Formula and Method Bank

Provenance: written 2026-09-28, grounded in research/veridian/WMA14-verified-facts.md (11 series with at least one primary document reviewed; 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) and cross-referenced against the 12 live lesson files under content/veridian/curriculum/wma14/ and content/veridian/technique-hub/wma14/forge-exam-technique-guide.md (completed the same day). No local WMA14 PDF archive exists on this machine — the standard ~/Downloads/PAPERI/IAL_Unit4_Complete_2024_2025/IAL_Materials/ has no Mathematics subfolder for Unit 4 — so every fact below traces through the verified-facts file's own independently-fetched primary sources (qualifications.pearson.com directly for the specification and formula booklet; a third-party mirror, cover-page-verified against Pearson's own publication codes, for the January 2026 series). Formulae are transcribed or restated as mathematical content, not as extended prose; any prose fragment quoted directly from a spec, booklet or mark scheme is under fifteen words with its own attribution, per this course's copyright discipline.

#WMA14 · CODEX — Formula & Identity Masterlist and Worked-Method Reference

#How to use this document

This sits in a different layer from the other two WMA14 documents. The FORGE guide (forge-exam-technique-guide.md) is exam craft — pacing, the M/A/B mark-coding system, a self-mark checklist. The 12 curriculum lessons teach the mathematics in full, each with its own worked chains and trap-taxonomy. This document is the reference layer between them: Part 1 is a masterlist of every real formula, identity and result the P4 spec actually requires, with each one flagged as booklet-supplied or memorize-only and its real domain restrictions stated. Part 2 is a set of short method skeletons — the reusable shape of a technique, compressed to the steps a student checks off mid-exam — organized by spec point, deliberately shorter than and distinct from the full worked-chain exemplars already in the curriculum lessons. Neither part re-teaches a topic from scratch; both assume the curriculum lesson has already been read once.

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#Part 1 — Formula & Identity Masterlist

Organized by the spec's own seven content areas and twelve numbered sub-points (verbatim numbering, Specification Issue 3, April 2019, pp.26–29, transcribed in full in the facts bank §1). Every entry is tagged:

• [GIVEN — Booklet, §X]: printed in Mathematical Formulae and Statistical Tables, Issue 2, January 2021, either in P4's own section or carried forward from P1–P3 (the booklet is explicitly cumulative — its own introduction states a P4 candidate may need formulae "first introduced in units P1 and P2").
• [MUST MEMORIZE]: spec P4.2.3's own list of formulae that will not appear in the booklet.
• [TECHNIQUE, NO DISCRETE FORMULA]: a real spec-tested skill with no single quotable identity to look up or memorize — the technique itself is what's examined.
• [STATUS AMBIGUOUS]: genuinely not resolved by the spec or booklet text — flagged rather than forced into one of the other three categories.

#1. Proof (spec 1.1)

No formula applies — spec 1.1 examines proof structure, consistent with AO2's "rigorous mathematical arguments and proofs" weighting (25–30 of P4's 75 marks, the same range as P1–P3 and FP1–FP3). The spec's own guidance names two specific results as worked templates: the irrationality of √2, and the infinitude of primes — both are "application to unfamiliar proofs" starting points, not formulae to recall.

[TECHNIQUE, NO DISCRETE FORMULA]. The four-part logical shape every real P4 proof-by-contradiction answer needs: assumption → algebraic derivation → identified impossibility → concluding sentence. See Part 2 §2.1 for the method skeleton and the three real contradiction mechanisms confirmed across series.

#2. Algebra and functions — partial fractions (spec 2.1)

[TECHNIQUE, NO DISCRETE FORMULA] — no booklet entry exists for partial-fraction decomposition itself; it is a real spec-tested skill with a fixed set of allowed forms, not a formula:

• Three distinct linear factors: $\dfrac{N(x)}{(ax+b)(cx+d)(ex+f)} = \dfrac{A}{ax+b}+\dfrac{B}{cx+d}+\dfrac{C}{ex+f}$
• One repeated linear factor: $\dfrac{N(x)}{(ax+b)(cx+d)^2} = \dfrac{A}{ax+b}+\dfrac{B}{cx+d}+\dfrac{C}{(cx+d)^2}$

Real condition, verbatim from spec guidance: "The degree of the numerator may equal or exceed the degree of the denominator" — a top-heavy fraction requires polynomial long division first, to reach a proper fraction plus a polynomial remainder, before either form above applies.

Genuinely excluded — verbatim: "Quadratic factors in the denominator such as (x² + a), a > 0, are not required." A denominator of the shape $(x^2+a)$ belongs to a different pure unit's spec point, not P4; a P4 lesson or mock question should never require this form.

#3. Coordinate geometry — parametric equations (spec 3.1)

[TECHNIQUE, NO DISCRETE FORMULA] — "conversion between cartesian and parametric forms" is elimination technique, not a quotable identity. See Part 2 §2.3 for the two real credited elimination routes and the domain/range method split.

#4. Binomial expansion for any rational n (spec 4.1)

[GIVEN — Booklet, P4 section, generalizing the natural-n binomial series already given from P2]:

$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots + \frac{n(n-1)\cdots(n-r+1)}{r!}x^r + \cdots, \quad |x|<1,\ n\in\mathbb{R}$$

The facts bank is explicit that this is the actual formula spec item 4.1 is built on, and it is given, not memorized — the tested skill is applying it, specifically to $(ax+b)^n$:

$$(ax+b)^n = b^n\left(1+\frac{a}{b}x\right)^n, \quad \text{valid for } |x| < \left|\frac{b}{a}\right|$$

— the factoring step, not the series itself, is what spec guidance actually targets ("for |x| < b/a, students should be able to obtain the expansion of (ax + b)ⁿ"). This factoring-and-restating-the-validity- condition is the single most consequential step in this topic: a validity condition copied from a different question's numbers, rather than re-derived from the actual $a$ and $b$ in front of the student, is a real, confirmed error class (Oct 2022, Q4).

#5. Differentiation (spec 5.1–5.2)

[TECHNIQUE, NO DISCRETE FORMULA for implicit differentiation itself — it is chain/product rule applied to an equation in two variables, both assumed P1 prerequisite knowledge, not re-given in the P4 booklet section]. The one genuine result spec 5.1 adds beyond P1–P3 differentiation rules:

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{dx}{dt}\neq 0$$

— parametric differentiation, differentiate $x(t)$ and $y(t)$ separately, then divide.

Tangent/normal at a point (P1 prerequisite, not re-stated in the P4 booklet section, but constantly used here): $y-y_1=m(x-x_1)$ for the tangent; normal gradient $=-\frac1m$ for the perpendicular line.

Connected rates of change (5.2) [TECHNIQUE, NO DISCRETE FORMULA] — chain-rule composition, e.g. $\dfrac{dV}{dt}=\dfrac{dV}{dr}\cdot\dfrac{dr}{dt}$, built from whichever two (or more) related quantities the modelling context supplies. The real tested skill is substituting every related geometric quantity as a function of the single differentiation variable before differentiating — holding a dependent quantity fixed instead is a confirmed, specific error (Jan 2024, Q4).

#6. Integration (spec 6.1–6.5)

Volume of revolution (6.1) [STATUS AMBIGUOUS]:

$$V = \pi\int y^2\,dx \quad\text{(about the x-axis only — } \pi\int x^2\,dy \text{ is explicitly NOT required by this spec)}$$ $$\text{parametric form: } V = \pi\int y^2\,\frac{dx}{dt}\,dt$$

This formula appears neither in spec P4.2.3's "must memorize" list (narrowly the dot-product formula only, see §7 below) nor in the formula booklet's P4 section — it is only descriptive guidance text ("π∫y²dx is required"). No examiner report reviewed penalises a candidate for not knowing the formula itself, only for misapplying it after writing it down correctly — treated here as resolved-by-inference (expected general knowledge) rather than fully spec-verified, exactly as the facts bank flags it. A real, confirmed mark-scheme distinction worth carrying as a formula-adjacent fact: using $2\pi\int y^2\,dx$ instead of $\pi\int y^2\,dx$ is scored as a wrong constant (partial credit possible), distinct from and better than setting up an entirely wrong integral (Jan 2021, Q9).

Two distinct trigonometric identities are needed when squaring a compound $y$ — both booklet-adjacent, neither separately listed under that exact name:

• A sum/difference $y$ (e.g. $y=\sin\theta+\cos\theta$) squares to include a cross term — the booklet's own double-angle identity $\sin2\theta=2\sin\theta\cos\theta$ resolves it directly.
• A bare $\sin^2\theta$ or $\cos^2\theta$ term needs the power-reduction form instead — a direct rearrangement of the booklet's own double-angle results, not itself a separate booklet line: $$\sin^2\theta=\frac{1-\cos2\theta}{2}, \qquad \cos^2\theta=\frac{1+\cos2\theta}{2}$$ A student must be able to produce this rearrangement from the given double-angle identity on demand — the FORGE guide's own trap-taxonomy flags conflating the two as a real, checkable error.

Integration by parts (6.2) [GIVEN — Booklet, P4 section]:

$$\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$$

$\int\ln x\,dx$ is explicitly spec-required (take $u=\ln x$, $\dfrac{dv}{dx}=1$). More than one application may be required (e.g. $\int x^2e^x\,dx$, $\int e^x\sin x\,dx$, both named in spec guidance) — the second application is a dependent method mark on the first, not an independent one.

Integration by substitution (6.2) [TECHNIQUE, NO DISCRETE FORMULA] — understood as the reverse of the chain rule. Spec guidance names $\int x\sqrt{x-2}\,dx$ as a self-devised-substitution example; more complicated integrands will have the substitution given.

Integration using partial fractions (6.3) [GIVEN — Booklet, cumulative $\ln$ integral, applied to a decomposed fraction]:

$$\int\frac{1}{ax+b}\,dx = \frac1a\ln|ax+b|+c$$

The $\frac1a$ scaling factor is the single most-repeated confirmed recurring slip in this entire facts bank — independently confirmed across two series (Oct 2022, Q2; Jan 2024, Q2b). A second, distinct family is explicitly spec-required alongside it: $\int\frac{1}{(ax+b)^n}\,dx$ for $n\geq2$ (e.g. $\frac{3}{(x-1)^2}$, $\frac{2}{(2x-1)^4}$) — this is a reverse-power-rule integral, not a logarithm, a genuinely distinct integral family sourced from the same partial-fraction decomposition.

Separable first-order differential equations (6.4) [TECHNIQUE, NO DISCRETE FORMULA]:

$$\frac{dy}{dx}=f(x)g(y) \;\Longrightarrow\; \int\frac{1}{g(y)}\,dy=\int f(x)\,dx\;(+c)$$

Both general and particular solutions are spec-required (a boundary condition solves for $c$).

Area under a parametric curve (6.5) [TECHNIQUE, NO DISCRETE FORMULA, but a genuine result worth stating in full since it has no other formula-booklet analogue]:

$$A = \int y\,dx = \int y\,\frac{dx}{dt}\,dt$$

— limits converted from $x$-values into $t$-values before integrating. Spec guidance explicitly states students will not be expected to sketch the curve from its parametric equations, so the area's shape must be reasoned about algebraically (via the given bounding curves/lines), not read off a sketch.

#7. Vectors (spec 7.1–7.7)

Magnitude (spec 7.2 guidance): $|\mathbf a| = \sqrt{a_1^2+a_2^2+a_3^2}$; unit vector $\hat{\mathbf a}=\dfrac{\mathbf a}{|\mathbf a|}$.

Position vectors (spec 7.4 guidance, verbatim structure): $\overrightarrow{OB}-\overrightarrow{OA}=\overrightarrow{AB}=\mathbf b-\mathbf a$.

Distance between two points (spec 7.5 guidance, verbatim): $d^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2$.

Vector equation of a line (spec 7.6 guidance, verbatim forms): $\mathbf r=\mathbf a+t\mathbf b$ (point + direction) or $\mathbf r=\mathbf c+t(\mathbf d-\mathbf c)$ (two points).

[MUST MEMORIZE — spec P4.2.3, the only item on this list for the entire unit]: the component-form scalar (dot) product —

$$\text{for }(x,y,z)\text{ and }(a,b,c):\quad (x,y,z)\cdot(a,b,c)=xa+yb+zc$$

Every other P4-specific formula in this document is booklet-supplied; only this one is spec-flagged as something a student must know cold, with nothing to fall back on in the exam room.

Angle between two vectors (a direct consequence of the dot-product definition, presented in spec/booklet as such rather than separately "must memorize" — but used constantly enough that it needs the same fluency in practice):

$$\cos\angle AOB=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}$$

Perpendicularity: $\mathbf a\cdot\mathbf b=0$ (for non-zero $\mathbf a,\mathbf b$) $\iff \mathbf a\perp\mathbf b$ (spec 7.7 guidance, verbatim consequence).

Parallel condition [TECHNIQUE, derived from 7.3's "algebraic operations...and their geometrical interpretations"]: $\mathbf a\parallel\mathbf b \iff \mathbf a=k\mathbf b$ for some real scalar $k$ — not merely $\mathbf a\neq\mathbf b$; a confirmed real trap is showing two direction vectors aren't identical and treating that as proof they aren't parallel (Oct 2022, Q9).

Skew condition (spec 7.6 guidance: "Conditions for two lines to be parallel, intersecting or skew"): two lines are skew $\iff$ they are not parallel AND do not intersect — both conditions required. A confirmed real trap substitutes "not perpendicular" for one of these two required conditions, which is irrelevant to skewness (Jan 2021, Q8).

Parallelogram area — two routes, both real-mark-scheme-credited on equal footing (Jan 2026, Q9, "Alt(c)"):

$$\text{Area} = |\mathbf a||\mathbf b|\sin\theta \qquad\text{or equivalently}\qquad \text{Area} = |\mathbf a\times\mathbf b|$$

[STATUS AMBIGUOUS for the cross-product route specifically]: the spec 7.7 guidance transcribed in the facts bank names only the $\sin\theta$ route via the dot-product-derived angle; it does not itself name a vector (cross) product. The cross-product route's real-exam credit is confirmed via a 2026 mark scheme's own explicit "Alt" method, but its formula is standard vector algebra reused from general mathematical literacy, not something the facts bank traces to explicit P4 spec or booklet text — flagged rather than silently presented as booklet-given.

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#Master GIVEN vs MEMORIZE table (consolidated)

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#Part 2 — Worked-Method Reference

Short method skeletons, organized by spec point. These compress a technique to the checklist a student runs through mid-exam — they are deliberately not full worked chains (those live in the curriculum lessons cited alongside each entry) and deliberately not a restatement of the FORGE guide's self-mark checklist (that audits a finished script; this is the procedure for producing one).

#2.0 Avoiding the fudged-reverse-fit pattern (cross-topic)

The FORGE guide names this pattern — three independently-sourced lessons each document a version of the same failure under the shared tag fudged-reverse-fit — but stops at naming it. Here is the check that actually prevents it.

The problem in one sentence: a "show that"/"hence show" question prints the destination. Grading targets whether every line is a genuine forward consequence of the line before it — not whether the final line matches the target. A candidate who reaches a wrong intermediate result and then silently adjusts a later step to land on the printed answer anyway has produced something that looks like a proof without being one, and examiners are specifically trained to catch this (FORGE guide §2).

The check, run before submitting, not after:

1. Find the last line you are genuinely confident is correct — call it line $k$. Do not look at the printed target while doing this; judge line $k$ purely on whether the step that produced it was valid.
2. Attempt to derive the printed target from line $k$ alone, as a short, independent step in its own right. If that step is a clean, justified algebraic or logical move, you're done.
3. If step 2 does not go cleanly — if reaching the target from line $k$ requires an unjustified jump, a sign that appears from nowhere, or a term that silently vanishes — the real error sits between your last confident line and the target, not necessarily at line $k$ itself. Go back further rather than forcing the jump.
4. Never let the printed target influence what you write on the line immediately before it. A genuine derivation is one you could have written with the target covered up. If you had to glance at the target to know what the next line "should" say, that is the fudge, even if the algebra you then wrote down is individually correct.

Three real question-shapes where this is the actual tested skill (compact anchors only — full derivations are in the cited curriculum lessons, not reproduced here):

• Proof by contradiction: a one-line assertion that reads like a derivation but states something false — "n is an integer so n² + 1 is odd" is grammatically a derivation step and is simply untrue (Jan 2021, Q3). The fix from step 1 above: that line has to be independently checkable as true, not just plausible-sounding, before it's allowed to stand as "line $k$."
• Parametric elimination: assuming the target Cartesian form and checking it against a couple of sample points, rather than completing a genuine coefficient-matching derivation through to the end (Jan 2021, Q4). The mark scheme's own credited alternative ("Alternative 1") is a legitimate assumed-form route — but only when the coefficient-matching is carried all the way through as a real identity, not eyeballed against one or two values.
• Separable differential equations: "adjusting" an intermediate rearrangement step after an earlier slip, specifically because the printed target made the error visible (Jan 2021, Q10). This is the clearest real instance of step 4 above being violated — the target was consulted mid-derivation, not after.

Full worked chains: proof-by-contradiction.ts, parametric-equations-conversion-and-domain-range.ts, separable-differential-equations.ts.

#2.1 Proof by contradiction — method skeletons (spec 1.1)

Generic skeleton, every item:

1. State "Assume [negation of the claim]" as an explicit sentence — real mark-scheme generosity: "no need for explicit statement of assumption... accept if just a suitable equation is set up" (Jun 2022, Q9 MS), but the final A1* mark needs "a clear assumption at the start that is contradicted."
2. Set up a genuine algebraic consequence of the assumption.
3. Derive forward, naming each intermediate result in words ("…so $p$ is even"), not left in bare algebra.
4. Identify the specific impossibility — don't stop at "this has no solutions."
5. Close with an explicit sentence tying the contradiction back to the original claim.

Three real mechanisms, confirmed across independent series (full chains in proof-by-contradiction.ts):

• Parity/divisibility contradiction: assume the target expression equals some multiple, derive a parity clash (Jun 2022, Q9: $n^2-2$ never divisible by 4).
• Forcing a real square negative: reduce the assumption to an equation of the shape $x^2=(\text{negative number})$; the closing sentence is "no real $x$ satisfies this, so [claim]." Confirmed as a real, reusable mechanism across two independent series: Jan 2024 Q8 (a cubic has no stationary points) and Oct 2024 Q2 (two curves don't intersect, reducing to $x^2=-5,-3$).
• Exhaustion nested inside a contradiction: factor to a fixed product (e.g. $(2p+q)(2p-q)=46$), list every possible positive-integer factor pair, check all of them (Jan 2026 Q8's own dM1 explicitly requires "attempts to solve both valid pairs", not one), and conclude none produces an integer solution. This mechanism is structurally different from the other two — it's a finite case-check, not a single derived impossibility — and it's real-mark-scheme-confirmed to accept three separate valid routes to the same conclusion (factor-pair exhaustion, an odd/even parity split, or a direct substitution $q=2m$) on exactly equal footing.

#2.2 Partial fractions — method choice skeleton (spec 2.1)

1. Compare numerator and denominator degree first. If numerator degree $\geq$ denominator degree, do polynomial long division before attempting decomposition, to reach a proper fraction plus a polynomial remainder term.
2. Match the decomposition form to the denominator's real factor structure — a repeated linear factor $(cx+d)^2$ needs both a $\frac{B}{cx+d}$ term and a $\frac{C}{(cx+d)^2}$ term.
3. Choose substitution (plug in the $x$-value that zeroes one factor) over full simultaneous equations where both are viable — real-mark-scheme-confirmed as less error-prone (Jan 2024, Q2a: simultaneous equations were "more prone to error than attempts via substitution"). Long division into the identity is a third legitimate route, with its own specific sign-tracking trap: setting a numerator equal to $-6$ but then re-substituting $+6$ (Oct 2021, Q3a).
4. When the decomposition feeds an integral, attach the $\frac1a$ scaling factor to every $\int\frac1{ax+b}dx$ term the moment it's written down — the single most-repeated confirmed slip in this whole facts bank (Oct 2022, Q2; Jan 2024, Q2b, independently, a year apart).

Full worked chains: partial-fractions-decomposition-and-integration.ts.

#2.3 Parametric conversion & domain/range — method skeleton (spec 3.1)

Elimination — two real credited routes:

• Direct substitution: solve one parametric equation for $t$, substitute into the other, simplify fully.
• Coefficient matching (only when direct substitution is intractable): build the target general algebraic form, substitute the parametric expressions for $x$ and $y$ into it, and match coefficients across a genuine identity carried all the way through — real-credited as "Alternative 1" (Jan 2021, Q4), but only when completed as a full identity, not checked against a couple of sample points (see §2.0).

Domain/range — method depends on the shape of the parameter interval:

• Closed and bounded ($t\in[p,q]$): substitute both endpoints into the Cartesian function directly.
• Open or unbounded ($t>0$, or similar): reason about the limiting behaviour as $t$ approaches the excluded or infinite end — substituting an endpoint value that is never actually attained scores nothing.
• Check for an interior extreme value before assuming the two endpoints bound the range: if the Cartesian function has a turning point inside the parameter's own domain, the true minimum/maximum sits there, not at either endpoint — a confirmed real trap where "the minimum value" was missing from answers that substituted only the domain's edges (Oct 2021, Q5c).

Full worked chains: parametric-equations-conversion-and-domain-range.ts.

#2.4 Binomial expansion — method skeleton (spec 4.1)

1. Factor $(ax+b)^n$ into $b^n\left(1+\frac{a}{b}x\right)^n$ — keep the sign of $\frac{a}{b}$ through every subsequent term; a lost or flipped sign on the linear coefficient is a confirmed recurring slip (Jan 2024, Q1).
2. Apply the booklet's general series to $\left(1+\frac{a}{b}x\right)^n$ term by term up to the power the question asks for.
3. State the validity condition as $|x|<\left|\frac{b}{a}\right|$, derived from this question's actual $a$ and $b$ — not copied from a different question's numbers (Oct 2022, Q4).
4. If the original expression is already in $x^2$ (e.g. $(1-4x^2)^n$), substitute $X=x^2$ into the general series directly — do not force a needless split into two separate $(1\pm kx)^{n/2}$ brackets (Oct 2021, Q4).
5. To use the expansion as a numeric approximation: substitute the same $x$-value into the original algebraic expression first, to confirm exactly what quantity the series approximates — it may need a reciprocal or another transform before it matches the target (Oct 2024, Q1b: the series approximated $6^{-1/3}$, and the target was $\sqrt[3]{6}$, its reciprocal — submitting the raw series value cost 2 of the question's 6 marks).
6. Give the final answer as an exact fraction/surd unless a decimal is explicitly requested (Jan 2021, Q1).

Full worked chains: binomial-expansion-for-rational-n.ts.

#2.5 Implicit & parametric differentiation, tangents/normals — method skeleton (spec 5.1)

Implicit:

1. Differentiate every term of $F(x,y)=c$ with respect to $x$, term by term.
2. Bare $y^n$ term: chain rule, $ny^{n-1}\frac{dy}{dx}$.
3. Mixed $x^my^n$ term: product rule first (splitting into two pieces), then chain rule on the $y$-factor of the relevant piece — both rules applied together in the same term is confirmed as the single hardest mechanical step in this whole topic (Jan 2024, Q3a).
4. An $x$-only term never picks up a $\frac{dy}{dx}$ factor — check this explicitly on any line mixing $x$-only and mixed terms; a confirmed real trap over-applies the "add $dy/dx$" rule reflexively even to a term with no $y$ in it at all (Oct 2021, Q1: "$4x^2+8\to8x+8\,dy/dx$").
5. Rearrange to make $\frac{dy}{dx}$ the subject last, once every term is differentiated.

Parametric: differentiate $x(t)$ and $y(t)$ separately first, then divide — $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$. Do not eliminate $t$ and differentiate the Cartesian form unless the question specifically permits it.

Tangent/normal: evaluate $\frac{dy}{dx}$ at the specific point or parameter value the question actually names — not a default point, and never the origin by default. A confirmed real trap: a "surprisingly large number" of candidates simply took the named point to be the origin, or found the correct point but still evaluated the gradient at $(0,0)$ (Jan 2021, Q6).

Full worked chains: implicit-and-parametric-differentiation-tangents-and-normals.ts.

#2.6 Volume of revolution — method skeleton (spec 6.1)

1. Quote $V=\pi\int y^2\,dx$ first, before manipulating $y$ — the scheme's own stated safety net for the method mark.
2. Square $y$ fully, including any cross term from a compound expression.
3. Identify which identity the squared expression needs: power-reduction for a bare $\sin^2\theta$ or $\cos^2\theta$, or double-angle for a cross term produced by squaring a sum/difference — these are two distinct identities, not interchangeable (Jan 2021, Q9 confirms candidates specifically failing to apply $\sin2\theta=2\sin\theta\cos\theta$ here).
4. If the integrand doesn't simplify with a trig identity, check whether it's a recognisable reverse-chain- rule form (e.g. the integrand of $(2x^2+3)^{-2}$'s own derivative) before defaulting to parts or substitution out of habit — a confirmed real "wasted much time pursuing incorrect methods" pattern (Oct 2022, Q5).
5. If parametric, convert $dx$ into $\frac{dx}{dt}\,dt$ and convert the $x$-limits into $t$-limits before integrating.
6. Use $\pi$, not $2\pi$, unless the solid genuinely requires the doubled constant — using $2\pi\int y^2dx$ is scored as a wrong-constant partial-credit error, distinct from an entirely wrong integral (Jan 2021, Q9).

Full worked chains: volume-of-revolution.ts.

#2.7 Integration by parts & substitution — method skeleton (spec 6.2)

Parts: quote $\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx$ first. Choose $u$ to be the factor that simplifies on differentiation (a polynomial, or $\ln x$ specifically, taking $\dfrac{dv}{dx}=1$). A second application is a dependent method mark on the first — a coefficient or differentiation slip in the first application costs both (Oct 2022, Q7ii: "a common error was then to integrate/differentiate the term cos2x incorrectly" during the second application, and "some candidates failed to realise they required a factor of 3" — a coefficient-tracking failure carried across the two-step derivation). The tabular "DI" layout is a real, equally-credited alternative for tracking a repeated application (Jan 2024, Q5a).

Substitution: rewrite the entire integrand in the new variable, converting $dx$ via $du=g'(x)\,dx$ — not relabelling $dx$ as $du$ directly (Oct 2021, Q6: "candidates who ignored the dx and simply wrote it as du"). Convert the limits of integration into the new variable before evaluating — reusing the original $x$-limits with a $u$-integrand is a separately confirmed, independent error (Oct 2022, Q7i: "use of the x limits...instead of the u limits").

Full worked chains: integration-by-parts-and-substitution.ts.

#2.8 Separable differential equations & rates of change — method skeleton (spec 5.2, 6.4)

1. Separate variables: every $y$-term (with $dy$) on one side, every $x$-term (with $dx$) on the other. Check the constant lands on the correct side/denominator — a confirmed real error "moves" a constant incorrectly mid-rearrangement, producing the wrong starting integral entirely (Oct 2021, Q2).
2. Integrate both sides, one constant of integration total.
3. Apply the boundary condition to solve for the constant.
4. Make the target variable the subject — watch specifically for the reciprocal-of-a-sum error, $\frac1{p+q}\neq\frac1p+\frac1q$, at this final rearrangement step (Oct 2021, Q2, confirmed at exactly this stage of an otherwise-correct solution).
5. For a rates-of-change setup: substitute every related geometric quantity as a function of the single differentiation variable before differentiating — don't hold a dependent quantity fixed (Jan 2024, Q4); and check the geometric formula actually fits the solid described — a confirmed real error applies the sphere-volume formula to a cylinder (Oct 2021, Q9).
6. For a limiting-value extension, once a closed rational solution is reached (e.g. $h=\dfrac{at^2}{(1+bt)^2}$): read the limit as the independent variable $\to\infty$ directly from the ratio of leading-order coefficients — not by differentiating to hunt for a stationary point, which can be a genuine dead end if the rate function stays one-signed throughout its own domain (Jan 2026, Q3b: $\frac{dh}{dt}>0$ for every finite $t>0$ in that question, so $h(t)$ never actually has a critical point, only an asymptotic limit — a real, checkable reason the "differentiate and set to zero" instinct fails here).
7. Give the final answer in the units the question actually asked for — a numerically-correct answer in the wrong unit is a confirmed, real, cost-real-marks error (Oct 2021, Q9: seconds given instead of minutes).

Full worked chains: separable-differential-equations.ts.

#2.9 Area under a parametric curve — method skeleton (spec 6.5)

1. Quote $A=\int y\,dx=\int y\,\dfrac{dx}{dt}\,dt$.
2. Convert the $x$-limits into $t$-limits (solve $x=\text{limit}$ for $t$) before integrating — using the $x$-limits directly as $t$-limits without converting is a confirmed real trap for this spec point.
3. If the bounded region includes a straight edge as well as the curve (a normal derived via 5.1, a chord, an axis segment): split the region into a straight-edge component — computed directly via elementary coordinate geometry (a trapezium or triangle area) — plus the curve's own parametric-integral component over its portion of the boundary, then add them. Do not attempt to force the whole region into a single parametric integral. This decomposition is a genuinely real structure confirmed in this paper's largest documented single question (Jan 2026, Q7, 16 marks: a trapezium under a normal line, area found by elementary means, added to a parametric integral over the curve from the named point back to where the curve meets an axis).
4. Track the sign of $\dfrac{dx}{dt}$ and the order of the $t$-limits carefully — a reversed limit pair or an unaccounted sign flips the computed area's sign; a decreasing $x(t)$ over the region of integration is exactly where this needs checking.

Full worked chains: area-under-a-parametric-curve.ts.

#2.10 Vectors — method skeletons (spec 7.1–7.7)

Angle between two lines/vectors meeting at a named vertex: identify the two vectors that both start from that vertex — e.g. angle $ABC$ needs $\overrightarrow{BA}$ and $\overrightarrow{BC}$, both sharing vertex $B$, never a vector from the wrong endpoint or a position vector substituted in by mistake. This exact error — using the wrong pair of vectors for the named angle, or substituting a position vector for the needed direction vector — is confirmed three times independently across different series (Jan 2021 Q2a; Oct 2022 Q3b; Jan 2024 Q6c), the single most robustly attested error pattern in this whole facts bank. Quote $\cos\theta=\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}$ first; then check whether the question wants the acute or the obtuse angle before reporting whatever $\arccos$'s raw output happens to be (Jan 2021, Q2a: "many candidates gave the acute angle...as their final answer" when the obtuse angle was required). Note the real mark-scheme nuance that reaching the supplementary angle and simply taking its acute complement, without a convincing geometric justification, does not earn the final accuracy mark on a between-two-sides-of-a-shape question (Jan 2026, Q9b) — unlike a between-two-lines question, where the acute value is often the one wanted by default; check which question type is in front of you.

Skew lines: (1) show the two direction vectors are not scalar multiples of one another — not merely that they are not identical (Oct 2022, Q9); (2) set the two line equations equal component-wise, solve the resulting system for the two parameters using any two of the three equations, then substitute into the third — an inconsistency there proves non-intersection; (3) conclude skew only once both (1) and (2) are shown. Neither alone is sufficient, and "not perpendicular" is never a valid substitute for either condition (Jan 2021, Q8).

Foot of perpendicular from external point $A$ onto line $l:\mathbf r=\mathbf c+\lambda\mathbf d$:

1. Form the line's general point $X=\mathbf c+\lambda\mathbf d$.
2. Form $\overrightarrow{AX}=X-\mathbf a$ — not the direction vector dotted directly against itself, and not the general point $X$ dotted directly against $\mathbf d$ either. Both are the same real, confirmed trap: "most incorrect responses attempted to set $(c+\lambda d)\cdot d=0$ rather than $AX\cdot d=0$" — the general point genuinely varies with $\lambda$, so the wrong equation is solvable and gives a plausible-looking wrong answer, not an obviously broken one (Oct 2021, Q7).
3. Set $\overrightarrow{AX}\cdot\mathbf d=0$, solve for $\lambda$.
4. Substitute $\lambda$ back into $\mathbf c+\lambda\mathbf d$ for $X$'s actual coordinates — this is its own dependent method mark, not folded into the mark for solving $\lambda$.

Reflection of $A$ in $l$, having found foot $X$: $B=2X-\mathbf a$ — the same position-vector midpoint identity from spec 7.4 ($\overrightarrow{AX}=\overrightarrow{XB}$), not a new formula (Oct 2021, Q7b).

Area of a vector-defined parallelogram: $\text{Area}=|\mathbf a||\mathbf b|\sin\theta$ (using the two side vectors sharing a vertex) or the equivalent $|\mathbf a\times\mathbf b|$ component cross-product — both real-mark-scheme-credited on equal footing (Jan 2026, Q9c). For a related composite shape built by extending one side (e.g. a trapezium from a parallelogram): find the base shape's area first, then scale by the ratio of the parallel sides' lengths — a confirmed real route, $\times1.5$ scaling for a side extended to twice the original's length by a stated midpoint condition (Jan 2026, Q9d).

Full worked chains: vectors-scalar-product-and-skew-lines.ts, position-vectors-distance-and-foot-of-perpendicular.ts.

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#Flagged Gaps

Stated plainly rather than resolved by assumption, matching the discipline both the facts bank and the FORGE guide already use:

• Volume-of-revolution formula's memorize/given status is genuinely unresolved. Neither the spec's P4.2.3 "must memorize" list (narrowly the dot-product formula) nor the booklet's P4 section contains $\pi\int y^2dx$; it appears only as descriptive guidance text. Treated here, as in the facts bank, as resolved-by-inference rather than spec-confirmed either way.
• The vector cross-product area route (Part 1, §7) is not traced to explicit P4 spec or booklet text. Only the $\sin\theta$ route is named in the spec guidance transcribed in the facts bank; the cross-product route's real-exam credit rests on one 2026 mark scheme's own "Alt" note, not on a spec citation.
• The P3 differentiation/integration table's exact typeset entries were not independently re-verified against the booklet PDF this session. The facts bank itemizes these by name only (tan kx, sec x, cot x, cosec x for differentiation; sec²kx, tan x, cot x for integration) without transcribing Pearson's own notation. The standard forms used implicitly in this document's worked-method skeletons match universal calculus convention and this course's own P1–P3 material, but a fresh pdftotext pass against the booklet itself would be needed before quoting them as verbatim Pearson typesetting.
• Evidence base is narrower than this course's Economics/Business precedent: 4 full examiner reports (Jan 2021, Oct 2021, Oct 2022, Jan 2024) against a 13–15-series precedent elsewhere in this course. Every January 2026 item used above (Q3, Q7, Q8, Q9) verifies real mark-scheme mechanics only — no examiner report has been published for that series yet, so none of those items contributes a documented candidate misconception, only a confirmed real method.
• Partial fractions (2.1) and parametric elimination (3.1) have no dedicated booklet formula at all — both are pure technique. The GIVEN/MEMORIZE framing genuinely does not apply to these two spec points; they are tagged [TECHNIQUE, NO DISCRETE FORMULA] rather than forced into one of the other categories.
• No local WMA14 PDF archive exists on this machine. Every fact above traces through WMA14-verified-facts.md's own independently-verified live fetches (qualifications.pearson.com for the spec and formula booklet; a cover-page-verified third-party mirror for the January 2026 series), not a local archive copy — confirmed directly, consistent with the FORGE guide's own opening note.

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#Sources Used

Primary Pearson documents (independently verified in research/veridian/WMA14-verified-facts.md, converted with pdftotext -layout and cross-checked against -raw; the January 2026 series' cover pages independently checked against Pearson's own publication codes since it was fetched from a third-party mirror):

• Specification, Issue 3, April 2019 (pp.26–29, "Unit P4: Pure Mathematics 4"; pp.68–69, 85–86).
• Mathematical Formulae and Statistical Tables, Issue 2, January 2021 — the P4 section directly, plus the cumulative P1–P3 sections it carries forward.
• Mark schemes: January 2021, June 2022, January 2024, October 2024 (Q1, Q2), January 2026 (Q3, Q7, Q8, Q9).
• Examiners' Reports (full): January 2021, October 2021, October 2022, January 2024. Examiner's Report (single-question extract): October 2024.

Real questions cited by series and number in this document: Jan 2021 Q1, Q2(a), Q3, Q4, Q6, Q8, Q9; Oct 2021 Q1, Q2, Q3(a), Q5(c), Q6, Q7, Q9; Jun 2022 Q9; Oct 2022 Q2, Q4, Q5, Q7(i)(ii), Q9; Jan 2024 Q1, Q2(a)(b), Q3(a), Q4, Q5(a), Q6(c), Q8; Oct 2024 Q1(b), Q2; Jan 2026 Q3, Q7, Q8, Q9.

Course files cross-referenced (not duplicated): forge-exam-technique-guide.md, and all 12 files under content/veridian/curriculum/wma14/ — paper-anatomy.ts, proof-by-contradiction.ts, partial-fractions-decomposition-and-integration.ts, parametric-equations-conversion-and-domain-range.ts, binomial-expansion-for-rational-n.ts, implicit-and-parametric-differentiation-tangents-and-normals.ts, volume-of-revolution.ts, integration-by-parts-and-substitution.ts, separable-differential-equations.ts, area-under-a-parametric-curve.ts, position-vectors-distance-and-foot-of-perpendicular.ts, vectors-scalar-product-and-skew-lines.ts.

Not affiliated with or endorsed by Pearson Edexcel. Every quoted prose fragment above is under fifteen words with its own series/question attribution; formulae are transcribed as mathematical content, not prose; all method-skeleton commentary is this course's own, built on — not copied from — the cited primary documents.