HKDSE · Analysis & Prediction

2021 HKDSE Mathematics M1 (Calculus and Statistics) Past Paper Analysis

The 2021 HKDSE Mathematics M1 paper presented a balanced yet highly rigorous set of questions. Section A tested fundamental competency in calculus and probability with standard but algebraically intensive problems. Section B elevated the difficulty significantly with complex application tasks, particularly the integration by substitution in rate of change contexts (Q12) and inequality proofs linking the trapezoidal rule to concavity (Q11).

Updated: 11 Jun 2026

Analysis Sample paper Solution

Difficulty 3.8/5

No official paper links

Total marks

100

Duration

150 mins

Most tested

Applications of Differentiation and Integration

Paper breakdown

Paper 1 (Calculus and Statistics): 100 marks · 150 mins

Top chapters

Applications of differentiation · 21 marksThe Poisson distribution · 14 marksApproximation of definite integrals using the trapezoidal rule · 11 marksApplications of the normal distribution · 11 marks

Examiner Verdict & Difficulty Profile

The 2021 HKDSE Mathematics Module 1 (Calculus and Statistics) paper maintains a challenging but fair standard, rated at 4 out of 5 stars in difficulty. While Section A contains accessible computational marks, Section B separates top-performing candidates with demanding questions that require conceptual integration. Specifically, the latter parts of Q11 (concavity-based error estimation in the trapezoidal rule) and Q12 (complex definite integration by substitution paired with related rates of change in an inverted cone) served as the ultimate differentiators.

Where the Marks are Won or Lost

In the Statistics section, high marks are achievable in the standard binomial, Poisson, and normal distribution applications (Q1, Q2, Q9, Q10). However, candidates frequently lost marks in conditional probability applications (such as Q10(e) and Q9(c)(ii)) due to incorrect sample space definitions. In Calculus, marks were easily secured in the basic differentiation (Q7(a)) and expansion (Q6). Conversely, the algebraic execution in Q12(b) and the justification of the inequality in Q11(b)(iii) using \( f''(x) < 0 \) were major areas where marks were lost.

Common Examiner Pitfalls

Truncation of Intermediate Values: Many candidates rounded intermediate probabilities to 2 or 3 decimal places, causing inaccuracies in final answers (which must be exact or correct to 4 decimal places).
Improper Notation in Integration: Forgetting to write the differential term (e.g., \( dx \) or \( du \)) or failing to change the integration limits during u-substitution (Q12(b)).
Inadequate Explanations: In Q11(b)(iii), failing to state that \( f''(x) < 0 \) implies a concave downward curve, which is necessary to establish that the trapezoidal approximation is an underestimation.

Preparation Strategy for Future Candidates

To excel in future sessions, candidates must focus heavily on the following:

Mastery of Related Rates & Modeling: Related rates in geometric solids (cones, spheres, cylinders) are recurrent. Memorizing formulas and practicing the chain rule \( \frac{dV}{dt} = \frac{dV}{dh} \cdot \frac{dh}{dt} \) is essential.
Integration by Substitution: Practice finding appropriate substitutions and performing algebraic simplifications carefully.
Rigorous Justifications: Always explicitly show the second derivative's sign when determining concavity or maximizing rates of change.

Charts

Marks by chapter

See where the marks were concentrated so revision time goes to the highest-value topics.

Question type mix

Compare the mark share of each paper section and question type.

Short Questions (Section A)50 marks · 8 questions
Structured Questions (Section B)50 marks · 4 questions

Mark accessibility

Estimate which marks were basic, mid-level, or high-difficulty.

76%within easy or medium reach

easy: 36 marksmedium: 40 markshard: 24 marks

Command word frequency

Spot common command words so answers match the expected response style.

Skill weighting

Shows the skill mix this paper tested most heavily.

Study ROI

Bigger bubbles recur more often; higher bubbles carry more marks, helping you rank revision priorities.

The Poisson distribution

14 marks · Difficulty 3.2 · recurrence 100%

Applications of the normal distribution

11 marks · Difficulty 3.4 · recurrence 100%

Applications of differentiation

21 marks · Difficulty 4.2 · recurrence 100%

Conditional probability and Bayes' theorem

6 marks · Difficulty 3.5 · recurrence 100%

Binomial expansion

5 marks · Difficulty 2.2 · recurrence 90%

Time vs marks

Compare marks with suggested time allocation to plan exam pacing.

marksmins

Next-year prediction

Topics worth watching next year, with the reason shown directly below each bar.

Sampling distribution and confidence interval for population mean90%

Confidence intervals for population proportion appeared in 2021, so a confidence interval for the population mean (using standard normal variable) is highly likely to return next year.

Exponential and logarithmic models (differentiation/integration applications)85%

Integration of exponential growth models or logarithmic models didn't feature heavily as a standalone long question in 2021, making it highly likely for future papers.

Cumulative marks ladder

The line is your running mark total question by question; dashed lines are the estimated grade cut-offs. See which question the line crosses your target grade at, so you know how far you must answer cleanly and which questions decide a band.

Examiner insights

Official report

Common pitfalls

Applying incorrect conditional probability formulas when computing multi-stage events.
Failing to write down dx/du during integration by substitution, leading to lost notation marks.
Truncating intermediate probability values, resulting in final answers that deviate from the 4 d.p. standard.

Misconceptions

Believing that the trapezoidal rule always overestimates or underestimates regardless of the concavity (second derivative).
Confusing standard normal variable transformation Z with the random variable X without standardising the mean and standard deviation.

Where marks are lost

In Q11(b)(iii), failing to use the second derivative f''(x) to justify concavity, which is required to prove the inequality.
In Q12(b), failing to change the integration limits from x-limits (0 to 8) to u-limits (1 to 2) during substitution.
In Q9(c)(ii), failing to correctly formulate the conditional probability where the denominator is the probability of exactly 1 big potato.

Estimated grade cut-off

Updated: 11 Jun 2026

Source: DSE Treasure — DSE Cut-off (2012–2025, all subjects)

Level	Mark	Percentage
5**	—	94%
5*	—	85%
5	—	74%
4	—	62%
3	—	58%

Estimated cut-offs only. The HKEAA does not publish official raw-mark cut-offs for the HKDSE; these figures are reconstructions from the cited source and may differ from the actual boundaries.