MATHEMATICS-M1-CALCULUS-AND-STATISTICS · HKDSE
MATHEMATICS-M1-CALCULUS-AND-STATISTICS/11
Paper 1
Mathematics M1 Calculus and Statistics · 2021 2021 · Variant 1
Relative difficulty
Analysis source: Hong Kong Examinations and Assessment Authority (HKEAA)
Analysis aligned to the official syllabus and assessment design.
3.8 / 5
100
150 min
Applications of Differentiation and Integration
Cohort performance
Session statistics from official examination reports
Total marks
100
Duration
150 min
Session difficulty
3.8 / 5
Level 5**
~94% of max
Level 5*
~85% of max
Level 5
~74% of max
Key examiner messages
Top priorities from the principal examiner before you revise
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
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 f''(x) < 0 f′′(x)<0 were major areas where marks were lost.
Question difficulty map
How candidates performed on each question in this series
No data available in official reports
Assessment objectives
Skill and AO weighting from official examiner commentary
Skill weighting
Shows the skill mix this paper tested most heavily.
Algebraic Manipulation
Weight: 9100%Statistical Modeling
Weight: 778%Geometric & Graphical Reasoning
Weight: 556%Relat
Weight: 444%Logical Justification
Weight: 333%Precision and
Weight: 111%
Method marks watchlist
Where working, steps, or method marks were commonly lost
No data available in official reports
Recurring mistakes across years
Themes examiners flag in multiple recent sessions for this subject
No data available in official reports
Question choice intelligence
Mean scores and popularity for optional questions (HKDSE electives)
No data available in official reports
Level exemplars
What candidate scripts at each grade level looked like
No data available in official reports
Grade & admission context
How marks relate to grade thresholds and entry standards
Reporting source
HKEAA Subject Examination Report — comments on candidates’ performance with marking schemes
Level 5**
Outstanding — competitive JUPAS programmes (medicine, law, top faculties)
Level 5*
Excellent — strong JUPAS profile for selective programmes
Level 5
Good — meets most university entrance requirements
Level 4
Satisfactory — foundation programmes or less selective routes
Level 3
Pass threshold for many sub-degree and vocational pathways
Admission context
Levels feed JUPAS and non-JUPAS university applications; 5** and 5* are most selective
Deep insights
What top candidates did
Techniques and approaches examiners rewarded in this series
No data available in official reports
Command word playbook
How to match each command word to the expected response style
Match the expected response style for “Find” questions.
Give reasons and link mechanism to outcome; each point needs a because/so chain.
Match the expected response style for “Expand” questions.
Match the expected response style for “Express” questions.
Match the expected response style for “Let” questions.
Time traps
Sections where candidates spent disproportionate time relative to marks
Min per mark: 2
Min per mark: 1.6
Min per mark: 1.6
Min per mark: 1.6
Min per mark: 1.6
Syllabus traceability
Topics linked to questions and mark weighting in this session
Applications of differentiation
21 marks this session
The Poisson distribution
14 marks this session
Approximation of definite integrals using the trapezoidal rule
11 marks this session
Applications of the normal distribution
11 marks this session
MCQ trap analytics
Commonly chosen wrong options from examiner commentary
No data available in official reports
Topic heatmap across years
Mark concentration by topic and exam year for this subject
Mark intensity
Applications of differentiation
Conditional probability and Bayes’ theorem
The Poisson distribution
Definite integration and its applications
Approximation of definite integrals using the trapezoidal rule
Applications of the normal distribution
Applications of the binomial and the Poisson distributions
Difficulty trend
How session difficulty has shifted across recent years
Paper comparison
Marks and duration breakdown across papers in this session
Paper 1 (Calculus and Statistics):
Marks you can still earn
Where valid approaches outside the mark scheme may still gain credit
No data available in official reports
Practise what examiners flagged
Target weak topics from this report inside the Revui app
Applications of differentiation
21 marks this session
Practise in RevuiThe Poisson distribution
14 marks this session
Practise in RevuiApproximation of definite integrals using the trapezoidal rule
11 marks this session
Practise in RevuiApplications of the normal distribution
11 marks this session
Practise in RevuiSelf-diagnostic checklist
Key actions before you sit this paper — copy and tick off as you revise
- 1Message
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
- 2Message
In the Statistics section, high marks are achievable in the standard binomial, Poisson, and normal distribution applications (Q1, Q2, Q9, Q10).
- 3Message
However, candidates frequently lost marks in conditional probability applications (such as Q10(e) and Q9(c)(ii)) due to incorrect sample space definitions.
- 4Message
In Calculus, marks were easily secured in the basic differentiation (Q7(a)) and expansion (Q6).
- 5Message
Conversely, the algebraic execution in Q12(b) and the justification of the inequality in Q11(b)(iii) using f′′(x)<0 f''(x) < 0 f′′(x)<0 were major areas where marks were lost.
Teacher briefing pack
One-page session summary for tutors and classroom review
2021 2021 2021
Mathematics M1 Calculus and Statistics
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
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
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.
- Total marks
- 100
- Duration
- 150 min
- Session difficulty
- 3.8 / 5
- Level 5**
- ~94% of max
- Level 5*
- ~85% of max
- Level 5
- ~74% of max
Session analysis
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 f''(x) < 0 f′′(x)<0 were major areas where marks were lost.
Updated Jun 11, 2026
Paper breakdown
Paper 1 (Calculus and Statistics):
Top chapters
Exam structure insights
Marks by chapter
See where the marks were concentrated so revision time goes to the highest-value topics.
Mark accessibility
Estimate which marks were basic, mid-level, or high-difficulty.
76% within easy or medium reach
Command word frequency
Spot common command words so answers match the expected response style.
Question type mix
Compare the mark share of each paper section and question type.
Short Questions
(Section A)
50·8·50%
Structured Questions
(Section B)
50·4·50%
Study ROI
Bigger bubbles recur more often; higher bubbles carry more marks, helping you rank revision priorities.
Time vs marks
Compare marks with suggested time allocation to plan exam pacing.
Section A Q1-Q4 (St
0.50 m/minSection B Q9 (Norma
0.61 m/minSection B Q10 (Pois
0.64 m/minSection B Q11 (Trap
0.61 m/minSection B Q12 (Subs
0.64 m/minTotal marks
55
Total time
90 min
Avg pace
0.61
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.
Next-year prediction
Topics worth watching next year, with the reason shown directly below each bar.
Sampling distribution and confidence interval for population mean
90%90%
Exponential and logarithmic models (differentiation/integration applications)
85%85%
Examiner notes & key calculations
- 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 dx dx or du du 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 f''(x) < 0 f′′(x)<0 implies a concave downward curve, which is necessary to establish that the trapezoidal approximation is an underestimation.
Analysis is paraphrased for study purposes. Always verify against the official examiner report and mark scheme.