(a) Let \(z = x + yi\), where \(x\) and \(y\) are real numbers. Then \(z^* = x - yi\).

Substitute into the equation: \(3(x + yi) - i(x - yi) = 1 + 5i\).

Simplify: \((3x + 3yi) - (xi + y) = 1 + 5i\).

Equate real and imaginary parts: \(3x - y = 1\) and \(3y - x = 5\).

Solve the system of equations:

1. \(3x - y = 1\)

2. \(3y - x = 5\)

From equation 1: \(y = 3x - 1\).

Substitute into equation 2: \(3(3x - 1) - x = 5\).

Simplify: \(9x - 3 - x = 5\) \(\Rightarrow 8x = 8\) \(\Rightarrow x = 1\).

Substitute \(x = 1\) into \(y = 3x - 1\): \(y = 3(1) - 1 = 2\).

Thus, \(z = 1 + 2i\).

(b) The region is a circle centered at the origin with radius 3, intersected with the half-plane \(y \geq 2\).

The line \(y = 2\) intersects the circle \(x^2 + y^2 = 9\) at \(x^2 + 4 = 9\), so \(x^2 = 5\), \(x = \pm \sqrt{5}\).

The points of intersection are \((\sqrt{5}, 2)\) and \((-\sqrt{5}, 2)\).

The greatest argument occurs at \((-\sqrt{5}, 2)\).

\(\arg z = \arctan\left(\frac{2}{-\sqrt{5}}\right)\).

Calculate: \(\arg z \approx 2.41\) radians.

(a) To express \(u\) in Cartesian form, multiply the numerator and denominator by the conjugate of the denominator:

\(u = \frac{(3 + 2i)(a + 5i)}{(a - 5i)(a + 5i)}\)

Using \(i^2 = -1\), the denominator becomes:

\(a^2 + 25\)

Expanding the numerator:

\((3 + 2i)(a + 5i) = 3a + 15i + 2ai + 10i^2\)

\(= 3a + 15i + 2ai - 10\)

\(= (3a - 10) + (2a + 15)i\)

Thus, \(u = \frac{3a - 10}{a^2 + 25} + \frac{2a + 15}{a^2 + 25}i\).

(b) Given \(\arg u = \frac{1}{4}\pi\), we have:

\(\arctan\left(\frac{2a + 15}{3a - 10}\right) = \frac{\pi}{4}\)

This implies:

\(\frac{2a + 15}{3a - 10} = 1\)

Solving for \(a\):

\(2a + 15 = 3a - 10\)

\(25 = a\)

Thus, \(a = 25\).

(i) To find \(uv\), substitute \(u = -3\sqrt{3} + i\) and \(v = \sqrt{3} + 2i\) into the product:

\((-3\sqrt{3} + i)(\sqrt{3} + 2i) = -3\sqrt{3} \cdot \sqrt{3} + (-3\sqrt{3}) \cdot 2i + i \cdot \sqrt{3} + i \cdot 2i\)

\(= -9 - 6\sqrt{3}i + \sqrt{3}i - 2\)

\(= -11 - 5\sqrt{3}i\)

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

\(\frac{-3\sqrt{3} + i}{\sqrt{3} + 2i} \cdot \frac{\sqrt{3} - 2i}{\sqrt{3} - 2i} = \frac{(-3\sqrt{3} + i)(\sqrt{3} - 2i)}{(\sqrt{3} + 2i)(\sqrt{3} - 2i)}\)

\(= \frac{-3\cdot3 + 6i + \sqrt{3}i - 2i^2}{3 + 4}\)

\(= \frac{-9 + 6i + \sqrt{3}i + 2}{7}\)

\(= \frac{-7 + 7\sqrt{3}i}{7}\)

\(= -1 + \sqrt{3}i\)

(ii) On the Argand diagram, point \(A\) represents \(u = -3\sqrt{3} + i\) and point \(B\) represents \(v = \sqrt{3} + 2i\). To find \(\angle AOB\), calculate \(\arg\left(\frac{u}{v}\right)\):

\(\arctan(-\sqrt{3}) = -\frac{2\pi}{3}\)

Thus, \(\angle AOB = \frac{2}{3}\pi\).

(i) To solve the quadratic equation \(z^2 + (2\sqrt{6})z + 8 = 0\), we use the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 2\sqrt{6}\), and \(c = 8\).

Calculate the discriminant: \(b^2 - 4ac = (2\sqrt{6})^2 - 4 \times 1 \times 8 = 24 - 32 = -8\).

Since the discriminant is negative, the roots are complex: \(z = \frac{-2\sqrt{6} \pm \sqrt{-8}}{2}\).

\(\sqrt{-8} = \sqrt{8}i = 2\sqrt{2}i\).

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

Therefore, the roots are \(-\sqrt{6} - \sqrt{2}i\) and \(-\sqrt{6} + \sqrt{2}i\).

(ii) On the Argand diagram, plot the points \(-\sqrt{6} - \sqrt{2}i\) and \(-\sqrt{6} + \sqrt{2}i\).

(iii) The distance \(OA = OB = \sqrt{(-\sqrt{6})^2 + (\pm \sqrt{2})^2} = \sqrt{6 + 2} = \sqrt{8} = 2\sqrt{2}\).

The angle \(AOB\) is the angle between the vectors \(OA\) and \(OB\), which are symmetric about the real axis. Therefore, \(\angle AOB = 60^\circ\).

(iv) Since \(OA = OB = AB = 2\sqrt{2}\), triangle \(AOB\) is equilateral.

\((i)(a) Substitute x = 1 + 2i into the equation 2x^3 - x^2 + 4x + k = 0.\)

\(Calculate (1 + 2i)^2 = 1 + 4i^2 + 4i = -3 + 4i.\)

\(Calculate (1 + 2i)^3 = (1 + 2i)(-3 + 4i) = -3 + 4i - 6i + 8i^2 = -11 - 2i.\)

\(Substitute into the equation: 2(-11 - 2i) - (-3 + 4i) + 4(1 + 2i) + k = 0.\)

\(Simplify: -22 - 4i + 3 - 4i + 4 + 8i + k = 0.\)

\(Combine: -15 + k = 0, so k = 15.\)

(i)(b) The conjugate of 1 + 2i is 1 - 2i, which is also a root.

\(Use the quadratic factor (x - (1 + 2i))(x - (1 - 2i)) = x^2 - 2x + 5.\)

\(Divide the cubic by this quadratic to find the third root: -\frac{3}{2}.\)

\((ii) The equation |z - (1 + 2i)| = 1 represents a circle with center 1 + 2i and radius 1.\)

The least value of arg z occurs at the point on the circle closest to the origin.

\(Calculate the angle using trigonometry: arg z = \tan^{-1}\left(\frac{2}{1}\right) - \sin^{-1}\left(\frac{1}{\sqrt{5}}\right).\)

Compute: arg z \approx 0.64 \, \text{radians}.