(a) 在 Argand 图的草图中，用阴影标出满足不等式 $|z-4-3\mathrm{i}|\le2$ 和 $\operatorname{Re}z\le3$ 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。

第一个条件是圆心 $(4,3)$、半径 $2$ 的闭圆盘；再与直线 $x=3$ 左侧的半平面取交集。

Write $z=x+y\mathrm{i}$. The required region satisfies

Draw the circle with centre $(4,3)$ and radius $2$, together with the vertical line $x=3$. Shade the part of the disc on or to the left of this line, including both boundaries.

从原点向圆作上方切线，其切点给出最大辐角。切点横坐标小于 $3$，所以确实位于涂色区域。

Let $C=(4,3)$ be the centre and $T$ the upper tangency point from the origin $O$. Then

Also $CT\perp OT$.

Thus

The upper tangency point is approximately $(2.260,3.986)$, so $x_T<3$ and it belongs to the shaded region. Therefore

将经过原点的切线设为 $y=mx$，与圆联立后使用判别式为零。较大的斜率对应上方切线。

For $y=mx$ to be tangent to the circle,

must have exactly one solution in $x$. Thus

has zero discriminant:

Use the larger slope. The contact point has

so it lies in the required region. Hence