtend equal angles of \(\frac{2}{3}\pi\) at the origin, forming an equilateral triangle.

(iv) To find the other vertices, multiply 4 + 2i by w and use \(i^2 = -1\).

\((4 + 2i) \times \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -2 - 2\sqrt{3}i + 2i + \sqrt{3} = -(2 + \sqrt{3}) + (2\sqrt{3} - 1)i\).

Divide 4 + 2i by w, multiplying numerator and denominator by the conjugate of w.

\(\frac{4 + 2i}{-\frac{1}{2} + i \frac{\sqrt{3}}{2}} \times \frac{-\frac{1}{2} - i \frac{\sqrt{3}}{2}}{-\frac{1}{2} - i \frac{\sqrt{3}}{2}} = -(2 - \sqrt{3}) - (2\sqrt{3} + 1)i\).