Formula and Method Bank

Provenance: compiled 2026-09-28. Primary compiled source: research/veridian/WMA13-verified-facts.md (13 real WMA13 series, Oct 2020–Jun 2026, mark schemes and examiner reports read against primary Pearson PDFs). Cross-referenced against the companion technique guide, content/veridian/technique-hub/wma13/forge-exam-technique-guide.md, and the 11 audited lessons in content/veridian/curriculum/wma13/ plus content/veridian/question-bank/wma13.ts — neither is reproduced here in full; both are cited by file and trap/question id. The formula-booklet claims in Part 1 were independently re-verified this pass directly against the primary PDF, ~/Downloads/IAL-Mathematics-Formula-Book.pdf (Mathematical Formulae and Statistical Tables, Issue 2, January 2021), converted with both pdftotext -layout and pdftotext -raw and cross-checked line-by-line against the research file's own §3 transcription — one genuine correction to that file's own claim was found in the process (the ∫cosec x dx entry; see §1.5 and the gaps section below). Mark-scheme structures in Part 2 were cross-checked against ~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_Mark_Schemes.md where a specific claim needed re-verification (see the June 2023 Q9 correction in §2.5).

#WMA13 (Pure Mathematics 3) — Formula & Identity Masterlist and Worked-Method Reference

#How to use this document

This is the "look it up" layer, not a lesson and not a technique narrative. forge-exam-technique-guide.md covers timing, the M/A/B mark-coding system, and the paper-wide calculator-only-method penalty — none of that is repeated here except where a formula's classification depends on it. The 11 lessons in content/veridian/curriculum/wma13/ teach each result from first principles, with full worked chains, diagrams and MCQ drilling — none of that is reproduced here either. What this document adds is the two things neither of those covers on its own:

• Part 1 is a single masterlist of every formula and identity spec P3 actually requires, spec-strand by spec-strand, each one flagged GIVEN (printed in Mathematical Formulae and Statistical Tables, "the Yellow booklet") or MEMORISE (the spec's own "will not appear in the booklet" list), with the real domain/range/validity condition attached where one exists. Use it to check, in seconds, whether a result needs to be recalled from memory under exam pressure or only found on the sheet.
• Part 2 is a bank of short, spec-topic-organised worked-method snippets — the generic M/A/B skeleton for a recurring real question shape, not a full worked exemplar. Where a lesson already has the full worked chain for a shape, this document names the lesson and gets out of the way; where the shape is one this course's question bank has a full real marked solution for (the two Oct 2024 items), this document gives the transferable method template and cites the full solution rather than repeating it.

Every claim below is either lifted directly from WMA13-verified-facts.md (itself primary-source-checked) or re-verified this pass against a primary PDF named at the point it's used. Nothing here is invented to fill a gap — where the real archive doesn't cover something, §3 says so.

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#PART 1 — Formula & Identity Masterlist

#1. Algebra and functions

1.1 Algebraic division and rational expressions. No new formula — this is a technique (factor/remainder theorem, carried in from the WMA12 prerequisite; long division or the equivalent-coefficients method), not a result to look up or memorise. The one real, checkable spec constraint: denominators tested are restricted to linear or quadratic, verbatim from the spec's own guidance column. GIVEN: nothing specific to this item; the general algebraic-manipulation baseline is assumed prior knowledge, not printed.

1.2 Functions. Composition notation ($fg$ means "do $g$ first, then $f$") and the inverse-function identity $f^{-1}f(x) = ff^{-1}(x) = x$ are definitions, not formulae to memorise or look up — they are simply known, per the spec's own guidance wording. Domain/range condition that actually costs marks: $f^{-1}$ exists only where $f$ is one-one; the domain of $f^{-1}$ equals the range of $f$, and a range/domain answer must be written in the notation the question's variable requires — a range is a set of outputs ($y$ or $f(x)$ notation), a domain a set of inputs ($x$ notation). Confirmed real-mark-scheme trap: Jan 2023 Q1(a) accepts $y \leqslant 9$ or $f \leqslant 9$ for a range but explicitly rejects $x \leqslant 9$ even though the number is correct.

1.3 The modulus function. $|x|$ is defined piecewise ($x$ for $x \geqslant 0$, $-x$ for $x < 0$) — a definition, not a formula to memorise separately. Two graph-transformation results the spec names directly and that must be reasoned from a sketch each time, not recalled as a boxed rule: $y = |f(x)|$ reflects in the $x$-axis any portion of $y=f(x)$ that lies below it (everything above stays fixed); $y = f(|x|)$ discards the original $x<0$ branch entirely and replaces it with a mirror image of the $x \geqslant 0$ branch reflected in the $y$-axis. Neither is printed in the booklet — both are worked from the graph of $y=f(x)$ every time.

1.4 Combinations of transformations. Four building-block transformations of $y=f(x)$, all definitional rather than "formulae" in the booklet sense — MEMORISE the mechanism, not a printed line:

Real, spec-verbatim exclusion: the guidance column explicitly states "the graph of $y = f(ax+b)$ will not be required" — a combined single-bracket transformation must be built from the four listed types applied in sequence (e.g. $y=2f(3x)$, $y=f(-x)+1$), never asked for as one fused rule.

#2. Trigonometry

2.1 Reciprocal trig — definitions (MEMORISE, not printed as such, though implied by other booklet entries):

$$\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$$

Domain restrictions (real, checkable, and mark-scheme-relevant): $\sec\theta$ is undefined wherever $\cos\theta = 0$ (odd multiples of $90°$/$\tfrac{\pi}{2}$); $\csc\theta$ and $\cot\theta$ are both undefined wherever $\sin\theta = 0$ (integer multiples of $180°$/$\pi$) — each function has a vertical asymptote exactly where the original ratio it reciprocates hits zero, not where the original ratio is itself undefined. Real, verbatim-confirmed error mode: writing $\csc\theta$ as the reciprocal of cosine rather than sine (the two three-letter reciprocal names are the single most confused pairing on this spec item — confirmed Jan 2022 Q2 examiner report). A second, distinct error confirmed on the same real question: folding a coefficient into the reciprocal itself, e.g. writing $3\csc\theta$ as $\frac{1}{3\sin\theta}$ rather than $\frac{3}{\sin\theta}$ — scored B0 on the real Jan 2022 Q2(a) mark scheme even though the underlying identity is "known," because the coefficient has been applied to the wrong side of the reciprocal.

Inverse trig — restricted domains (real spec requirement, verbatim: "Understanding of their graphs and appropriate restricted domains"). MEMORISE — not printed:

2.2 $\sec^2\theta \equiv 1+\tan^2\theta$ and $\csc^2\theta \equiv 1+\cot^2\theta$ — MEMORISE, on the spec's own "will not appear in the booklet" list. Both follow directly from $\cos^2\theta+\sin^2\theta\equiv1$ divided through by $\cos^2\theta$ or $\sin^2\theta$ respectively — a derivation, not two unrelated facts to hold separately.

2.3 Double angle, sum/difference, and harmonic form.

GIVEN, printed verbatim in the booklet's P3 section:

$$\sin(A\pm B) \equiv \sin A\cos B \pm \cos A\sin B, \qquad \cos(A\pm B) \equiv \cos A\cos B \mp \sin A\sin B, \qquad \tan(A\pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$

plus the four factor (sum-to-product) formulae — $\sin A + \sin B \equiv 2\sin\left(\tfrac{A+B}{2}\right)\cos\left(\tfrac{A-B}{2}\right)$ and the corresponding $\sin A - \sin B$, $\cos A + \cos B$, $\cos A - \cos B$ forms. Flagged gap, not previously in this course's material: none of the 13 series reviewed for WMA13-verified-facts.md contains a question that visibly requires the factor formulae specifically — they are genuinely on the printed sheet, but their examinability was not independently confirmed against a real question in this pass; treat them as "on the sheet, unconfirmed as directly tested" rather than assert they never appear.

MEMORISE — will not appear in the booklet, verbatim from the spec's own list:

$$\cos^2A+\sin^2A\equiv1, \quad \sin2A\equiv2\sin A\cos A, \quad \cos2A\equiv\cos^2A-\sin^2A\equiv2\cos^2A-1\equiv1-2\sin^2A, \quad \tan2A\equiv\frac{2\tan A}{1-\tan^2A}$$

The three equivalent forms of $\cos2A$ are not three facts, they are one identity substituted with $\cos^2A+\sin^2A\equiv1$ two different ways — but a real, well-attested error is reaching for the wrong one of the three mid-equation (confirmed Oct 2020 Q1, Q5). Real spec-boundary note, checked directly against a real past question: the half-angle tangent substitution ($t=\tan\tfrac{\theta}{2}$) is explicitly excluded by the spec's guidance ("the $t = \tan(\tfrac{1}{2}\theta)$ formulae will NOT be required"), but the triple-angle identity $\sin3x \equiv 3\sin x - 4\sin^3x$ is not named anywhere in the spec's guidance column and is nonetheless real, assessed content — confirmed Oct 2020 Q5(a), a genuine 4-mark "show that" derived from $\sin(2x+x)$ plus the double-angle results above. Do not treat "not named in the guidance column" as "safely excludable" for compound/multiple-angle derivations built from the given sum formulae.

Harmonic form — genuinely no formula to look up or memorise. There is no boxed "$R=\sqrt{a^2+b^2}$, $\alpha=\arctan(b/a)$" line printed anywhere in the P3 section, and this is a real, checkable fact worth stating plainly: $R$ and $\alpha$ are derived by expanding $R\cos(\theta\mp\alpha)$ (or $R\sin(\theta\pm\alpha)$) with the compound-angle formulae above and matching coefficients against the given expression $a\cos\theta+b\sin\theta$, every single time — the "formula" is the method (expand, match, solve for $R$ and $\alpha$), not a fact to memorise in closed form. Real, mark-scheme-confirmed condition: on the construction marks themselves (not just the final solved angle), $R$ must be left exact (a surd, not a rounded decimal) and $\alpha$ must be given in the unit the question names — verified verbatim, Oct 2020 Q7(a): a numerically correct $\alpha = \text{awrt } 75.96°$ scores A0 when the question asked for radians to 3 d.p.

#3. Exponentials and logarithms

3.1–3.2. $e^x$ and $\ln x$ are mutual inverses; $\ln x$'s own graph and domain ($x>0$) follow directly from that. GIVEN, printed in the P3 section: $e^{x\ln a} = a^x$ — the single identity that lets $a^x$ be rewritten in base $e$ for differentiation or integration by the memorised $e^{kx}$ results. Solving $e^{ax+b}=p$ (take $\ln$ of both sides: $ax+b=\ln p$) and $\ln(ax+b)=q$ (exponentiate both sides: $ax+b=e^q$) are direct consequences of the inverse relationship, not separate formulae. Real, verified domain trap: an exponential model $y = c + de^{-kt}$ with $d,k>0$ has a genuine floor at $y=c$ (as $t\to\infty$, $e^{-kt}\to0^+$ but never reaches $0$) — asking the model to reach a target below that floor produces an equation requiring $\ln(\text{negative or zero})$, which is a real, correctly-reasoned "no solution" answer, not an error to fix. Confirmed verbatim, Jan 2024 Q5(d): the accepted reasoning is that the log of a negative number cannot be found — not "logs can't be negative" (a log's output can be negative; what can never happen is a non-positive argument), which the real mark scheme explicitly refuses as ambiguous.

3.3 Log-linear graph parameter estimation. GIVEN, carried forward from P2: $\log_a x = \dfrac{\log_b x}{\log_b a}$ (change of base). The two linearisations spec 3.3 names verbatim are mechanisms, not printed formulae, derived by taking logs of both sides:

• $y = ax^n \Rightarrow \log y = n\log x + \log a$ — plotting $\log y$ against $\log x$ gives a straight line, gradient $n$, intercept $\log a$.
• $y = kb^x \Rightarrow \log y = x\log b + \log k$ — plotting $\log y$ against $x$ (not $\log x$) gives a straight line, gradient $\log b$, intercept $\log k$.

The base of the log used to plot is whatever the question states (base 6, base 10 — both real, confirmed contexts); mixing it up with base 10 or base $e$ mid-question is a real, confirmed error (Jun 2023 Q2 examiner report). Real, previously-undocumented nuance worth carrying into a method-reference: a further question part can ask for the constant's verbal interpretation with no further calculation at all — e.g. "interpret the value of $b$" once $a$ and $b$ are already found numerically — and the real mark scheme (Jun 2026 Q5(b)) requires three components stated together for the one mark: a direction word, the percentage to 2 s.f., and a per-year unit; six near-miss phrasings that drop exactly one of the three are named and explicitly not credited.

#4. Differentiation

4.1 — five standard results. Sum/difference rule and the power rule (both from WMA11) are assumed. Five new results, all MEMORISE — none of these five appear in the booklet:

$$\frac{d}{dx}[e^{kx}]=ke^{kx}, \qquad \frac{d}{dx}[\ln kx]=\frac{1}{x} \; (\text{no } k), \qquad \frac{d}{dx}[\sin kx]=k\cos kx, \qquad \frac{d}{dx}[\cos kx]=-k\sin kx, \qquad \frac{d}{dx}[\tan kx]=k\sec^2kx$$

Real, checkable reason $\ln kx$ has no $k$: $\ln(kx)=\ln k+\ln x$ by the log-of-a-product law, and $\ln k$ is a constant once $k$ is fixed — its derivative is $0$, so the $k$ never survives into the result. This is a derivation, not a special case to memorise separately.

4.2 — product, quotient, chain, and the three further trig derivatives.

$$\text{Product rule (MEMORISE): } (uv)'=u'v+uv' \qquad\qquad \text{Chain rule (MEMORISE): } \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

GIVEN, quotient rule, printed verbatim: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v-uv'}{v^2}$. The quotient rule is not a fourth independent rule — it is the product rule applied to $u\cdot v^{-1}$, with the chain rule supplying $\frac{d}{dx}[v^{-1}]=-v^{-2}v'$; both routes reach the identical result, and this course's method-comparison block on $\frac{e^{3x}}{x}$ (differentiation-rules.ts) proves it side by side.

GIVEN, printed verbatim: $\sec x \to \sec x\tan x$; $\cot x \to -\csc^2x$; $\csc x \to -\csc x\cot x$; $\tan kx \to k\sec^2kx$. All four can be rebuilt on the spot from $\frac{\sin kx}{\cos kx}$ etc. and the quotient rule if the booklet page isn't found fast enough under pressure — worth knowing how to rebuild them, not just that they're printed.

4.3 $\dfrac{dy}{dx}=\dfrac{1}{dx/dy}$ — MEMORISE the relationship (not printed as its own line; it is a direct consequence of the chain rule applied to $g(y(x))=x$, differentiated both sides). Valid wherever $\frac{dx}{dy}\neq0$. Used whenever differentiating with respect to the other variable is structurally easier — the spec's own example is $x=\sin3y$. Real, repeatedly-confirmed condition, not a footnote: an answer left in terms of $y$ (or in terms of the "wrong" variable generally) after inverting is a genuinely incomplete answer if the question asks for the result in terms of $x$ — confirmed across multiple series (Oct 2020 Q8(ii); the reciprocal-trig/differentiation-rules lessons' own trap taxonomies both name this independently).

4.4 Exponential growth/decay. MEMORISE — verbatim from the spec's own list, and not printed anywhere in the P3 section: $\dfrac{d}{dx}[a^x]=a^x\ln a$. Rebuildable on demand via the P3-section-given identity $e^{x\ln a}=a^x$: rewrite $a^x$ as $e^{x\ln a}$, then apply the memorised $e^{kx}\to ke^{kx}$ result with $k=\ln a$ (a constant, since $a$ is fixed) — $\frac{d}{dx}[e^{x\ln a}] = (\ln a)e^{x\ln a} = a^x\ln a$. Real, confirmed error distinct from a sign slip: applying the power rule ("bring down the power, reduce by one") to a term where the variable is the exponent, not the base — e.g. differentiating $N=100\times2.24^t$ as though $t$ were a fixed index rather than the thing varying, confirmed Oct 2024 Q4(c), scoring nothing. Name which variable is in the exponent before choosing a rule.

#5. Integration

5.1 — four standard results, plus the sum rule. MEMORISE (verbatim from the spec's "will not appear" list):

$$\int\cos kx\,dx=\frac{1}{k}\sin kx+c, \quad \int\sin kx\,dx=-\frac{1}{k}\cos kx+c, \quad \int e^{kx}\,dx=\frac{1}{k}e^{kx}+c, \quad \int\frac{1}{x}\,dx=\ln|x|+c \; (x\neq0)$$

also $\int a^x\,dx=\dfrac{a^x}{\ln a}+c$ — MEMORISE, the reverse of 4.4's own memorised derivative.

GIVEN, printed verbatim in the P3 integration table: $\int\sec^2kx\,dx=\dfrac{1}{k}\tan kx+c$; $\int\tan x\,dx=\ln|\sec x|+c$; $\int\cot x\,dx=\ln|\sin x|+c$.

5.2 Integration by recognition. MEMORISE — the two general forms, verbatim from the spec's content list:

$$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c \qquad\qquad \int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+c$$

Both are the chain rule run backwards, not new content — the mechanism blocks in integration-by-recognition.ts derive each from the corresponding differentiation result. Real, well-confirmed error, checked across two series: dropping the modulus in the log form, ending up trying to evaluate $\ln$ of a negative number instead of recognising the domain issue — confirmed Oct 2020 Q9(c), Jun 2023 Q9(c). Integrating $\sin^2x$, $\tan^2x$, $\cos^23x$ and similar squared-trig integrands requires converting via the double-angle identities from 2.3 first (spec's own guidance example) — these do not fit either recognition pattern directly and are a real, separate skill from spotting $f'(x)$ inside an expression.

§1.5 — genuinely new finding this pass, on $\int\csc x\,dx$. WMA13-verified-facts.md flagged this integral's exact booklet wording as unverified. Independently re-checked this pass, directly against ~/Downloads/IAL-Mathematics-Formula-Book.pdf (pdftotext -layout and -raw, cross-checked): the result

$$\int\csc x\,dx = -\ln|\csc x+\cot x|+c \quad\left(\equiv \ln\left|\tan\tfrac{x}{2}\right|+c\right)$$

is printed — but under the "Pure Mathematics P4" heading of the booklet, not P3, and the P3 section's own opening line states only that P3 candidates "may also require those formulae listed under Pure Mathematics P1 and P2" — a backward reach into earlier units, never a forward one into P4. The booklet's own general note (p.1) frames cross-unit access the same way: "a candidate sitting a unit may be required to use formulae... first introduced in [a] preceding unit." So, strictly by the booklet's own stated convention, a WMA13 (P3) candidate is not guided to this line — though the physical Yellow booklet is one shared document covering every unit, so a candidate who knows to flip past their own section could still find it. This resolves — and partially corrects — the research file's own account: cross-checking the real June 2023 Q9(c) mark scheme directly (Mathematics_Paper3_Mark_Schemes.md) shows that question is not $\int\csc x\,dx$ at all; it is $\int\csc x\cot x\,dx$, credited via the P3-printed derivative $\csc x \to -\csc x\cot x$ read in reverse (spec 5.2's own recognition mechanism, applied to a booklet-given result rather than a memorised one) — a materially different, and entirely P3-resident, fact from the harder $\int\csc x\,dx$ result. Treat $\int\csc x\,dx$ itself as a genuine open question for WMA13 teaching purposes: present on the shared physical sheet, not confirmed as ever directly tested on a real WMA13 paper in the 13 series reviewed, and not officially "given" under the P3 candidate's own stated booklet section.

#6. Numerical methods

6.1 Root location by sign change. Not a formula — the Intermediate Value Theorem, informally: if $f$ is continuous on $[a,b]$ and $f(a)$, $f(b)$ have opposite signs, at least one root of $f(x)=0$ lies in $(a,b)$. The single most mark-scheme-enforced word on this entire paper: the reasoning mark requires the word continuity/continuous stated explicitly, not just "sign change" — confirmed, verbatim mark-scheme wording (Oct 2023 Q1(a)): three separate components required — both function values correct, a reason that names continuity and states the sign change, and a "hence root" conclusion. Confirmed as the single most repeated specific loss across five-plus series.

6.2 Iteration. $x_{n+1}=g(x_n)$, converging (where it does) to a root of $x=g(x)$, i.e. a root of the original $f(x)=0$ once rearranged into that form — a definition and a method, not a formula to look up. Real, confirmed secondary errors: giving an iterate to fewer decimal places than asked, or rounding prematurely mid-chain; and embedding the iteration via a calculator's ANS key with the actual substitution never written down, which the mark scheme treats as insufficient method evidence even when every printed value is correct.

#Summary table — carried-forward P1/P2 formulae actually usable on WMA13

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#PART 2 — Worked-Method Reference

Organised by spec item, matching Part 1's numbering. Each entry is a short, generic method skeleton for a recurring real question shape — the transferable pattern, with the real M/A/B structure attached — not a full worked exemplar. Where a lesson file already carries the complete worked chain, it is named rather than repeated; where the question bank carries a full real marked solution, it is cited rather than reproduced.

#2.1 Algebraic division — two distinct "show a given constant" patterns

These look identical on the page (division producing polynomial + remainder, then "show [constant] = [value]") but are mark-scheme-distinct, and treating them as the same method is a real, documented error this course's own research corrected.

Pattern A — linear divisor, a genuine factor to cancel (real anchor: Oct 2020 Q9(a), $\frac{x^4-x^3-10x^2+3x-9}{x^2-x-12}\equiv x^2+P+\frac{Q}{x-4}$, "show $Q=5$"). Method: divide to reach the remainder over the given quadratic denominator → factorise that denominator (here $x^2-x-12=(x-4)(x+3)$) → cancel the common factor between the remainder and the newly-factorised denominator → the value that survives is the constant to be shown. Skeleton: M1 (writes the expression in the required form using the factorised denominator) → A1 (the stated constant recovered from that cancellation, not merely asserted). Full worked chain: content/veridian/curriculum/wma13/rational-expressions-and-algebraic-division.ts, trap cancel-and-simplify-step-abandoned.

Pattern B — quadratic divisor with no real factor, nothing to cancel (real anchor: Jan 2025 Q4(a), $\frac{4x^3+2x^2+3x+8}{x^2+4}\equiv Ax+B+\frac{Cx+D}{x^2+4}$, "show $D=0$"). $x^2+4$ has no real linear factor, so Pattern A's cancel-and-simplify route is structurally unavailable. Method: set up the identity $4x^3+2x^2+3x+8 \equiv (Ax+B)(x^2+4)+Cx+D$, compare coefficients to form at least two equations, and solve to show the stated constant directly — or, via long division, carry the division through in full until the remainder is genuinely constant-degree, then read the value off. Skeleton: a single independent B1* ("show that" mark) awarded only for a complete, substitution-verified derivation — not for the value simply appearing. Real, checked-against-primary-source distinction: a lesson or mock that teaches "factorise the denominator and cancel" as the universal fix for this question shape is wrong for any quadratic divisor with no real roots — check the divisor's discriminant before choosing the method.

#2.2 Functions — composition/inverse, and reading the domain constraint into an equation

Standard inverse-with-domain pattern (real anchor: Jan 2023 Q1(c), $g(x)=\frac{3}{2x+1} \to g^{-1}(x)=\frac{3-x}{2x}$, domain $0<x\leqslant3$). Method: swap $x\leftrightarrow y$ (or rearrange for $x$ first, then relabel — both score the same generous M1) → state the inverse attached to the correct function's letter → state its domain separately, as its own mark, equal to the range of the original function. Skeleton: M1 (rearranges/swaps correctly) → A1 (correct algebra, cao, and attached to the right letter — a correct $g^{-1}(x)$ mislabelled $f^{-1}(x)$ in a question defining both functions scores A0) → B1 (domain, independent of the algebra). Full worked chain: content/veridian/curriculum/wma13/functions-composition-and-inverses.ts.

Composed-algebraic-argument pattern, genuinely distinct from a composed number (real anchor: Jun 2026 Q2(d), $fg(a^2-5)=g(59)$, solve for $a>0$ in the form $\ln p$). Method: evaluate the numerical side first ($g(59)$) → substitute the algebraic argument into the inner function and use the question's own stated constraint (here $a>0$) to collapse a square root to a clean expression in $a$ before an equation can even be formed → substitute into the outer function → solve. Skeleton: B1 (sets up the correct equation, fully processed — not left with an unsimplified root/exponential mix) → M1 (isolates the exponential/log term and takes logs correctly) → A1 (exact form in the notation the question specifies). This is the only real, located WMA13 question where the composed input is itself an expression rather than a bare number — a lesson teaching composition-with-numbers only will not cover it.

#2.3 Modulus equations and inequalities — the two-branch, one-region method

"Solve $|\text{linear}| > \text{linear}$" pattern (real anchor: Jan 2022 Q7(b), the spec's own named example shape). Method: solve one case's equation (either by splitting into the two linear cases, or by squaring both sides into one quadratic — both valid) → both critical values normally fall out of a single accuracy mark, not one per case → select the outside (or inside) region the inequality's direction actually requires → state both bounds together as the final answer. Skeleton: M1 (one correct case attempted through to a value) → A1 (both critical values, "may be part of an incorrect inequality" at this stage) → dM1 (region selection, dependent on the M1 — singly dependent because only one method mark genuinely precedes it) → A1 (final combined answer, do not isw after a correct region is stated). Real, checked distinction: when a modulus equation is nested inside another operation (e.g. a modulus of a logarithm), two separate method marks can precede the region-selection step instead of one, which escalates the dependency to ddM1 — confirmed Oct 2023 Q9(c). Count the method marks that genuinely precede a step in the specific question before labelling it dM1 or ddM1; don't copy a label from a differently-structured question. Full worked chain: content/veridian/curriculum/wma13/modulus-and-graph-transformations.ts.

#2.4 Transformations applied to a named point — check which point, not just how well

Method skeleton, real anchor Summer 2026 Q3(c): identify the point the question stem itself names for the transformation — not a point computed in an earlier part of the same question, even one still fresh in working memory. Apply the transformation's input/output rule directly: for $y=f(x)\to y=f(ax)+b$, a point $(p,q)$ maps to $\left(\frac{p}{a}, q+b\right)$. Skeleton: B1 (correct $x$-coordinate of the image) → dB1 (correct $y$-coordinate, dependent on transforming the right point in the first place — the real mark scheme scores this B0dB0 outright for a correctly-executed transformation applied to the wrong point). The failure mode here is not arithmetic; it is which of two candidate numbers on the page gets picked up.

#2.5 Trig identity proofs and double/triple-angle equation-solving

"Show that [trig identity]" proof pattern (real anchors: Oct 2020 Q1 — solve $2\cos2x=7\cos x$ for $0\leqslant x<360°$; Oct 2020 Q5(a) — show $\sin3x\equiv3\sin x-4\sin^3x$). Method for an equation-solving version: substitute the double-angle form of $\cos2x$ (or $\sin2x$) that matches what the rest of the equation already contains, in brackets — a missing bracket around a substituted $(2\cos^2x-1)$-type term is a real, named mark-scheme trap, scored M0 unless the correct identity was stated first — → collect into a three-term-quadratic in the single remaining ratio → solve → convert back from the substituted variable ($y=\cos x$, say) to the actual angle before stopping. Skeleton: M1 (correct substitution, bracketed) → M1 (attempts to solve the resulting 3TQ) → A1 (correct ratio value, not left as the substituted variable) → dM1 (inverse-trig step to at least one angle in range, dependent on the second M1) → A1 (all solutions in range, none outside). For a "show that" identity proof specifically (no numbers to solve for): every intermediate line must actually appear on the page — reaching the printed answer with a step skipped is confirmed, across four-plus series, as a real, repeated way to lose the final mark even when the maths is entirely correct. Where a proof needs two separate prior method marks combined (e.g. Oct 2020 Q5(a)'s $\cos^2x=1-\sin^2x$ substitution, dependent on both the compound-angle expansion and the double-angle substitution before it), the combining step is a ddM1, independently confirmed on this identity as a genuine, non-modulus example of the same doubly-dependent convention. Full worked chains: content/veridian/curriculum/wma13/compound-and-double-angle-formulae.ts.

Reciprocal-identity-embedded-in-an-equation pattern (real anchor: Jan 2022 Q2, "show $8\cos\theta=3\csc\theta$ can be written $\sin2\theta=k$; hence find the smallest positive solution"). Skeleton, part (a): B1 (states/uses $\csc\theta=\frac{1}{\sin\theta}$ — note the real mark scheme explicitly refuses this mark, B0, if a coefficient has been folded into the reciprocal rather than stated cleanly) → M1 (uses $\sin2\theta=2\sin\theta\cos\theta$ to reach $\sin2\theta=k$, $k\neq1$) → A1 (the constant $k$, cso). Skeleton, part (b): M1 (correct order of operations to find any value satisfying $\sin2\theta=k$) → A1, marked "ONLY" for the value the question actually names. Two real, confirmed arithmetic-recovery traps, distinct from a sign or quadrant error: (i) reporting the doubled angle (e.g. $48.6°$) instead of dividing back by 2 to recover $\theta$ from a solved $2\theta$; (ii) finding every solution correctly but failing to discard the ones the question's own wording ("the smallest," "the first") didn't ask for — doing strictly more correct mathematics than needed still costs the final mark if the surplus solution isn't actively rejected.

Harmonic-form construction-then-application pattern (real anchor: Oct 2020 Q7 — the only located WMA13 question testing $R\cos(\theta-\alpha)$ head-on). Skeleton for the construction itself: B1 (exact $R$, no decimals) → M1 (attempts $\tan\alpha=\pm\frac{b}{a}$, or the sine/cosine-of-$R$ equivalent) → A1 (exact $\alpha$, in the unit named, cso). Skeleton for a "minimise/maximise a fraction with this harmonic-form denominator" hence-part: recognise that minimising $\frac{k}{a+R\cos(\ldots)}$ (positive $k$) requires maximising the denominator's cosine term, not the fraction directly — a real, confirmed point of confusion — then state the extreme value directly from $R$ and the constant, without needing calculus. Skeleton for a "solve for $t$ when $H=$ [value]" hence-part, when the harmonic-form argument is itself a half-angle: M1 (uses the part-(a) $R$ and $\alpha$ to reach $\cos(\beta t\pm\alpha)=k$) → A1 ($k$ evaluated) → M1 (full method to make $t$ the subject — look for $2\times(\arccos k \pm \alpha)$ if the argument was $\tfrac12t$) → A1 (final $t$, extra solutions outside the domain ignored, but not if genuine extras remain inside it). Full worked chain: content/veridian/curriculum/wma13/harmonic-form.ts.

#3.1–3.3 Exponential/log modelling and log-linear graphs

Real-world exponential model, standard-value pattern. Method for "initial value" (evaluate at $t=0$), "long-run/limiting value" (let $t\to\infty$, the decaying exponential term vanishes), and "domain restriction" (a value beyond the model's floor/ceiling has no solution — explain via the log-of-a-non-positive-number reasoning in Part 1 §3, not via an ad hoc "can't be negative" phrase the mark scheme explicitly rejects). The single most consistently lost mark across this whole spec area, confirmed across four-plus series with near-identical wording each time: a numerically correct modelled answer loses its accuracy mark for missing the unit and/or a direction word ("£", "tonnes", "m²", "loss," "increasing") — always check the last line of a modelling answer carries both.

Log-linear parameter-estimation pattern (real anchor: Jun 2023 Q2, $\log_6T$ vs $\log_6x$ through $(0,4)$ and $(2,0)$). Method: read gradient and intercept directly off the two given points → write the linear log-equation → either substitute a given $x$-value directly into that linear equation and rearrange for the un-logged quantity (a separate, generous mark) or eliminate the logs algebraically first, using a log law as the first move (e.g. $-2\log_6x=-\log_6x^{-2}$), to reach the non-log relationship $T=k x^{\pm n}$ in one continuous derivation. Skeleton: B1 (the linear log-equation itself) → M1, A1 (substitute-and-solve route to a single value) → M1 (first correct log-law move toward eliminating logs entirely) → dM1 (full, correct elimination to the non-log form, dependent on the M1) → A1 (final relationship, exact, isw). Real, confirmed finding: the dM1 elimination step, not the graph-reading itself, is where marks are most often lost — reported directly by the examiner as the point "log rules [were] not well understood by many students," on a question nearly half of candidates otherwise scored full marks on. Full worked chain: content/veridian/curriculum/wma13/exponentials-and-logarithmic-graphs.ts.

Interpret-the-constant-in-words pattern (real anchor: Jun 2026 Q5(b), no calculation, $H=ab^t$ with $a,b$ already found). Skeleton: a single B1ft, requiring a direction word, the percentage to 2 s.f. (computed as $(b-1)\times100$, not $b\times100$ — conflating "the new value is $119\%$ of the old" with "the increase is $19\%$" is a real, named near-miss the mark scheme rejects), and a per-year time unit, all three present together. This is a distinct skill from the units-omission trap above: there is nothing to compute here, the mark is earned entirely on correctly reading what a multiplicative constant means.

#4.2 Differentiation — the product-rule-into-single-fraction pattern

Real anchors: Oct 2020 Q3 ($f(x)=(2x+3)(4x-1)^{-1/2}$) and Oct 2024 Q7(a) ($f(x)=x^3\sqrt{4x+7}$, "show $f'(x)=kx^2(2x+3)/\sqrt{4x+7}$") — the same shape, four years apart, both confirming the identical mark-losing step. Full real marked solution for the second: content/veridian/question-bank/wma13.ts, id wma13-oct24-q7a-product-chain-rule. Method skeleton, generalised: identify the product as $u\cdot v^{1/2}$ or $u\cdot v^{-1/2}$ (a polynomial times a root or reciprocal-root factor) → apply the product rule with the chain rule folded into $v'$ → this first stage is the accessible half — real examiner-report evidence across both series states plainly that most candidates carry the differentiation itself out accurately → the genuinely discriminating step is combining the two unsimplified terms (one with a positive index on $v$, one with a negative) over a common denominator of $\sqrt{4x+7}$ (or equivalent), then factorising the numerator down to the form the question names. Skeleton: M1 (product rule attempted, chain rule correctly folded into the root/reciprocal-root factor) → A1 (correct unsimplified two-term derivative) → dM1 (dependent — correctly combines into a single fraction over the shared root denominator) → A1 (final factorised single-fraction form, cao). Real, quotable statistic worth internalising: on the Oct 2020 occurrence, only about a third of candidates scored the combining-mark, and only a quarter scored all four — the differentiation is not where this question is hard.

#4.3 The $\frac{dy}{dx}=\frac{1}{dx/dy}$ pattern — and the one invalid shortcut it tempts

Standard pattern: differentiate $x$ with respect to $y$ directly (often far easier than the reverse) → invert → convert back into the variable the question actually asked for, using whatever identity connects the two (commonly $\cos^2+\sin^2\equiv1$ if $x$ was defined via a trig function of $y$). Stopping at an inverted-but-unconverted expression is a real, repeatedly-confirmed way to lose the final mark with otherwise flawless working (Oct 2020 Q8(ii)).

The invalid shortcut, real anchor Oct 2024 Q2: on a turning-point question of the form $x=f(y)$, the turning point occurs where $\frac{dx}{dy}=0$ — which is exactly where the inverted $\frac{dy}{dx}$ is undefined, not where it is "equal to zero." A candidate who inverts first and then sets $\frac{1}{4y+5}$ (say) "equal to zero" and solves via the denominator lands, for this particular curve, on the same numerical answer as the valid method — but the real mark scheme states this scores zero marks outright, calling it "an incorrect assumption": a fraction is never made equal to zero by zeroing its denominator; that operation finds where the fraction is undefined. Method, correctly ordered: find $\frac{dx}{dy}$ → set that (before inverting) equal to zero → solve for $y$ → only then invert if the gradient itself is still needed elsewhere in the question.

#4.4 The two-exponential-term maximum/minimum pattern

Real anchor: Oct 2024 Q8(c), $H=32+40e^{-0.2t}-20e^{-0.9t}$, find the time of maximum heart rate. Full real marked solution: content/veridian/question-bank/wma13.ts, id wma13-oct24-q8c-two-exponential-maximum; the technique is also built into content/veridian/curriculum/wma13/exponential-growth-and-decay.ts. Method skeleton, generalised — the technique to reuse on any sum of two differently-rated exponential terms: differentiate term-by-term (each term's own chain rule, kept separate, not combined yet) → set the derivative to zero → rearrange to isolate one exponential term per side of the equation → divide one side by the other to collapse the two different rate constants into a single exponential term (the step with no equivalent in a single-exponential-term model) → only then take logs → solve for the variable. Skeleton: M1 (differentiates to the correct two-term form) → A1 (exact coefficients) → dM1 (sets to zero and rearranges to a single-exponential equation, dependent on the M1 — this is the combining step named above) → M1 (solves the single-exponential equation with correct log work) → A1 (final value, awrt). The real, exact cost of skipping the algebra and reaching for a calculator's solver instead — not a vague deduction: a numerically perfect final answer with no shown rearrangement scores the differentiation marks only, 2 of the 5 available, because the three marks attached to solving the equation are chained to method the working never demonstrated. A second, separately-named failure mode on the same real question: taking logs before the two exponential terms are combined — mathematically invalid, not merely under-shown, and distinct from the calculator-only failure.

#5.2 Integration by recognition — the two forms, and where dM1 genuinely belongs

Power-form pattern, $\int f'(x)[f(x)]^n\,dx$ (real anchors: Oct 2021 Q5(ii), Jan 2022 Q3(i)). Method: spot $f'(x)$ (up to a constant) sitting alongside $[f(x)]^n$ → raise the power by one, divide by the new power and by whatever constant makes the derivative check out → for a definite integral, substitute the limits as a separate, later step. Skeleton (indefinite): M1 (reaches $k[f(x)]^{n+1}$ for any constant $k$ — generous on the constant) → A1 (the exact constant, together with $+c$, as one combined requirement — a missing $+c$ costs the entire A1, not a fraction of it). Skeleton (definite, with limits): the antiderivative pair above, plus dM1 (substitutes both limits into the antiderivative and subtracts, either way round, dependent on the antiderivative's own M1) plus A1 (final evaluated value). Real, confirmed structural fact: dM1 is reserved for the substitute-the-limits step, never used to separate "spot the pattern" from "get the constant right" within the antiderivative itself — those two live inside one M1/A1 pair.

Log-form pattern, $\int\frac{f'(x)}{f(x)}\,dx$ (real anchor: Jan 2022 Q3(ii), with a genuine coefficient mismatch between the numerator and the true $f'(x)$). Method: identify $f(x)$ in the denominator → check whether the numerator matches $f'(x)$ exactly or only up to a constant multiple → write $b\ln|f(x)|$ with $b$ as that constant → for a definite integral, substitute limits as a separate step exactly as above. Skeleton is structurally identical to the power-form pattern (M1/A1 for the antiderivative, generous on the constant at the M-stage; dM1/A1 for limit substitution) — the only difference between the two patterns is which reverse-chain-rule form applies, not the mark-scheme shape around it. Recurring trap, distinct from the mechanics above: forgetting the modulus inside the log, which surfaces as attempting to evaluate $\ln$ of a negative number rather than a genuine domain check.

#6.1–6.2 Numerical methods — the two-part opener, in skeleton form

Sign-change location, method skeleton: evaluate $f$ at both given endpoints, to the precision the question allows → state the change of sign explicitly → state that $f$ is continuous on the interval, by name — the mark scheme requires this word or a clear paraphrase, not "sign change" alone → conclude "hence a root lies in [interval]." All three components (values, continuity-named reason, conclusion) are required for the one accuracy mark; omitting "continuity" specifically is the single most repeated individual error found across the entire archive, confirmed across five-plus series and only showing signs of improving in the most recent series reviewed.

Iteration, method skeleton: rearrange $f(x)=0$ into $x=g(x)$ (usually given, or a lead is provided per the spec's own wording) → substitute the given $x_0$ into $g$, writing the substitution itself as a visible line, not just the calculator's running output → repeat to the number of decimal places the question specifies, without rounding intermediate values → state the converged value to the precision asked. Real, confirmed trap distinct from an arithmetic slip: embedding the iteration silently via a calculator's ANS key, with no substitution line ever written — the mark scheme treats this as insufficient evidence of method even when every displayed value is correct.

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#Genuine gaps flagged

• The factor (sum-to-product) trig formulae are printed in the booklet's P3 section but their direct examinability was not confirmed against any real question in the 13 series WMA13-verified-facts.md reviewed, nor in the wider primary-source check for this pass. Treat "on the sheet" and "confirmed as directly tested" as separate facts here — the first is verified, the second is not.
• $\int\csc x\,dx$ is printed in the Yellow booklet, but under the "Pure Mathematics P4" heading, not P3 — a genuinely new finding this pass (§1.5 above), and a partial correction to WMA13-verified-facts.md's own earlier account, which had attributed a real WMA13 question's use of "the Formula Booklet" to this specific integral. The real question in question (Jun 2023 Q9(c)) turns out to use a different, P3-resident booklet entry (the reversed derivative of $\csc x$) instead. Whether $\int\csc x\,dx$ itself has ever been directly required on a real WMA13 paper remains unconfirmed.
• Section 2.5's harmonic-form-with-half-angle-argument skeleton is sourced to one real question only (Oct 2020 Q7) — it is the sole located WMA13 question testing $R\cos(\theta\mp\alpha)$ head-on rather than as one step inside a differentiation or integration question, per WMA13-verified-facts.md's own flagged limitation. Treat the half-angle recovery step as evidenced-once, not multiply-confirmed.
• Several Part 2 patterns are single-series findings carried over directly from the FORGE guide's own §7 flag (the "reject the extra solution," "halve $2\theta$ back to $\theta$," "$y$/$f(x)$-notation for a range," "$dx/dy=0$ not the reciprocal," "wrong-rule-on-the-exponent," and "terminate-the-division-early" patterns) — real and verbatim-sourced, but not yet cross-checked against a second occurrence the way the continuity-omission or "quote the formula" findings have been across five-plus series each.
• Question-count and per-question topic-ordering are not fixed by the spec (9 or 10 questions depending on series) — nothing in this document should be read as implying a fixed question number or position for any topic above.
• No WMA13 past question was located testing the booklet's carried-forward P1 cosine rule or sphere/cone mensuration results directly within P3's own six content strands; they are listed in the summary table as available, not as confirmed-examined.

#Sources

• research/veridian/WMA13-verified-facts.md — primary compiled source, 13 real series (Oct 2020–Jun 2026).
• content/veridian/technique-hub/wma13/forge-exam-technique-guide.md — companion guide; paper structure, M/A/B conventions, and the calculator-only-method penalty are covered there and not repeated here.
• content/veridian/curriculum/wma13/*.ts — all 11 lessons, cited by file and trap id throughout Part 2 rather than reproduced: rational-expressions-and-algebraic-division.ts, functions-composition-and-inverses.ts, modulus-and-graph-transformations.ts, compound-and-double-angle-formulae.ts, harmonic-form.ts, reciprocal-and-inverse-trigonometric-functions.ts, exponentials-and-logarithmic-graphs.ts, differentiation-rules.ts, exponential-growth-and-decay.ts, integration-by-recognition.ts, numerical-methods.ts.
• content/veridian/question-bank/wma13.ts — the two Oct 2024 real marked solutions cited in full rather than reproduced (§4.2, §4.4 above).
• ~/Downloads/IAL-Mathematics-Formula-Book.pdf (Mathematical Formulae and Statistical Tables, Issue 2, January 2021) — independently re-downloaded copy, converted with pdftotext -layout and -raw this pass to verify every GIVEN/MEMORISE classification in Part 1 directly, including the P3/P4 boundary finding on $\int\csc x\,dx$.
• ~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_Mark_Schemes.md — consulted directly this pass to re-verify the real June 2023 Q9 structure against WMA13-verified-facts.md's own account (§1.5 above).