Argand 图中的阴影区域表示由两个不等式定义的复数 $z$ 所对应的点。阴影区域由一个圆和一条平行于实轴的直线围成。该区域的边界包含在阴影区域内。

(a) 求定义该阴影区域的两个关于 $z$ 的不等式。

(b) 求该区域内的点所对应的 $|z|$ 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。

从图上读出圆心与半径，再根据阴影在水平线的哪一侧确定虚部不等式。

The circle has centre $(-2,1)$ and radius $3$. The shaded region is inside this circle and below the line with imaginary coordinate $-1$.

Including the boundaries, the inequalities are

最大模长在下方圆帽的左上端点。先由水平边界求该点坐标，再求到原点的距离。

Write $z=x+y\mathrm{i}$. On the horizontal boundary $y=-1$,

so $x=-2\pm\sqrt5$. The farther endpoint from $O$ is $(-2-\sqrt5,-1)$.

To check maximality, every point in the cap has $y\leqslant-1$, so $(y-1)^2\geqslant4$. The disk inequality therefore gives

and hence $x\geqslant-2-\sqrt5$. Expanding the same disk inequality,

Equality is attained at the identified endpoint. Therefore