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HKDSE M1 Mock Paper (2027): a Single 100% Paper with 8 Short and 3 Long Questions
TUTORZONE ORIGINAL MOCK PAPER · HKDSE MATHEMATICS EXTENDED PART MODULE 1
Written to the HKEAA 2027 framework: Mathematics Extended Part Module 1 (Calculus and Statistics) is a single paper worth 100% over 2 h 30 min, with Section A carrying 50 marks in 8-12 short questions and Section B 50 marks in 3-5 long questions, all compulsory, with statistical tables provided. This paper has 8 short and 3 long questions with answers and mark allocation.
What is in HKDSE M1?
Direct answer: Mathematics Extended Part Module 1 is a single paper worth 100% over 2 h 30 min, all compulsory: Section A 50 marks (8-12 short questions) and Section B 50 marks (3-5 long questions), with statistical tables provided and no SBA.
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Section A (short questions) | 8-12 short questions worth 50 marks | 50 | 60-70 min |
| Section B (long questions) | 3-5 long questions worth 50 marks | 50 | 70-80 min |
| In the exam room | Statistical tables are provided; calculators are allowed | — | — |
Two and a half hours: about 60-70 minutes for Section A and 70-80 for Section B. M1 marks follow model, substitution, answer – on distribution questions write X ~ B(n, p) explicitly, or a right answer may not score.
How this paper was built
This is a TutorZone original mock paper: the questions follow the 2027 framework’s structure (one paper at 100% over 2 h 30 min, Section A 50 marks and Section B 50) and the answers are our own, not HKEAA past papers or marking references.
HKEAA publishes no fixed question count, so this paper sets its own: 8 short and 3 long questions. Every numeric answer, including the binomial, normal and geometric probabilities, was recomputed programmatically.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Section A (short questions) | 50 | about 60-70 min | Differentiation, integration and binomial expansion must be reflexive |
| Section B (long questions) | 50 | about 70-80 min | Formula first, then substitute; keep units and significant figures |
| In the exam room | — | Check df and distribution type before using tables | State the distribution and parameters on every distribution question |
How should the 150 minutes be split?
Direct answer: about 60-70 minutes on Section A and 70-80 on Section B. Section A should be reflex; in Section B write the formula and the distribution notation (X ~ B(n, p)) first, then substitute, then give the answer with its accuracy.
Section A: short questions (8 questions, 50 marks)
Aim for 60-70 minutes; state the formula before differentiating or integrating.
Question 1 (5 marks)
Let y = 3x³ − 2x + 5. (a) Find dy/dx; (b) find dy/dx at x = −1.
Question 2 (6 marks)
Evaluate ∫₀² (6x² − 4x + 1) dx.
Question 3 (6 marks)
Find the coefficient of x³ in the expansion of (1 + 2x)⁶.
Question 4 (6 marks)
A radioactive substance has a half-life of 20 years. Starting from 80 mg, find the mass remaining after 60 years.
Question 5 (6 marks)
Let X ~ B(10, 0.3). Find P(X = 3) to 4 decimal places.
Question 6 (7 marks)
Let X ~ N(60, 8²). Find P(X > 76).
Question 7 (7 marks)
For a geometric distribution with success probability p = 0.2, find P(X = 4).
Question 8 (7 marks)
Find the minimum point and the minimum value of y = x² − 6x + 5.
Section B: long questions (3 questions, 50 marks)
Aim for 70-80 minutes. On distribution questions write X ~ B(n, p) and state the formula used.
Question 1 (17 marks)
The volume V (litres) of water in a tank after t years is V = 100(1 − e−0.05t).
(a) Find the limit of V as t → ∞; (b) find V when t = 10 (3 s.f.); (c) find dV/dt and its value at t = 10 (3 s.f.); (d) state whether that rate is falling or rising and why.
Question 2 (17 marks)
A machine produces defective items at a rate of 5%; 20 items are drawn at random.
(a) State a suitable distribution and why; (b) find the probability of exactly 2 defectives (4 d.p.); (c) find the probability of at least 1 defective (4 d.p.); (d) why is an approximation convenient if 40 items are drawn?
Question 3 (16 marks)
The curve y = 4 − x² encloses a region with the x-axis.
(a) Find the intersections; (b) find the area of the region; (c) explain why no sign correction is needed for this area.
Where are the answers, and do they show the steps?
Direct answer: each question is followed by an answer-and-marks block giving the formula, the substitution, the answer and the marks each step earns (the normal question: 3 marks for the z value, 4 for the table lookup), and long questions break down part by part.
Section A answers and mark allocation
Question 1
(a) dy/dx = 9x² − 2 (3 marks); (b) 9(−1)² − 2 = 7 (2 marks).
Question 2
Antiderivative 2x³ − 2x² + x (3 marks); at 2: 16 − 8 + 2 = 10; at 0: 0 → 10 (3 marks).
Question 3
General term C(6,r)(2x)ʳ; with r = 3, C(6,3) × 2³ = 20 × 8 = 160 (6 marks).
Question 4
60 years is three half-lives (3 marks); 80 × (1/2)³ = 10 mg (3 marks).
Question 5
P = C(10,3)(0.3)³(0.7)⁷ (3 marks) = 0.2668 (3 marks).
Question 6
z = (76 − 60) ÷ 8 = 2 (3 marks); from tables P(Z > 2) = 0.0228 (4 marks).
Question 7
P(X = 4) = (0.8)³ × 0.2 (4 marks) = 0.1024 (3 marks).
Question 8
dy/dx = 2x − 6 = 0 gives x = 3 (3 marks); y = 9 − 18 + 5 = −4 (3 marks), so the minimum point is (3, −4) (1 mark).
Section B answers and mark allocation
Question 1
(a) As t → ∞, e−0.05t → 0, so V → 100 litres (3 marks); (b) V = 100(1 − e−0.5) = 100(1 − 0.6065) = 39.3 litres (4 marks); (c) dV/dt = 100 × 0.05 e−0.05t = 5e−0.05t (5 marks), so at t = 10 it is 5 × 0.6065 = 3.03 litres per year (3 marks); (d) the rate is falling, because e−0.05t decreases with t, so dV/dt also decreases towards 0 (2 marks).
Question 2
(a) Binomial B(20, 0.05) (3 marks) because the number of trials is fixed, trials are independent and each has the same two-outcome probability (2 marks); (b) C(20,2)(0.05)²(0.95)¹⁸ = 0.1887 (5 marks); (c) 1 − 0.95²⁰ = 0.6415 (4 marks); (d) with large n and small p a Poisson approximation (λ = np = 2) is simpler to compute (3 marks).
Question 3
(a) 4 − x² = 0 gives x = ±2, so the intersections are (−2, 0) and (2, 0) (4 marks); (b) area = ∫₋₂² (4 − x²) dx = [4x − x³/3]₋₂² = (8 − 8/3) − (−8 + 8/3) = 32/3 ≈ 10.67 (8 marks); (c) because between −2 and 2 the curve lies entirely above the x-axis (4 − x² > 0), so the definite integral is already positive and no splitting or absolute values are needed (4 marks).
Where are M1 marks most easily lost?
Direct answer: distribution questions without notation or parameters; normal questions without the z calculation; definite integrals without substituted limits; and geometric and binomial formulas swapped. All four are right-answer, wrong-format losses.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| No distribution notation | Straight to a probability | The question asks for the distribution and its parameters | Write X ~ B(20, 0.05), then P(X = 2) = … |
| Skipping the z conversion | Reading tables without standardising | Normal questions require the z calculation | Write z = (x − μ) ÷ σ before using the table |
| Definite integrals without limits | Only the antiderivative | A definite integral must substitute both limits and subtract | Write [F(x)] and substitute immediately |
| Geometric vs binomial | Swapping the formulas | Geometric counts trials to the first success; binomial counts successes in n trials | Identify which question is asked first |
How to use this paper
First sit: no references beyond the statistical tables, the full 150 minutes.
Mark it: on distribution questions check notation, parameters, formula and answer, in that order.
Second sit: redo it three to seven days later, inside 120 minutes with all notation written.
What should I do after this paper?
Direct answer: make differentiation, integration and binomial expansion reflexive; put the three distributions (binomial, geometric, normal) with their notation on one page; drill past-paper questions by topic; then redo this paper inside 120 minutes.
FAQ
Is this an HKEAA paper?
No. The questions and answers are TutorZone originals; only the framework figures (100%, 2 h 30 min, 50 + 50 marks) come from the HKEAA 2027 assessment framework.
Does M1 have SBA?
No. The Extended Part is assessed by one public paper worth 100%.
Are statistical tables provided?
Yes – M1 papers come with distribution tables and calculators are allowed. Check the degrees of freedom and the distribution type before reading a value.
Official references
| Resource | What to check |
|---|---|
| HKEAA 2027 HKDSE Mathematics Assessment Framework (with Extended Part) | Confirm M1’s single paper at 100% over 2 h 30 min with 50 + 50 marks |
| https://www.tutorzone.com.hk/en-dse-m1-paper-drill-guide/ | M1 drilling guide |
Related: practice area · M1 drilling guide · M2 mock
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