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100+ Free HKDSE Maths (Core) Practice Questions

Pass your HKDSE Mathematics Compulsory Part exam on the first try — instant access, no signup required.

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A cone has base radius 3 cm and height 4 cm. Find its slant height.

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Key Facts: HKDSE Maths (Core) Exam

The HKDSE Mathematics Compulsory Part is a two-paper exam (Paper 1 conventional 65% over 2.25 hours, Paper 2 with 45 MCQs 35% over 1.25 hours) graded on the 5**-to-1 / U scale by the HKEAA.

Sample HKDSE Maths (Core) Practice Questions

Try these sample questions to test your HKDSE Maths (Core) exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Simplify (3a^2b)(2ab^3) and express the answer in index form.
A.6a^3b^4
B.6a^2b^3
C.5a^3b^4
D.6a^3b^3
Explanation: Multiply the coefficients: 3 x 2 = 6. Add the indices of like bases: a^(2+1) = a^3 and b^(1+3) = b^4. So the product is 6a^3b^4.
2Round 0.0498765 to 3 significant figures.
A.0.0498
B.0.0499
C.0.050
D.0.0499000
Explanation: Leading zeros are not significant, so the first significant figure is 4. The first three significant figures are 4, 9, 8; the next digit is 7, so the 8 rounds up to 9. The result is 0.0499.
3A jacket is sold at $480 after a 20% discount. What was the marked price?
A.$576
B.$560
C.$600
D.$400
Explanation: After a 20% discount, the selling price is 80% of the marked price. So 0.8 x marked price = 480, giving marked price = 480 / 0.8 = $600.
4Factorize x^2 - 9x + 20.
A.(x + 4)(x + 5)
B.(x - 2)(x - 10)
C.(x - 4)(x + 5)
D.(x - 4)(x - 5)
Explanation: Find two numbers whose product is +20 and whose sum is -9. These are -4 and -5, since (-4)(-5) = 20 and (-4) + (-5) = -9. So x^2 - 9x + 20 = (x - 4)(x - 5).
5Solve the quadratic equation x^2 - 5x + 6 = 0.
A.x = 2 or x = 3
B.x = -2 or x = -3
C.x = 1 or x = 6
D.x = -1 or x = -6
Explanation: Factorize: x^2 - 5x + 6 = (x - 2)(x - 3) = 0. Setting each factor to zero gives x = 2 or x = 3, since these multiply to 6 and add to 5.
6Using the quadratic formula, find the discriminant of 2x^2 - 4x + 5 = 0 and state the nature of its roots.
A.Discriminant = 24; two distinct real roots
B.Discriminant = -24; no real roots
C.Discriminant = 0; one repeated root
D.Discriminant = -16; two real roots
Explanation: The discriminant is b^2 - 4ac = (-4)^2 - 4(2)(5) = 16 - 40 = -24. Since the discriminant is negative, the equation has no real roots.
7The quadratic function f(x) = -2x^2 + 8x - 3 attains its maximum value at x equals which value?
A.x = -2
B.x = 4
C.x = 2
D.x = -4
Explanation: For f(x) = ax^2 + bx + c, the vertex occurs at x = -b/(2a). Here a = -2 and b = 8, so x = -8 / (2 x -2) = -8 / -4 = 2. Since a < 0, this gives a maximum.
8If the graph of y = x^2 + kx + 9 touches the x-axis at exactly one point, find the positive value of k.
A.3
B.9
C.18
D.6
Explanation: Touching the x-axis at one point means the discriminant equals zero: k^2 - 4(1)(9) = 0, so k^2 = 36 and k = +/-6. The positive value is 6.
9Given (x - 2) is a factor of P(x) = x^3 - 4x^2 + ax - 6, use the factor theorem to find a.
A.7
B.5
C.-7
D.3
Explanation: By the factor theorem, P(2) = 0. Substituting: 2^3 - 4(2^2) + 2a - 6 = 8 - 16 + 2a - 6 = 2a - 14 = 0, so a = 7.
10Find the remainder when 2x^3 - 3x^2 + x - 5 is divided by (x + 1).
A.-5
B.-11
C.1
D.-9
Explanation: By the remainder theorem, the remainder is P(-1). Substituting: 2(-1)^3 - 3(-1)^2 + (-1) - 5 = -2 - 3 - 1 - 5 = -11.

About the HKDSE Maths (Core) Exam

The HKDSE Mathematics Compulsory Part is the core mathematics subject of the Hong Kong Diploma of Secondary Education, taken by Secondary-6 students and administered by the HKEAA. It is assessed through two written papers: Paper 1 (conventional questions, 65%, 2.25 hours) carries about 105 marks across Section A(1), Section A(2) and Section B, while Paper 2 (35%, 1.25 hours) consists of 45 multiple-choice questions. Content is organized into three strands: Number and Algebra, Measures, Shape and Space, and Data Handling. Results are reported on a five-level scale where Level 5** is the highest grade and U is unclassified. The subject is used for university admission through JUPAS in Hong Kong and is recognized internationally via UCAS tariff points.

Questions

100 scored questions

Time Limit

Paper 1: 2 hours 15 minutes; Paper 2: 1 hour 15 minutes

Passing Score

5-level scale (1 to 5), with 5* and 5** for top Level-5 candidates and U (Unclassified) below Level 1

Exam Fee

HK$519 per subject for school/local-resident candidates (2026); private candidates pay a flat HK$595 application fee (Hong Kong Examinations and Assessment Authority (HKEAA))

HKDSE Maths (Core) Exam Content Outline

45%

Number and Algebra

Numbers and estimation, percentages, polynomials, factorization and identities, quadratic equations and functions, exponential and logarithmic functions, variation, arithmetic and geometric sequences, inequalities and linear programming.

30%

Measures, Shape and Space

Mensuration of solids, coordinate geometry of straight lines and circles, trigonometry including sine and cosine rules and 3-D problems, and basic properties of circles.

25%

Data Handling and Probability

Permutation and combination, probability laws, measures of dispersion including standard deviation, basic normal distribution, and statistics.

How to Pass the HKDSE Maths (Core) Exam

What You Need to Know

  • Passing score: 5-level scale (1 to 5), with 5* and 5** for top Level-5 candidates and U (Unclassified) below Level 1
  • Exam length: 100 questions
  • Time limit: Paper 1: 2 hours 15 minutes; Paper 2: 1 hour 15 minutes
  • Exam fee: HK$519 per subject for school/local-resident candidates (2026); private candidates pay a flat HK$595 application fee

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

HKDSE Maths (Core) Study Tips from Top Performers

1Master Number and Algebra first, since it is the largest strand and underpins functions, sequences and inequalities questions in both papers.
2Practice Paper 2 multiple-choice questions under timed conditions; 45 questions in 75 minutes means roughly 100 seconds each, so speed and accuracy matter.
3Memorize and rehearse the sine rule, cosine rule and the angle properties of circles, which appear reliably every year.
4For standard deviation questions, remember that adding a constant leaves the spread unchanged while multiplying scales it by that factor.
5Always show full working in Paper 1, because method marks are awarded even when the final numerical answer is wrong.

Frequently Asked Questions

How is the HKDSE Mathematics Compulsory Part structured?

It has two papers. Paper 1 (65%, 2 hours 15 minutes) is conventional questions in Sections A(1), A(2) and B requiring full working. Paper 2 (35%, 1 hour 15 minutes) is 45 multiple-choice questions split into a Foundation Section A and a Non-foundation Section B.

How is the exam graded?

Results are reported on a five-level scale from Level 1 (lowest) to Level 5 (highest). The top candidates within Level 5 are awarded 5* and 5**, and performance below Level 1 is recorded as Unclassified (U).

What are the three content strands?

The Compulsory Part covers Number and Algebra (the largest strand), Measures, Shape and Space, and Data Handling. A Further Learning Unit asks students to apply skills across these strands to real-life and mathematical problems.

How much does the 2026 HKDSE cost?

For 2026, school candidates and local-resident self-study candidates pay HK$519 per non-language subject, after a roughly 4% fee increase. Private candidates pay a flat HK$595 application fee, with higher charges for non-permanent residents.

Is a calculator allowed?

Yes. Candidates may use HKEAA-approved calculators in both Paper 1 and Paper 2, though many short questions are designed to be solved efficiently without lengthy calculation.