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100+ Free HKDSE Maths M2 Practice Questions

Pass your HKDSE Mathematics Extended Part Module 2 (Algebra and Calculus) exam on the first try — instant access, no signup required.

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What is the coefficient of x^4 in the expansion of (1 - 2x)^7?

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Key Facts: HKDSE Maths M2 Exam

HKDSE Mathematics Extended Part Module 2 is a single 2.5-hour paper of 100% conventional questions, split into Section A and Section B (each roughly 50 marks), graded on the HKDSE Level 1 to 5** scale.

Sample HKDSE Maths M2 Practice Questions

Try these sample questions to test your HKDSE Maths M2 exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Simplify the surd expression sqrt(50) + sqrt(18) - sqrt(8).
A.5*sqrt(2)
B.4*sqrt(2)
C.6*sqrt(2)
D.7*sqrt(2)
Explanation: Write each surd in terms of sqrt(2): sqrt(50) = 5*sqrt(2), sqrt(18) = 3*sqrt(2), and sqrt(8) = 2*sqrt(2). Then 5*sqrt(2) + 3*sqrt(2) - 2*sqrt(2) = 6*sqrt(2).
2Rationalise the denominator of 1/(sqrt(5) - sqrt(3)).
A.(sqrt(5) - sqrt(3))/2
B.(sqrt(5) + sqrt(3))/2
C.sqrt(5) + sqrt(3)
D.(sqrt(5) + sqrt(3))/8
Explanation: Multiply numerator and denominator by the conjugate sqrt(5) + sqrt(3). The denominator becomes (sqrt(5))^2 - (sqrt(3))^2 = 5 - 3 = 2, giving (sqrt(5) + sqrt(3))/2.
3In a proof by mathematical induction of a proposition P(n) for all positive integers n, what must be shown in the inductive step?
A.P(1) is true
B.P(n) is true for n = 1, 2, 3 only
C.P(k) implies P(k-1)
D.If P(k) is true for a positive integer k, then P(k+1) is true
Explanation: The inductive step assumes P(k) holds for some positive integer k (the inductive hypothesis) and proves that P(k+1) then follows. Together with the base case P(1), this establishes truth for all positive integers.
4By mathematical induction the sum 1 + 2 + 3 + ... + n equals which closed form for all positive integers n?
A.n(n-1)/2
B.n(n+1)/2
C.n^2/2
D.(n+1)(n+2)/2
Explanation: The base case n = 1 gives 1 = 1(2)/2 = 1. Assuming the formula for k, adding (k+1) gives k(k+1)/2 + (k+1) = (k+1)(k+2)/2, matching the formula for k+1. Hence the sum is n(n+1)/2.
5Using the Binomial Theorem, what is the coefficient of x^3 in the expansion of (1 + x)^6?
A.15
B.20
C.6
D.30
Explanation: The coefficient of x^r in (1+x)^n is C(n, r). Here C(6, 3) = 6!/(3!*3!) = 20.
6In the expansion of (2 + x)^5, what is the coefficient of x^2?
A.40
B.10
C.80
D.160
Explanation: The general term is C(5, r) * 2^(5-r) * x^r. For r = 2: C(5, 2) * 2^3 = 10 * 8 = 80.
7What is the constant term in the expansion of (x + 2/x)^6?
A.240
B.160
C.60
D.320
Explanation: The general term is C(6, r) * x^(6-r) * (2/x)^r = C(6, r) * 2^r * x^(6-2r). The constant term needs 6 - 2r = 0, so r = 3: C(6, 3) * 2^3 = 20 * 8 = 160.
8Using the compound angle formula, sin(A + B) equals which expression?
A.sin A cos B - cos A sin B
B.cos A cos B + sin A sin B
C.sin A cos B + cos A sin B
D.cos A cos B - sin A sin B
Explanation: The compound angle identity is sin(A + B) = sin A cos B + cos A sin B, one of the standard formulas in the M2 trigonometry unit.
9Using the double angle formula, cos(2A) can be written as which of the following?
A.1 + 2 sin^2 A
B.2 sin A cos A
C.2 cos^2 A + 1
D.1 - 2 sin^2 A
Explanation: cos(2A) = cos^2 A - sin^2 A = 1 - 2 sin^2 A (using cos^2 A = 1 - sin^2 A). It can also be written as 2 cos^2 A - 1.
10Express sin(5x) + sin(3x) using the sum-to-product formula.
A.2 cos(4x) sin(x)
B.2 sin(4x) sin(x)
C.2 sin(4x) cos(x)
D.2 cos(4x) cos(x)
Explanation: The sum-to-product identity is sin P + sin Q = 2 sin((P+Q)/2) cos((P-Q)/2). With P = 5x, Q = 3x: (P+Q)/2 = 4x and (P-Q)/2 = x, giving 2 sin(4x) cos(x).

About the HKDSE Maths M2 Exam

Module 2 (Algebra and Calculus) is one of two Extended Part modules of HKDSE Mathematics, taken alongside the Compulsory Part. It is assessed by a single 2.5-hour public examination consisting entirely of conventional questions arranged in two sections (A and B), with all questions to be attempted. The syllabus covers three areas: a Foundation Knowledge Area (surds, mathematical induction, the Binomial Theorem, advanced trigonometry, and the number e), a Calculus Area (limits, differentiation, integration and their applications), and an Algebra Area (determinants, matrices, systems of linear equations, and vectors). M2 is the more theoretical and pure-mathematics-oriented module, favoured by students aiming for science, engineering, and mathematics programmes. Results are graded on the standard HKDSE seven-level scale.

Questions

100 scored questions

Time Limit

2 hours 30 minutes

Passing Score

Reported on the HKDSE 7-level scale (Level 1 to 5, with 5* and 5** within Level 5); there is no fixed numeric pass mark.

Exam Fee

Included in HKDSE subject fees; the 2026 HKDSE non-language subject fee is HK$519 for school candidates (HK$1,119 for private candidates without a permanent HKIC). (Hong Kong Examinations and Assessment Authority (HKEAA))

HKDSE Maths M2 Exam Content Outline

30%

Foundation Knowledge Area

Surds, mathematical induction, the Binomial Theorem, compound and double angle plus sum-to-product trigonometric identities, and the number e and natural logarithm.

40%

Calculus Area

Limits and continuity, differentiation from first principles and by rules (product, quotient, chain, implicit, parametric, higher derivatives), and indefinite and definite integration including substitution, by parts, and applications to area and volume.

30%

Algebra Area

Determinants, matrices and inverses, systems of linear equations via Cramer's rule and the inverse method, 2D and 3D vectors, scalar and vector products, and geometric applications of vectors.

How to Pass the HKDSE Maths M2 Exam

What You Need to Know

  • Passing score: Reported on the HKDSE 7-level scale (Level 1 to 5, with 5* and 5** within Level 5); there is no fixed numeric pass mark.
  • Exam length: 100 questions
  • Time limit: 2 hours 30 minutes
  • Exam fee: Included in HKDSE subject fees; the 2026 HKDSE non-language subject fee is HK$519 for school candidates (HK$1,119 for private candidates without a permanent HKIC).

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 M2 Study Tips from Top Performers

1Master the Foundation Knowledge Area first: surds, the Binomial Theorem, and trigonometric identities recur throughout calculus and algebra questions.
2Practise mathematical induction proofs to a fixed structure (base case, inductive hypothesis, inductive step) since proof questions are common in Section B.
3Drill calculus techniques such as the chain, product, and quotient rules, implicit and parametric differentiation, integration by parts, and substitution until they are automatic.
4Learn the standard limits (sin x)/x to 1 and (e^x - 1)/x to 1, which underpin differentiation from first principles.
5For the Algebra Area, be fluent with 2x2 and 3x3 determinants, Cramer's rule, the inverse-matrix method, and 3D vector scalar and cross products.
6Always show full working and keep answers exact (surds, logarithms, fractions) unless the question asks for a decimal, as marks are awarded for method.

Frequently Asked Questions

How long is the HKDSE Mathematics M2 examination?

The M2 paper lasts 2 hours 30 minutes and is a single public examination consisting entirely of conventional (long-answer) questions arranged in two sections, A and B.

Does HKDSE M2 have multiple-choice questions?

No. Unlike Paper 2 of the Compulsory Part, the M2 examination is 100% conventional questions; candidates must show all working in a Question-Answer Book.

How is HKDSE M2 graded?

M2 is reported on the standard HKDSE seven-level scale: Levels 1 to 5, with 5* and 5** awarded to the highest performers within Level 5, and U (Unclassified) below Level 1.

What topics are covered in HKDSE M2?

M2 covers a Foundation Knowledge Area (surds, mathematical induction, binomial theorem, advanced trigonometry, the number e), a Calculus Area (limits, differentiation, integration), and an Algebra Area (determinants, matrices, systems of linear equations, vectors).

What is the difference between HKDSE M1 and M2?

M1 (Calculus and Statistics) is more applied and statistics-focused, while M2 (Algebra and Calculus) is more theoretical, emphasising pure algebra, linear algebra, vectors, and rigorous calculus techniques.

How much does it cost to take HKDSE M2 in 2026?

M2 is a non-language HKDSE subject; the 2026 per-subject fee is HK$519 for school candidates and HK$1,119 for private candidates without a permanent Hong Kong identity card.