(a) First, calculate \(u^2\):

\(u = 2e^{\frac{1}{4} \pi i}\)

\(u^2 = (2e^{\frac{1}{4} \pi i})^2 = 4e^{\frac{1}{2} \pi i}\)

Now, calculate \(\frac{u^2}{w}\):

\(w = 3e^{\frac{1}{3} \pi i}\)

\(\frac{u^2}{w} = \frac{4e^{\frac{1}{2} \pi i}}{3e^{\frac{1}{3} \pi i}} = \frac{4}{3} e^{(\frac{1}{2} \pi i - \frac{1}{3} \pi i)} = \frac{4}{3} e^{\frac{1}{6} \pi i}\)

Thus, \(r = \frac{4}{3}\) and \(\theta = \frac{1}{6} \pi\).

(b) For \(w^n\) to be real and positive, \(n \theta\) must be a multiple of \(2\pi\), where \(\theta = \frac{1}{3} \pi\).

\(n \cdot \frac{1}{3} \pi = 2k\pi\) for some integer \(k\).

\(n = 6k\)

The smallest positive integer \(n\) is \(n = 6\).

1. The inequality \(|z| \leq 3\) represents a circle with radius 3 centered at the origin on the Argand diagram.

2. The inequality \(\text{Re} \, z \geq -2\) represents the region to the right of the vertical line \(x = -2\).

3. The inequality \(\frac{1}{4}\pi \leq \arg z \leq \pi\) represents the region between the angles \(\frac{1}{4}\pi\) and \(\pi\) from the positive real axis.

4. The solution is the intersection of these regions: inside the circle, to the right of the line \(x = -2\), and between the specified angles.

(a) On the Argand diagram, point A represents \(u = 3 - i\), point B represents \(u^* = 3 + i\), and point C represents \(u^* - u = 2i\). The quadrilateral OABC is a parallelogram because opposite sides are equal and parallel.

(b) To express \(\frac{u^*}{u}\), calculate \(\frac{3+i}{3-i}\). Multiply numerator and denominator by the conjugate of the denominator: \(\frac{(3+i)(3+i)}{(3-i)(3+i)} = \frac{9 + 6i + i^2}{9 + 1} = \frac{8 + 6i}{10} = 0.8 + 0.6i\).

(c) The argument of \(\frac{u^*}{u}\) is \(\arg(u^*) - \arg(u) = \arctan\left(\frac{1}{3}\right) - (-\arctan\left(\frac{1}{3}\right)) = 2 \arctan\left(\frac{1}{3}\right)\). Thus, \(\arctan\left(\frac{3}{4}\right) = 2 \arctan\left(\frac{1}{3}\right)\).

1. The inequality \(|z - 2| \leq 1\) represents a circle centered at (2, 0) with radius 1.

2. The inequality \(|z - 1 + 2i| \leq |z|\) represents the region below the perpendicular bisector of the line joining the point (1, -2) and the origin (0, 0).

3. The perpendicular bisector can be found by determining the midpoint of the line segment joining (1, -2) and (0, 0), which is \((0.5, -1)\), and finding the line perpendicular to the segment through this point.

4. The correct region to shade is the intersection of the circle and the region below the perpendicular bisector.

(a) Substitute \(x = -1 + \sqrt{7}i\) into the equation \(2x^3 + 3x^2 + 14x + k = 0\). Calculate \(x^2\) and \(x^3\) using \(i^2 = -1\):

\((-1 + \sqrt{7}i)^2 = -1 - 2\sqrt{7}i - 7 = -8 - 2\sqrt{7}i\)

\((-1 + \sqrt{7}i)^3 = (-1 + \sqrt{7}i)(-8 - 2\sqrt{7}i) = 20 - 4\sqrt{7}i\)

Substitute into the equation:

\(2(20 - 4\sqrt{7}i) + 3(-8 - 2\sqrt{7}i) + 14(-1 + \sqrt{7}i) + k = 0\)

Simplify to find \(k = -8\).

(b) The conjugate root is \(-1 - \sqrt{7}i\). Use these roots to find a quadratic factor:

\((x - (-1 + \sqrt{7}i))(x - (-1 - \sqrt{7}i)) = x^2 + 2x + 8\)

Divide the cubic by this quadratic to find the remaining root:

\(2x^3 + 3x^2 + 14x + 8 = (x^2 + 2x + 8)(2x - 1)\)

The remaining root is \(\frac{1}{2}\).

(c) The locus \(|z - u| = 2\) is a circle with center \(-1 + \sqrt{7}i\) and radius 2.

(d) The maximum value of \(\arg z\) is calculated using the geometry of the circle, resulting in 2.72 radians.