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Free Practice Questions for GCE A/L Combined Mathematics

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Key Facts: GCE A/L Combined Mathematics Exam

30 Problems

Official Question Count Across Papers I & II

Department of Examinations, Sri Lanka

6 Hours

Total Examination Time (Two 3-Hour Papers)

Department of Examinations, Sri Lanka

Grade S

Minimum Passing Standard

Department of Examinations, Sri Lanka

Subject 12

Official National Subject Code

Department of Examinations, Sri Lanka

3 Languages

Administered in Sinhala, Tamil, and English

Ministry of Education, Sri Lanka

Sri Lanka GCE A/L Combined Mathematics (Subject Code 10) is the benchmark examination for Physical Science stream students. It consists of two 3-hour papers (Pure and Applied Mathematics), demanding deep analytical problem-solving and rigorous multi-step derivations for state university engineering and physical science admissions.

Sample GCE A/L Combined Mathematics Practice Questions

Try these sample questions to review concepts for the GCE A/L Combined Mathematics exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Evaluate the limit lim_{x -> 0} [sin(3x) / (5x)].
A.3/5
B.5/3
C.1
D.0
Explanation: Rewrite the expression using the fundamental standard trigonometric limit lim_{u -> 0} [sin(u) / u] = 1. We have lim_{x -> 0} [sin(3x) / (5x)] = lim_{x -> 0} (3/5) * [sin(3x) / (3x)] = (3/5) * 1 = 3/5.
2Given f(x) = (2x - 1)^3, evaluate the derivative f'(x) at x = 2.
A.54
B.27
C.18
D.108
Explanation: Apply the chain rule: f'(x) = 3(2x - 1)^2 * d/dx(2x - 1) = 3(2x - 1)^2 * 2 = 6(2x - 1)^2. Substituting x = 2: f'(2) = 6(2(2) - 1)^2 = 6(3^2) = 6 * 9 = 54.
3Determine the indefinite integral ∫ (4x^3 - 6x^2 + 2x - 7) dx.
A.x^4 - 2x^3 + x^2 - 7x + C
B.4x^4 - 6x^3 + 2x^2 - 7x + C
C.12x^2 - 12x + 2 + C
D.x^4 - 3x^3 + 2x^2 - 7x + C
Explanation: Integrate term-by-term using the power rule ∫ x^n dx = x^{n+1}/(n+1): ∫ 4x^3 dx = 4(x^4 / 4) = x^4; ∫ -6x^2 dx = -6(x^3 / 3) = -2x^3; ∫ 2x dx = 2(x^2 / 2) = x^2; ∫ -7 dx = -7x. Combining these yields x^4 - 2x^3 + x^2 - 7x + C.
4Find the gradient of the tangent to the curve y = x^3 - 3x^2 + 5 at the point where x = 3.
A.9
B.0
C.5
D.18
Explanation: Differentiate y with respect to x: dy/dx = 3x^2 - 6x. At x = 3, the gradient is dy/dx|_{x=3} = 3(3)^2 - 6(3) = 3(9) - 18 = 27 - 18 = 9.
5Determine the maximal real domain of the function f(x) = √(9 - x^2) / (x - 1).
A.[-3, 3]
B.(-3, 3) \ {1}
C.[-3, 1) U (1, 3]
D.(-inf, 1) U (1, inf)
Explanation: For the square root in the numerator to be defined in real numbers, 9 - x^2 ≥ 0 ⇒ x^2 ≤ 9 ⇒ -3 ≤ x ≤ 3. For the denominator to be non-zero, x - 1 ≠ 0 ⇒ x ≠ 1. Combining both conditions gives [-3, 1) ∪ (1, 3].
6Evaluate the limit lim_{x -> 2} [(x^3 - 8) / (x^2 - 4)].
A.3
B.2
C.0
D.4
Explanation: Factorize both numerator and denominator: x^3 - 8 = (x - 2)(x^2 + 2x + 4) and x^2 - 4 = (x - 2)(x + 2). Cancelling the common factor (x - 2) for x ≠ 2: lim_{x -> 2} [(x^2 + 2x + 4) / (x + 2)] = (2^2 + 2(2) + 4) / (2 + 2) = (4 + 4 + 4) / 4 = 12 / 4 = 3.
7Evaluate the definite integral ∫_0^{π/4} sec^2(x) dx.
A.1
B.1/2
C.sqrt(2)
D.π/4
Explanation: The standard antiderivative of sec^2(x) is tan(x). Applying the limits: [tan(x)]_0^{π/4} = tan(π/4) - tan(0) = 1 - 0 = 1.
8The radius of a circular oil slick expands at a constant rate of 0.5 m/s. At what rate is the area of the slick increasing when the radius is 10 m?
A.10π m^2/s
B.5π m^2/s
C.20π m^2/s
D.100π m^2/s
Explanation: Area of a circle is A = π r^2. Differentiating with respect to time t gives dA/dt = 2π r (dr/dt). Substituting r = 10 m and dr/dt = 0.5 m/s: dA/dt = 2π(10)(0.5) = 10π m^2/s.
9If y = ln[(1 + x) / (1 - x)] for -1 < x < 1, find dy/dx.
A.2 / (1 - x^2)
B.1 / (1 - x^2)
C.2x / (1 - x^2)
D.2 / (1 + x^2)
Explanation: Use the logarithm quotient rule: y = ln(1 + x) - ln(1 - x). Differentiating term-by-term: dy/dx = 1/(1 + x) - [(-1)/(1 - x)] = 1/(1 + x) + 1/(1 - x) = [(1 - x) + (1 + x)] / [(1 + x)(1 - x)] = 2 / (1 - x^2).
10Find the equation of the normal to the parabola y = x^2 - 4x + 3 at the point where x = 4.
A.x + 4y - 16 = 0
B.4x - y - 13 = 0
C.x - 4y + 8 = 0
D.4x + y - 19 = 0
Explanation: At x = 4, y = 4^2 - 4(4) + 3 = 3, so the point is (4, 3). The derivative is dy/dx = 2x - 4. At x = 4, tangent gradient m_t = 2(4) - 4 = 4. The gradient of the normal is m_n = -1 / m_t = -1/4. The equation of the normal line is y - 3 = (-1/4)(x - 4) ⇒ 4(y - 3) = -(x - 4) ⇒ 4y - 12 = -x + 4 ⇒ x + 4y - 16 = 0.

About the GCE A/L Combined Mathematics Exam

The Sri Lanka General Certificate of Education (Advanced Level) Combined Mathematics (Subject Code 10) is the premier mathematics examination administered by the Department of Examinations, Sri Lanka. It serves as the primary evaluation standard for senior secondary students pursuing the Physical Science stream, qualifying candidates for competitive university faculties such as engineering, physical sciences, computer science, and architecture. The examination demands comprehensive problem-solving proficiency, rigorous deductive reasoning, and precise multi-step algebraic derivations across pure mathematics (calculus, functions, algebra, trigonometry, matrices, complex numbers) and applied mathematics (kinematics, Newton's laws, impulse-momentum, work-energy, circular motion, statics, equilibrium, friction, probability, and statistics). Performance across both papers determines the candidate's national Z-score used for merit-based state university allocation. This study resource provides 100 carefully constructed multiple-choice practice items with complete step-by-step mathematical solutions to help candidates develop problem-solving speed, conceptual precision, and rigorous analytical habits.

Exam sponsor: Department of Examinations, Sri Lanka (doenets.lk). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Combined Mathematics (Subject Code 10) consists of two three-hour written examination papers: Paper I (Pure Mathematics) and Paper II (Applied Mathematics). Each paper consists of Part A (10 compulsory short-answer questions, 25 marks total) and Part B (7 structured essay questions where candidates must choose and answer 5, 75 marks total). Total marks for each paper is 100, combined equally into an aggregate score out of 100 for final grading and Z-score computation.

Time Limit

Paper I: 3 hours plus 10 minutes' reading time; Paper II: 3 hours plus 10 minutes' reading time (total 6 hours 20 minutes)

Passing Score

Grade S or higher (grades: A, B, C, S, F; university admission requires high Z-score)

Exam / Certification Fees

Free for school candidates; nominal fee for private candidates

Exam sponsor website

Reported exam pass rate: Approximately 50% to 60% of candidates achieve a passing grade (S or higher) annually.. This describes exam candidates, not OpenExamPrep users or results from using our resources. Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

25%

Pure Mathematics: Calculus and Functions

Limits, real functions, continuity, differential calculus (product/quotient/chain rules, implicit differentiation, tangents and normals, rates of change, maxima and minima), integral calculus (standard integrals, substitution, integration by parts, partial fractions, definite integrals, and area under curves).

25%

Pure Mathematics: Algebra and Trigonometry

Quadratic equations and roots, polynomial division and remainder theorem, permutations and combinations, binomial theorem, principle of mathematical induction, complex numbers (Argand diagram, modulus-argument form, de Moivre's theorem), matrices, trigonometric identities, trigonometric equations, and inverse trigonometric functions.

30%

Applied Mathematics: Mechanics and Kinematics

Rectilinear motion with constant and variable acceleration, velocity-time graph analysis, relative velocity and closest approach, 2D projectile motion under gravity, Newton's laws of motion, connected particles (pulley systems), impulse and linear momentum, work, energy, power, and circular motion in horizontal and vertical planes.

20%

Applied Mathematics: Statics and Probability

Coplanar forces acting on a particle and rigid body, conditions of equilibrium, Lami's theorem, moments, friction and limiting equilibrium, centers of gravity of composite bodies and wires, basic probability rules, conditional probability, discrete random variables, expectation, variance, and descriptive statistics.

Preparing for the GCE A/L Combined Mathematics Exam

What You Need to Know

  • Passing score: Grade S or higher (grades: A, B, C, S, F; university admission requires high Z-score)
  • Assessment: Combined Mathematics (Subject Code 10) consists of two three-hour written examination papers: Paper I (Pure Mathematics) and Paper II (Applied Mathematics). Each paper consists of Part A (10 compulsory short-answer questions, 25 marks total) and Part B (7 structured essay questions where candidates must choose and answer 5, 75 marks total). Total marks for each paper is 100, combined equally into an aggregate score out of 100 for final grading and Z-score computation.
  • Time limit: Paper I: 3 hours plus 10 minutes' reading time; Paper II: 3 hours plus 10 minutes' reading time (total 6 hours 20 minutes)
  • Exam / certification fees: Free for school candidates; nominal fee for private candidates Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

Frequently Asked Questions

What is the format and duration of the GCE A/L Combined Mathematics examination?

Combined Mathematics consists of two separate 3-hour papers: Paper I (Pure Mathematics) and Paper II (Applied Mathematics). Each paper has Part A (10 short-answer questions, 25 marks) and Part B (7 structured essay questions where candidates must answer 5, 75 marks).

Are calculators allowed in the GCE A/L Combined Mathematics examination?

No. Electronic calculators of any type, electronic devices, and formula sheets are strictly forbidden in the examination hall. All arithmetic and algebraic steps must be completed manually.

In which languages is Combined Mathematics offered?

The examination is provided simultaneously in three official languages: Sinhala, Tamil, and English. Candidates register and sit the exam in their designated medium of instruction.

How are grades awarded and how does it affect university admissions?

Candidates receive letter grades: A (Distinction, typically 75+), B (Very Good, 65-74), C (Credit, 55-64), S (Ordinary Pass, 35-54), and F (Fail, below 35). State university admissions for engineering and physical science programs are determined by the candidate's national Z-score calculated across their three core subjects.

Can private candidates sit the Combined Mathematics exam?

Yes. Private candidates who have completed GCE O/L requirements or previous A/L attempts can apply through the Department of Examinations portal upon payment of the gazetted subject fee.