IAL Pure Mathematics 3 (WMA13): analysis of 19 past papers with October 2026 predictions
Edexcel IAL Pure Mathematics 3 (WMA13). 19 papers, updated 3 October 2026.
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What this is: a question-by-question analysis of all 19 Edexcel IAL P3 papers, January 2020 to June 2026, ending with a prediction for October 2026.
The 30-second version
- Biggest topic: differentiation, about 12 of the 75 marks a paper on average.
- In every single paper: iteration, a modulus graph and some integration (19 of 19).
- In almost every paper: an inverse function and a trig “show that, then solve” (18 of 19).
- The A boundary: 57/75 in June 2026 (A* was 63).
- October 2026 prediction: the same core question types, a bigger modulus question than June’s, and the R form slightly better than even.
Jump to
- The paper at a glance: structure and boundaries, 1 minute
- Topic weighting: where the marks go, 1 minute
- Marks per topic in every paper: the full grid
- Question-by-question breakdown: reference only, skim the paper you need
- Technique-level analysis: which variant of each method appears
- How the mark schemes award marks: the rules that cost marks
- Trends and the October papers: what is changing
- Predictions for October 2026: the final section
Scope and method
All 19 main P3 papers (WMA13/01), January 2020 to June 2026, with their mark schemes. Every question part was tagged by topic and by the exact technique it tests, then counted in marks.
- Papers: the January, June and October sessions from January 2020 to June 2026. The October 2020 paper carries a June 2020 date on its cover; its mark scheme is the October 2020 one.
- Unit of analysis: the question part. Parts that mix topics were split by marking point using the mark scheme, so topic totals are approximate, roughly ±2 marks per paper.
- Technique level: each part is tagged at the depth of “which case of the method”, for example a product rule pairing a polynomial power with an exponential, or sec2θ = 1 + tan2θ turned into a quadratic in sec θ.
- Not included: the 01A alternative papers some centres sit, and the specimen paper.
Sources
- Pearson Edexcel IAL Mathematics specification (Issue 3), for P3 content and the formulae students must learn.
- Pearson IAL grade boundaries, June 2026, for the WMA13 boundaries.
- Pearson Edexcel WMA13/01 question papers and mark schemes, January 2020 to June 2026.
The paper at a glance
75 marks in 1 hour 30 minutes, 9 or 10 questions, median question 8 marks. In June 2026 the boundaries were A* 63, A 57, B 51 and C 45 out of 75 (Pearson, June 2026 IAL grade boundaries).
| Question size | Questions (of 180) | Share |
|---|---|---|
| 3–5 marks | 21 | 12% |
| 6–7 marks | 56 | 31% |
| 8–9 marks | 69 | 38% |
| 10–11 marks | 24 | 13% |
| 12–14 marks | 10 | 6% |
- Ten papers have 9 questions and nine have 10.
- Question 1 is short: 3–9 marks, median 6. Iteration opened 5 of the last 8 papers.
- The last question carries 7–14 marks, median 9. It is most often x = f(y) with dy/dx = 1 ÷ dx/dy (6 papers) or a trig identity, equation or R-form question (6 papers).
- The largest single questions, at 13–14 marks, were October 2020 Q9, June 2021 Q8, January 2024 Q4, October 2024 Q9 and October 2025 Q8.
Topic weighting
Differentiation is the biggest topic at 12.3 marks a paper. With functions, trig and modulus it makes up 53% of the average paper. Only the R form (11 of 19) and x = f(y) (14 of 19) are ever left out.
| Topic | Average marks per paper | Range | Papers it appears in |
|---|---|---|---|
| Differentiation | 12.3 | 4–25 | 19 of 19 |
| Functions and transformations | 9.3 | 4–14 | 19 of 19 |
| Trig identities and equations | 9.3 | 3–15 | 19 of 19 |
| Modulus | 8.6 | 4–13 | 19 of 19 |
| Integration and division | 8.3 | 4–13 | 19 of 19 |
| Exponential models | 7.2 | 0–13 | 18 of 19 |
| Log graphs | 5.7 | 0–8 | 18 of 19 |
| x = f(y), dy/dx | 5.3 | 0–11 | 14 of 19 |
| Iteration and sign change | 5.0 | 3–7 | 19 of 19 |
| R form | 4.1 | 0–9 | 11 of 19 |
Differentiation also sits inside other topics: rates of change in models, stationary points that feed an iteration, and calculus proofs that a function is increasing.
Marks per topic in every paper
The same ten topics every time; only their sizes change. Differentiation, for example, ran from 4 to 25 marks. Every row totals 75; newest first.
| Session | Differentiation | Functions | Trig | Modulus | Integration | Exp. models | Log graphs | x = f(y) | Iteration | R form |
|---|---|---|---|---|---|---|---|---|---|---|
| Jun 2026 | 9 | 11 | 9 | 5 | 11 | 12 | 7 | 6 | 5 | 0 |
| Jan 2026 | 6 | 8 | 7 | 10 | 10 | 9 | 6 | 4 | 6 | 9 |
| Oct 2025 | 9 | 9 | 9 | 13 | 5 | 12 | 5 | 0 | 7 | 6 |
| Jun 2025 | 6 | 11 | 8 | 10 | 4 | 8 | 5 | 11 | 5 | 7 |
| Jan 2025 | 12 | 10 | 9 | 8 | 9 | 9 | 3 | 9 | 6 | 0 |
| Oct 2024 | 17 | 12 | 13 | 7 | 8 | 2 | 6 | 5 | 5 | 0 |
| Jun 2024 | 12 | 9 | 11 | 4 | 11 | 3 | 6 | 10 | 3 | 6 |
| Jan 2024 | 18 | 14 | 3 | 11 | 4 | 7 | 7 | 0 | 6 | 5 |
| Oct 2023 | 15 | 10 | 10 | 9 | 4 | 7 | 6 | 9 | 5 | 0 |
| Jun 2023 | 11 | 8 | 15 | 9 | 5 | 8 | 6 | 8 | 5 | 0 |
| Jan 2023 | 14 | 6 | 9 | 8 | 12 | 9 | 5 | 0 | 6 | 6 |
| Oct 2022 | 13 | 7 | 9 | 12 | 9 | 4 | 0 | 5 | 7 | 9 |
| Jun 2022 | 25 | 9 | 8 | 8 | 10 | 0 | 7 | 0 | 3 | 5 |
| Jan 2022 | 8 | 11 | 13 | 10 | 6 | 7 | 8 | 7 | 5 | 0 |
| Oct 2021 | 17 | 6 | 9 | 10 | 11 | 6 | 6 | 7 | 3 | 0 |
| Jun 2021 | 4 | 10 | 9 | 8 | 8 | 13 | 7 | 5 | 3 | 8 |
| Jan 2021 | 4 | 14 | 9 | 9 | 7 | 11 | 7 | 10 | 4 | 0 |
| Oct 2020 | 17 | 4 | 9 | 4 | 13 | 3 | 6 | 4 | 6 | 9 |
| Jan 2020 | 17 | 8 | 8 | 9 | 10 | 6 | 5 | 0 | 5 | 7 |
June 2022’s 25 differentiation marks include a velocity model maximised by calculus and a trig quotient, which shows how often calculus sits inside other topics.
Question-by-question breakdown
Reference section: jump to the paper you need rather than reading it straight through. Every question of all 19 papers, newest first, with the technique each part tests and its marks in brackets. Ambiguous notation was checked against the mark scheme.
June 2026
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 8 | Differentiation → iteration | f(x) = ⅔x3 − 3x2 + 5x + e2x(1 − 2x): f′(x) with the product rule on the exponential term (3); sign change for f′(x) = 0 on [0, 1] (2); iterate x = (2x2 + 5)/(4e2x + 6) from x1 = 0 for x3 and α to 4 d.p. (3) |
| 2 | 9 | Functions | f(x) = 3 + 4e2x, g(x) = √(x + 5): range of an exponential (1); g−1 with domain (3); gf(ln 7) exactly (2); solve fg(a2 − 5) = g(59) for a = ln p, p irrational (3) |
| 3 | 8 | Differentiation, transformation | f(x) = (4x2 − 14x + 15)/(x − 3), x < 3: quotient rule to (ax2 + bx + c)/(x − 3)2 (4); turning point (3/2, −2), rejecting x = 9/2 by the domain (2); image of P(5/2, −10) under y = f(2x) + 3 (2) |
| 4 | 5 | Integration (trig identity) | Exact ∫ of (sin 3x + cos 3x)2 from π/18 to π/12: expand, 2 sin 3x cos 3x = sin 6x, integrate (5) |
| 5 | 7 | Log graph (y = abx) | log10H against t through (0, 2.5) and (10, 3.25): a and b to 3 s.f. (4); interpret b, the yearly growth factor (1); dH/dt at t = 5 using ln b (2) |
| 6 | 6 | x = f(y) | x as a function of y with a sin 2y term; tangent at a point with given y meets the x-axis at Q; find Q’s x-coordinate, unstructured (6) |
| 7 | 12 | Exponential models (two) | Car price 7 + 16e^(−0.1t) in £000s: T when £15 000 (3); dP/dt at t = 5 (2); van 4 + me^(−0.2t): m from the initial value and the initial difference (3); prices equal, a quadratic in e^(−0.1t), α = 4.38 (4) |
| 8 | 9 | Trig identities and equation | Show sec θ cosec θ ÷ 4 cosec 4θ ≡ cos 2θ (2); turn an equation in cosec 4θ, tan2 2θ, sec θ cosec θ into 5cos2 2θ − 13 cos 2θ − 6 = 0 (3); solve in degrees on 0 < x < 180°, with multiples of 45° excluded (4) |
| 9 | 11 | Integration, modulus, area | ∫ of an exponential term e^(x−3) and a 1/(2x − 5) term (2); maximum point of y = a − |bx − 1| in terms of a, b (2); a, b from the intersection at x = 3 and the y-intercept 9 (3); area bounded by both graphs and the axes, to 3 s.f. (4) |
January 2026
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Iteration | g(x) = 3x − 5 + 5 sin(x2), radians: sign change on [0.7, 0.8] (2); rearrange to x = √(arcsin(1 − px)), find p (1); x2 and α to 4 d.p. (3) |
| 2 | 6 | Differentiation | Product rule on x2 ln 2x (2); quotient rule on e2x ÷ (4x − 3)2, shown as 2e2x(ax + b)/(4x − 3)3 (4) |
| 3 | 10 | Integration, division | Reverse chain rule on a power of (3x + 5) (2); f(x) = (4x3 + x2 − 7x + 14)/((x + 2)(x − 1)): cancel the factor (x + 2), then divide to ax + b + c/(x − 1) (4); integrate to an ln answer (4) |
| 4 | 6 | Log graph (y = axn) | log10h = 2.4 − 0.25 log10m: h when m = 3 (2); rewrite as h = p/m^q, p = 251, q = 0.25 (3); interpret p (1) |
| 5 | 8 | Functions | f(x) = (2x + 16)/(x − 4), g(x) = ln x: ff(x) as one fraction (3); f−1(12) (2); fg(a) = g(a), a quadratic in ln a giving exact a (3) |
| 6 | 7 | Trig (compound angle → tan) | 2 cos(θ − 60°) = 3 sin θ written as tan θ in surd form (4); hence solve a shifted double-angle version in degrees (3) |
| 7 | 10 | Modulus | f(x) = 3|x − 11| − 4: vertex (11, −4) (2); solve f(x) = 8 (3); values of m for which y = mx meets the graph exactly once (3); a and b from the vertex of af(x + b) (2) |
| 8 | 4 | x = f(y) | x = e^(2 tan y): show dy/dx = a/(x(b + (ln x)n)), using sec2 = 1 + tan2 and ln x = 2 tan y (4) |
| 9 | 9 | R form (double angle first) | 8 + 3 sin x cos x − 2 sin2 x rewritten in sin 2x and cos 2x, then P + R sin(2x + α) with exact P and R (6); range of h (1); x of the last minimum point (2) |
| 10 | 9 | Exponential models (two) | Rate of growth of population A at t = 5 (2); B = 8000 + Pe^(kt): exact P and k from two data points (4); T when the populations are equal (3) |
October 2025
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 9 | Functions | f(x) = ln(x2 + 3), g(x) = (3 + 5x)/(x + 2): range of f, f(x) ≥ ln 3 (1); g−1 (3); fg(0) (2); exact a from g(e^(2a)) = f(√(e4 − 3)) (3) |
| 2 | 7 | Iteration | f with an ex denominator and a 1/(2x) term: sign change on [1, 2] (2); show the rearrangement (2); x3 and α to 4 d.p. (3) |
| 3 | 9 | Log graph + rate | log10V against t through (0, 2) and (5, 2.25): line (2); V = abt with exact a = 100 and b to 3 s.f. (3); T when dV/dt = 50, solved with ln (4) |
| 4 | 6 | R form | a sin 2x − b cos 2x as R sin(2x − α), R and α exact (3); exact minimum of a reciprocal function of f(3x) and the smallest x giving it (3) |
| 5 | 8 | Exponential model (fraction) | N = 4000e^(0.1t)/(19 + e^(0.2t)): initial value (1); dN/dt by quotient rule (2); show e^(0.2T) = A at the maximum (2); maximum N (3) |
| 6 | 7 | Differentiation, integration | d/dx ln(2x2 + 5) (2); ∫ 21x/(3x2 + k) with k in a limit, compared with 7 ln 8 to find the greatest integer k (5) |
| 7 | 7 | Differentiation (unstructured) | y = 2x e^(x2 + (3k − 2)x) has two turning points: dy/dx = 0 gives a quadratic, discriminant > 0, range of k (7) |
| 8 | 13 | Modulus with a cubic | y = −2x3 + 5x2 + 4x − 3 and y = a + |5x + b| meet at P(−2, 25), Q and R(2, 9): show a = 15 + b (2); a and b (3); vertex (2); Q by solving a cubic with known roots (6) |
| 9 | 9 | Trig identities and equation | 6 sin2θ cot 2θ + 4 sin θ cos θ in sin 2θ and cos 2θ (3); show 5 sin2 2θ + 14 sin 2θ − 3 = 0 (3); solve on 0 < x < 90° to 1 d.p. (3) |
June 2025
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 8 | Functions | f(x) = 2x/(3x + 1), g(x) = 4 − x2: gf(1) (2); range of f (2); f−1(x) (2); solve f−1(x) = f(x) (2) |
| 2 | 7 | R form | 7 cos x − 24 sin x = 25 cos(x + 1.287) (3); minimum of 90/(constant + 3f(2x)) as a fraction, and the smallest positive x giving it (4) |
| 3 | 5 | Log graph (y = axn) | log10y against log10x through (−3.5, 0) and (0, −2): equation (2); y = px^q with rational p and q (3) |
| 4 | 5 | Rational expressions, differentiation | 49x/(x2 + x − 12) + 7x/(x + 4) simplified to 7x/(x − 3) (3); f′(x) by quotient rule (2) |
| 5 | 4 | Integration | ∫ sin 3x (2); reverse chain rule on x(x2 + 4)n (2) |
| 6 | 8 | Exponential model | θ = 21 + Ae^(−kt): A from θ(0) = 75 (2); k from 75 → 25 °C in 5 minutes (3); T when cooling at 9 °C per minute (3) |
| 7 | 9 | Differentiation → iteration | y = e^(−x2) sin 3x: stationary point gives x = ⅓ arctan(3/(2x)) (4); x2 and x4 (3); show 0.430 to 3 d.p. with a chosen interval and function (2) |
| 8 | 8 | Trig identities and equation | Prove tan 3x ≡ (3 tan x − tan3x)/(1 − 3 tan2x) (3); hence solve an equation in tan 3θ and sec2 3θ, radians, 2 d.p. (5) |
| 9 | 11 | x = f(y) | x = k sin(3y − π/4): exact y at the y-axis crossings A and B (3); show dy/dx = 1/√(p − qx2) (4); normal at A meets tangent at B, exact x (4) |
| 10 | 10 | Modulus (parameter) | y = |kx − 10| + k: y-intercept and vertex in terms of k (3); solve |kx − 10| + k ≤ 2k in terms of k (3); y = 3x + 1 meets the graph twice: range of k (4) |
January 2025
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Iteration | f(x) = 2 sec x + 6x − 3: sign change for 0.1 < α < 0.2 (2); show α = ½ − 1/(3 cos x) (1); x2 and α from x1 = 0.15 (3) |
| 2 | 3 | Log model (y = abx) | log10A = 1 + 0.03t: initial area (1); T for 25 m2 (2) |
| 3 | 6 | Differentiation | Quotient rule on a linear function over a power of (x + 3); solve the inequality for the range where y is increasing (6) |
| 4 | 9 | Division, integration | Cubic over x2 + 4 written as Ax + B + (Cx + D)/(x2 + 4), showing D = 0 (4); definite integral as p + q ln 2, the Cx/(x2 + 4) term giving ln (5) |
| 5 | 9 | Exponential model | H = 280e^(−0.05t) + 24: initial value (1); sketch with asymptote H = 24 (2); t for H = 144 (3); show dH/dt = a + bH (3) |
| 6 | 8 | Functions, transformation | f(x) = (4x + 3)/(x − 2): f−1 (3); ff(x) = (ax + b)/(cx + d) (3); image of P(3, 15) under y = 2f(3x) + 8 (2) |
| 7 | 8 | Modulus (parameters) | Sketch y = |3x − a| and y = |3x − a| − b with the minimum point and y-intercept in terms of a, b (6); solve |3x − a| − b = 5x in terms of a, b (2) |
| 8 | 9 | Trig equations | 3 cosec θ = 8 cos θ via sin 2θ = ¾, radians to 3 s.f. (5); recognise tan(2x + 70°) from the tan(A + B) form and solve in degrees (4) |
| 9 | 8 | Differentiation, range | f(x) = 6x ln 4x: x-intercept (1); stationary point exactly by the product rule, (1/(4e), −3/(2e)) (5); range of g(x) = −2f(x) (2) |
| 10 | 9 | x = f(y) | x = 3 cos 2y: dx/dy (2); show dy/dx = k/√(9 − x2) (3); exact coordinates of the point with a given gradient (4) |
October 2024
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 5 | Trig equation | 3 tan2θ + 7 sec θ − 3 = 0 → 3 sec2θ + 7 sec θ − 6 = 0, reject sec θ = ⅔, solve cos θ = −⅓ in degrees (5) |
| 2 | 5 | x = f(y) | x = 2y2 + 5y − 6: dy/dx in terms of y (2); point where the tangent is parallel to the y-axis (3) |
| 3 | 7 | Modulus with a quadratic | f(x) = 2x2 − 10x: solve f(|x|) = 48 (3); solve an inequality between |f(x)| and x (4) |
| 4 | 6 | Log model (y = abx) | log10N = 0.35t + 2: initial value (1); N = abt with b = 2.24 (3); rate of growth at t = 5 (2) |
| 5 | 8 | Trig identity and equation | sin 3x = P sin x + Q sin3x from sin(2x + x) (4); solve 2 sin 3θ = 5 sin 2θ in degrees, including sin θ = 0 solutions (4) |
| 6 | 8 | Functions | f(x) = 6/(2x + 3), x ≥ 0, g(x) = x2 + 5: gf(2) (2); f−1 (3); solve gg(x) = 126 (3) |
| 7 | 10 | Differentiation, range, transformation | f(x) built from x3 and (4x + 7), x ≥ −7/4: show f′(x) in a given factorised form (4); minimum point (2); range of g = −4f (2); image of a point under a stretch and translation (2) |
| 8 | 12 | Exponential model → iteration | H = 32 + 40e^(−0.2t) − 20e^(−0.9t): initial value (1); long-term value (1); time of the maximum, e^(0.7t) = 2.25 (5); show M satisfies a ln iteration formula (2); t2 and M (3) |
| 9 | 14 | Differentiation, division, area | f(x) = (6x2 + 4x − 2)/(2x + 1): f′(x) (3); normal at P(2, 6) is 16y + 5x = 106 (3); Ax + B + D/(2x + 1) (3); area bounded by C, the normal and the x-axis as P + Q ln 3 (5) |
June 2024
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Modulus, transformation | f(x) = 2|x − 5| + 10: vertex (2); solve 2|x − 5| + 10 > 6x (2); image of the vertex under y = 3f(x − 2) (2) |
| 2 | 7 | Division, integration | g(x) as Ax + B + C/(linear) with integers (3); definite integral = α + β ln 3 (4) |
| 3 | 6 | Log graphs (both types) | Sketch log10y against log10x for y = 106/x3 with intercepts (3); log3N against t through (−2, 0) and (0, 4) gives N = 81 × 9t (3) |
| 4 | 9 | R form (double angle first) | 8 sin x cos x + 4 cos2x − 3 = 4 sin 2x + 2 cos 2x − 1 (3); R sin(2x + α) − 1 with R = √20 (3); maximum value and the second smallest x where it occurs (3) |
| 5 | 10 | Functions, differentiation | f(x) = 2 + 5 ln x: f−1(22) (2); prove g(x) = (6x − 2)/(2x + 1) is increasing with the quotient rule (3); g−1 (3); range of fg (2) |
| 6 | 9 | Differentiation, area | y = √(4x − 7): normal at P(8, 5) is 5x + 2y − 50 = 0 (5); exact area bounded by the curve, the x-axis and the normal, using ∫(4x − 7)^½ (4) |
| 7 | 8 | Trig (compound angle → tan) | 2 sin(x + 45°) = cos(x − 60°) → tan x = −2 − √3 (4); hence solve 2 sin 2θ = cos(2θ − 105°) on 0 ≤ θ < 180° (4) |
| 8 | 10 | Exponential model → iteration | Golf ball h = 1.5x − 0.5xe^(0.02x): d = 50 ln 3 (3); show the maximum satisfies x = 50 ln(150/(x + 50)) (4); x2 and the root to 2 d.p. (3) |
| 9 | 10 | x = f(y) | x = 4 sin2y − 1: verify P(2, π/3) (1); dx/dy, then dy/dx in terms of x (6); exact area of triangle OPN with the normal, as aπ + bπ2 (3) |
January 2024
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 4 | Transformations | Image of P(−4, −3) under y = f(2x) (1), y = 3f(x − 1) (2) and y = |f(x)| (1) |
| 2 | 6 | Iteration | f(x) = x4 − 5x2 + 4x − 7: sign change on [2, 3] (2); rearrange to x = ∛((5x2 − 4x + 7)/x) (1); x2 and α (3) |
| 3 | 7 | Log model (y = abx) | log10D = 1.04 + 0.38t written as D = abt to 4 s.f. (3); T for £45 000 (2); test whether £350 000 is reached in 12 months (2) |
| 4 | 13 | Rational expressions, functions | (2x2 − 32)/((3x − 5)(x + 4)) + 8/(3x − 5) simplified to 2x/(3x − 5) (3); show f is decreasing, f′(x) = −10/(3x − 5)2 (3); g(x) = 3 + 2 ln x: g−1 (3); exact a with gf(a) = 5 (4) |
| 5 | 7 | Exponential model | T = 10 + Ae^(−Bt): A from 18 °C (1); B from 16 °C at 45 minutes (3); rate of change at t = 2 (2); explain why T cannot reach 5 °C (1) |
| 6 | 7 | Differentiation (product + chain, trig) | f(x) = 2e^(3 sin x) cos x: x-intercept (1); turning points give 3 sin2x + sin x − 3 = 0 via cos2x = 1 − sin2x (4); x of the second turning point (2) |
| 7 | 12 | Differentiation, integration | y = 16/(9(3x − k)): dy/dx (2); k = 7/3 or 11/3 from the gradient −12 at x = 1 (3); normal at P with integer coefficients (3); ∫ of 16/(9(3x − k)) as λ ln 10 (4) |
| 8 | 11 | Modulus (parameters) | y = a − |2x − b|: maximum point, y-intercept and x-intercepts in terms of a, b (5); sketch y = |x + 1| (2); a and b from intersections at x = −3 and x = 5 (4) |
| 9 | 8 | Trig identity → R form | Reduce an equation in sin θ cos θ, sec 2θ and (cos θ ± sin θ) to 3 sin 2θ − 4 cos 2θ = 2 (3); solve on π < x < 3π/2 to 3 s.f., expected method R = 5 (5) |
October 2023
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 5 | Iteration | f(x) = x2 − 5x + ex: sign change on [1, 2] (2); x2 = √(5x1 − e^(x1)) = 1.5105 from x1 = 1 (2); α = 1.7340 (1) |
| 2 | 8 | Functions | f(x) = (x + 3)/(x − 4): ff(6) = 15 (2); f−1(x) = (4x + 3)/(x − 1), x ≠ 1 (3); solve (f(a))2 + 5 = 7, giving a = 11 + 7√2 (3) |
| 3 | 6 | Trig proof, integration | Prove cos 2A ≡ 2cos2A − 1 from cos(A + A) and sin2 + cos2 = 1 (2); integrand in cos2 3x rewritten as 3 − 2 cos 6x, exact integral from π/12 to π/8 (4) |
| 4 | 7 | Exponential model | N = 125 − Ae^(−0.109t) (thousands): A = 93 (1); T = 12.05 when N = 100 (3); dN/dt at t = 7, 4730 a month (2); explain why 150 000 is never reached (1) |
| 5 | 7 | Differentiation (quotient, ln) | y = ln(x2 + k)/(x2 + k): dy/dx = 2x(1 − ln(x2 + k))/(x2 + k)2 (3); stationary points x = 0, ±√(e − k) (3); upper limit k < e (1) |
| 6 | 6 | Log model (y = abx) | log10S = 4.5 − 0.006t: S at t = 2, 30 800 km2 (2); S = 31 600 × 0.986t (3); interpret q as the yearly proportion kept (1) |
| 7 | 10 | Differentiation (product + chain), range | f(x) = e^(−x2)(2x2 − 3)2: range 0 ≤ f(x) ≤ 9 (2); f′(x) = 2xe^(−x2)(2x2 − 3)(7 − 2x2) (4); range of k for a given number of roots of f(x) = k, 16e^(−7/2) < k < 9 (4) |
| 8 | 8 | Trig identity and equation | Prove 2 cosec2 2θ (1 − cos 2θ) ≡ 1 + tan2θ (4); hence sec2x − 3 sec x − 4 = 0, cos x = ¼, x = 75.5°, 284.5° (4) |
| 9 | 9 | Modulus with ln | f(x) = |2 − 4 ln(x + 1)|: domain bound k = −1 (1); y-intercept (1); x-intercept e^½ − 1 (2); f(x) > 3 solved exactly in two ranges (5) |
| 10 | 9 | x = f(y) | x = sin2 4y: y at x = ¼ (2); dx/dy = 8 sin 4y cos 4y (2); dy/dx = 1/√(16 − 64(x − ½)2) (3); least gradient ¼ at x = ½ (2) |
June 2023
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 5 | Iteration | g(x) = x6 + 2x − 1000: sign change on [3, 4] (2); iterate x = (1000 − 2x)^(1/6) for x2 and α (3) |
| 2 | 6 | Log graph (y = axn, base 6) | log6T against log6x through (0, 4) and (2, 0): equation and exact T at x = 216 (3); T = 1296/x2 (3) |
| 3 | 7 | Differentiation, integration | d/dx ln(sin2 3x) = 6 cot 3x (2); differentiate a power of (3x2 − 4) (2); hence a definite integral of x times a power of (3x2 − 4) equals an integer (3) |
| 4 | 6 | Functions (inverse graph) | f(x) = 2x2 − 5, x ≥ 0: range (1); sketch f−1 by reflection in y = x (2); exact x where f meets f−1, by solving f(x) = x (3) |
| 5 | 7 | Trig equations | A linear factor times a sec x factor equal to 0 on 0 < x < π (3); 10 sin θ = 3 cos 2θ via 1 − 2 sin2θ, a quadratic in sin θ (4) |
| 6 | 9 | Modulus, functions | f(x) = 3|x − 2| − 10: vertex (2); ff(0) (2); 3|x − 2| − 10 < 5x + 10 (2); solve f(|x|) = 0 (3) |
| 7 | 8 | Exponential models (two) | N = Ae^(kt): exact A and k from 2500 and 10 000 at t = 8 (3); rate of decrease of 60 000e^(−0.6t) at t = 5 (2); T when equal (3) |
| 8 | 9 | Differentiation (product + chain), transformation | f(x) = (2x + 1)3e^(−4x): f′(x) = A(2x + 1)2(1 − 4x)e^(−4x) (4); exact stationary points (3); maximum of g(x) = 8f(x − 2) (2) |
| 9 | 10 | Trig identity, equation, integration | Prove cos 2x/sin x + sin 2x/cos x ≡ cosec x (3); hence solve an equation in θ, 3 s.f. (5); exact ∫ of the same expression times cot x, which becomes ∫ cosec x cot x (2) |
| 10 | 8 | x = f(y) | x = (2y2 + 6)/(3y + 3): dx/dy as a simplified fraction (4); equations of the two tangents parallel to the y-axis, from dx/dy = 0 (4) |
January 2023
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Functions | Range of f, f(x) ≤ 9 (1); fg(1.5) (2); g−1 (3) |
| 2 | 6 | R form | cos x + 2 sin x = √5 cos(x − 1.107) (3); exact maximum of a reciprocal function of f(2x) and the smallest x giving it (3) |
| 3 | 5 | Log graph (y = kbx) | log10y against x through (0, 1.5) and (−4.8, 0): equation (2); y = kbx with k = 31.6, b = 2.05 (3) |
| 4 | 7 | Division, integration | A quartic fraction reduced to Ax2 + Bx + C + D/(x + 3), with A = 2, B = 3, C = −1, D = 5 (4); integrate, the D term giving ln (3) |
| 5 | 9 | Trig identity and equation | Prove an identity linking cot x − tan x with double angles, using cos4x − sin4x = cos 2x (4); hence solve, 2 d.p. (5) |
| 6 | 8 | Modulus (parameter) | y = 3|x − 5a| − 2a: y-intercept, x-intercepts and minimum in terms of a (4); solve the inequality with a line in terms of a (4) |
| 7 | 9 | Differentiation (inverse trig via x = f(y)) | Curve defined with arctan: x = 3 tan(y − π/6) gives dy/dx = a/(x2 + b) (4); tangent at P crosses the x-axis at Q, exact x (5) |
| 8 | 5 | Integration (trig identity) | ∫ (2 cos x − sin x)2 dx: expand, then cos2x = ½(1 + cos 2x), sin2x = ½(1 − cos 2x) and 2 sin x cos x = sin 2x (5) |
| 9 | 11 | Differentiation → iteration | A curve involving e^(x2), x ≥ 0: dy/dx (2); tangent through the origin gives an equation in α (3); sign change on [1, 2] (2); rearrange (1); x3 and α (3) |
| 10 | 9 | Exponential model (fraction) | F = 350e^(kt)/(9 + e^(kt)): initial value (1); show k = (ln 12)/15 (3); both values of T when F grows at 10 a day, via the quotient rule and a quadratic (5) |
October 2022
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 9 | Division, integration | 2x3 − 4x − 15 ≡ (Ax + B)(x2 + 3x + 4) + C(2x + 3): A = 2, B = −6, C = 3 (4); ∫ from 3 to 5 = 4 + ln 8, the C(2x + 3) term giving ln (5) |
| 2 | 7 | Functions | f(x) = 5 − 4/(3x + 2), x ≥ 0: range 3 ≤ f(x) < 5 (2); f−1 and its domain (3); fg(−π) with g(x) = |4 sin(x/3 + π/6)| (2) |
| 3 | 7 | Differentiation (product + chain), range of k | f(x) = (x − 2)2e3x: stationary point (4/3, 4e4/9) by product rule (5); range of k read from the stationary values, 0 < k ≤ 4e4/9 (2) |
| 4 | 5 | x = f(y) with logs | y = log10(2x + 1): x in terms of y (2); dy/dx = 2/((2x + 1) ln 10) via dx/dy (3) |
| 5 | 11 | Model with ln → iteration | Profit P(t) = (4t − 1)/10 + ¾ ln((t + 1)/(2t + 1)2): P(1) as a loss of about £830 000 (2); sign change on [6, 7] (2); rearrange for t (2); t2 and t6 (3); convert to months, 76 (2) |
| 6 | 6 | Differentiation (trig quotient) | y = (2 + 3 sin x)/(cos x + sin x): show dy/dx = (2 tan x + 3 sec x − 2)/(sec x + 2 sin x), using sin2 + cos2 = 1 and multiplying by sec x (6) |
| 7 | 12 | Modulus with a quadratic | y = 5 − |3x − 22|: maximum (22/3, 5) and y-intercept (2); x-intercepts 9 and 17/3 (2); sketch with y = x2/9 − 9 (3); both intersections, (9, 0) and (3, −8) (5) |
| 8 | 9 | R form | 8 sin x − 15 cos x = 17 sin(x − 1.081) (3); minimum of 15/(41 + 16 sin x − 30 cos x) is 1/5, at x = 2.65 (4); follow-through for y = 2f(x) − 5 and y = −f(2x) (2) |
| 9 | 9 | Trig identity and equation | Prove cos2θ/(cos 2θ − sin 3θ) ≡ (1 + sin θ)/(1 − 2 sin θ − 4 sin2θ), factor 1 − sin θ (4); hence 9 sin2θ + 5 sin θ − 2 = 0, four solutions in degrees (5) |
June 2022
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Differentiation (chain), normal | y = (3x − 2)6: dy/dx (2); normal at P(1/3, 1) with integer coefficients (4) |
| 2 | 9 | Functions | f(x) = (5 − x)/(3x + 2), g(x) = 2x − 7: fg(5) (2); f−1 (3); solve f(1/a) = g(a + 3), a quadratic (4) |
| 3 | 6 | Division, integration | ∫ 9x/(3x + k) dx after dividing (2); find k given the definite integral equals ln 8 (4) |
| 4 | 7 | Log graph (y = abx) | log10N against t through (0, 3.08) and (5, 3.85): line (2); a, b to 3 s.f. (3); T for 500 000 subscribers (2) |
| 5 | 8 | Modulus (parameter) | f(x) = |kx − 9| − 2: y-intercept and x of the minimum (2); solve f(x) < 0 in terms of k (3); range of k for which y = 3 − 2x meets the graph twice (3) |
| 6 | 10 | Differentiation (product, surd), range | f(x) with a (4x + 9)^½ factor: show f′(x) = k(5x2 + bx + c)/(4x + 9)^½ (4); f′(x) = 0 (1); exact local maximum (2); range of g, a transformation of f (3) |
| 7 | 8 | Trig identity and equation | 2 sin θ(3 cot2 2θ − 7) = 13 sec θ → 3 cosec2 2θ − 13 cosec 2θ − 10 = 0 (4); solve on 0 < θ < π/2 to 3 s.f. (4) |
| 8 | 12 | Model, calculus, iteration | v = 12 − e^(t−10) − 12e^(−0.75t): maximum velocity by calculus (5); ∫0T v dt = 100 gives T = (1/12)(116 − 16e^(−0.75T) + e^(T−10) − e^(−10)) (4); T2 and T (3) |
| 9 | 9 | Differentiation (trig quotient) → R form | y = (1 + 2 cos x)/(1 + sin x): the minimum M satisfies 2 sin x + cos x = −2 (4); solve with √5 sin(x + 0.464) = −2, x of M to 3 s.f. (5) |
January 2022
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 4 | Differentiation (product) | y = (2x + 5)e3x: stationary point x = −17/6 (4) |
| 2 | 5 | Trig equation | 8 cos θ = 3 cosec θ → sin 2θ = ¾ (3); θ = 24.3° (2) |
| 3 | 6 | Integration | ∫ (2x − 5)7 dx (2); ∫0^(π/3) 4 sin x/(1 + 2 cos x) dx = ln(9/4), the f′/f type (4) |
| 4 | 7 | Exponential model (fraction) | A = 80pe^(0.15t)/(pe^(0.15t) + 4): show p = 2.4 (2); T = 6.8 for A = 50 (4); limit 80 m2 (1) |
| 5 | 9 | Iteration, differentiation | y = 6 ln(2x + 3) − ½x2 + 4: sign change on [−1.25, −1.2] (2); x = √(12 ln(2x + 3) + 8), x2 and x6 (3); turning point x = (−3 + √105)/4 (4) |
| 6 | 11 | Functions (calculus) | f(x) = (5x − 3)/(x − 4): f′(x) = −17/(x − 4)2 < 0 so decreasing (3); f−1 with domain x > 5 (3); ff(x) = (22x − 3)/(x + 13) (3); range 5 < ff(x) < 22 (2) |
| 7 | 10 | Modulus (W shape) | y = ½|2x + 7| − 10: vertex (2); inequality with the line y = x/3 + 1, x ≤ −87/8 or x ≥ 45/4 (4); sketch y = |f(x)|, a W with maximum (−7/2, 10) and minima at −27/2 and 13/2 (4) |
| 8 | 8 | Log model (y = pq−t) | log10x = 2.74 − 0.079t → x = pq−t, p = 550, q = 1.2 (4); interpret p as the initial dose (1); rate dx/dt at t = 5 (3) |
| 9 | 8 | Trig equation and proof | 2 sec2x − 3 tan x = 2 → tan x(2 tan x − 3) = 0, x = 0.983, π (4); prove sin 3θ/sin θ − cos 3θ/cos θ ≡ 2 (4) |
| 10 | 7 | x = f(y) | x = ye^(2y): show dy/dx = y/(x(1 + 2y)) (4); range of k for which x = k meets C twice, −1/(2e) < k < 0, from dx/dy = 0 at y = −½ (3) |
October 2021
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 9 | Rational expressions, functions | 5x/(x2 + 7x + 12) + 5x/(x + 4) simplified to 5x/(x + 3) (3); f−1 (3); f′(x) and whether f is increasing, with a reason (3) |
| 2 | 10 | Modulus, functions | f(x) = |3x − 13| + 5: vertex (2); range and ff(4) (2); 16 − 2x > |3x − 13| + 5 (4); a, b from the vertex of af(x + b) at (4, 20) (2) |
| 3 | 6 | Exponential model | G = 40 − 30e^(1 − 0.05t): k where G = 0, as a year and month (3); G at 1870 (2); limit 40 tonnes (1) |
| 4 | 7 | Trig (compound angle → tan) | 2 sin(θ − 30°) = 5 cos θ → tan θ = 2√3 (4); hence 2 sin(x − 10°) = 5 cos(x + 20°) on 0 ≤ x < 360° (3) |
| 5 | 6 | Integration | Exact definite integral of a reciprocal square of a linear term (4); reverse chain rule on x times a power of (x2 − 3) (2) |
| 6 | 8 | Differentiation | y = 3 ln(x2 − 5) − 4x2 + 15: stationary point in surd form (4); y = 4x − 12 sin2x: dy/dx = 4 − 12 sin 2x and the maximum gradient 16 (4) |
| 7 | 6 | Log model (y = axn) | log10M = 1.93 log10r + 0.684: M at r = 45 (2); M = prφ, p = 4.83, q = 1.93 (3); interpret p (1) |
| 8 | 7 | Inverse trig, x = f(y) | Sketch y = arcsin(x/2) (1); with x = 2 sin y, show dy/dx = 1/√(A − x2) (3); tangent at the point with given y (3) |
| 9 | 9 | Differentiation → iteration | f(x) = x(x2 − 4)e^(−x/2): f′(x) (2); the normal at O meets the curve at P, giving an equation for x (4); x2 and P’s x-coordinate (3) |
| 10 | 7 | Trig identity, integration | (1 + 2 cos 2x)2 ≡ p + q cos 2x + r cos 4x (2); exact total area up to the second point where the curve touches the x-axis (5) |
June 2021
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 7 | Differentiation → iteration | y = x2 cos(x/2): product rule, dy/dx = 0 gives x = 2 arctan(4/x) (4); x2 = 2.214 and x6 = 2.155 (3) |
| 2 | 9 | Trig identity and equation | Prove (1 − cos 2x)/sin 2x ≡ tan x (3); use it to reach 4 tan2θ − 9 tan θ + 4 = 0 via sec2 = 1 + tan2, θ = 31.4°, 58.6° (6) |
| 3 | 8 | Integration, division | ∫ 12/(2x − 1)2 dx = −6/(2x − 1) (2); (4x + 3)/(x + 2) = 4 − 5/(x + 2) (2); ∫ from −8 to −5 = 12 + 5 ln 2, needing |x + 2| in the ln (4) |
| 4 | 10 | Functions | f(x) = (4x + 6)/(x − 5), g(x) = 5 − 2x2: solve fg(x) = 3, x = −√13 only (4); f−1(x) = (5x + 6)/(x − 4) (3); sketch g and g−1 with intercepts (3) |
| 5 | 7 | Log model + rate | Line of log10A against t gives A = pqt, p = 2.089, q = 1.072 (4); dA/dt = pqt ln q at t = 6, 0.22 (3) |
| 6 | 8 | Modulus (parameter) | Sketch y = k − 2|x| and y = |2x − k/3| with intercepts (4); solve k − 2|x| = |2x − k/3|, x = −k/6 or k/3 (4) |
| 7 | 5 | x = f(y) | x = 6 sin2 2y: dx/dy = 24 sin 2y cos 2y, flip, then sin 2y = √(x/6) to give dy/dx = 1/(4√(6x − x2)) (5) |
| 8 | 13 | Exponential model (fraction) | N = 600e^(0.3t)/(2 + e^(0.3t)): initial 200 (1); limit 600 (1); N = 500 at 7 years 8 months (4); dN/dt = 360e^(0.3t)/(2 + e^(0.3t))2 (3); dN/dt = 8 gives a quadratic in e^(0.3t), T = 12.4 (4) |
| 9 | 8 | R form | 12 sin θ − 5 cos θ type as 13 sin(θ − 0.395) (3); minimum −3 of a shifted double-angle version and where (3); range of h = 10 − 169 sin2(β − 0.395), −159 ≤ h ≤ 10 (2) |
January 2021
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 3 | Integration | ∫ (x2 − 5)/(2x3) dx: split into ½ ln x + 5/(4x2) + c (3) |
| 2 | 6 | Transformations (sketches) | From a polynomial graph with turning points (1, 2) and (3, 0): sketch y = 3f(2x) (3) and y = f(−x) − 1 (3) |
| 3 | 8 | Rational expressions, functions | 3 − (x − 2)/(x + 1) + (5x + 26)/(2x2 − 3x − 5) as (ax + b)/(cx + d) (4); f−1 (2); domain of f−1 (2) |
| 4 | 9 | Modulus (parameter) | f(x) = |3x + a| + a: vertex (−a/3, a) (2); sketch g(x) = |x + 5a| (2); both intersections in terms of a (5) |
| 5 | 11 | Exponential model | θ = A − 180e^(−kt): A = 198 (2); k = ⅕ ln(5/3) (4); θ at t = 9 (2); rate of increase at t = 9 (3) |
| 6 | 8 | Differentiation → iteration | f(x) = x cos(x/3): f′(x) (2); f′(x) = 0 as x = 3 arctan(3/x) (2); x2 and x6 (2); show 2.581 to 3 d.p. with a chosen interval and function (2) |
| 7 | 9 | Trig identity and equation | Prove sin 2x/cos x + cos 2x/sin x ≡ cosec x (3); hence solve an equation in 4θ, 2θ and cot2 2θ, radians (6) |
| 8 | 7 | Log model (y = abx) | Gradient 0.09, intercept 0.68: P = abt, a = 4.79, b = 1.23 (4); interpret a (1); estimate for 2015 (2) |
| 9 | 4 | Integration (f′/f, f′fn) | ∫ (3x − 2)/(3x2 − 4x + 5) dx = ½ ln(3x2 − 4x + 5) (2); ∫ e2x/(e2x − 1)3 dx (2) |
| 10 | 10 | x = f(y) | x = 3 sec2 2y: dx/dy = 12 sec2 2y tan 2y (2); dy/dx = p/(qx√(x − 3)), p irrational (3); exact normal at a given y (5) |
October 2020
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 5 | Trig equation | 2 cos 2x = 7 cos x → 4cos2x − 7 cos x − 2 = 0, cos x = −¼, x = 104.5°, 255.5° (5) |
| 2 | 6 | Log model (y = abx) | log10N = 0.0646t + 1.478 → N = 30 × 1.16t (4); N at t = 30, about 2600 (2) |
| 3 | 7 | Differentiation (quotient, surd), range | y = (2x + 3)/√(4x − 1): f′(x) = (4x − 8)/(4x − 1)^(3/2) (4); turning point at x = 2 gives the range f(x) ≥ √7 (3) |
| 4 | 8 | Modulus, functions | f(x) = 21 − 2|2 − x|, x ≥ 0: ff(6) = −1 (2); f(x) = 5x, x = 25/7 (2); 17 ≤ k < 21 for exactly two roots (2); a = 1/7, b = 4 from a vertex (2) |
| 5 | 8 | Trig identity, integration | Prove sin 3x ≡ 3 sin x − 4 sin3x (4); ∫0^(π/3) sin3x dx = 5/24 by rearranging the identity (4) |
| 6 | 9 | Exponential equation, iteration | 5e^(x−1) + 3 = 18 gives x = ln 3e (3); sign change on [1.1335, 1.1345] to show α = 1.134 (3); iterate x = −√(7 − 5e^(x−1)) to β = −2.620330 (3) |
| 7 | 9 | R form in context | cos x + 4 sin x = √17 cos(x − 1.326) (3); minimum height 3.37 m of a seabird modelled by H = 24/(3 + cos(t/2) + 4 sin(t/2)) (2); t when H = 10 (4) |
| 8 | 9 | Differentiation, x = f(y) | g(x) = e3x sec 2x: product rule (2); stationary point via tan 2x = −1.5, x = −0.491 (3); x = ln(sin y): dy/dx = ex/√(1 − e2x) (4) |
| 9 | 14 | Division, tangent, area | (x4 − x3 − 10x2 + 3x − 9)/(x2 − x − 12) ≡ x2 + 2 + 5/(x − 4) (4); tangent at x = 2, y = 11x/4 − 2 (5); area 20/3 − 5 ln 2 (5) |
January 2020
| Q | Marks | Topic | What each part tests |
|---|---|---|---|
| 1 | 6 | Exponential model (fraction) | N = 900e^(0.12t)/(2e^(0.12t) + 1): initial 300 (1); t when N = 420 (4); never reaches 500 because the limit is 450 (1) |
| 2 | 8 | Functions | f(x) = (…)/(x + 1), g(x) = ln x: fg(e2) (2); f−1 (3); all real solutions of f−1(x) = f(x) (3) |
| 3 | 5 | Log graph (y = axn) | log10y against log10x through (0, 4) and (6, 0): equation (2); y = 104x^(−2/3) (3) |
| 4 | 11 | Differentiation | Quotient rule with f′(x) = P(x)/Q(x), both fully factorised quadratics, then where f is increasing (6); product rule with sin 4x: the maximum satisfies tan 4x + kx = 0 (5) |
| 5 | 8 | Trig equation | Substitute t = tan x into 12 tan 2x + 5 cot x sec2x = 0 to get 5t4 − 24t2 − 5 = 0 (4); solve in degrees (4) |
| 6 | 9 | Modulus | f(x) = 2|2x − 5| + 3, x ≥ 0: vertex (2); f(x) = 3x − 2 (4); range of k for which f(x) = kx + 2 has exactly two roots (3) |
| 7 | 11 | Iteration, differentiation (trig) | y = 2 cos 3x − 3x + 4: sign change on [0.8, 0.9] (2); iterate x = ⅓ arccos(1.5x − 2) for x2 and x5 (3); exact x of the first two minima from sin 3x = −½ (6) |
| 8 | 10 | Integration, division | Exact definite integral of a reciprocal involving (3x − 1) (4); h(x) = (2x3 − 7x2 + 8x + 1)/(x − 1)2 as Ax + B + C/(x − 1)2, then ∫ h(x) dx (6) |
| 9 | 7 | R form, transformations | 5 cos θ − 4 sin θ = √41 cos(θ + 0.675) (3); describe the stretch and translation from y = cos θ (2); range of g(θ) = 2/(4 + (f(θ))2) (2) |
Technique-level analysis by topic
Which variant of each method appears, and in which papers. Counts are papers, not marks; one paper can count under several variants.
Differentiation
Product rule pairings are dominated by a polynomial with an exponential; the quotient rule is mostly rational functions, with trig quotients reserved for “show that” proofs.
| Variant | Papers | Count |
|---|---|---|
| Product: polynomial or linear power × e^(kx) or e^(f(x)) | Oct 2021 Q9, Jan 2022 Q1, Oct 2022 Q3, Jun 2023 Q8, Oct 2023 Q7, Oct 2025 Q7, Jun 2026 Q1 | 7 |
| Product: polynomial × trig | Jan 2020 Q4, Jan 2021 Q6, Jun 2021 Q1 | 3 |
| Product: exponential × trig | Oct 2020 Q8, Jan 2024 Q6, Jun 2025 Q7 | 3 |
| Product: polynomial × ln | Jan 2025 Q9, Jan 2026 Q2 | 2 |
| Product: polynomial × surd | Jun 2022 Q6, Oct 2024 Q7 | 2 |
| Quotient: rational function | Jan 2020, Oct 2021, Jan 2022, Jan 2024, Jun 2024, Oct 2024, Jan 2025, Jun 2025, Jun 2026 | 9 |
| Quotient: exponential (incl. fraction models) | Jun 2021 Q8, Jan 2023 Q10, Oct 2025 Q5, Jan 2026 Q2 | 4 |
| Quotient: trig, as a “show that” | Jun 2022 Q9, Oct 2022 Q6 | 2 |
| Quotient: with ln or a surd | Oct 2020 Q3, Oct 2023 Q5 | 2 |
| Chain: ln of a function | Oct 2021, Jan 2022, Jun 2023, Oct 2023, Oct 2025 | 5 |
| Chain: ax or bt (rates in log models) | Jun 2021, Jan 2022, Oct 2024, Oct 2025, Jun 2026 | 5 |
| Calculus feeding an iteration | Jan 2021, Jun 2021, Oct 2021, Jan 2023, Jun 2024, Jun 2025, Jun 2026 | 7 |
Applications: stationary points in 17 papers, a tangent or normal in 11, “increasing or decreasing” in 6, a range from a turning point in 5, and a discriminant condition once (October 2025).
x = f(y)
Half of these are x as a trig function of y, which forces a Pythagorean identity to get dy/dx in terms of x.
| Variant | Papers | Count |
|---|---|---|
| x = trig of y (sin2, cos 2y, sec2 2y, sin(3y − π/4)) | Jan 2021, Jun 2021, Oct 2021, Oct 2023, Jun 2024, Jan 2025, Jun 2025 | 7 |
| x = exponential or log of y | Oct 2020 (ln sin y), Jan 2022 (ye^(2y)), Oct 2022 (log10), Jan 2026 (e^(2 tan y)) | 4 |
| x = polynomial or rational in y | Jun 2023, Oct 2024 | 2 |
| x = mixed trig and linear in y | Jun 2026 | 1 |
Tasks: dy/dx in terms of x in 11 papers; a tangent or normal in 5; tangents parallel to the y-axis (dx/dy = 0) in 2; a least gradient, a point with a given gradient and a range of k once each. January 2023 Q7 is the same skill in disguise: an arctan curve differentiated through x = 3 tan(y − π/6).
Exponential models
The “constant plus decaying exponential” model is the staple; fraction-shaped models appear about once every four sessions.
| Variant | Papers | Count |
|---|---|---|
| A ± Be^(−kt), decaying to a limit | Jan 2021, Oct 2021, Oct 2023, Jan 2024, Jan 2025, Jun 2025, Jun 2026 | 7 |
| Fraction (logistic) Ae^(kt)/(B + e^(kt)) | Jan 2020, Jun 2021, Jan 2022, Jan 2023, Oct 2025 | 5 |
| Two models compared or set equal | Jun 2023, Jan 2026, Jun 2026 | 3 |
| Two exponential terms in one model | Jun 2022 (velocity), Oct 2024 (heart rate) | 2 |
| x × exponential (projectile) | Jun 2024 | 1 |
| Model with ln (profit) | Oct 2022 | 1 |
Tasks: a rate of change in 13 papers (including log models); a limit or “why can it never reach” in 8; a maximum found by calculus in 4; an iteration set inside the model in 4; “show dH/dt = a + bH” once (January 2025). Every two-model question dates from 2023 or later.
Log graphs
Two in three are y = abx from log y against x; the rest are y = axn from a log–log line.
| Variant | Papers | Count |
|---|---|---|
| y = abx (log y against x) | Oct 2020, Jan 2021, Jun 2021, Jan 2022, Jun 2022, Jan 2023, Oct 2023, Jan 2024, Jun 2024, Oct 2024, Jan 2025, Oct 2025, Jun 2026 | 13 |
| y = axn (log y against log x) | Jan 2020, Oct 2021, Jun 2023, Jun 2024, Jun 2025, Jan 2026 | 6 |
| Base other than 10 | Jun 2023 (base 6), Jun 2024 (base 3) | 2 |
| Line given by two points (rather than an equation) | Jan 2020, Jun 2021, Jun 2022, Jan 2023, Jun 2023, Jun 2024, Jun 2025, Oct 2025, Jun 2026 | 9 |
| Interpret a constant in context | Jan 2021, Oct 2021, Jan 2022, Oct 2023, Jan 2026, Jun 2026 | 6 |
| Rate of change via ln b | Jun 2021, Jan 2022, Oct 2024, Oct 2025, Jun 2026 | 5 |
Iteration and sign change
Iteration formulas fall into four families; inverse-trig and root forms make up most of them.
| Variant | Papers | Count |
|---|---|---|
| Inverse trig (arccos, arctan, arcsin) | Jan 2020, Jan 2021, Jun 2021, Jun 2025, Jan 2026 | 5 |
| Square, cube or sixth root | Oct 2020, Oct 2021, Jan 2022, Jan 2023, Jun 2023, Oct 2023, Jan 2024, Oct 2025 | 8 |
| ln form | Oct 2022, Jun 2024, Oct 2024 | 3 |
| Algebraic with trig or exponential terms | Jun 2022, Jan 2025, Jun 2026 | 3 |
| Root to n d.p. proved with a chosen interval and function | Oct 2020, Jan 2021, Jun 2025 | 3 |
A sign change appears in 14 papers and always needs both values, “continuous” and a conclusion. Answers are usually to 4 d.p.; October 2022 and January 2021 used 3 d.p., June 2024 used 2 d.p. and October 2020 used 6 d.p.
Functions
Rational functions dominate. Since 2022 the question nearly always ends with an equation built from composites or inverses.
| Variant | Papers | Count |
|---|---|---|
| Rational function (linear over linear, or similar) | Jan 2020, Jan 2021, Jun 2021, Oct 2021, Jan 2022, Jun 2022, Oct 2022, Oct 2023, Jan 2024, Jun 2024, Oct 2024, Jan 2025, Jun 2025, Oct 2025, Jan 2026 | 15 |
| Exponential or log function | Jan 2020, Jan 2024, Jun 2024, Oct 2025, Jan 2026, Jun 2026 | 6 |
| Quadratic or surd function | Jun 2021, Jun 2023, Oct 2024, Jun 2025, Jun 2026 | 5 |
| Solve an equation in fg, gf or gg | Jun 2021, Jun 2022, Oct 2023, Jan 2024, Oct 2024, Oct 2025, Jan 2026, Jun 2026 | 8 |
| f−1(x) = f(x), or where f meets f−1 | Jan 2020, Jun 2023, Jun 2025 | 3 |
| Simplify to one fraction first | Jan 2021, Oct 2021, Jan 2024, Jun 2025 | 4 |
| ff(x) as a single fraction | Jan 2022, Jan 2025, Jan 2026 | 3 |
| Increasing or decreasing proved by calculus | Jan 2020, Oct 2021, Jan 2022, Jan 2024, Jun 2024, Jan 2025 | 6 |
An inverse function appears in 18 of 19 papers; October 2020 is the exception.
Transformations
Transformations are short add-ons, worth 1–2 marks a part, attached to modulus, functions or differentiation questions.
| Variant | Papers | Count |
|---|---|---|
| Map a point, e.g. y = 2f(3x) + 8 | Jun 2023, Jan 2024, Jun 2024, Oct 2024, Jan 2025, Jun 2026 | 6 |
| Vertex after y = af(x + b) | Oct 2020, Oct 2021, Jan 2026 | 3 |
| Sketch or describe a transformed graph | Jan 2020, Jan 2021 | 2 |
| Follow-through from an R form | Oct 2022 | 1 |
Modulus
Most questions use a single V or inverted V; the hardest pair the modulus with a quadratic, a cubic or a log.
| Variant | Papers | Count |
|---|---|---|
| V shape, p|x − q| + r | Jan 2020, Jan 2021, Oct 2021, Jan 2022, Jun 2022, Jan 2023, Jun 2023, Jun 2024, Jan 2025, Jun 2025, Jan 2026 | 11 |
| Inverted V, a − |bx − c| | Oct 2020, Jun 2021, Oct 2022, Jan 2024, Jun 2026 | 5 |
| Answers in terms of letters (a, b or k) | Jan 2021, Jun 2021, Jun 2022, Jan 2023, Jan 2024, Jan 2025, Jun 2025, Jun 2026 | 8 |
| f(|x|) or |f(x)| of a quadratic, or a W shape | Jan 2022, Jun 2023, Oct 2024 | 3 |
| Paired with a quadratic, cubic, log or exponential curve | Oct 2022, Oct 2023, Oct 2025, Jun 2026 | 4 |
| Inequality | Oct 2021, Jan 2022, Jun 2022, Jan 2023, Jun 2023, Oct 2023, Jun 2024, Oct 2024, Jun 2025 | 9 |
| Range of k or m for a number of intersections | Jan 2020, Oct 2020, Jun 2022, Jun 2025, Jan 2026 | 5 |
Three of the four “paired with another curve” questions were October papers (2022, 2023 and 2025).
Trig identities and equations
sec2 = 1 + tan2 and the double-angle formulae carry most equations; triple angles keep returning in proofs.
| Variant | Papers | Count |
|---|---|---|
| sec2 = 1 + tan2 (or cosec2 = 1 + cot2) to form a quadratic | Jan 2020, Jan 2021, Jun 2021, Jan 2022, Jun 2022, Oct 2023, Oct 2024, Jun 2025, Jun 2026 | 9 |
| cos 2x or sin 2x turning an equation into a quadratic | Oct 2020, Jun 2023, Oct 2025, Jun 2026 | 4 |
| sin 2θ = k from a product of sin and cos | Jan 2022, Jan 2025 | 2 |
| Compound angle to tan θ = k | Oct 2021, Jun 2024, Jan 2026 (and tan(A + B) recognised, Jan 2025) | 4 |
| Triple angle: sin 3x or tan 3x | Oct 2020, Jan 2022, Oct 2022, Oct 2024, Jun 2025 | 5 |
| Other substitution or factor (t = tan x, cos4 − sin4) | Jan 2020, Jan 2023 | 2 |
| Product of a linear factor and a trig factor | Jun 2023 | 1 |
Solutions are asked in degrees about twice as often as in radians. Excluded values (such as multiples of 45°) appeared in June 2026.
R form
The R form is used far more for maximising a reciprocal than for solving an equation.
| Variant | Papers | Count |
|---|---|---|
| R cos(x ± α) | Jan 2020, Oct 2020, Jan 2023, Jun 2025 | 4 |
| R sin(x ± α), including 2x arguments | Jun 2021, Jun 2022, Oct 2022, Jan 2024, Jun 2024, Oct 2025, Jan 2026 | 7 |
| Double-angle rewrite before the R form | Jun 2024, Jan 2026 | 2 |
| Maximum or minimum of a reciprocal such as c/(d + f(2x)) | Jan 2020, Oct 2022, Jan 2023, Jun 2025, Oct 2025 | 5 |
| Solve an equation via the R form | Oct 2020, Jun 2022, Jan 2024 | 3 |
| Context model (seabird height) | Oct 2020 | 1 |
| Range of a squared version | Jun 2021 | 1 |
α is normally asked in radians to 3 d.p.; October 2022 asked for 4 s.f., June 2024 for 3 s.f., and October 2025 for an exact value.
Integration
Integration is the topic most likely to finish with an exact “p + q ln r” answer.
| Variant | Papers | Count |
|---|---|---|
| Divide first, then ln | Jan 2020, Oct 2020, Jun 2021, Jun 2022, Oct 2022, Jan 2023, Jun 2024, Oct 2024, Jan 2025, Jan 2026 | 10 |
| f′(x)/f(x) → ln, without division | Jan 2021, Jan 2022, Jan 2024, Oct 2025 | 4 |
| Reverse chain rule on (ax + b)n | Jan 2020, Oct 2021, Jun 2021, Jan 2022, Jun 2024, Jan 2026 | 6 |
| f′(x)[f(x)]n | Jan 2021, Oct 2021, Jun 2023, Jun 2025 | 4 |
| Trig integrand needing an identity | Oct 2020, Oct 2021, Jan 2023, Jun 2023, Oct 2023, Jun 2026 | 6 |
| Area under a curve, or with a normal or second graph | Oct 2020, Oct 2021, Jun 2024, Oct 2024, Jun 2026 | 5 |
| Find a constant from an integral’s value | Jun 2022, Oct 2025 | 2 |
Divisors range from linear (most), to a repeated factor (x − 1)2 in January 2020, to a quadratic (October 2020, October 2022, January 2025). January 2026 needed a common factor cancelled before dividing.
How the mark schemes award and withhold marks
The schemes give method marks generously and accuracy marks strictly. The same rules recur scheme after scheme; the examples below come from the 2020–2023 schemes.
| Rule | How the scheme applies it | Examples |
|---|---|---|
| Calculator-only answers | Where the paper says solutions relying on calculator technology are not acceptable, an unsupported answer scores 0 | Oct 2020 Q5(b): 5/24 with no working scores nothing; Jan 2022 Q3(ii): a calculator value of the integral scores 0; Oct 2023 Q1: 1.7340 alone scores M0A0A0 |
| “Show that” answers (A1*, cso) | The final mark needs a complete solution with no errors and correct notation | Oct 2020 Q5(a): sin x2 written for sin2x loses the A mark; Jan 2022 Q6(a): “−17 < 0” is not enough without stating f′(x) < 0 |
| Sign change | Both values correct to 1 s.f., the words sign change and continuous, and a conclusion | Oct 2023 Q1(a); Jan 2022 Q5(a); Oct 2022 Q5(b) |
| Iteration | At least one iterate must be shown before the root | Jan 2022 Q5(b): the root needs evidence of the iteration; Oct 2023 Q1(b) |
| R form | R must be exact; α in degrees scores A0 when radians are asked | Oct 2020 Q7(a): 4.12 for √17 is B0, 75.96° is A0; Jun 2021 Q9(a); Oct 2022 Q8(a) |
| Solutions in a range | All solutions and no extras; an extra value inside the range loses the final A1 | Oct 2020 Q1; Jun 2021 Q2(b); Oct 2022 Q9(b) |
| Range notation | Ranges must use f(x) or y; using x scores A0 | Oct 2020 Q3(b): x ≥ √7 not allowed; Oct 2022 Q2(a); Jan 2022 Q6(c): 5 < x < 22 scores B1 B0 |
| Inverse functions | The domain is part of the answer | Jun 2021 Q4(b): final A1 needs both rule and domain; Oct 2023 Q2(b): B1 for x ≠ 1 |
| ln with negative arguments | Missing modulus signs cost the final A mark | Jun 2021 Q3(ii)(b): limits −8 to −5 without |x + 2| lose the last A1 |
| Modulus inequalities | “or” (∪) for two regions; ∩ or a single joined inequality scores A0 | Jan 2022 Q7(b); Oct 2023 Q9(c) |
| Units and context | Units must be right and the context stated | Oct 2020 Q7(b): 3.37 cm is A0; Oct 2022 Q5(a): must say loss and include £; Oct 2023 Q4(c): 4.73 alone is A0 |
| Interpreting constants | Must convey the proportional change in context; vague answers score B0 | Oct 2023 Q6(c): “rate of change” or “q is the gradient” rejected; Oct 2023 Q4(d): “the number is too large” is B0 |
| Sketches | Key points must be exact and on the graph; the graph overrides the text | Jan 2022 Q7(c); Jun 2021 Q4(c): exact intercepts required |
| Exact answers | The requested form, with common factors cancelled | Oct 2022 Q1(b): 4 + ln 8; Oct 2023 Q2(c): 11 + 7√2, factors cancelled |
| Dependent and follow-through marks | dM needs the earlier M; ft marks follow a candidate’s own earlier values | Oct 2020 Q1 (dM1 needs the solved quadratic); Oct 2022 Q8(c)–(d) and Jun 2021 Q9(b) (B1ft on their R) |
The general instructions are the same in every scheme: a misread costs 2 of the A or B marks in that part, and when several attempts are left uncrossed, the best single attempt is marked.
Trends from 2020 to 2026, and the October papers
Fewer hints and more combined questions, while the core question types stay fixed.
- Composite-function equations are now standard. Seven of the last nine papers included an equation in fg, gf, gg or f−1, against four of the first ten.
- x = f(y) has become regular. It appeared in 8 of the 10 papers since June 2023, often as the final question.
- Exponential questions are growing. Two-model comparisons arrived in June 2023 and returned in both 2026 papers; October 2025 and June 2026 gave the topic 12 marks each.
- Modulus is being paired with other curves. A quadratic (October 2022), a log (October 2023), a cubic (October 2025) and an exponential-and-log curve (June 2026) all came after 2021.
- Less scaffolding. October 2025 Q7 (7 marks) and June 2026 Q6 (6 marks) gave a single instruction with no sub-steps.
- Iteration has moved to the front. It was Question 1 in 5 of the last 8 papers.
- Steady, not trending: algebraic division before integrating (10 of 19 papers, every year since 2020) and the R form (11 of 19).
The R form has never been missing from more than two papers in a row. When it was missing, it came back in the very next paper four times out of seven.
The six October papers
| October | Largest question | Modulus | R form | Exponential model | x = f(y) | Distinctive feature |
|---|---|---|---|---|---|---|
| 2025 | Q8, 13 marks: modulus with a cubic | V with a cubic | Yes, exact α | Fraction model, 8 marks | No | Discriminant for two turning points (Q7) |
| 2024 | Q9, 14 marks: quotient, normal, division, area | f(|x|) and |f(x)| of a quadratic | No | Two-exponential heart-rate model with iteration, 12 marks | Yes (polynomial in y) | sin 3x identity (Q5) |
| 2023 | Q7, 10 marks: e^(−x2)(2x2 − 3)2 | Modulus of a ln function | No | A − Be^(−kt), 7 marks | Yes (sin2 4y) | cos 2A proof feeding an integral (Q3) |
| 2022 | Q7, 12 marks: modulus with a quadratic | Inverted V with a quadratic | Yes, reciprocal minimum | Profit model with ln and iteration | Yes (log10 inverse) | Trig proof using sin 3θ (Q9) |
| 2021 | Q2, 10 marks: modulus and functions | V with an inequality | No | Constant minus exponential, 6 marks | Yes (arcsin) | Compound angle to tan θ (Q4) |
| 2020 | Q9, 14 marks: division, tangent, area | Inverted V with k range | Yes, seabird model | Exponential equation only | Yes (ln sin y) | sin 3x proof and ∫ sin3x (Q5) |
Three patterns stand out in October papers. Every one since 2022 had a modulus question beyond a plain V. Three of the six used sin 3x or sin 3θ. Their largest question averaged 12 marks, above the all-session median of 11 for a paper’s largest question.
Predictions for October 2026
Eight question types are near-certain or very likely, because each appeared in at least 17 of the 19 papers. The open questions are the R form (slightly better than even), x = f(y) (likely) and whether integration is a division question or a trig-identity one.
The likelihoods below are judgement-based estimates from frequency, recency and October patterns; they are not statistical forecasts.
Near-certain (about 95%)
- Iteration with a sign change: x2 and the root to 4 d.p. It opened 5 of the last 8 papers.
- Inverse and composite functions: f−1 with its domain, then an equation in fg or gf (7 of the last 9 papers).
- A modulus graph: expect 8–12 marks after June’s 5, probably with letters in it or a second curve.
- A trig “show that”, then “hence solve”: usually ending in a quadratic in one trig function, in degrees.
- Integration of some kind: division then ln (about 50%) or a trig-identity integral (about 35%).
Very likely (about 90%)
- Product or quotient rule with stationary points: a “show f′(x) = …” in factorised form.
- An exponential model: a constant, a time, a rate and a limit; possibly two models compared.
- A log graph turned into y = abx: a constant to interpret and a rate via ln b.
Likely (about 75%)
- x as a function of y: dy/dx in terms of x through a Pythagorean identity, then a tangent or normal.
Slightly better than even (55–60%)
- The R form: missing in June 2026, but never absent three papers running, and back straight after an absence 4 times in 7.
Outside bet (about 30%)
- sin 3x or tan 3x: used in 3 of the 6 October papers.
Expect one hard question (about 60%)
- A 6–7 mark part with no steps given, like October 2025 Q7 and June 2026 Q6.
Predicted paper blueprint
A plausible October 2026 paper built from the most likely form of each topic, at typical sizes. Question order varies from paper to paper; this follows a common order.
| Q | Marks | Topic | Predicted content |
|---|---|---|---|
| 1 | 6 | Iteration | Sign change on a given interval; show a rearrangement; x2 and the root to 4 d.p. |
| 2 | 8 | Functions | A rational or log function: range, f−1 with domain, a composite value, an equation in fg or gf |
| 3 | 6 | Log graph | Line of log10y against t to y = abt; interpret b; rate of change at a given time |
| 4 | 8 | Differentiation | Product rule on a polynomial and an exponential; show a factorised f′(x); exact stationary points; a range |
| 5 | 7 | Integration | Divide, then integrate to p + q ln r |
| 6 | 9 | Exponential model | A ± Be^(−kt): initial value, a constant, a time, a rate and a limit |
| 7 | 7 | x = f(y) | x as a trig function of y; dy/dx in terms of x; a tangent or normal |
| 8 | 9 | Modulus | A V or inverted V with letters, or paired with another curve; vertex, intercepts, an inequality, a range of k |
| 9 | 8 | Trig | Prove a double-angle or reciprocal identity; hence solve a quadratic in one trig function in degrees |
| 10 | 7 | R form | Express in R form; extreme value of a reciprocal; the x where it occurs |
If the R form is left out, its marks go to other topics: in June 2026 the exponential models (12 marks) and integration (11) both ran larger than usual.