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HKDSE M2 Mock Paper (2027): a Single 100% Paper with 8 Short and 3 Long Questions
TUTORZONE ORIGINAL MOCK PAPER · HKDSE MATHEMATICS EXTENDED PART MODULE 2
Written to the HKEAA 2027 framework: Mathematics Extended Part Module 2 (Algebra and Calculus) 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 trigonometric formulas provided. This paper has 8 short and 3 long questions with answers and mark allocation.
What is in HKDSE M2?
Direct answer: Mathematics Extended Part Module 2 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 trigonometric formulas 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 | Trigonometric formulas are provided; calculators are allowed | — | — |
Two and a half hours: about 60-70 minutes for Section A and 70-80 for Section B. M2 marks follow definition, computation, conclusion – proofs and induction must show the structure (base, assumption, step, conclusion), or the method marks are gone.
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 integrals, vector angles and determinants, was recomputed programmatically.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Section A (short questions) | 50 | about 60-70 min | Complex numbers, matrices and vectors must be exact |
| Section B (long questions) | 50 | about 70-80 min | Show every step in proofs; write the substitution out in integrals |
| In the exam room | — | Write the defining formula first on matrices and vectors | Induction needs all four steps |
How should the 150 minutes be split?
Direct answer: about 60-70 minutes on Section A and 70-80 on Section B. Section A rewards accuracy; Section B rewards written structure – the four induction steps and the u, dv, du, v of integration by parts.
Section A: short questions (8 questions, 50 marks)
Aim for 60-70 minutes; show the working on complex-number and matrix items.
Question 1 (5 marks)
Simplify (1 − 2i)(3 + i).
Question 2 (6 marks)
Solve log₂ x + log₂ (x − 2) = 3.
Question 3 (6 marks)
Let A = [[2, 1], [0, 3]]. Find A².
Question 4 (6 marks)
Let a = (2, −1, 3) and b = (1, 2, −1). (a) Find a·b; (b) find |a|.
Question 5 (7 marks)
Solve 2sin²θ − sinθ − 1 = 0 for 0 ≤ θ ≤ 2π.
Question 6 (7 marks)
Find lim(x→0) (sin 3x)/x.
Question 7 (7 marks)
Let y = x³ex. Find dy/dx.
Question 8 (7 marks)
Prove by mathematical induction that 1 + 2 + … + n = n(n + 1)/2 for all positive integers n (the induction steps suffice).
Section B: long questions (3 questions, 50 marks)
Aim for 70-80 minutes. Write u, dv, du and v on integration-by-parts; write the cosθ expression on vector items.
Question 1 (17 marks)
Solve 2x + y = 7 and x + 3y = 11. (a) Solve by elimination or matrices; (b) find the determinant of the coefficient matrix; (c) explain what a non-zero determinant means for the solution.
Question 2 (17 marks)
(a) Use integration by parts to evaluate ∫₀¹ x ex dx; (b) state the formula you used; (c) explain the usual principle for choosing u and dv.
Question 3 (16 marks)
Let a = (2, −1, 3) and b = (1, 2, −1). (a) Find the angle between them (1 d.p.); (b) explain what a negative dot product means geometrically; (c) what would the dot product be if a and b were perpendicular?
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 or definition, the working, the answer and the marks per step (matrix item: 6 marks for the row-by-column computation; integral: 4 marks for u/dv/du/v and 6 for the substitution).
Section A answers and mark allocation
Question 1
= 3 + i − 6i − 2i² (3 marks) = 3 − 5i + 2 = 5 − 5i (2 marks).
Question 2
log₂[x(x − 2)] = 3 → x(x − 2) = 8 (3 marks) → x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0; since x must exceed 2, x = 4 (3 marks).
Question 3
A² = [[2×2 + 1×0, 2×1 + 1×3], [0×2 + 3×0, 0×1 + 3×3]] = [[4, 5], [0, 9]] (6 marks).
Question 4
(a) a·b = 2 − 2 − 3 = −3 (3 marks); (b) |a| = √(4 + 1 + 9) = √14 (3 marks).
Question 5
(2sinθ + 1)(sinθ − 1) = 0 (3 marks) → sinθ = 1 or sinθ = −1/2; θ = π/2, 7π/6, 11π/6 (4 marks).
Question 6
= lim (sin 3x)/(3x) × 3 (4 marks) = 1 × 3 = 3 (3 marks).
Question 7
By the product rule: dy/dx = 3x²ex + x³ex = x²ex(3 + x) (7 marks).
Question 8
(i) Base: for n = 1 the left side is 1 and the right is 1×2/2 = 1 (2 marks). (ii) Assume it holds for n = k, i.e. 1 + … + k = k(k+1)/2 (2 marks). (iii) Then 1 + … + k + (k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2, so it holds for n = k+1 (2 marks). (iv) By induction it holds for all positive integers n (1 mark).
Section B answers and mark allocation
Question 1
(a) Elimination: from the first equation y = 7 − 2x, so x + 21 − 6x = 11 → −5x = −10 → x = 2, y = 3, giving x = 2, y = 3 (8 marks); (b) det = 2×3 − 1×1 = 5 (5 marks); (c) a non-zero determinant means the lines are not parallel, so the system has a unique solution and the matrix is invertible (4 marks).
Question 2
(a) Take u = x and dv = exdx, so du = dx and v = ex (4 marks); ∫x exdx = x ex − ∫exdx = x ex − ex (6 marks); substituting 0 and 1 gives (e − e) − (0 − 1) = 1 (4 marks); (b) the formula ∫u dv = uv − ∫v du (1 mark); (c) choose u as the factor that simplifies when differentiated (a polynomial, say) and dv as the factor that is easy to integrate (2 marks).
Question 3
(a) a·b = −3, |a| = √14 and |b| = √6 (4 marks); cosθ = −3 ÷ √84 = −0.3273 (5 marks), so θ ≈ 109.1° (4 marks); (b) a negative dot product means the angle is greater than 90° (obtuse), i.e. the vectors point broadly opposite ways (2 marks); (c) for perpendicular vectors the dot product is 0 (1 mark).
Where are M2 marks most easily lost?
Direct answer: conclusions without proof steps; integration by parts without writing u, dv, du and v; matrix multiplication in the wrong order; and log equations where the negative root is kept without a domain check.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Integration without substitution | The final answer only | Integration by parts needs u, dv, du and v written out | List the four, then uv − ∫v du, then substitute the limits |
| Matrix order | Mixing A×B and B×A | Matrix multiplication is not commutative, so the order decides the answer | Write the row-by-column computation explicitly |
| Log equations without domain checks | Keeping a negative root | Logarithms require a positive argument | Test every root against the domain |
| Skipping proof steps | The conclusion only | Proofs and induction are marked per step | Base, assumption, induction step, conclusion |
How to use this paper
First sit: no references beyond the trigonometric formulas, the full 150 minutes.
Mark it: check all four induction steps and the choice of u and dv on integrals.
Second sit: redo it three to seven days later inside 120 minutes with every step written.
What should I do after this paper?
Direct answer: put the standard complex-number, matrix, vector and calculus results on one page; template the four-step induction and proof structure; 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 M2 have SBA?
No. The Extended Part is assessed by one public paper worth 100%.
What is provided in the exam?
M2 papers include trigonometric formulas and calculators are allowed, but the standard matrix, vector and calculus results must be memorised.
Official references
| Resource | What to check |
|---|---|
| HKEAA 2027 HKDSE Mathematics Assessment Framework (with Extended Part) | Confirm M2’s single paper at 100% over 2 h 30 min with 50 + 50 marks |
| https://www.tutorzone.com.hk/en-dse-m2-paper-drill-guide/ | M2 drilling guide |
Related: practice area · M2 drilling guide · M1 mock
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