(i) To express \(u\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator:

\(u = \frac{(7 + 4i)(3 + 2i)}{(3 - 2i)(3 + 2i)}\)

\(= \frac{21 + 14i + 12i + 8i^2}{9 + 6i - 6i - 4i^2}\)

\(= \frac{21 + 26i - 8}{9 + 4}\)

\(= \frac{13 + 26i}{13}\)

\(= 1 + 2i\)

(ii) On the Argand diagram, plot the point \(U = 1 + 2i\). The locus \(|z - u| = 2\) is a circle centered at \(U\) with radius 2.

(iii) The greatest value of \(\arg z\) occurs when \(z\) is at the point on the circle furthest from the real axis. This is when the line from the origin is tangent to the circle. The angle is given as 126.9° or 2.21 radians.

(i) The complex number w has modulus 1 and argument \(\frac{2}{3} \pi\), so \(w = \cos \frac{2}{3} \pi + i \sin \frac{2}{3} \pi = -\frac{1}{2} + \frac{\sqrt{3}}{2}i\).

To find uw, multiply u and w: \(uw = 2i \times \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = -\sqrt{3} - i\).

To find \(\frac{u}{w}\), multiply numerator and denominator by the conjugate of w: \(\frac{u}{w} = \frac{2i}{-\frac{1}{2} + \frac{\sqrt{3}}{2}i} \times \frac{-\frac{1}{2} - \frac{\sqrt{3}}{2}i}{-\frac{1}{2} - \frac{\sqrt{3}}{2}i} = \sqrt{3} - i\).

\((ii) On the Argand diagram, point U is at (0, 2), point A is at (-\sqrt{3}, -1), and point B is at (\sqrt{3}, -1).\)

(iii) To prove triangle UAB is equilateral, show that all sides are equal: \(AB = UA = UB\). Alternatively, show that angles are equal: \(\angle AUB = \angle ABU = \angle BAU = 60^\circ\).

(a) Substitute \(x = -3\) into \(p(x)\):

\(p(-3) = (-3)^3 + 5(-3)^2 + 31(-3) + 75 = -27 + 45 - 93 + 75 = 0\).

Since \(p(-3) = 0\), \((x + 3)\) is a factor of \(p(x)\).

(b) Substitute \(z = -1 + 2\sqrt{6}i\) into \(p(z)\):

Calculate \(z^2 = (-1 + 2\sqrt{6}i)^2 = -1 - 4\sqrt{6}i - 24 = -25 - 4\sqrt{6}i\).

Calculate \(z^3 = z \cdot z^2 = (-1 + 2\sqrt{6}i)(-25 - 4\sqrt{6}i) = 25 + 4\sqrt{6}i + 50\sqrt{6}i + 48 = 73 + 54\sqrt{6}i\).

Substitute into \(p(z)\):

\(p(z) = z^3 + 5z^2 + 31z + 75 = 73 + 54\sqrt{6}i + 5(-25 - 4\sqrt{6}i) + 31(-1 + 2\sqrt{6}i) + 75 = 0\).

Thus, \(z = -1 + 2\sqrt{6}i\) is a root of \(p(z) = 0\).

(c) Let \(z^2 = w\), then \(p(w) = w^3 + 5w^2 + 31w + 75 = 0\).

The roots of \(p(w) = 0\) are \(w_1 = 3i, w_2 = -3i, w_3 = -1 + 2\sqrt{6}i\).

For \(w_1 = 3i\), \(z = \pm \sqrt{3}i\).

For \(w_2 = -3i\), \(z = \pm \sqrt{6}i\).

For \(w_3 = -1 + 2\sqrt{6}i\), \(z = -1 \pm 2\sqrt{6}i\).

Thus, the roots are \(z = \pm \sqrt{3}i, \pm \sqrt{6}i, -1 \pm 2\sqrt{6}i\).

(a) To find the square roots of \(-3 + 4i\), express \(x + iy\) and equate real and imaginary parts:

\(x^2 - y^2 = -3\) and \(2xy = 4\).

From \(2xy = 4\), we have \(xy = 2\).

Substitute \(y = \frac{2}{x}\) into \(x^2 - y^2 = -3\):

\(x^2 - \left(\frac{2}{x}\right)^2 = -3\).

Simplify to get \(x^4 + 3x^2 - 4 = 0\).

Solving gives \(x = 1, y = 2\) and \(x = -1, y = -2\).

Thus, the square roots are \(1 + 2i\) and \(-1 - 2i\).

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

\(z = \frac{(-1 + 3i)(2 - i)}{(2 + i)(2 - i)} = \frac{-2 + i + 6i - 3}{4 + 1} = \frac{-5 + 7i}{5}\).

Simplify to get \(z = 0.2 + 1.4i\).

(b)(ii) Plot the points \(-1 + 3i\), \(2 + i\), and \(0.2 + 1.4i\) on an Argand diagram.

(b)(iii) The equation relating the lengths is \(OC = \frac{OA}{OB}\).

(i) The complex number \(u = 1 + i \sqrt{3}\) can be expressed in polar form as \(r(\cos \theta + i \sin \theta)\).

Calculate the modulus: \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2\).

Calculate the argument: \(\theta = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}\).

Thus, \(u = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})\).

For \(u^2\):

Modulus: \(2^2 = 4\).

Argument: \(2 \times \frac{\pi}{3} = \frac{2\pi}{3}\).

For \(u^3\):

Modulus: \(2^3 = 8\).

Argument: \(3 \times \frac{\pi}{3} = \pi\).

(ii) Substitute \(u = 1 + i \sqrt{3}\) into the equation \(z^2 - 2z + 4 = 0\):

\((1 + i \sqrt{3})^2 - 2(1 + i \sqrt{3}) + 4 = 0\).

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

\(-2 + 2i \sqrt{3} - 2 - 2i \sqrt{3} + 4 = 0\).

Thus, \(u\) is a root. The other root is \(1 - i \sqrt{3}\).

(iii) On the Argand diagram, plot \(i\) at \((0, 1)\) and \(u\) at \((1, \sqrt{3})\).

Draw a circle centered at \(i\) with radius 1.

Draw a line from the origin with angle \(\frac{\pi}{3}\) (the argument of \(u\)).

Shade the region inside the circle and above the line.