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CIE 9709 2022 November Paper 11 Q3

A Level / CIE / P1

CIE 9709 2022 Nov Paper 11 Paper · Question 3

题目

Problem

A curve has equation y=ax12−2xy=ax^{\frac{1}{2}}-2x, where x>0x>0 and aa is a constant. The curve has a stationary point at the point PP, which has xx-coordinate 9.

Find the yy-coordinate of PP.

(5)
题目中文翻译

一条曲线的方程为 y=ax12−2xy=ax^{\frac{1}{2}}-2x,其中 x>0x>0,aa 为常数。曲线在点 PP 处有驻点,PP 的 xx 坐标为 9。

求 PP 的 yy 坐标。

(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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驻点处一阶导数为零。先用横坐标确定 aa,再代回原方程。

答题过程

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The derivative is a/(2x)−2a/(2\sqrt x)-2. At x=9x=9 it is zero, so a/6−2=0a/6-2=0 and a=12a=12. The height is then 12(3)−2(9)=1812(3)-2(9)=18.