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CIE 9709 2022 November Paper 31 Q2

A Level / CIE / P3

CIE 9709 2022 Nov Paper 31 Paper · Question 2

题目

Problem

On a sketch of an Argand diagram shade the region whose points represent complex numbers zz satisfying the inequalities ∣z∣≤3|z|\le3, Re⁡z≥−2\operatorname{Re}z\ge-2 and 14π≤arg⁡z≤π\frac14\pi\le\arg z\le\pi.

(4)
题目中文翻译

在阿尔冈图的草图上,用阴影表示由满足不等式 ∣z∣≤3|z|\le3、Re⁡z≥−2\operatorname{Re}z\ge-2 和 14π≤arg⁡z≤π\frac14\pi\le\arg z\le\pi 的复数 zz 所对应的点组成的区域。

(4)
原卷通用作答要求

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。

解答

解法一

思路

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三个条件分别给出闭圆盘、直线 x=−2x=-2 右侧,以及辐角介于 π/4\pi/4 与 π\pi 的扇形。涂色须同时满足三个条件。

答题过程

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Write z=x+yiz=x+y\mathrm{i}. Draw the circle x2+y2=9x^2+y^2=9, the vertical line x=−2x=-2, and the ray y=xy=x from the origin into the first quadrant.

Shade the points satisfying

x2+y2≤ 9,x≥ −2,π4≤ arg⁡z≤π.\begin{align*} x^2+y^2\le&\,9,\\[4mm] x\ge&\,-2,\\[4mm] \frac\pi4\le&\,\arg z\le\pi. \end{align*}

Equivalently, shade the part inside the circle, to the right of x=−2x=-2, above the real axis and above the ray y=xy=x. Include the boundaries, except the origin where the argument is undefined.