Pure Mathematics 3

Battle Card

Provenance: written 2026-09-29, against research/veridian/WMA13-verified-facts.md (read in full — the primary provenance-checked source for every spec/booklet/mark-scheme fact below) and the three existing technique-hub documents for this paper, forge-exam-technique-guide.md, codex-formula-and-method-bank.md and signal-frequency-and-boundaries.md (each read in full). Those three already carry the exam-craft, formula/method and frequency/boundary material in depth — this document adds nothing new to any of them; it re-packages their already-cited findings into a single dense page for last-look use, and every line below traces back to a citation that already exists in one of those four documents. content/veridian/curriculum/wma13/paper-anatomy.ts was also re-read to cross-check the paper-vitals numbers for consistency. No formula, mark-scheme line, or exam question below is drawn from Pearson's own textbook — spec, formula booklet, and freely-published mark schemes/examiner reports only.

#WMA13 — Battle Card

#Pure Mathematics 3 (Pearson Edexcel IAL) — last-look reference, not a lesson

What this is. The one page to scan in the last ten minutes before the exam, or between questions if you freeze. It is deliberately not a re-teach — no derivations, no worked examples, nothing you haven't already met in the curriculum lessons or the technique-hub documents above. If a claim below needs its full citation, it's already sourced in full in one of those.

#Paper vitals

90 minutes, 75 marks, 9 or 10 questions — not fixed by the spec. Confirmed directly: Jan 2023 had 10 (6+6+5+7+9+8+9+5+11+9=75), Oct 2020 had 9, Oct 2021 had 10. Budget off the number printed next to the question, not memory of a practice paper.
Calculator allowed on the entire paper, but ordinary-scientific only — Appendix 6 bars "symbolic algebra, differentiation and/or integration." Specific questions additionally carry a bold instruction that solutions "relying entirely on calculator technology are not acceptable" (verified verbatim, Jan 2023 Q5; recurs Jun 2022, Jan 2024 Q8(c), Jan 2025 Q3).
Yellow booklet = Mathematical Formulae and Statistical Tables, Issue 2 (Jan 2021). P3's own section opens by saying you "may also require" the P1 and P2 sections too.
Average pace 72 sec/mark — real per-question spread runs 1 to 12 marks, so budget off each question's own printed tariff.
"Students must answer all questions" — no choice, no sections, one flat sequence.

#Formula & definition-cue masterlist

GIVEN = printed in the booklet for P3 (or the P1/P2 sections a P3 candidate may also draw on). MEMORIZE = not printed anywhere for this paper.

1. Algebra & functions

1.1 No formula — factor/remainder theorem and division, carried in from P2. Denominators tested are linear or quadratic only.
1.2 MEMORIZE: domain of f⁻¹ = range of f. Write a range in y/f(x) notation, a domain in x notation — a numerically correct range stated as "x⩽9" is rejected (Jan 2023 Q1(a)).
1.3 |x| piecewise, not a lookup formula. y=|f(x)| reflects any below-axis part in the x-axis; y=f(|x|) discards the x<0 branch and mirrors the x⩾0 branch in the y-axis. Neither printed.
1.4 MEMORIZE the four building blocks: af(x) vertical stretch sf a; f(x)+a vertical translation a up; f(x+a) horizontal translation a in the negative x-direction; f(ax) horizontal stretch sf 1/a. y=f(ax+b) explicitly not required as one fused rule — build combinations from the four in sequence.

2. Trigonometry

2.1 MEMORIZE: sec=1/cos, csc=1/sin, cot=1/tan=cos/sin. sec undefined where cos=0; csc/cot undefined where sin=0. MEMORIZE restricted domains/ranges: arcsin dom [−1,1] rng [−π/2,π/2]; arccos dom [−1,1] rng [0,π]; arctan dom ℝ rng (−π/2,π/2) open.
2.2 MEMORIZE: sec²θ≡1+tan²θ; csc²θ≡1+cot²θ.
2.3 GIVEN: sin(A±B)≡sinAcosB±cosAsinB; cos(A±B)≡cosAcosB∓sinAsinB; tan(A±B)≡(tanA±tanB)/(1∓tanAtanB); the four sum-to-product factor formulae.
MEMORIZE (spec's own "will not appear" list): cos²A+sin²A≡1; sin2A≡2sinAcosA; cos2A≡cos²A−sin²A≡2cos²A−1≡1−2sin²A; tan2A≡2tanA/(1−tan²A).
Excluded: t=tan(θ/2) substitution NOT required. Not excluded despite silence in the guidance column: sin3x≡3sinx−4sin³x is real, assessed content (Oct 2020 Q5(a), 4 marks).
Harmonic form: no boxed R/α formula — expand Rcos(θ∓α) or Rsin(θ±α) and match coefficients, every time. R stays exact (surd), α in the unit asked — a numerically-right α in degrees scored A0 when radians were required (Oct 2020 Q7(a)).

3. Exponentials & logs

GIVEN: e^(x ln a)=aˣ.
A model y=c+de^(−kt), d,k>0, has a genuine floor at y=c — a target below it gives a correctly-reasoned "no solution," argued as "log of a negative number," never "logs can't be negative" (that phrasing is explicitly rejected, Jan 2024 Q5(d)).
3.3 GIVEN (P2): logₐx=log_bx/log_ba. Linearisations (mechanisms, not printed): y=axⁿ → log y=n log x+log a (plot log y vs log x, gradient n, intercept log a); y=kbˣ → log y=x log b+log k (plot log y vs x, gradient log b, intercept log k). A later part can ask you to interpret a found constant in words, no further calculation — three components required together: direction, percentage to 2 s.f., "per year" (Jun 2026 Q5(b)).

4. Differentiation

4.1 MEMORIZE (none printed): e^kx→ke^kx; ln kx→1/x (no k survives); sin kx→k cos kx; cos kx→−k sin kx; tan kx→k sec²kx.
4.2 MEMORIZE: product rule (uv)'=u'v+uv'; chain rule dy/dx=dy/du·du/dx. GIVEN: quotient rule (u/v)'=(u'v−uv')/v²; sec x→sec x tan x; cot x→−csc²x; csc x→−csc x cot x.
4.3 MEMORIZE the relationship dy/dx=1/(dx/dy), valid where dx/dy≠0. An answer left in y when the question wants it in x is incomplete (Oct 2020 Q8(ii)).
4.4 MEMORIZE: d/dx[aˣ]=aˣln a — rebuild via e^(x ln a) if forgotten under pressure.

5. Integration

5.1 MEMORIZE: ∫cos kx=(1/k)sin kx+c; ∫sin kx=−(1/k)cos kx+c; ∫e^kx=(1/k)e^kx+c; ∫1/x=ln|x|+c (x≠0); ∫aˣ=aˣ/ln a+c. GIVEN: ∫sec²kx=(1/k)tan kx+c; ∫tan x=ln|sec x|+c; ∫cot x=ln|sin x|+c.
5.2 MEMORIZE the two recognition forms: ∫f'(x)/f(x)dx=ln|f(x)|+c; ∫f'(x)[f(x)]ⁿdx=[f(x)]ⁿ⁺¹/(n+1)+c. sin²x/tan²x/cos²3x-type integrands need a double-angle rewrite first — neither form applies to them directly.

6. Numerical methods

6.1 Not a formula — the sign-change/IVT idea, informally. The word continuity/continuous must be stated, not just "sign change" (see danger warning #1).
6.2 xₙ₊₁=g(xₙ). Not a formula to look up — a rearrangement-and-iterate method.

P1/P2 carried forward, real but not P3's own six strands: cosine rule a²=b²+c²−2bc cosA (P1); arithmetic/geometric series formulae, binomial series, change-of-base (P2). Trapezium rule is P2 content, not P3 — WMA13's own "Numerical methods" strand is root-location and iteration only; don't expect it here.

#Tariff & mark-code reminders

M = a correct method or a recognisable attempt. A = dependent — scores only once its M is already scored ("M0 A1 is impossible"). B = independent, no method needed.
dM1 / ddM1: the depth of dependency tracks how many method marks genuinely precede that step in that specific question, not a fixed per-topic label — confirmed singly-dependent (Jan 2022 Q7(b)) and doubly-dependent (Oct 2023 Q9(c); Oct 2020 Q5(a), an unrelated topic) on real papers.
Quote the formula before substituting — a method mark can sometimes be gained by implication from correct working, but is far more securely earned, and can be lost outright if a slip follows an unquoted formula (verbatim general marking principle; echoed in Oct 2020 Q3's examiner report on the quotient rule).
ft applies to A/B marks only, never M.
Misreads cost 2 marks off the A/B total for that part, never an M mark; subsequent A marks become "A ft."
Multiple attempts: cross out the wrong one. If none, or all, are crossed out, the marker scores your highest surviving attempt (verbatim Jan 2023 general guidance).
Exact answers: a correctly-rounded decimal still loses the final accuracy mark where an exact form or surds are required — repeatedly reconfirmed, not a style preference.
Calculator-only working on a flagged question scores zero, not partial credit — because every A depends on its own M, an unshown method caps the whole question's method-chain, even with the exactly right final number.
Glossary: bod · ft · cao · cso (no errors anywhere in that part) · isw · awrt · oe · SC · dep/indep · dp/sf · ★ (answer printed on paper) · → (second mark dependent on the first).

#Top-5 danger warnings (ranked by how repeatedly each is confirmed)

"Sign change" without the word "continuity" scores nothing on a root-location question. The single most repeated specific error in the whole verified archive — named across Oct 2022 ("the past few series"), Jan 2022 Q5(a), and Jan 2024 ("despite it being highlighted in every appropriate series"); Jan 2025 is the first series where the pattern finally seems to improve. The mark scheme needs three things together: both function values, a reason that names continuity and states the sign change, and "hence root."
Domain of f⁻¹ omitted, even with fully correct algebra. Confirmed near-identically across Jun 2022 Q2 ("the majority... still failed to state the domain"), Jan 2022 Q6 (modal score 2/3), and Oct 2021 Q1(b). Remember it's the range of f, in x-notation — a separately confirmed mix-up (Oct 2022 Q2(b)).
Units or context word dropped from a correct modelling answer. Confirmed across at least 4 series with near-identical wording: Oct 2021 Q3(b) ("tonnes... often omitted"), Jan 2022 Q4(c), Oct 2022 Q5(a) ("£" or "loss"), Jan 2025 Q2(a) ("m²"). The number is right; the accuracy mark isn't earned without it.
Calculator-only method on a flagged question. Literal rubric text bans it (Jan 2023 Q5), independently echoed in Jun 2022, Jan 2024 Q8(c), and Jan 2025 Q3 — a correct typed-in answer with no algebra shown scores zero, not partial credit.
Wrong angle unit, or a missing/extra solution, on "solve in this range." Confirmed across 5+ series (Oct 2020, Jan 2022, Jun 2022, Jun 2023, Jan 2024): degrees where radians were wanted (or the reverse), a second in-range solution never found, or an out-of-range one left in. On a doubled-angle equation, remember to halve back to θ — reporting the doubled value is its own separately-confirmed slip (Jan 2022 Q2(b)).

#Sources

research/veridian/WMA13-verified-facts.md, §1–§5 — spec content, booklet contents, mark-scheme conventions, per-topic verified findings.
content/veridian/technique-hub/wma13/forge-exam-technique-guide.md, codex-formula-and-method-bank.md, signal-frequency-and-boundaries.md — read in full; every citation above already exists in one of these three, condensed here into single-page form, nothing new asserted.
content/veridian/curriculum/wma13/paper-anatomy.ts — standing spec/timing reference, cross-checked for consistency on the paper-vitals section.