WMA14 P4: full past-paper analysis, Oct 2020 to Jun 2026 with October 2026 predictions

Headline findings

WMA14 is highly predictable by topic: seven of the eleven topics scored marks in all 18 papers, and even the least regular, connected rates of change, scored in 13. The strongest single signal for October 2026 is a volume of revolution, because June 2026 was the first paper without one after 17 in a row.

• Vectors are the largest topic at 10.7 marks a paper (range 8–16), and five routines cover almost every part: lines meeting, angle, foot of a perpendicular, area, reflection.
• Close to 30 of the 75 marks need integration, spread across substitution, parts, partial fractions, volumes and differential equations.
• LIATE picks u correctly in all 19 integration-by-parts integrals: logarithm 4, algebraic 11, and 4 e^x-times-trig loops where either choice works.
• Binomial opens 11 of the 18 papers, and since June 2022 the bracket has started with a number other than 1 in 11 of 13.
• Question styles are shifting: gradient-condition follow-ups in implicit differentiation (7 of the last 13), rational parametric curves (4 of the 6 since June 2025), cross-topic proofs (5 of the last 9), and twice as many no-calculator notices since January 2025 (3.6 a paper, up from 1.85).
• October-only pattern: all three improper partial-fraction questions fell in October papers (2020, 2021, 2024).
• Grade boundary for reference: in October 2024 an A needed 52/75 and the a* 58/75 (Pearson).

Scope and method

This analysis covers every WMA14 paper from October 2020 to June 2026: 18 papers, 166 questions and 1,350 marks, each read with its mark scheme.

• Sessions: October 2020; January, June and October of 2021 to 2025; January and June 2026. There are no gaps in that window.
• One relabelled paper: the question paper labelled “2020 May” is printed June 2020, but its mark scheme is the October 2020 one (WMA14_01_2010_MS). It is treated here as October 2020.
• Tagging: every question part was assigned to the skill its marks test, using the part marks printed on the paper. Every paper sums to exactly 75.
• Mixed parts: where one part tests two skills (a volume finished by integration by parts, say), its marks went to the skill doing most of the work. Read per-topic mark figures as roughly ±2.
• Technique detail (the choice of u in parts, the substitution, the binomial power, the form of a differential equation, the vector task) comes from each mark scheme’s main method, not from alternatives.
• Eleven topic rows are used throughout: vectors, parametric equations, implicit differentiation, differential equations, volumes and areas, binomial expansion, integration by substitution, partial fractions, proof by contradiction, connected rates of change, and integration by parts.

Paper structure

Every paper is 75 marks over 8 to 11 questions (166 in all, 8.1 marks on average). Binomial expansion opens 11 of the 18 papers, and the last question is parametric, proof or a differential equation in 16 of 18.

• Question 1: binomial in 11 papers (twice preceded by partial fractions), implicit differentiation in 3 (Oct 2021, Jan 2022, Jun 2025), and once each proof, a parametric-to-Cartesian conversion, integration by parts and a volume.
• Largest question: parametric work (including parametric areas and volumes) is the largest, or joint largest, in 9 papers; vectors in 5. The single biggest question in the set is October 2025 Q9 at 15 marks.
• Last question: parametric in 6 papers, proof in 6 (Jun 2021’s vectors question ends in a proof part), a differential equation in 4, implicit and a volume once each.
• Question sizes run from 2 marks (Jan 2021 Q3, no greatest odd integer) to 15. Papers with 8 questions (Jun 2023, Oct 2023, Jun 2026) carry more 10+ mark questions.
• No-calculator notices count the “Solutions relying (entirely) on calculator technology are not acceptable” lines. They averaged 1.85 a paper from Oct 2020 to Oct 2024, and 3.6 a paper across the five papers since January 2025.

Marks by topic in every paper

Read across a row for how steady a topic is, and down a column for one paper’s mix. Rates of change is missing from five papers, integration by parts from two (both times it sat inside a differential equation), and volumes and areas only from June 2026.

Binomial expansion

A binomial question appears in all 18 papers (6.8 marks on average, range 4–10), and it is Question 1 in 11 of them. Since June 2022 the bracket has started with a number other than 1 in 11 of 13 papers, so taking out aⁿ is now routine.

• Powers: −1/2 four times, 1/2 three, −2 three, −1 twice (via partial fractions), and 1/3, −1/3, −3, −3/2, 4/3 and 5/2 once each. Positive integer powers never appear here; they belong to P2.
• Unknown constants in 5 papers (Oct 2020, Jun 2021, Jun 2022, Jan 2025, Jan 2026). Jan 2026 is the hardest form: two unknowns, eliminated by dividing (na)² into n(n − 1)a².
• Approximations in 7 papers, always a root: √5, √1.15, √3 twice, ∛31, ∛6, √135. Students must pick the x that turns the bracket into the target and check it is in range.
• Validity is asked in 7 papers, including 4 of the last 5. The rest print the range in the stem.
• Inner term in x² or x³ (Oct 2021, Jan 2022, Oct 2022) changes “first n terms” into “first n non-zero terms” and doubles or triples the powers.
• Near-repeat: Oct 2023 Q1 is (2 − 5x)^−2 and Oct 2025 Q1 is (2 + 5x)^−2.
• Combined uses: partial fractions then expansion (Jan 2023, Jun 2026), a product with a polynomial (Oct 2020), a product of two expansions (Jun 2025), and an expansion feeding a proof (Jun 2024).

Partial fractions

Partial fractions appear in all 18 papers: 19 decompositions in total, examined as their own part in 17 papers (June 2021 hides one inside a substitution). Fourteen are two distinct linear factors; the decomposition is almost never the end goal, because 17 of the 19 feed an integral, a differential equation or an expansion.

• Forms: 14 two-factor distinct, 3 improper (Oct 2020, Oct 2021, Oct 2024), 1 repeated factor (Jan 2024), 1 with three distinct factors (Jun 2025). All three improper cases fell in October sessions.
• What it fed: a definite or indefinite integral ending in logs 11 times, a differential equation 4, a binomial expansion 2, and once each a derivative argument and a Cartesian form.
• In a new variable: four decompositions were in u or t (Jun 2021, Jun 2024, Oct 2025, Jan 2026), so students meet partial fractions mid-way through a substitution or a parametric area, not only as part (a).
• The log step carries the marks: ∫1/(ax + b) dx = (1/a) ln|ax + b| and combining logs into the requested form (p ln q + r, ln k) are where the A marks sit.

Implicit differentiation

Implicit differentiation is in all 18 papers (8.4 marks on average, range 6–10). The product rule on an x-and-y term is needed in 17 of them, and since June 2022 the follow-up has moved from “tangent or normal at a point” towards “find the point where the gradient meets a condition” (7 of the last 13).

• Follow-up tasks: a tangent or normal at a point 8 times (two of them then meet an axis); dy/dx = 0 four times (Oct 2020, Jun 2022, Oct 2022, Jan 2024); a vertical tangent twice (Jan 2023, Jun 2026); a given gradient twice (Oct 2023, Oct 2025); a gradient product once (Jan 2026).
• The condition questions share one route: set the numerator, the denominator or the gradient expression to the condition, get a link between x and y, then substitute that link into the curve. The last step carries 2–3 marks and is the one weaker students skip.
• a^x terms (2^x, 8^x) appear three times since June 2023, so d/dx(a^x) = a^x ln a from P3 is assumed, and answers come out in ln 2.
• Exponentials of x (e^(−2x), e^(2x−1)) appear three times (Jan 2021, Jun 2021, Jun 2025), always multiplied by y or on the right-hand side.
• Unknown constants in the curve (k) appear three times; each is found from the follow-up condition.
• Points on the y-axis (Jan 2021, Jun 2021, Jan 2025) need x = 0 in the curve before any differentiation.

Parametric equations

Every paper has a parametric question, 25 in all (9.1 marks a paper, range 2–13). Trigonometric parametrisations dominated to October 2024 (10 of 18 curves). Since June 2025, four of the six curves have been rational, such as x = 4/(t + 2), and three of those converted to a rational Cartesian form.

• Types: 12 trigonometric, 9 rational or rational-and-polynomial, one each polynomial, roots, logarithmic, and polynomial with trigonometric.
• Cartesian conversions (11 curves): by identity when trigonometric (cos 2t = 1 − 2sin²t in Jun 2021, Jan 2022 and Oct 2024; sec²t = 1 + tan²t in Oct 2021 and Jan 2024; compound angle in Jan 2024), otherwise by making t the subject. Domain or range is asked 8 times, twice with no explicit conversion first (Oct 2022 Q6, Jan 2026 Q5).
• Tangent meets the curve again (Oct 2020, Jun 2022, Jan 2025): substitute x(t), y(t) into the line. The contact point gives a repeated root, so (t − t₀)² is a factor; Jun 2022 turns the same idea into a quadratic in cos t.
• Range traps: the extreme value can sit inside the interval, not at an end (t = 0 in Oct 2021; t = 1 in Jan 2026).
• Parametric areas and volumes are set in 10 of these curves; they are analysed in Volumes and areas.

Connected rates of change

Connected rates appear in 13 of 18 papers (4.1 marks a paper, range 0–8). Spheres and balloons are the most common shape (5 times), and June 2026 Q4 re-ran June 2021 Q3 almost exactly, with a square root added to the volume formula.

• Missing in October 2020, January 2021, October 2022, June 2023 and October 2025. No paper has gone two sessions in a row without one since 2021.
• Shapes: spheres and balloons 5, cones 2, cylinders and circles 2, containers with a given V(h) 2, a cube, an icosahedron and a circle segment once each.
• Formula given vs built: the question supplies the volume or area formula in 7 of the 13. Students build it themselves for the cube, the cylinder pieces, the segment area, the cone (similar triangles) and, in January 2026, the sphere, whose formula was not printed.
• Three rates questions start a differential equation (Oct 2021, Jan 2023, Jan 2026): the chain rule produces dh/dt or dr/dt, which the next part separates and solves.
• Calculus beyond the chain rule: the June 2026 volume needs the product rule on h²√(h + 4); the January 2024 surface area needs the chain rule on √(25 + r²).

Integration by substitution

A substitution integral appears in all 18 papers (6.2 marks a paper, range 3–8), 18 in total. The substitution is printed in 16 of them; only October 2020 and October 2023 left the choice to the student. The real test is usually what comes after the change of variable: partial fractions, parts, a trig identity or a reverse chain rule.

• Substitution types: roots 6, trig functions inside (u = 3 + 4 sin x and similar) 4, trig substitutions (x = a sin θ, tan u = √x) 4, exponential 2, linear or polynomial 2.
• What follows the change of variable: powers of u 8 times, partial fractions 2, division of a fraction 2, parts 1, trig integrals 3, e^u once, a reverse chain rule once.
• Definite 15, indefinite 3 (Jun 2023, Oct 2024, Jun 2025). In definite ones the limits must be changed: the mark schemes award an M mark for using the u-limits, and returning to x is never needed.
• Context: 14 standalone, 2 areas (Oct 2021; Jan 2022, parametric), 2 volumes (Jan 2024, Jun 2024).
• Trig substitutions are rising: x = 2 sin u (Jun 2022), x = 4 sin θ (Jan 2025), 2x = sin u (Jun 2026). Each relies on 1 − sin²θ = cos²θ to clear a root.
• The step the schemes police hardest: converting dx. The June 2021 scheme states that dx cannot simply be replaced by du; doing so loses the method mark for changing the integral.

Integration by parts

Integration by parts appears in every paper: as its own part in 16, and inside a differential equation in the other two (June 2021, October 2025). LIATE gives a correct choice of u in all 19 integrals: u = a logarithm 4 times, u = a power of x 11 times, and the 4 e^x-times-trig loops work with either choice.

• LIATE counts: L (logarithm) 4, A (algebraic) 11, loops 4. I (inverse trig) never appears as u. Every choice that LIATE makes is one the mark schemes accept.
• The loop is the one place to choose freely. LIATE says u = trig (T before E); Jan 2021’s scheme does that, Oct 2022’s and Jun 2026’s main methods take u = e^(ax), and Jun 2025’s shows both. What matters is keeping the same choice for the second application; swapping it undoes the first step.
• Applications: once 8, twice 7, loop 4. Every “twice” case is x² times an exponential or a cosine (or the quadratic in Jan 2022).
• Exponents and arguments seen: e^x, e^(2x), e^(4x), e^(−x), e^(−2x), e^(−t); cos 2x, cos 3x, cos 4x, sin x, cos x; ln x, (ln x)², ln(x/2). The linear coefficient is the main source of slips (v = (1/4)e^(4x), not e^(4x)).
• Context: 7 standalone, 5 inside volumes (4 papers), 3 inside areas, 3 inside differential equations, 1 after a substitution. Integration by parts is the main tool for finishing volume questions.
• ∫ln x dx on its own (u = ln x, dv/dx = 1) has not been set in this period.
• Marking points: M1 for parts “the right way round” to the correct form, dM1 for the second application, A1 for the result, with +c usually required on indefinite answers. The June 2022 scheme explicitly credits the tabular (DI) method, row by row.

Volumes and areas

A volume of revolution appeared in 17 of the 18 papers; June 2026 is the only paper with no volume or area at all. Most volumes are really integration-technique questions in disguise: setting up π∫y² dx is worth 1–3 marks, and finishing the integral (parts, substitution, partial fractions, a trig identity) carries the rest.

• Volumes: 13 Cartesian and 4 parametric (Jan 2021, Jun 2023, Oct 2023, Jan 2026). Areas: 6 parametric and 3 Cartesian.
• Finishing technique for the 17 volumes: parts in 4 (Oct 2021, Jan 2022, Jun 2022, Jun 2025), a trig identity in 3, the reverse chain rule in 2, the power rule in 2, a substitution in 2, and once each a log by division, f′/f, plain expansion and a parametric power integral. October 2025’s is a plain algebraic integral.
• Composite regions (curve plus a cylinder, cone, triangle or rectangle) in Jun 2021, Jan 2023, Jun 2023 and Jun 2024.
• Real objects (Jan 2022 doorknob, Jun 2022 density, Jun 2025 paperweight) add a final step that uses the volume, worth 1–4 marks.
• Working backwards from a given volume to a limit happened once (Oct 2022).
• Parametric volumes always reduce to a product of sin and cos powers, cleared by a double-angle identity or by writing an odd power as one factor times (1 − sin²) or (1 − cos²).

Differential equations

Every paper has a separable differential equation: 19 in all (October 2021 has two), worth 8.4 marks a paper on average (range 5–12). 13 of the 19 are set in context, and since January 2024 the integration on one side has often needed a P4 technique: partial fractions, parts or a trig result.

• Integration on the variable side: a log of a linear expression 4 times, powers 8, partial fractions 3, cot y once, a signposted composite once, and 2 direct integrations (Jan 2023, Oct 2025). On the independent side: parts 3 (Jun 2021, Jan 2024, Oct 2025), partial fractions once (Jun 2022), trig 3 (Jun 2025, Jan 2024, Oct 2024).
• Context vs pure: 13 in context, 6 pure (Jun 2021, Oct 2021 Q2, Jan 2022, Jun 2022, Jan 2024, Jun 2025).
• Built from words three times: “inversely proportional to r²” (Oct 2022), rates in and out of a tank (Oct 2021), a constant rate of volume loss (Jan 2026).
• Follow-ups: a long-term or limiting value 8 times; a time 8 times; one sketch (Oct 2022). Two papers (Oct 2023, Jun 2026) open with a non-DE model to compare against.
• Where marks are lost (from the scheme notes): one constant of integration, found from the starting values before rearranging; logs combined before exponentiating; and the final “in context” answer with units and the stated accuracy.

Vectors

Vectors are the largest topic: every paper, 10.7 marks on average (range 8–16), across 24 questions. Almost every part is one of five routines: where two lines meet, an angle by the scalar product, the foot of a perpendicular, an area, or a reflection. Six papers split vectors over two questions.

• Routine counts: an angle by the scalar product 10 times; the foot of a perpendicular (nearest point, shortest distance) 9; an area of a triangle or parallelogram 8; lines meeting, including finding a constant so they do, 6; proving lines skew or non-intersecting 4; a reflection 2.
• Unknown constants in a line or point (a, b, p, q, α, β) appear in 9 questions. Each is found from one condition: the lines meet, two vectors are perpendicular, or an angle is given.
• Two-answer parts (both positions of a point, or two values of a constant) appear 5 times, all since October 2023 (Oct 2023, Jun 2024 twice, Jan 2025, Jun 2025). They come from a quadratic in λ or from a ratio either side of a point.
• Accuracy demands are shifting to exact forms: exact cos θ in Oct 2023, Oct 2024 and Oct 2025, and exact surd lengths or areas in Jun 2022, Jun 2023, Oct 2024 and Oct 2025. Degrees to 1 d.p. remain common.
• Notation is marked: the October 2020 scheme rejects coordinates where a position vector is asked for, and penalises the wrong notation once.
• Proof meets vectors in June 2026 Q2, where “the lines do not intersect” had to be written as a proof by contradiction.

Proof by contradiction

Proof by contradiction is in every paper (4.6 marks on average, range 2–8). Five families account for all 18; until mid-2023 they were almost all number theory, but since October 2023 five of nine have been tied to another topic: calculus, curves, trigonometry or vectors.

• Families: parity and divisibility 6, never meets or no real solution 5, irrational roots 3, no integer solutions 2, inequalities 2. Families repeat in runs of up to three (parity from Oct 2020 to Jun 2021; never-meets from Jan 2024 to Oct 2024), so there is no reliable rotation.
• The assumption line is a mark on its own (B1) in every scheme checked. It must negate the statement precisely: “there exists n such that n³ is even and n is odd”, not “for all n”.
• Scaffolded “complete the proof” questions appear 4 times (Oct 2021, Oct 2022, Jan 2023, Oct 2025). Three of them pair a lemma with an application in the second part.
• Cross-topic proofs since October 2023: a derivative (Jan 2024), a binomial expansion (Jun 2024), two curves (Oct 2024), trigonometry (Jun 2025) and vectors (Jun 2026).
• Two-case arguments (3k + 1 and 3k + 2) are required in Jun 2021 and Jan 2023; missing a case caps the marks.

Trends from 2020 to 2026

The topic mix has been stable for six years; what has changed is how each topic is asked. Recent papers demand more unaided algebra, more exact forms and more “find the condition” reasoning.

• What has not changed: every topic scores in nearly every paper, binomial usually opens, vectors stay the largest topic, and question count stays between 8 and 11.
• Implication for teaching: method fluency without a calculator, exact-form answers, and the “condition, then substitute back” pattern now carry more marks than they did in 2021.

October 2026 predictions

The single strongest prediction is a volume of revolution: it appeared in 17 consecutive papers before June 2026 had none. These are patterns, not leaks; Pearson does not rotate P4 topics on a fixed cycle, and every paper covers almost the whole specification.

• A likely paper shape: 9 or 10 questions, opening with binomial (6–9 marks), with implicit (8–10), partial fractions or substitution (6–8 each), parametric (9–12), vectors (8–12), a differential equation (6–10), a volume (6–9), rates (4–7) and proof (4–6).
• Less likely, but not to be skipped: a Cartesian area under a curve (3 times, last in October 2021), reflection of a point in a line (twice, last in June 2023), log differentiation of x^(f(x)) (once, October 2020).
• Exam technique that the recent papers reward: full algebraic working on no-calculator questions (3–4 a paper now), exact forms, and the final substitution step in condition questions.

Appendix: every question

All 166 questions with their marks, topic tags and a one-line summary, newest session first.

January 2024 to June 2026

October 2020 to October 2023