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CIE 9709 2023 June Paper 33 Q1

A Level / CIE / P3

CIE 9709 2023 June Paper 33 Paper · Question 1

题目

Problem

Solve the equation ln⁡(x+5)=5+ln⁡x\ln(x+5)=5+\ln x. Give your answer correct to 33 decimal places.

(4)
题目中文翻译

解方程 ln⁡(x+5)=5+ln⁡x\ln(x+5)=5+\ln x。答案保留 33 位小数。

(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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先确认 x>0x>0,再把两个对数合并为商的对数,指数化得到一次方程。

答题过程

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The domain requires x>0x>0. Rearranging,

ln⁡(x+5)−ln⁡x= 5,ln⁡x+5x= 5.\begin{align*} \ln(x+5)-\ln x=&\,5,\\[4mm] \ln\frac{x+5}{x}=&\,5. \end{align*}

Exponentiating,

x+5= e5x,(e5−1)x= 5.\begin{align*} x+5=&\,e^5x,\\[4mm] (e^5-1)x=&\,5. \end{align*}

Hence

x=5e5−1≈0.034x=\frac5{e^5-1}\approx\boxed{0.034}

to 3 decimal places. This value is positive, as required.