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CIE 9709 2023 June Paper 33 Q4

A Level / CIE / P3

CIE 9709 2023 June Paper 33 Paper · Question 4

题目

Problem

The parametric equations of a curve are

x=cos⁡θ2−sin⁡θ,y=θ+2cos⁡θ.x=\frac{\cos\theta}{2-\sin\theta},\quad y=\theta+2\cos\theta.

Show that dydx=(2−sin⁡θ)2\dfrac{\mathrm{d}y}{\mathrm{d}x}=(2-\sin\theta)^2.

(5)
题目中文翻译

一条曲线的参数方程为

x=cos⁡θ2−sin⁡θ,y=θ+2cos⁡θ.x=\frac{\cos\theta}{2-\sin\theta},\quad y=\theta+2\cos\theta.

证明 dydx=(2−sin⁡θ)2\dfrac{\mathrm{d}y}{\mathrm{d}x}=(2-\sin\theta)^2。

(5)
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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分别对参数求导。求 xx 的导数时用商法则，再用 sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 合并分子，最后取两导数之比。

答题过程

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Differentiating y=θ+2cos⁡θy=\theta+2\cos\theta,

dydθ=1−2sin⁡θ.\frac{\mathrm{d}y}{\mathrm{d}\theta}=1-2\sin\theta.

By the quotient rule,

dxdθ= −(2−sin⁡θ)sin⁡θ+cos⁡2θ(2−sin⁡θ)2= −2sin⁡θ+sin⁡2θ+cos⁡2θ(2−sin⁡θ)2= 1−2sin⁡θ(2−sin⁡θ)2.\begin{align*} \frac{\mathrm{d}x}{\mathrm{d}\theta} =&\,\frac{-(2-\sin\theta)\sin\theta+\cos^2\theta} {(2-\sin\theta)^2}\\[4mm] =&\,\frac{-2\sin\theta+\sin^2\theta+\cos^2\theta} {(2-\sin\theta)^2}\\[4mm] =&\,\frac{1-2\sin\theta}{(2-\sin\theta)^2}. \end{align*}

At regular parameter values, where 1−2sin⁡θ≠01-2\sin\theta\ne0,

dydx= dy/dθdx/dθ= 1−2sin⁡θ(1−2sin⁡θ)/(2−sin⁡θ)2.\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} =&\,\frac{\mathrm{d}y/\mathrm{d}\theta}{\mathrm{d}x/\mathrm{d}\theta}\\[4mm] =&\,\frac{1-2\sin\theta} {(1-2\sin\theta)/(2-\sin\theta)^2}. \end{align*}

Hence

dydx=(2−sin⁡θ)2.\boxed{\frac{\mathrm{d}y}{\mathrm{d}x}=(2-\sin\theta)^2}.

When sin⁡θ=1/2\sin\theta=1/2, both parameter derivatives vanish; the tangent-gradient limit from nearby regular points is (3/2)2=9/4(3/2)^2=9/4.