HKDSE Mathematics Practice Paper: Indices, Logarithms And Sequences (12 Questions, 100 Marks, 75 Minutes)

HKDSE Mathematics Practice Paper: Indices, Logarithms and Sequences (12 Questions, 100 Marks, 75 Minutes)

2026-11-04

HKDSE Mathematics: Indices, Logarithms and Sequences

HKDSE Mathematics paper: 12 questions, 100 marks, 75 minutes, covering index laws, logarithms, arithmetic and geometric sequences, infinite sums and compound interest.

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Paper details　Level: S6 | Subject: Mathematics | Time: 75 minutes | Marks: 100 | Questions: 12

Section A: Indices and Logarithms

Short-answer and calculation questions

#	Question	Answer and marking notes
1	Simplify, giving fractions or integers: (a) 2^5 x 2^-8; (b) 27 to the power 2/3; (c) 81 to the power -1/2.	(a) 2^5 x 2^-8 = 2^-3 = 1/8. (b) 27^(2/3) = (3^3)^(2/3) = 3^2 = 9. (c) 81^(-1/2) = 1 / 9.
2	Evaluate: (a) log base 2 of 32; (b) log base 3 of 1/9; (c) solve log base 10 of x plus log base 10 of 2 equals 3.	(a) 5. (b) log base 3 of 1/9 = -2. (c) log base 10 of 2x = 3, so 2x = 1000 and x = 500.
3	Solve these exponential equations: (a) 2^(x+1) = 32; (b) 5^(2x) = 125; (c) state which of the two solutions is larger on the number line.	(a) x + 1 = 5, so x = 4. (b) 2x = 3, so x = 3/2. (c) 4 is greater than 1.5, so the solution in (a) is larger.
4	An arithmetic sequence has first term 7 and common difference 4. (a) Find the 20th term. (b) Find the sum of the first 20 terms. (c) Find the least n for which the sum of the first n terms is at least 1000.	(a) T20 = 7 + 19 x 4 = 83. (b) S20 = 20 x (7 + 83) / 2 = 900. (c) Sn = n(4n + 10)/2 = 2n^2 + 5n; setting this at least 1000 gives n = 22.

Section B: Arithmetic and Geometric Sequences

Application and context questions

#	Question	Answer and marking notes
5	A geometric sequence has first term 3 and common ratio 2. (a) Find the 10th term. (b) Find the sum of the first 10 terms. (c) State the units digit of that sum and justify it.	(a) T10 = 3 x 2^9 = 1536. (b) S10 = 3 x (1024 – 1) = 3,069. (c) The units digit is 9, because 3 x 1023 gives a units digit of 9.
6	An infinite geometric series has first term 8 and common ratio 0.6. (a) Find the third term. (b) Explain why it converges and find the sum to infinity. (c) Find the sum of the first four terms.	(a) T3 = 8 x 0.36 = 2.88. (b) The common ratio has modulus below 1, so S = 8 / 0.4 = 20. (c) S4 = 8 x (1 – 0.1296) / 0.4 = 17.408.
7	A deposit of 50,000 dollars earns 4 percent per year compounded annually for 3 years. (a) Find the amount. (b) Find the total interest. (c) How much less interest would simple interest give?	(a) Amount = 50,000 x 1.04^3 = 56,243.20 dollars. (b) Interest = 6,243.20 dollars. (c) Simple interest = 6,000 dollars, so 243.20 dollars less.
8	A machine costs 240,000 dollars and depreciates 15 percent a year on the reducing-balance method. (a) Find its value after 4 years. (b) Find the total depreciation over the 4 years. (c) In which year does its value first fall below 150,000 dollars?	(a) Value = 240,000 x 0.85^4 = 125,281.50 dollars. (b) Total depreciation = 114,718.50 dollars. (c) Year 3, when the value is 147,390 dollars, the first figure below 150,000.

Section C: Series, Compound Interest and Exponential Equations

Multi-step computation questions

#	Question	Answer and marking notes
9	Evaluate: (a) the sum from k = 1 to 10 of (3k – 2); (b) the sum from k = 1 to 8 of 2^(k-1); (c) compare how the two sums grow.	(a) First term 1, last term 28, ten terms: 10 x 29 / 2 = 145. (b) First term 1, ratio 2, eight terms: 2^8 – 1 = 255. (c) (a) grows linearly and (b) grows exponentially.
10	In a geometric sequence the third term is 12 and the sixth term is 96. (a) Find the common ratio. (b) Find the first term. (c) Find the ninth term.	(a) T6 / T3 = r^3 = 8, so r = 2. (b) First term = 12 / 4 = 3. (c) T9 = 3 x 256 = 768.
11	A culture has N = 500 x 3^t bacteria after t hours. (a) Find t when N = 40,500. (b) Find t when N = 4,500. (c) Use (a) and (b) to state how N changes each hour.	(a) 3^t = 81, so t = 4. (b) 3^t = 9, so t = 2. (c) Over two hours N rises from 4,500 to 40,500, a factor of 9, so N is multiplied by 3 each hour.
12	Solve 3^(2x) – 4 x 3^x + 3 = 0 by letting y = 3^x. Give all real solutions and check one of them.	Let y = 3^x: y^2 – 4y + 3 = 0, so y = 1 or y = 3, giving x = 0 or x = 1. Check x = 0: 1 – 4 + 3 = 0.

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