(a) Let \(z = x + iy\). Then \(z^* = x - iy\). Substitute into the equation:

\(x - iy + 1 = 2i(x + iy)\)

Equate real and imaginary parts:

Real: \(x + 1 = -2y\)

Imaginary: \(-y = 2x\)

From \(-y = 2x\), we have \(y = -2x\).

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

\(x + 1 = 4x\)

\(3x = 1\)

\(x = \frac{1}{3}\)

Then \(y = -2x = -\frac{2}{3}\).

Thus, \(z = \frac{1}{3} - \frac{2}{3}i\).

(b)(i) The region is a circle with center \(-1 + 3i\) and radius 1, intersecting the line \(\text{Im } z = 3\). Shade the region inside the circle and above the line.

(b)(ii) The greatest and least values of \(\arg z\) are determined by the intersection points of the circle and the line. The difference in \(\arg z\) is 0.588 radians (accept 33.7°).