AceDSE
← All subjects
AceDSE · Paper · 2026.07

數學
Mathematics

Compulsory Part Multi-step Reasoning Test

minutes85 minutes
marks40 marks
Attempted componentEnglish
Candidate name
Date

Sample-paper basis

the mathematics archive contains three official HKEAA HTML sample pages but no local PDF; the blueprint therefore follows the current conventional/MCQ split recorded in the assessment profile

Scope of this paper

Compulsory Part only. M1 and M2 are deliberately excluded because they are separate examination entries.

Candidate instructions

  1. Give exact values unless a numerical approximation is requested.
  2. Show essential working. An unsupported answer may not receive full marks.
  3. Diagrams are not necessarily drawn to scale.

Section A — Conventional questions

37 marks
1
The roots of x²−7x+k=0 differ by 3. Find the two roots and the value of k. Verify your result using the sum and product of roots.
[5 marks]
2
A theatre has 18 seats in the first row. Each succeeding row has 4 more seats than the preceding row. There are 690 seats altogether. Find the number of rows and the number of seats in the last row.
[5 marks]
3
A(2, −1) and B(8, 7). (a) Find the equation of the perpendicular bisector of AB. (b) It meets the x-axis at P. Find P and verify that PA=PB.
[6 marks]
4
Two observation points A and B lie on level ground in line with the foot of a vertical tower. B is 30 m closer to the tower. The angles of elevation of the top are 28° at A and 45° at B. Find the height of the tower, correct to 3 significant figures.
[5 marks]
5
A test score x has mean 62 and standard deviation 8. A moderated score is y=1.1x−4. Find the mean and standard deviation of y. Explain why the ranking of candidates is unchanged.
[5 marks]
6
A bag contains 4 red, 3 blue and 2 green counters. Two counters are drawn without replacement. (a) Find the probability that they have the same colour. (b) Given that at least one counter is red, find the probability that both are red.
[6 marks]
7
A rectangular garden is built against a straight wall, so fencing is needed for only three sides. Exactly 60 m of fencing is available. Find the dimensions that maximise the area, and state the maximum area.
[5 marks]

Section B — Short objective check

3 marks
8
For f(x)=2x²−12x+11, which statement is correct?
  1. The minimum value is −7.
  2. The axis of symmetry is x=−3.
  3. The graph has no x-intercepts.
  4. f(0)=−11.
[2 marks]
9
A fair die is rolled 120 times. Which is the best interpretation of an observed relative frequency of 0.225 for rolling a six?
  1. The die must be biased.
  2. Exactly 27 sixes will occur in every 120 rolls.
  3. The result can occur by random variation and should be tested with more trials.
  4. The theoretical probability has changed to 0.225.
[1 marks]

Every question and figure is original AceDSE work; official samples were used only to calibrate format, complexity and reasoning depth.
Marking guidance is hidden by default and prints only after you reveal it.