Pure Mathematics 4

Battle Card

Provenance: written 2026-09-29. This is the fourth WMA14 technique-hub document, deliberately built as a condensation-and-ranking layer on top of the other three rather than a fourth independent research pass — no new primary Pearson document was fetched for it. Every fact below traces to research/veridian/WMA14-verified-facts.md (11 series with at least one primary document, 4 full Examiners' Reports — January 2021, October 2021, October 2022, January 2024 — plus mark-scheme-only verification for January 2026, no examiner's report published for that series yet) via the three existing WMA14 technique-hub documents this card reuses and cites rather than restates: codex-formula-and-method-bank.md (formula status and method skeletons), forge-exam-technique-guide.md (tariffs, M/A/B mechanics, common-error digest) and signal-frequency-and-boundaries.md (topic frequency, and the confirmed absence of any real grade-boundary source). What is genuinely new here is the format — a single dense last-look page, not a fourth discursive layer — and the explicit top-5 danger ranking in §4, which does not exist as a ranked list anywhere else in this course's WMA14 material. No claim below is re-derived from a primary PDF independently of those three documents; where one of them flags a status as ambiguous or unresolved, this card repeats that flag rather than resolving it. Not affiliated with or endorsed by Pearson Edexcel.

#WMA14 · BATTLE CARD — Last-Look Reference

Pure Mathematics 4. Read this in the last five minutes before you go in, or between questions if you freeze. It is not a lesson — if a line here doesn't make sense, that's the cue to go back to the named lesson file or technique-hub document, not to guess.

#1. Paper facts, at a glance

Fact	Value	Source
Total marks	75	Spec P4.2.2; printed on every cover page
Duration	1 hr 30 min	Spec P4.2.2
Questions	All compulsory, no choice	Spec
Question count	Not fixed — confirmed 9 (Jun 2022, Jan 2026), 11 (Oct 2022)	facts bank §2
Calculator	Allowed throughout, no non-calculator section exists for this spec	facts bank §2
Calculator ban	Ordinary scientific only — no symbolic algebra/calculus, no stored formulae, no machine-to-machine link	Appendix 6, spec pp.85–86
AO mark ranges (of 75)	AO1 25–30 · AO2 25–30 · AO3 5–10 · AO4 5–10 · AO5 5–10	Spec p.69
Grade boundaries	No primary source exists in this course's review — none stated here, none implied	SIGNAL §2

Mark-code decode: M1 = method attempted and applied (can be earned by implication from working, but safer if the formula is quoted first). A1 = accuracy, cannot stand without its M mark. B1 = independent, unconditional. dM1 = a method mark dependent on an earlier mark — a distinct, separately-checkable stage, not folded into it. An asterisked final mark on a "show that" line (A1*) requires every prior line to be a genuine, correct consequence of the one before — see §4.1. (FORGE §2, paper-anatomy.ts)

#2. Formula & definition-cue quick-fire

Status tags: G = given in the formula booklet · M = must memorize (spec's own list) · T = technique, no discrete formula exists · A = status genuinely ambiguous/unresolved in the source review.

Spec	Cue	Status
1.1 Proof	Contradiction shape: assume → derive → name the impossibility → conclude. No formula.	T
2.1 Partial fractions	3 linear factors → 3 terms; 1 repeated factor $(cx+d)^2$ → extra term $\dfrac{C}{(cx+d)^2}$ . Degree(num) ≥ degree(denom) → long divide first. Quadratic-factor denominators $(x^2+a)$ are not P4.	T
3.1 Parametric conversion	Eliminate $t$ by substitution or full coefficient-matching — never an assumed target form checked loosely against a couple of points.	T
4.1 Binomial, rational $n$	$(1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots$ , valid $\lvert x\rvert<1$ . For $(ax+b)^n$ : factor to $b^n\left(1+\dfrac{a}{b}x\right)^n$ , valid $\lvert x\rvert<\lvert b/a\rvert$ — re-derive the validity condition from the actual $a,b$ in front of you, don't reuse a previous question's.	G
5.1 Parametric differentiation	$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ , $dx/dt\neq0$ . Tangent: $y-y_1=m(x-x_1)$ . Normal gradient: $-1/m$ .	G/T
5.2 Connected rates	Chain-rule composition, e.g. $\dfrac{dV}{dt}=\dfrac{dV}{dr}\cdot\dfrac{dr}{dt}$ — every related quantity must be substituted as a function of the diff. variable before differentiating, never held fixed.	T
6.1 Volume of revolution	$V=\pi\int y^2\,dx$ (x-axis only — not $\pi\int x^2\,dy$ ). Parametric: $V=\pi\int y^2\dfrac{dx}{dt}\,dt$ . Status unresolved either way — not on the spec's memorize list, not in the booklet's P4 section, guidance text only.	A
6.1 Squaring a compound $y$	Sum/difference $y$ → cross term → double-angle $\sin2\theta=2\sin\theta\cos\theta$ . Bare $\sin^2\theta$ / $\cos^2\theta$ → power-reduction $\dfrac{1\mp\cos2\theta}{2}$ . Two different identities — don't conflate them.	G (derived)
6.2 Integration by parts	$\int u\,\dfrac{dv}{dx}\,dx=uv-\int v\,\dfrac{du}{dx}\,dx$ . $\int\ln x\,dx$ : take $u=\ln x$ , $dv/dx=1$ . A second application is a dependent mark on the first.	G
6.3 $\ln$ integral	$\int\dfrac{1}{ax+b}\,dx=\dfrac1a\ln\lvert ax+b\rvert$ — the $1/a$ factor is a real, two-series-confirmed drop (Oct 2022 Q2; Jan 2024 Q2b).	G (cumulative)
6.4 Separable DE	Separate variables, keep every constant on the correct side; target-variable-as-subject at the end — no illegal $\frac1{p+q}\to\frac1p+\frac1q$ .	T
6.5 Area, parametric curve	$A=\int y\,\dfrac{dx}{dt}\,dt$ — convert $x$ -limits to $t$ -limits before integrating, don't reuse them raw.	T (derived)
7.6–7.7 Scalar product	$\mathbf a\cdot\mathbf b=x_ax_b+y_ay_b+z_az_b$ — the only P4 item on the spec's own must-memorize list. $\cos\theta=\dfrac{\mathbf a\cdot\mathbf b}{\lvert\mathbf a\rvert\lvert\mathbf b\rvert}$ .	M
7.7 Skew lines	Not skew until both shown: direction vectors not scalar multiples (not just "not identical"), and the two line equations give an inconsistent system. "Not perpendicular" proves neither.	T
7.4–7.5 Foot of perpendicular	From $A$ onto $\mathbf r=\mathbf c+\lambda\mathbf d$ : form $\overrightarrow{AX}=(\mathbf c+\lambda\mathbf d)-\mathbf a$ first, then set $\overrightarrow{AX}\cdot\mathbf d=0$ — never dot the general point straight against $\mathbf d$ . Substituting $\lambda$ back for $X$ 's coordinates is its own dependent mark.	T
7.4 Reflection	Given foot $X$ : $B=2X-\mathbf a$ .	T (derived)
7.7 Vector-area	Parallelogram: $\lvert\mathbf a\rvert\lvert\mathbf b\rvert\sin\theta$ or $\lvert\mathbf a\times\mathbf b\rvert$ — both credited. Cross-product route's spec/booklet status is not traced, credited on one 2026 scheme's "Alt" line only.	A

Full derivations and worked chains: the 12 files in content/veridian/curriculum/wma14/; full status reasoning: codex-formula-and-method-bank.md Parts 1–2.

#3. Tariff reminders

Real confirmed single-question sizes — not a per-paper weighting; this spec publishes none beyond the AO table in §1. Read as scale evidence, not a promise for your sitting.

Topic	Seen in	Real confirmed size
Proof by contradiction	6 of 7 series mined	4–5 marks (Jun 2022 Q9 = 4; Jan 2026 Q8 = 5)
Binomial expansion	5 of 7	6 marks (Oct 2024 Q1, 4+2)
Implicit/parametric differentiation	4 of 7	9 marks (Jan 2021 Q6, 5+4)
Scalar product / skew lines	4 of 7	11 marks (Jan 2026 Q9, 3+4+2+2)
Separable DE / rates	4 of 7	8 marks (Jan 2026 Q3, 6+2)
Partial fractions	3 of 7	no single confirmed total captured
Parametric conversion	3 of 7	7 marks (Jan 2021 Q4, 5+2)
Volume of revolution	3 of 7	10 marks (Jan 2021 Q9, 7+3)
Integration by parts	3 of 7	part of a 10-mark question (Jun 2022 Q8b)
Position vectors / foot of ⟂	1 of 7	9 marks (Oct 2021 Q7, 7+2)
Area, parametric curve	1 of 7	16 marks — the largest confirmed item in the whole facts bank (Jan 2026 Q7, 2+5+6+3)

The bottom two rows are thin evidence, not rarity — area under a parametric curve is compulsory, spec-listed content that simply had zero citations until the Jan 2026 series was reviewed. Don't deprioritize either. (SIGNAL §1)

#4. Top 5 danger warnings, ranked

Ranked by how many independent topics/series each pattern is confirmed across — this ranking itself is new synthesis, not copied from any single existing document.

1. The fudged-reverse-fit pattern — any "show that" or "hence show" line. Any question with a printed target is graded on whether every line is a genuine forward consequence of the one before it, not on whether your last line matches the target. Confirmed independently in three different topics: proof (Jan 2021 Q3), parametric elimination (Jan 2021 Q4), separable DE (Jan 2021 Q10). If your working doesn't reach the given answer, find the real error — never write the target down anyway. (FORGE §4)

2. Rounding an exact answer early. Verbatim mark-scheme rule: marks are "normally lost" for resorting to rounded decimals where an exact or surd answer was required — independent of whether your method was correct. Keep exact values until the question itself asks for a decimal. (FORGE §2)

3. Substituting the wrong reference point. Two independent topics, same underlying instinct — reaching for a convenient nearby object instead of the one the question actually names: evaluating a gradient at the origin by default instead of the stated point (Jan 2021 Q6), and dotting a line's general point directly against the direction vector instead of first forming $\overrightarrow{AX}$ (Oct 2021 Q7). (FORGE §4)

4. Wrong pair of vectors for a named angle. The single most robustly attested specific error in the whole facts bank — confirmed across three independent series (Jan 2021 Q2a; Oct 2022 Q3b; Jan 2024 Q6c). Angle $ABC$ needs $\overrightarrow{BA}$ and $\overrightarrow{BC}$ — both from vertex $B$ — never a position vector substituted in, and never a vector from the wrong endpoint. Then check acute vs. obtuse before reporting whatever $\arccos$ returns. (CODEX §2.10)

5. A question-specific rubric can override the paper's general calculator policy. WMA14 is calculator-allowed throughout, but a specific real question can print its own stricter ban. Jan 2026 Q7 — the paper's largest confirmed item at 16 marks — has "Solutions relying on calculator technology are not acceptable" printed twice, covering all four parts. A numerically-correct final answer with no algebraic working scores nothing on a question like that. (FORGE §2, §1)

#5. Honesty check

No grade-boundary figure appears anywhere above, and none should be inferred — confirmed absent from this course's entire WMA14 source review (SIGNAL §2).
Every January 2026 item cited (Q3, Q7, Q8, Q9) verifies real mark-scheme mechanics only — no examiner's report has been published for that series yet, so none of it is a documented candidate error, only a confirmed real method or rubric line.
Evidence base: 11 series with at least one primary document, only 4 with a full examiner's report. "Seen in N of 7" above is scoped to the 7 series that received real question-by-question mining (SIGNAL §1) — not a probability, and not every series this course holds a document for.
This card adds no new primary-source fact beyond what the three documents below already established; its additive content is the single-page format and the ranked top-5 in §4.

#Sources used

research/veridian/WMA14-verified-facts.md (via the three documents below, not re-fetched independently); content/veridian/technique-hub/wma14/codex-formula-and-method-bank.md (Parts 1–2, formula status and method skeletons); content/veridian/technique-hub/wma14/forge-exam-technique-guide.md (§1 tariffs, §2 M/A/B mechanics, §4 common-error digest); content/veridian/technique-hub/wma14/signal-frequency-and-boundaries.md (§1 frequency, §2 grade-boundary gap). Curriculum files referenced by name, not duplicated: all 12 files under content/veridian/curriculum/wma14/.

Not affiliated with or endorsed by Pearson Edexcel. Every quoted fragment above is under fifteen words with its own attribution; all else is this course's own condensation of material already verified elsewhere in this repository.