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HKDSE Mathematics Paper 1 Mock Paper (2027): 105 Marks with Worked Answers
TUTORZONE ORIGINAL MOCK PAPER · HKDSE MATHEMATICS PAPER 1
Written to the HKEAA 2027 framework: Paper 1 conventional questions, 65%, 2 h 15 min, 105 marks split 35 + 35 + 35 with every question compulsory. This paper has 18 questions with question-by-question answers and mark allocation, ready to sit under time.
What is in HKDSE Mathematics Paper 1?
Direct answer: Paper 1 uses conventional questions and is 65% of the subject, 2 hours 15 minutes, 105 marks, all compulsory: Section A(1) 35 marks (8-11 elementary questions), A(2) 35 marks (4-7 harder questions) and B 35 marks (4-7 of the hardest). There is no SBA in Mathematics.
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Section A(1) | 8-11 elementary questions; foundation topics of the Compulsory Part and S1-S3 | 35 分 | 35-40 min |
| Section A(2) | 4-7 harder questions; Compulsory Part plus foundation and non-foundation S1-S3 | 35 分 | 45-50 min |
| Section B | 4-7 of the hardest questions; Compulsory Part plus foundation and non-foundation S1-S3 | 35 分 | 40-45 min |
Cut the 135 minutes as 40 / 48 / 47 (A1 / A2 / B) plus 8-10 minutes to check – closer to reality than an even split, since A(1) is fastest and B needs the buffer.
How this paper was built
This is a TutorZone original mock paper: the questions were written to the 2027 framework’s structure (sections, question-count ranges, mark split) and the answers and mark allocation are our own model answers, not HKEAA past papers or official marking references.
18 questions in total: Section A(1) nine questions for 35 marks, A(2) five for 35, Section B four for 35. Every numerical answer was recomputed programmatically.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Section A(1) | 35 分 | 35-40 min (target full marks) | Take what you can at once, skip and return |
| Section A(2) | 35 分 | 45-50 min (aim 28+) | Show every step; method marks are the cheap marks |
| Section B | 35 分 | 40-45 min plus 8-10 min checking | Start with your two strongest; stop a hard question once the key steps are down |
How many questions does this paper have and how should the time be split?
Direct answer: 18 questions: A(1) nine questions for 35 marks, A(2) five for 35, Section B four for 35. Split the 135 minutes 40 / 48 / 47 and keep 8-10 minutes to check units, significant figures and unanswered parts.
Section A(1): elementary short questions (9 questions, 35 marks)
Aim for 35-40 minutes. All compulsory; take the ones you can do first.
Question 1 (3 marks)
Evaluate (3/4 − 5/6) ÷ 1/12.
Question 2 (3 marks)
Simplify (2x³y)² ÷ 4xy².
Question 3 (4 marks)
Factorise 6x² + 7x − 20.
Question 4 (4 marks)
Solve 3(x − 2) = 2(x + 1).
Question 5 (4 marks)
A good costs $240, is marked up 15% and then discounted 15%. Find the final price.
Question 6 (4 marks)
Find the size of each interior angle of a regular 12-sided polygon.
Question 7 (4 marks)
In a right-angled triangle the opposite side is 7 cm and the hypotenuse 12 cm. Find the acute angle at the hypotenuse (1 d.p.) and the adjacent side (1 d.p.).
Question 8 (4 marks)
Data: 12, 15, 11, 19, 15, 18. Find the mean, median and mode.
Question 9 (5 marks)
A sector has radius 6 cm and angle 150°. Find its area in terms of π and correct to 3 significant figures.
Section A(2): harder questions (5 questions, 35 marks)
Aim for 45-50 minutes. Method marks are most often lost here.
Question 1 (6 marks)
Solve the simultaneous equations 2x + 3y = 12 and x − y = 1.
Question 2 (7 marks)
Given y = x² − 4x − 5: (a) find where the graph cuts the x-axis; (b) find the vertex; (c) solve x² − 4x − 5 = 7.
Question 3 (7 marks)
A geometric sequence has first term 3 and common ratio 2. Find (a) the 6th term and (b) the sum of the first 6 terms.
Question 4 (7 marks)
Line L passes through A(1, 2) and B(5, 10). (a) Find the equation of L; (b) find where L meets y = x + 6.
Question 5 (8 marks)
A bag holds 5 red and 7 blue balls. Two balls are drawn without replacement. Find the probability that (a) the first is red; (b) the first is red and the second is blue; (c) at least one is red.
Section B: long questions (4 questions, 35 marks)
Aim for 40-45 minutes: start with your two strongest and keep 8-10 minutes to check.
Question 1 (8 marks)
Circle C has equation x² + y² − 6x + 8y − 11 = 0. (a) Find its centre and radius; (b) find where C cuts the x-axis and the distance between those two points (3 s.f.).
Question 2 (9 marks)
Solve 3^(2x + 1) = 27^(x − 1).
Question 3 (9 marks)
(a) Solve x² − x − 6 ≤ 0. (b) A rectangle has perimeter 40 cm; find its greatest possible area.
Question 4 (9 marks)
From a point A on the ground the angle of elevation of a tower’s top is 30°. After walking 40 m towards it to B, the angle becomes 60°. Find the height of the tower (3 s.f.).
Where are the answers, and do they show the working?
Direct answer: every question is followed by an answer-and-marks block giving each intermediate result with the marks it earns (question 1: 2 marks for the common denominator, 1 for the division) plus the usual losses. Not just final answers.
Section A(1) answers and mark allocation
Question 1
3/4 − 5/6 = 9/12 − 10/12 = −1/12 (2 marks); −1/12 ÷ 1/12 = −1 (1 mark). The usual loss is cross-multiplying instead of taking a common denominator.
Question 2
(2x³y)² = 4x⁶y² (1 mark); 4x⁶y² ÷ 4xy² = x⁵ (2 marks). Powers multiply: (x³)² = x⁶, not x⁹.
Question 3
Two numbers with product 6 × (−20) = −120 and sum 7: 15 and −8 (2 marks); 6x² + 15x − 8x − 20 = 3x(2x + 5) − 4(2x + 5) (1 mark) = (2x + 5)(3x − 4) (1 mark).
Question 4
3x − 6 = 2x + 2 (2 marks); 3x − 2x = 2 + 6 (1 mark); x = 8 (1 mark).
Question 5
240 × 1.15 = 276 (2 marks); 276 × 0.85 = $234.6 (2 marks). The two percentages apply to different bases, so the price does not return to $240.
Question 6
Exterior angle = 360° ÷ 12 = 30° (2 marks); interior angle = 180° − 30° = 150° (2 marks).
Question 7
sin θ = 7/12 (1 mark); θ ≈ 35.7° (2 marks); adjacent side = √(12² − 7²) = √95 ≈ 9.7 cm (1 mark).
Question 8
Mean = 90 ÷ 6 = 15 (2 marks); sorted (11, 12, 15, 15, 18, 19) median = (15 + 15) ÷ 2 = 15 (1 mark); mode = 15 (1 mark).
Question 9
Area = 150/360 × π × 6² (2 marks) = 15π (1 mark) ≈ 47.1 cm² (2 marks).
Section A(2) answers and mark allocation
Question 1
From x − y = 1, x = y + 1 (1 mark); substitute: 2(y + 1) + 3y = 12 (2 marks); 5y = 10 → y = 2 (2 marks); x = 3, y = 2 (1 mark).
Question 2
(a) x² − 4x − 5 = (x − 5)(x + 1) = 0 → (5, 0) and (−1, 0) (3 marks); (b) axis x = 2, y = 4 − 8 − 5 = −9 → (2, −9) (2 marks); (c) x² − 4x − 12 = 0 → (x − 6)(x + 2) = 0 → x = 6 or x = −2 (2 marks).
Question 3
(a) T₆ = 3 × 2⁵ = 96 (3 marks); (b) S₆ = 3(2⁶ − 1) ÷ (2 − 1) = 3 × 63 = 189 (4 marks).
Question 4
(a) Gradient = (10 − 2) ÷ (5 − 1) = 2 (2 marks); y − 2 = 2(x − 1) → y = 2x (2 marks); (b) 2x = x + 6 → x = 6, y = 12 → (6, 12) (3 marks).
Question 5
(a) 5/12 (2 marks); (b) 5/12 × 7/11 = 35/132 (3 marks); (c) 1 − P(both blue) = 1 − 7/12 × 6/11 = 1 − 42/132 = 15/22 (3 marks). Without replacement the second denominator must change to 11.
Section B answers and mark allocation
Question 1
(a) (x − 3)² + (y + 4)² = 3² + 4² + 11 = 36 → centre (3, −4), radius 6 (4 marks); (b) put y = 0: x² − 6x − 11 = 0 → x = 3 ± 2√5 ≈ 7.47 and −1.47 (3 marks); distance = 4√5 ≈ 8.94 (1 mark).
Question 2
27 = 3³ (2 marks); 3^(2x + 1) = 3^(3x − 3) (2 marks); 2x + 1 = 3x − 3 (3 marks); x = 4 (2 marks). Both sides must share the same base before the indices can be equated.
Question 3
(a) (x − 3)(x + 2) ≤ 0 → −2 ≤ x ≤ 3 (4 marks); (b) let the length be x cm, so the width is (20 − x) cm and the area is x(20 − x) = −(x − 10)² + 100 (3 marks) → greatest area 100 cm² (2 marks).
Question 4
Let the height be h and the distance from A to the foot be d: tan 30° = h/d, tan 60° = h/(d − 40) (3 marks); so h cot 30° − h cot 60° = 40 (2 marks); h(√3 − √3/3) = 40 → h(2√3/3) = 40 (2 marks); h = 20√3 ≈ 34.6 m (2 marks).
Where do Paper 1 marks actually sit?
Direct answer: in Sections A(2) and B, 70 marks between them – and the losses there are mostly unwritten method marks: skipped steps, unnamed rules, missing units. Mark yourself against the allocation, not against the final answers.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Answers only | Stopping when the number is right | Method marks lost, one to two levels lower | Write every step, including the rule and its condition |
| Fighting one Section B question | Fifteen minutes on one item | No time left to check or return | Stop at the key steps, flag it and move on |
| Units and accuracy | Bare numbers | Paper 1 marks units and accuracy | Check units and significant figures at the end |
How to use this paper
First sit: no reference materials, full 135 minutes, no formula sheet or textbook.
Mark it: checking answers alone is useless – work through the mark allocation and count the steps you omitted.
Second sit: redo the same paper three to seven days later, aiming for fewer minutes and more complete working.
What should I do after finishing this paper?
Direct answer: three things: log every question where a method mark was missed and name the missing step; redo the paper three to seven days later with more complete working; then drill the matching past-paper questions by topic until that topic is error-free under time.
FAQ
Is this an HKEAA paper?
No. The questions, answers and mark allocation are TutorZone originals; only the framework figures come from the HKEAA 2027 assessment framework.
What is a good score?
This is not a real paper, so the score does not convert to a level. The useful number is how many method marks you failed to write down.
Can this paper be my only practice?
No. Its job is timed rehearsal and marking-scheme work; topic drilling and past papers still do the teaching.
Official references
| Resource | What to check |
|---|---|
| HKEAA 2027 HKDSE Mathematics Assessment Framework | Confirm Paper 1’s 65%, 2 h 15 min, 105 marks and the three 35-mark parts |
| https://www.tutorzone.com.hk/en-dse-maths-paper-1-lq-drill-guide/ | Paper 1 drilling guide: 135-minute pacing and method-mark writing |
| https://www.tutorzone.com.hk/en-dse-maths-paper-2-mc-drill-guide/ | Paper 2 (MC) drilling guide |
Related: practice area · IB Maths AA SL mock paper
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