(a) 在 Argand 图的草图中，用阴影标出满足不等式 $|z-4-2\mathrm{i}|\le3$ 和 $|z|\ge|10-z|$ 的复数 $z$ 所对应的点组成的区域。

(b) 求此区域内各点所对应的 $\arg z$ 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。

第一个条件是闭圆盘。第二个条件表示到原点的距离不小于到 $(10,0)$ 的距离，可平方化成半平面。

Write $z=x+y\mathrm{i}$. The first inequality gives

the closed disc with centre $(4,2)$ and radius $3$.

For the second inequality, both moduli are non-negative, so it is equivalent to

Draw the circle and the vertical line $x=5$. Shade the part inside the circle and on or to the right of $x=5$, including the boundaries.

涂色部分在 $x\ge5$，最大辐角出现在直线 $x=5$ 与圆的上方交点。须使用涂色区域，不能直接取整个圆的切线方向。

On $x=5$, the circle gives

The upper intersection is $T=(5,2+2\sqrt2)$.

For $5\le x\le7$, the upper boundary is

As $x$ increases, this positive $y$ decreases while $x$ increases, so $y/x$ decreases. Points below this boundary have smaller arguments. Thus the greatest argument occurs at $T$: