Exam guide

1. Paper structure

• All questions are compulsory, and you answer in the booklet. Calculators are allowed and the MF19 formula list is supplied. Non-exact answers go to 3 significant figures.
• For the full A Level you sit all four papers. The raw total is 75 + 75 + 50 + 50 = 250, and the overall thresholds are set out of 250.
• From June 2024 the threshold tables also list AS Further Mathematics options worth 125 marks: P1 with P3, or P1 with P4.
• Changes over 2022–2026. Times, marks and the number of questions stayed the same. A fourth variant (14, 24, 34, 44) was added in June 2025. From 2025 the papers carry new barcode and margin layouts ("Do not write in this margin"). The question style did not change.

2. Topic weights

Share of paper = the topic's marks as a percentage of that paper's 9-session total. Share of A Level = share of paper × 25%, because each paper counts for a quarter of the qualification. "Sessions" = how many of the 9 sessions tested the topic in variant 2.

P1 Further Pure Mathematics 1 (9 papers, 675 marks)

P2 Further Pure Mathematics 2 (9 papers, 675 marks)

P3 Further Mechanics (9 papers, 450 marks)

P4 Further Probability & Statistics (9 papers, 450 marks)

What this means: 9231 is very even. Every topic except the stand-alone hyperbolic parts appeared in every session's variant 2 paper. You cannot safely skip any syllabus section, but the tables show where the most marks are.

3. Bankers

These subtopics appeared in at least 8 of the 9 sessions (variant 2).

P1 Further Pure 1

• Rational function question (9/9), usually the last or second-last question and worth 14–16 marks. It runs in a fixed order: asymptotes, stationary points (or show there are none), a sketch with axis crossings, a related curve (modulus or reciprocal, 7/9), then an inequality solved with the sketches (8/9).
• Polar curve (9/9), 12–15 marks: sketch, area by integration (9/9), then the point furthest from the pole or the initial line, usually ending with "verify there is a root between ..." (8/9).
• Method of differences (9/9), usually after partial fractions. It is often paired with a standard MF19 sum (8/9) and a sum to infinity (8/9).
• Roots of polynomials (9/9): symmetric functions and power sums such as Σα², Σ1/α², Σα⁻³. Finding a transformed equation came up 6/9.
• Proof by induction (9/9), a single 5–8 mark item. The content changes (see Trends).
• Matrices: invariant lines through the origin (8/9) and describing a pair of transformations in order (8/9).
• Vectors: distances (point to plane, or the shortest distance between skew lines, 8/9) and plane equations in the form ax + by + cz = d (7/9).

P2 Further Pure 2

• Second-order differential equation with constant coefficients (9/9), usually a single 10-mark question with initial conditions.
• Rectangles under a curve giving upper and lower bounds (9/9), often with a limit as n → ∞ or "find the least N".
• Diagonalisation, A = PDP⁻¹ or a power of A (8/9), plus using the characteristic equation (Cayley-Hamilton) to write A⁴ or A⁻¹ in terms of lower powers (8/9).
• First-order differential equation with an integrating factor (7/9), often 7–10 marks with a given point.
• De Moivre: multiple-angle or power formulae (7/9).

P3 Further Mechanics

• All six topics appear in every session. Circular motion is the biggest: horizontal circles, such as conical pendulums, turntables and two-string systems, came up 9/9.
• Elastic strings and springs with energy (8/9).
• A projectile question (9/9), usually involving the trajectory equation, two particles, or the direction of motion.
• An oblique collision of spheres or a collision with a barrier (9/9 combined).

P4 Further Probability & Statistics

• A confidence interval (9/9), usually t-based. Some ask for the interval in reverse, or for a difference of two means.
• Expectation and variance of a continuous random variable (9/9), plus finding a PDF or CDF (8/9) and medians or percentiles (7/9).
• PGFs: mean and variance (9/9) and the PGF of a sum of independent variables (8/9).
• A chi-squared test (9/9): goodness of fit 6/9, contingency table 3/9.
• At least one non-parametric test (9/9) and at least one t-test or two-sample test (9/9).

4. Rarely tested

Seen in 3 or fewer of the 9 variant-2 papers. They are all still on the syllabus, and may have been tested in other variants.

5. Trends

Topic shares, 2022–23 against 2025–26 (% of each paper):

• Induction has changed type. In 2022–23 the induction questions were about nth derivatives and matrix powers. Since 2024 they have been about recurrence sequences (show uₙ equals a formula, or uₙ stays below a bound) and divisibility. Practise all four types, but expect a sequence or divisibility question.
• The P2 "substitution to reduce a DE" question has gone quiet. It appeared in all four 2022–23 papers (y = vx, v = y⁴, u = cosh y and similar) and in none from 2024 to 2026. Integrating-factor questions took its place in every session from J24. Because it was so regular before, it is a strong candidate to come back.
• Reduction formulae appear in the June papers. In variant 2 they came up in J23, J24, J25 and J26, and in no November paper from N22 to N25.
• Complex roots are rising. Questions like "solve z³ = ..." or "z⁴ = ..." in exponential form appeared in J24, J25, N25 and J26. Before that, de Moivre multiple-angle proofs followed by solving a polynomial were the main pattern.
• P3 mixes topics more often. A particle leaves a sphere or loses string tension and then moves as a projectile (J24, N25, J26). Resistance that depends on v is the usual variable-force model in N24, N25 and J26.
• The P4 non-parametric questions now favour the sign test (J24, J25, N25, J26). They often add a 1-mark question on why a Wilcoxon test or a paired t-test would or would not suit the data. Rank-sum tests have not appeared in variant 2 since N23.
• Thresholds: P3 has become the hardest paper. Its A threshold in variant 2 was 34/50 in J22 but has stayed at 23–28/50 since J24. P2 thresholds stay high (A at 61–69/75).

6. Question styles

• Command words. The most common are "Find", "Show that" (given answers, which need every step), "Hence" and "Deduce" (use the previous part), and "Sketch" (asymptotes, intercepts and behaviour at the ends must be labelled). You will also see "State", "Prove by mathematical induction", "Use the method of differences", "Verify that the equation has a root between ...", "Give full details of the transformation" and "Interpret geometrically".
• P1 has 7 long multi-part questions. Each question builds step by step, for example partial fractions, then a sum, then a sum to infinity. The last two questions (rational function and polar) are usually worth 13–17 marks each. You must quote MF19 results correctly.
• P2 has 8 questions. At least one is a single unstructured 9–11 mark question, usually a differential equation with initial conditions. Matrix questions end with "find P and D" and "use the characteristic equation to show ...". Integration questions often give a diagram of rectangles under a curve.
• P3 has 7 questions, usually with a diagram. Answers are asked for "in terms of m, g and a". There are many "show that" steps such as "show that cos θ = ...". Energy and Newton's second law along the radius are the core tools. Collision questions involve components along and perpendicular to the line of centres.
• P4 has 5–7 questions built on data tables or summary statistics (Σx, Σx²). Hypothesis tests are worth 6–9 marks and need hypotheses, a test statistic, a critical value, a decision and a conclusion in context. Short 1-mark parts ask for an assumption ("the population of differences is normal") or a reason for choosing a test.

7. Common mistakes

Taken from the J22, N22 and J23 examiner reports (variant 2 sections).

P1

• Method of differences: too few terms written out to show the cancellation pattern. Show at least three consecutive terms at each end, otherwise a term of the answer often goes missing.
• Matrices: multiplying in the wrong order. The matrix applied first goes on the right.
• Roots: using the same letters for the roots of the old and new equations and mixing them up, or dividing where you should multiply.
• Polar: forgetting to give the coordinates of the point furthest from the pole, or writing them in the wrong order (r, θ).
• Rational functions: in modulus sketches, leaving part of the curve below the x-axis. In inequalities, missing a critical value such as x = 0.
• Vectors: confusing position vectors with direction vectors, arithmetic slips in cross products, giving a position vector when coordinates were asked for, and using the wrong direction for the common perpendicular.

P2

• Not justifying "show that" results fully, especially when the answer is printed in the question.
• Second-order DEs: errors when solving for the particular-integral constants, and writing the answer as an expression instead of "y = ...".
• Integrating factor: slips with constants, and forgetting the arbitrary constant before using the initial condition.
• Substitution DEs: not separating the variables after substituting, or not giving the answer in the form asked.
• Systems of equations: stopping at "no unique solution" without describing the geometry. Name it: a sheaf, or parallel planes that are not identical.

P3

• No diagram, which leads to sign errors and to mixing up sin and cos when resolving.
• Vertical circles: assuming the speed is zero when the string goes slack. The condition is that the tension is zero.
• Elastic strings: forgetting that the string starts stretched at the equilibrium position, so the energy equation needs two elastic-energy terms.
• Collisions: leaving the mass out of the momentum equation, and forgetting the velocity component perpendicular to the line of centres (which stays the same).
• Moments: equations that are not dimensionally consistent, and leaving the boundary value out of an equilibrium inequality.
• Projectiles: using the wrong time for the second particle when one is launched later.

P4

• Wrong critical value: using a z-value (1.645, 1.96) when the sample is small and the variance unknown, or reading the t-table at the wrong level or degrees of freedom.
• Paired tests: working with "before − after" in one place and "after − before" in another. Stating "the data is normal" instead of "the population of differences is normal".
• Two-sample t-tests: not using a pooled variance when the question says the variances are equal.
• Wilcoxon and sign tests: omitting the continuity correction in normal approximations, and writing hypotheses about the mean when the test is about the median.
• Wording: conclusions written as certain ("the test proves ..."). Always include doubt: "there is (in)sufficient evidence that ...".
• Rounding: rounding to 3 significant figures in the middle of a long calculation, which changes the final answer. Keep at least 4 significant figures until the end.

8. Grade thresholds

Variant 2 component thresholds (raw marks, A / E), and the overall A Level option AY (components 12, 22, 32, 42, out of 250):

• Reading the table: an A* has needed 200–224 out of 250 (80–90%), and an A 172–199.
• P2 has the highest component thresholds. Students score well on it, so lost marks there cost a lot relative to other candidates.
• P3 Mechanics has the lowest thresholds (A at about 50–56% since 2024). Partial method marks on P3 are valuable.

9. Revision priorities for Oct/Nov 2026 and 2027

Do first (big, regular marks)

1. P1 rational functions: asymptotes (including oblique), stationary points or showing there are none, sketches of y = f(x), y = |f(x)| and y = 1/f(x), and inequalities solved with those sketches. Expect around 15 marks.
2. P1 polar curves: sketching, area (including with a substitution), and the furthest-point condition followed by "verify a root between ...".
3. P2 differential equations: second-order constant-coefficient DEs with initial conditions, integrating-factor DEs, and substitution DEs. Substitution DEs have not appeared in variant 2 since 2023 and could return.
4. P2 rectangles and bounds, plus eigenvalues, PDP⁻¹ (including for powers or shifted matrices like A − 2I) and Cayley-Hamilton.
5. P3 circular motion: horizontal circles, vertical circles with energy, and the moment the particle leaves the surface or the string goes slack. Practise follow-on projectile parts.
6. P4 t-distribution work: confidence intervals (including reverse problems), paired and two-sample t-tests with pooled variance, and the right critical values.

Then (steady, mid-size)

• P1: method of differences with partial fractions, symmetric functions and power sums of roots, invariant lines and transformation descriptions, and plane equations, distances and angles.
• P2: de Moivre multiple angles and nth roots in exponential form, reduction formulae (they have come up in every June since 2023), and implicit or parametric second derivatives and Maclaurin series.
• P3: oblique collisions, centre of mass with toppling or suspension, elastic energy, and variable forces with v-dependent resistance (use v dv/dx versus dv/dt correctly).
• P4: CDF and PDF work including Y = g(X), PGFs of sums, goodness-of-fit tests (combine classes when expected frequencies are below 5), the sign test, and Wilcoxon signed-rank.

Do not skip (rare in variant 2 but on the syllabus)

• Induction on derivatives or matrices, polar–Cartesian conversion, surface area and arc length, summing series with complex numbers, hyperbolic identities from exponentials, rank-sum tests, contingency tables, direct collisions and barrier collisions.

Exam technique

• Show every step in "show that" questions.
• Label every sketch fully.
• Keep 4+ significant figures in statistics.
• Draw a diagram for every mechanics question.
• Write test conclusions with doubt, in context.