Cambridge O Level

Additional Mathematics 4037

Advanced algebra, functions, trigonometry, vectors and calculus for strong O Level mathematicians.

Exam year

Syllabus version: 2025–2027 syllabusChecked against the Cambridge syllabusExam year 2027 is covered by the 2025–2027 syllabus.

Subject review complete (21 Oct 2026). All 40 Additional Mathematics questions were re-solved and checked against the syllabus and Cambridge mark-scheme conventions, and their auto-marking was tested with correct and incorrect answers: 11 approved as written, 29 corrected.

Paper 1 Non-calculator50%
Paper 2 Calculator50%

Original questions40

Topics covered14/14

Papers2

Mock papers2

What is assessed

2 components. Question formats: short answer, structured.

Paper 1 Non-calculator

2 h80 marksWeighting 50%

Structured and unstructured questions on any part of the subject content, all compulsory, answered without a calculator with all working shown.

What the paper looks like

About 11–12 compulsory questions, 80 marks in 2 hours, no calculator. Short opening questions (sketching a cubic, polynomial division, logarithm laws, binomial terms) of 3–6 marks lead to longer multi-part questions and one or two unstructured problems of 7–9 marks such as a coordinate geometry or area problem. Answers must be exact (fractions, surds, π, e, ln) with every step shown; “Show that”, “Hence” and “Write down” are common.

Short answerStructuredCalculators not allowed; List of formulas printed in the paper (not every formula needed is given)

Checked against the Cambridge syllabus

Paper 2 Calculator

2 h80 marksWeighting 50%

Structured and unstructured questions on any part of the subject content, all compulsory; non-exact answers to 3 significant figures and angles in degrees to 1 decimal place.

What the paper looks like

About 11–12 compulsory questions, 80 marks in 2 hours, scientific calculator required. Mixes modulus equations, discriminant problems, trigonometric equations in degrees and radians, permutations and combinations, progressions, vectors in context, rates of change and practical optimisation, often 6–9 marks each. Non-exact answers to 3 s.f. (angles 1 d.p.); the calculator is a checking tool, so method must still be shown. A formula list is printed in the paper.

Short answerStructuredScientific calculator required; List of formulas printed in the paper (not every formula needed is given)

Checked against the Cambridge syllabus

Assessment overview

From the 2025–2027 syllabus. Checked against the Cambridge syllabus

All candidates

Paper 1Paper 2

Both papers are compulsory; grades A* to E are available.

Assessment objective	O Level weighting
AO1 Knowledge and understanding of mathematical techniques	45–55%
AO2 Analyse, interpret and communicate mathematically	45–55%

Changes in this syllabus version

Paper 1 became a non-calculator paper (2 hours, 80 marks, 50%); a scientific calculator is still required for Paper 2 (2 hours, 80 marks, 50%).
Indices and surds removed as a separate topic (now assumed knowledge) and a new topic, Coordinate geometry of the circle, added.
Learning outcomes reworded with a Notes/Examples column to show depth; other content added, removed or clarified within topics.
List of formulas, command words and assessment objective wording updated; a section on mathematical conventions added.
Mark schemes award more marks for working where appropriate; a February 2024 update made minor amendments to the marking principles.

Command words used in Additional Mathematics papers (13)

Calculate: work something out from the facts, figures or information given
Describe: give the main points or characteristics of something
Determine: establish an answer with certainty, with supporting working
Explain: give reasons or make relationships clear, saying why or how with evidence
Give: produce an answer from a given source or from memory
Plot: mark one or more points accurately on a graph
Show (that): set out structured working that leads to a result you have been given
Sketch: draw a simple freehand graph or diagram showing the key features (from 2028, taking care over proportions)
State: express clearly and briefly
Verify: confirm that a given statement or result is true
Work out: calculate from given information, with or without a calculator
Write: give an answer in a particular form
Write down: give an answer with little or no working

Examiner guidance from recent reports (14)

Functions. Composite functions work right to left: fg(x) is f(g(x)). To explain why there is no inverse, give a concrete reason (e.g. two x-values with the same output); “not one–one” alone is not enough. Give the inverse in terms of x with its domain.

Quadratic functions. Use the discriminant directly rather than the quadratic formula when asked about the number of roots, and state the conclusion (e.g. “b² − 4ac < 0 so no real roots”). Write ranges with f(x), not x.

Factors of polynomials. “Product of linear factors” means factorise completely with no constant left over. If you use synthetic division with a factor such as 2x − 1, remember to divide the quadratic coefficients by 2.

Equations, inequalities and graphs. When sketching cubics, mark every intercept as a number on the axis and get the end behaviour right from the sign of the x³ term. Use the sketch to solve the inequality instead of long algebra; modulus graphs need sharp cusps.

Simultaneous equations. Choose the substitution that keeps the algebra simplest, and pair each x with its own y-value from the linear equation. Check the final pairs in both equations.

Logarithmic and exponential functions. Change of base and the power law are where marks are lost: log_b a = 1/log_a b. Do not reject negative roots of a quadratic in log x automatically, and give answers in the exact form requested.

Straight-line graphs. In straight-line form questions label the axes as the transformed variables (e.g. ln y) and write the line as ln y = …, not y = …. Use points that lie on the line, far apart, for the gradient.

Coordinate geometry of the circle. Complete the square carefully to find the centre and radius. A tangent is perpendicular to the radius; to show a point is inside, compare its distance from the centre with r.

Circular measure. Arc length rθ and sector area ½r²θ need θ in radians and are not on the formula list. Do not confuse an arc with a chord, and keep 4 or more significant figures until the end.

Trigonometry. Find all solutions in the interval: transform the interval for compound angles and consider ± square roots. In proofs change to sin and cos, and always write the angle (sin x, not sin).

Permutations and combinations. Decide first whether order matters. For restricted arrangements fill restricted positions first, and use “total minus unwanted” for “not together” or “at least one” problems.

Series. Read whether the question wants an exact answer. For a sum to infinity the condition is −1 < r < 1, written as one continuous inequality; in the binomial theorem n is a positive integer.

Vectors in two dimensions. Distance is the magnitude of a displacement vector, not the vector itself. Velocity = speed × unit vector; show that particles collide by finding one t that satisfies both components.

Calculus. “Hence” means use the previous result, e.g. reverse a derivative to integrate. Include + c in indefinite integrals, use radians for trigonometric calculus, and find when v = 0 before working out distance travelled.

Topic map

Practice types

Free sample questions

FAQs

Are these official 4037 past-paper questions?

No. They are original questions written to match the Additional Mathematics syllabus topics. Official past papers are available to schools through Cambridge.

Why choose an exam year?

Cambridge updates syllabuses periodically. Your exam year selects the matching configuration so topics and components follow the right version.

Is a calculator allowed?

It depends on the paper. Each component card shows the calculator rule from this configuration.