What to Study in Year 2 of A-Level Mathematics: Nine Pure Mathematics Topics in Cambridge Paper 3, and a Point-by-Point Look at Edexcel P3 and P4

The only "choices" in A-Level Mathematics lie in the applied modules: Cambridge 9709 requires choosing between Mechanics (Paper 4) and Probability & Statistics 2 (Paper 6), while Edexcel IAL requires selecting two units from M1, S1, D1, M2, and S2 in specified combinations. We have already covered these combinations and the content of the applied papers in How A-Level Math is Tested, How to Study A-Level Statistics, and How to Study A-Level Mechanics. This article focuses on the mandatory core for the second year: Pure Mathematics. Cambridge Paper 3 (Pure Mathematics 3) and Edexcel P3 and P4 are mandatory for A-Level and carry the most weight. This article is organized point by point based on the Cambridge 9709 syllabus (for the 2026 and 2027 exams) and the Pearson IAL Mathematics syllabus (3rd edition), comparing the differences between the two.

1. Which Papers Are Taken in the Second Year

Cambridge 9709: If taken in stages, students take Paper 1 and one applied paper (Paper 4 or Paper 5) in the first year, and Paper 3 and the other applied paper in the second year. The syllabus outlines three arrangements:

• Year 1: Paper 1 + Paper 4; Year 2: Paper 3 + Paper 5;
• Year 1: Paper 1 + Paper 5; Year 2: Paper 3 + Paper 4;
• Year 1: Paper 1 + Paper 5; Year 2: Paper 3 + Paper 6.

Paper 3 Pure Mathematics 3: 1 hour 50 minutes, 75 marks, 9–11 structured questions. It is only used for A-Level and accounts for 30% of the A-Level grade. The syllabus notes that the content of Paper 1 is assumed knowledge and may be tested in Paper 3.

Edexcel IAL: P3 and P4 are mandatory units for A-Level, each lasting 1 hour 30 minutes with 75 marks. All questions are compulsory. Calculators are allowed (without symbolic algebra capabilities), and a formulae and statistical tables booklet is provided. P3 assumes knowledge of P1 and P2, while P4 assumes knowledge of P1–P3. As verified in Article 35, achieving an A* in IAL Mathematics requires a combined UMS score of at least 180 out of 200 in P3 and P4.

2. Cambridge Paper 3: Nine Topics

1. Algebra: Absolute values (graphing y = |ax + b|, solving absolute value equations and inequalities; the syllabus notes that y = |f(x)| and y = f(|x|) for non-linear functions are not tested); polynomial division (up to degree 4); factor and remainder theorems; partial fractions (denominators up to (ax+b)(cx+d)(ex+f), (ax+b)(cx+d)², and (ax+b)(cx²+d); numerators with a degree higher than the denominator are not tested); binomial expansion of (1 + x)ⁿ for rational n and its range of validity (finding the general term is not required);
2. Logarithmic and Exponential Functions: Laws of logarithms (change of base is not tested); properties, graphs, and inverse relationship of eˣ and ln x; solving equations and inequalities with unknowns in the exponent; using logarithms to linearize relationships and finding constants from the gradient and intercept;
3. Trigonometry: sec, cosec, cot, and the graphs and properties of all six trigonometric functions; simplifying identities and solving equations: sec²θ ≡ 1 + tan²θ, angle addition formulas, double angle formulas, expressing a sin θ + b cos θ in the form R sin(θ ± α) or R cos(θ ± α);
4. Differentiation: Derivatives of eˣ, ln x, sin x, cos x, tan x, and tan⁻¹x (sin⁻¹x and cos⁻¹x are not required); differentiation of products and quotients; first derivatives of parametric equations and implicit functions, including finding tangents and normals;
5. Integration: Integrals of e^(ax+b), 1/(ax+b), sin, cos, sec²(ax+b), and 1/(x²+a²); integration using trigonometric identities (e.g., using double angle formulas to integrate sin²x); integration using partial fractions; recognizing the kf′(x)/f(x) form; integration by parts; integration using given substitutions;
6. Numerical Solutions of Equations: Locating roots using graphs or sign changes; the relationship between the iterative formula xₙ₊₁ = F(xₙ) and the equation, using iteration to find a root to a specified accuracy (convergence conditions are not required, but students should know that iteration may not converge);
7. Vectors: Notation and operations for 2D and 3D vectors; magnitude, unit vectors, position vectors; equation of a line r = a + tb; determining whether two lines are parallel, intersecting, or skew, and finding points of intersection; scalar product, finding the angle between two lines and the foot of the perpendicular from a point to a line. The syllabus notes that the shortest distance between skew lines and the vector product are not required;
8. Differential Equations: Formulating differential equations from descriptions of rates of change (including introducing a constant of proportionality); finding the general solution by separating variables and the particular solution using initial conditions; interpreting results in practical contexts;
9. Complex Numbers: Real and imaginary parts, modulus, argument, conjugate; arithmetic operations in Cartesian form; non-real roots of polynomial equations with real coefficients occurring in conjugate pairs; Argand diagrams; multiplication and division in polar form; finding the two square roots of a complex number; geometric interpretations of complex number operations; representing loci on an Argand diagram, such as |z − a| < k, |z − a| = |z − b|, arg(z − a) = α.

3. Edexcel P3 and P4

P3 (Six Topics):

• Algebra and Functions: Simplifying rational expressions and algebraic division; domain and range of functions, composite and inverse functions; absolute value functions, including sketching y = |f(x)| and y = f(|x|) from the graph of y = f(x); combined transformations of y = af(x), f(x) + a, f(x + a), and f(ax);
• Trigonometry: sec, cosec, cot, and arcsin, arccos, arctan; Pythagorean identities; double angle (including half-angle applications) and angle addition formulas, expressing a cos θ + b sin θ in the form r cos(θ ± α) or r sin(θ ± α);
• Exponentials and Logarithms: Graphs and inverse relationship of eˣ and ln x; using logarithmic graphs to estimate parameters in y = axⁿ and y = kbˣ;
• Differentiation: Derivatives of e^(kx), ln kx, sin kx, cos kx, and tan kx; product, quotient, and chain rules (also requiring derivatives of cosec x, cot x, and sec x); dy/dx = 1/(dx/dy); exponential growth and decay models;
• Integration: Integration of basic functions; recognizing the f′(x)/f(x) and f′(x)[f(x)]ⁿ forms; integration using trigonometric identities;
• Numerical Methods: Locating roots using sign changes; finding approximate solutions using iteration.

P4 (Seven Topics):

• Proof: Proof by contradiction, including proving that √2 is irrational and that there are infinitely many primes, and applying it to unfamiliar proofs;
• Partial Fractions: Denominators up to repeated linear factors; the degree of the numerator can be equal to or higher than the denominator; quadratic factors like (x² + a) are not required;
• Parametric Equations: Converting between parametric and Cartesian equations;
• Binomial Expansion: Expansion for any rational n, including expanding rational functions using partial fractions;
• Differentiation: Differentiating implicit functions and parametric equations, including tangents and normals; setting up simple differential equations, which may include related rates of change;
• Integration: Volumes of revolution (required about the x-axis using π∫y²dx, not π∫x²dy, including curves given by parametric equations); integration by substitution and by parts (may require integration by parts twice in succession); integration using partial fractions; solving first-order differential equations by separating variables; finding the area under a curve given by parametric equations;
• Vectors: 2D and 3D vectors; magnitude and unit vectors; position vectors and distance between two points; vector equations of lines, conditions for parallel, intersecting, and skew lines; scalar product and the angle between two lines.

4. Differences Between the Two Exam Boards

The practical takeaway is: the choice of textbooks and practice problems must align with your specific syllabus. For example, for Cambridge students, complex numbers are a key chapter in the second year; for Edexcel Mathematics students, dedicated practice in proof by contradiction and areas and volumes using parametric equations is essential.

5. How to Study Second-Year Pure Mathematics

Here are some practical recommendations based on experience:

1. Solidify your Year 1 foundation first: Both exam boards specify that Year 1 Pure Mathematics is assumed knowledge and may be tested directly. If you are not proficient in functions, trigonometry, and basic calculus, Year 2 will be very challenging;
2. Calculus is the main thread: In Year 2, differentiation rules, integration methods (substitution, by parts, partial fractions, trigonometric identities), and differential equations are interconnected. Practice them in the order of 'master differentiation → master integration → master solving differential equations';
3. Master trigonometric identities: They are used in simplification, solving equations, and integration. Practice the identities listed in the syllabus until you can confidently choose the right formula;
4. Draw diagrams for vectors: For the positional relationships, angles, and perpendicular feet of 3D lines, draw a sketch before setting up equations;
5. Cambridge students should practice complex numbers separately: Complex numbers are new content. Practice algebraic operations, polar form, and Argand diagram loci. The syllabus requires showing full working for multiplication, division, and finding roots;
6. Edexcel students should practice proofs: For proof by contradiction, you must be able to write out the complete logical structure (see article 195 for how to write proof questions);
7. Past papers by topic, then by full paper: After finishing a chapter, do past paper questions for that specific chapter. Finally, complete full past papers under timed conditions, strictly following the mark scheme.

Final Thoughts

In the second year of A-Level Mathematics, the applied papers are optional, but Pure Mathematics is the mandatory core: Cambridge Paper 3 accounts for 30% of the A Level, while Edexcel P3 and P4 determine the A*. The Pure Mathematics content across both exam boards is broadly similar, but there are notable differences in areas such as complex numbers, proof by contradiction, volumes of revolution, and partial fractions. Go through your specific syllabus point by point, and you will have a clear checklist for your Year 2 revision. If you need A-Level Mathematics tutoring, feel free to chat with our teachers.

Verification note: This article was verified in September 2026: the three staged route arrangements for Cambridge 9709, the duration, marks, number of questions, and weighting of Paper 3, the note that Paper 1 is assumed knowledge, the content and annotations of the nine topics in Paper 3 (including untested content), and the fact that volumes of revolution belong to Paper 1, are sourced from Cambridge International's "Mathematics 9709 syllabus for 2026 and 2027" (4th edition); the prerequisite knowledge, duration, marks, calculator and formula booklet regulations, topic content and annotations for Edexcel P3 and P4, and the fact that complex numbers belong to Further Mathematics FP1, are sourced from Pearson's IAL Mathematics, Further Mathematics, and Pure Mathematics specifications (3rd edition); the IAL Mathematics A* rules have been verified in article 35 on this site. The content is our abridged translation. The study methods are our experience-based recommendations.

References:

• Cambridge International: AS & A Level Mathematics (9709)
• Pearson: International Advanced Level Mathematics
• Cambridge International: Mathematics 9709 Syllabus 2026–2027 (PDF)
• Pearson: IAL Mathematics Specification (PDF)