(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.