HKDSE Master 香港中學文憑大師

Honesty, Modesty & Perseverance 誠實，謙遜，毅力

Appreciation of Other Public Examination Questions 其他公開試題目賞識Chemistry 化學Economics 經濟Enriched Learning 增潤學習Free challenging questions 免費挑戰性題目Little wisdom 小智慧Mathematics Compulsory 數學 必修部分Mathematics Extended Part Modules 1 數學 延伸部分 單元（一）Mathematics Extended Part Modules 2 數學 延伸部分 單元（二）Physics 物理

Finding \(a\) When Two Functions Have the Same Minimum Value

Problem Statement: We are given the functions \[f(x)=e^x−ax\] and \[g(x)=ax−\ln x\] with \(x>0\) for \(g(x)\) It is given that \(f(x)\) and \(g(x)\) have the same minimum value. (1) Find \(a\) (2) Prove that there exists a line \(y=b\) intersecting both curves \(y=f(x)\) and \(y=g(x)\) at three distinct points whose \(x\)-coordinates form an arithmetic progression Answer: (1) \(f'(x)=e^x-a\) \(f′(x)=0\) ⟹ \(0=e^x-a\) ∴ \(x= \ln a\) The minimum value... » read more

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Free Challenging Questions : M2-04-01

Challenging Questions Mathematics – Module 2 (Algebra and Calculus) Learning Unit: More about trigonometric functions Question M2-04-01 The smallest angles of three distinct primitive Pythagorean triangles have a sum of π/2 radians. Two of these triangles are given by the triples (3, 4, 5) and (5, 12, 13). Identify the third primitive Pythagorean triple. #... » read more

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Free Challenging Questions : M0-03-07

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-07 Given that \( \log_{10}2 = a \) and \( \log_{10}3 = b \), express the following in terms of \( a \) and \( b \): # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-06

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-06 Solve $$ x^{\log x \, + 1}=100 $$ # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-05

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-05 Solve $$ \log_2x + \log_4x = 6 $$ # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-04

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-04 Solve $$ 9^x – 2 \cdot 3^{x+1} -27 = 0 $$ # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-03

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-03 Given that \( x^2 + x^{-2} = 3 \) and \( x>0 \), find the value of \( x + x^{-1} \). # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-02

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-02 Evaluate the following expression for \( a = 2 \) and \( b = 3 \): $$ \left( \frac{a^{1/2}b^{-1/3}}{a^{-3/4}b^{2/3}} \right)^{4} $$ # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-03-01

Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-01 Simplify the following expression and express your answer with positive indices: $$ \sqrt[4]{a^{-3}\sqrt[3]{a^{-2}\sqrt{a^{-1}}}} $$ # More challenging questions will be updated # Answer

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Free Challenging Questions : M0-07-10

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-10 A person takes out a loan of $100,000 from a bank and agrees to repay it in 24 equal monthly installments. The bank charges an annual interest rate of 6%, which is applied monthly on the remaining balance... » read more

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Free Challenging Questions : M0-07-09

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-09 Consider three sequences \(f(n)\), \(g(n)\) and \(h(n)\), # More challenging questions will be updated # Answer

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