Argand 图上的阴影区域表示由两个不等式定义的复数 $z$ 所对应的点。阴影区域由图中的圆和一条平行于虚轴的直线围成，并包含其边界。

(a) 求用 $z$ 表示的两个不等式，以定义这个阴影区域。

(b) 计算阴影区域内各点所对应的 $|z|$ 的最大值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

从图中读出圆心、半径与竖直直线的位置。阴影在圆内且在直线右侧，边界均包含。

The circle has centre $-3-\mathrm{i}$ and radius $2$. The vertical line is $\operatorname{Re}z=-2$, with the shaded side to its right. Since both boundaries are included, the inequalities are

最大模长在离原点最远的阴影边界点。先求直线与圆的两个交点，再比较它们到原点的距离。

Write $z=x+\mathrm{i}y$. The left boundary of the shaded region is $x=-2$, where the circle gives

Throughout the shaded region, $-2\le x\le-1$. Its lower circular boundary is $y=-1-\sqrt{4-(x+3)^2}$. As $x$ increases through this interval, both $|x|$ and the largest possible $|y|$ decrease. Thus the farthest point is $(-2,-1-\sqrt3)$. The greatest modulus is