(a) The region is defined by the argument inequalities \(-\frac{1}{3}\pi \leq \arg(z - 1 - 2i) \leq \frac{1}{3}\pi\), which represent half-lines from the point \((1, 2)\) on the Argand diagram. These lines are symmetrical about the line \(y = 2\) and are drawn between angles \(\frac{\pi}{4}\) and \(\frac{5\pi}{12}\). Additionally, the line \(x = 3\) is drawn, extending in both quadrants. The correct region is shaded between these lines.

(b) To find the least value of \(\arg z\), we use the formula \(-\arctan\left(\frac{2\sqrt{3} - 2}{3}\right)\). Calculating this gives \(-0.454\) radians, correct to 3 decimal places.