Technique Masterclasses

Pure Mathematics 3

Vocabulary Precision List

Provenance: written 2026-09-29, grounded in research/veridian/WMA13-verified-facts.md (read in full, then re-grepped specifically for command-word and term-precision material — the primary provenance-checked source for every citation below). The four main existing technique-hub documents for this paper (forge-exam-technique-guide.md, forge-battle-card.md, codex-formula-and-method-bank.md, signal-frequency-and-boundaries.md) were read in full to establish what they already cover, so this document does not restate it; the eleven per-topic signal-topic-brief-*.md/codex-deep-dive-*.md files were grepped (not fully read) for vocabulary/command-word/"initial" hits specifically, which is how entry 10's second half was found. Where a real finding below overlaps a point those four main documents make in passing, the overlap is named explicitly and this entry's actual addition — usually the exact accepted/rejected wording, which none of them spell out in full — is stated separately. No Pearson textbook was used; only the spec, the formula booklet, and freely-published mark schemes/examiner reports, transcribed in WMA13-verified-facts.md from qualifications.pearson.com and its documented mirrors. Direct quotations below are kept short (under ~15 words) with attribution; longer mark-scheme or examiner-report passages are paraphrased rather than reproduced.

WMA13 — Vocabulary Precision List

Pure Mathematics 3 (Pearson Edexcel IAL) — command words and technical terms this paper's mark schemes are exact about

How to use this

This is a glossary, not a technique guide or a battle card — it is organised by term, not by trap or by spec point, because that is the one organising structure the other four documents in this folder don't already use. Each entry names a word or a pair of words candidates treat as interchangeable, states the real distinction a mark scheme draws between them, and gives the series and quote that prove it. Where the underlying error pattern already has a full worked treatment elsewhere (a lesson's trap taxonomy, the exam-technique guide's self-mark checklist, the battle card's danger warnings), this document says so and does not re-derive it — the value added here is the word-level precision: which exact phrasings score and which are named as scoring nothing, catalogued in one place.

Deliberately not repeated here (already covered in full in forge-exam-technique-guide.md §2 and forge-battle-card.md's tariff section): the M/A/B mark-type definitions, the dM1/ddM1 dependency convention, "quote the formula before substituting," the rule that ft applies only to A/B marks and never M, the misread and multiple-attempts conventions, and the full abbreviation list (bod, cao, cso, isw, awrt, oe, SC, dep/indep, dp/sf). Those are mark-scheme notation, not the paper's own command-word/technical-term vocabulary, which is this document's actual subject.

1. "Continuous" / "continuity" — the one word "sign change" cannot substitute for

This is the single most repeated word-level finding in the whole archive, so it is worth being precise about exactly which words score, not just that the general trap exists (already named in the exam-technique guide's checklist and the battle card's danger warning #1).

A root-location accuracy mark bundles three things together, and the mark scheme is explicit that a correct sign change alone is not one of them: the reason "must mention continuity and state or indicate sign change in some way" (Oct 2023 Q1(a) notes). Accepted reason-phrasings actually printed in real schemes: "sign change and continuous" and "y(−1.25)y(−1.2)<0y(-1.25)y(-1.2)<0 and continuous" (Jan 2022 Q5(a)). Accepted minimal conclusions: "hence proven", "hence root", or "hence f(x)=0f(x)=0 between x=1x=1 and x=2x=2" (Oct 2023 Q1(a)). None of these accepted forms mentions a numerical sign-change test without also naming continuity in the same breath.

The word itself, not the underlying reasoning, is what repeatedly costs the mark: examiner reports name "the omission of reference to 'continuity' or 'continuous'" as the most common error across "the past few series" (Oct 2022), still recurring in Jan 2022 Q5(a) and named again in Jan 2024 as happening "despite it being highlighted in every appropriate series." Jan 2025 is the first series where the wording finally starts appearing reliably. A candidate who writes a numerically flawless sign-change argument and omits the one word "continuous" loses the whole bundled mark, not a fraction of it.

2. "Hence" — the method is wired to the specific result just found, not to the original problem

A "hence" instruction ties the required method to the preceding part's own stated result, not to any independently valid route to the same underlying equation. Jan 2022 Q2 shows the wiring directly: part (a) asks to show 8cos⁡θ=3csc⁡θ8\cos\theta=3\csc\theta can be written as sin⁡2θ=k\sin2\theta=k; part (b), "hence find the smallest positive solution," has its method mark defined in the scheme as reaching a value of θ\theta "that works for their sin⁡2θ=k\sin2\theta=k" — the mark is keyed to the part-(a) result by name, not to a fresh solve of the original cosecant equation.

Worth stating the strength of this evidence honestly: this is read from how the mark scheme itself is structured (the M-mark's own wording), not from an examiner-report sentence naming candidates who were penalised for solving the original equation independently instead. No such sentence was found in this pass. Treat the distinction as real and mark-scheme-grounded, but as an inference from structure rather than a directly quoted warning.

3. "State" vs "Find" / "Show" / "Determine"

"State" signals an independent B-mark for a direct read-off — no method needs to be shown, and none is credited toward it. "Find," "show," and "determine" signal that a value must be derived, with the accuracy mark chained to a method mark that has to be earned first.

The clearest real contrast sits inside one question. Question 8 (Oct 2024), the two-exponential-term heart-rate model: "State the initial heart rate of the horse" is worth 1 mark (read HH at t=0t=0 off the given equation); "State the value of LL" (the long-run limit) is also 1 mark (read the equation's surviving constant as t→∞t\to\infty); "Find the value of TT" (the time of maximum heart rate) is worth 5 marks, running a full differentiate-then-solve M/dM/A chain confirmed verbatim in the mark scheme. Three consecutive parts of the same question, two of them answerable by inspection with "state" and one requiring five marks' worth of shown calculus with "find" — the command word alone tells you which is which before you've read the mark tariff. (The question paper's own tariff — [1], [1], [5] — is directly verified; the two "state" parts read as B-mark-shaped from that tariff and the general B-mark convention, but this pass did not independently re-locate the scheme's own letter-code for parts (a)-(b) specifically, only for part (c).)

One softening worth knowing precisely: inside the continuity reason specifically (entry 1 above), the scheme accepts "state or indicate" sign change — a rare case where the mark scheme itself treats a slightly weaker verb as equivalent to "state," but only in that one bundled context, not as a general rule.

4. "The smallest" / "the first" / "the only" — command words that require you to reject solutions, not just find them

A plain "solve...in the interval" instruction wants the full solution set in range, nothing more, nothing less — well documented elsewhere as the "missing/extra solution" trap (5+ series; see the exam-technique guide's checklist and the battle card's danger warning #5). A question that instead names a specific member of that set — "the smallest," "the first," "the only" — draws a different, narrower line: finding every solution is necessary but not sufficient, because the accuracy mark is reserved for that one named value alone.

Jan 2022 Q2(b) asks for "the smallest positive solution" of 8cos⁡θ=3csc⁡θ8\cos\theta=3\csc\theta; the scheme's accuracy mark for this part is noted "ONLY" — awarded solely for that one value. The examiner report confirms candidates who correctly found both solutions to the underlying equation but failed to discard the larger one "lost the accuracy mark which was even more of a shame," despite having done strictly more correct mathematics than the question asked for. The failure mode is the mirror image of the missing-solution trap: here the candidate has too much, not too little, and the word "smallest" is exactly what should have triggered discarding the rest.

5. "Exact value" / "exact form" — not every algebraically equivalent expression is accepted

"Exact" does not mean "any correct equivalent expression left unevaluated" — mark schemes name specific accepted forms and specific superficially-equivalent forms that are explicitly refused, usually because the refused form leaves an operation the scheme wants fully carried out.

The clearest catalogued case: Summer 2026 Q2(d) asks for an answer "in the form ln⁡p\ln p" with pp irrational. The scheme accepts ln⁡(52)\ln\left(\frac{\sqrt5}{2}\right) and stated equivalents such as ln⁡1.25\ln\sqrt{1.25} or ln⁡54\ln\frac{\sqrt5}{\sqrt4} — but not ln⁡(54)1/2\ln\left(\frac54\right)^{1/2} and not 12ln⁡1.25\frac12\ln1.25, even though both are algebraically and numerically identical to the accepted forms. The line the scheme draws is specific: the square root has to sit inside the single ln⁡(…)\ln(\ldots) the question named, not survive as an outer exponent or get pulled out as a coefficient in front of the log.

A second, independently confirmed case sits one step earlier in a derivation, not at the final answer: Oct 2020 Q7(a)'s harmonic-form construction marks require RR left as the exact surd — "do not allow decimals for this mark" — and α\alpha given in the unit the question actually named; a numerically correct α\alpha converted into degrees when radians were asked for is marked "A0" even though the value itself is right. "Exact," on this evidence, is a constraint on the form the scheme will accept, checked as precisely as the number itself.

6. "Domain" vs "Range" — one substitution, and a separate labelling trap

Two distinct precision failures sit under this pair, and they are worth separating because they are different mistakes even though both involve identical correct algebra.

The substitution itself. The domain of f−1f^{-1} is, by definition, the range of ff — not a second quantity to be worked out from scratch. Examiner reports note candidates independently re-deriving the inverse's domain rather than recognising it as the already-known range of ff (Oct 2022 Q2(b)). This sits alongside the much more heavily documented "domain of the inverse omitted entirely" trap (4+ series, already the subject of its own lesson trap and battle-card danger warning #2) — that trap is about the mark being missing altogether; this one is about candidates doing unnecessary extra work because they don't treat domain-of-inverse and range-of-original as the same fact under two names.

The labelling trap. A separate, previously undocumented failure surfaces in a question that defines two functions at once. Jan 2023 Q1 defines both ff and gg; part (c) inverts g(x)=32x+1g(x)=\frac{3}{2x+1}. The scheme allows the correct expression written as g−1(x)=3−x2xg^{-1}(x)=\frac{3-x}{2x} but explicitly refuses the identical expression labelled f−1f^{-1}: "NOT f−1=3−x2xf^{-1}=\frac{3-x}{2x}." A candidate who inverts the right function but writes the wrong function's name in front of a flawless answer scores nothing on that mark — a pure labelling precision issue, distinct from any algebra error.

7. "In terms of xx" (or "in terms of yy") — zero residual instances of the other variable, not "mostly rewritten"

This phrase means the final expression must contain no surviving trace of the variable it names as excluded — not "predominantly in the right variable," not "correct except for one leftover term."

Oct 2020 Q8(ii) differentiates x=ln⁡(sin⁡y)x=\ln(\sin y) implicitly and then asks for the result in terms of xx. Candidates who correctly reach dydx=excos⁡y\frac{dy}{dx}=\frac{e^x}{\cos y} — a mixed-variable expression that is numerically and structurally on the right track — are reported to have "proceeded no further to replace cos⁡y\cos y in terms of xx." The calculus is finished; the phrase "in terms of xx" names one further, separately credited step (recognising cos⁡y=1−sin⁡2y=1−e2x\cos y=\sqrt{1-\sin^2y}=\sqrt{1-e^{2x}} from the original relation) that a mixed expression does not satisfy no matter how far the working has otherwise progressed.

8. "Cosecant" and "secant" — precisely defined reciprocals, and where a coefficient may legally stand

Two distinct precision issues sit under the reciprocal trigonometric functions, both confirmed on the same real question (Jan 2022 Q2).

The name itself. Even weak scripts are credited for correctly stating csc⁡θ=1sin⁡θ\csc\theta=\frac{1}{\sin\theta} as an accessible first mark — but the examiner report separately records candidates "struggling writing csc⁡θ=1cos⁡θ\csc\theta=\frac{1}{\cos\theta}," swapping cosecant's reciprocal for secant's. The two names are precisely defined against different base ratios, and the error is a term-definition slip, not an algebra slip.

Coefficient placement. A second, distinct error the same series confirms: writing 3csc⁡θ3\csc\theta as 13sin⁡θ\frac{1}{3\sin\theta} — folding the coefficient into the denominator alongside the sine, rather than leaving it multiplying the whole reciprocal from outside. The scheme is explicit that this scores nothing: "3csc⁡θ=13sin⁡θ3\csc\theta=\frac{1}{3\sin\theta} is B0 unless there is an aside that does state csc⁡θ=1sin⁡θ\csc\theta=\frac{1}{\sin\theta}." A coefficient in front of a reciprocal trig function is not itself reciprocated — a genuine notational precision point distinct from the name-swap above, on the identical real question.

9. "Identity" (≡\equiv) vs "equation" (==) — the spec's own notational split, and why it drives two different completeness standards

The spec's own guidance for this content states the two side by side, one clause apart, and switches symbols to mark the difference: candidates should be able to solve equations such as acos⁡θ+bsin⁡θ=ca\cos\theta+b\sin\theta=c in a given interval, and separately, prove identities such as "cos⁡xcos⁡2x+sin⁡xsin⁡2x≡cos⁡x\cos x\cos2x+\sin x\sin2x\equiv\cos x" (spec p.21, item 2.3 guidance). An equation is true only for specific values, found and reported; an identity is asserted true for every value of the variable, and a "prove" or "show" instruction against one asks for an unbroken algebraic chain, not a value.

This single notational distinction is the reason two of this paper's best-evidenced error patterns look different on the surface but share one root cause. A "prove"/"show" identity question loses its final mark not through wrong mathematics but through skipped intermediate lines — confirmed across 4+ series (Oct 2020, Jan 2022, Jun 2023, Jan 2024) — because an identity's proof is only complete when every line of the chain is shown, and there is no "answer" short of the full chain to check instead. A "solve...in a given interval" equation question, by contrast, loses marks through an incomplete solution set — a missing second solution, an extra out-of-range one, or the wrong angle unit (5+ series) — because an equation's completeness standard is about which values made the list, not about how each line of algebra was justified. Neither trap is new here (both already have full treatment elsewhere in this folder); what is new is naming the single spec-level distinction, ≡\equiv against ==, that these two otherwise-unconnected traps both trace back to.

10. "Interpret...in words" — the constant's own kind decides how many components the sentence needs

Two real questions ask a candidate to interpret a model constant "in words" with no further calculation, and the required sentence is a genuinely different shape depending on what kind of constant it is — a distinction the constants' own names already flag, but candidates conflate.

An initial-value (additive) constant. Jan 2022 Q8(b): log⁡10x=2.74−0.079t\log_{10}x=2.74-0.079t, converted to x=pq−tx=pq^{-t}; interpreting pp only needs one idea — that it is the quantity present at t=0t=0 (the mark scheme condones "the dose given," or equivalent). This is the same "initial" recognition named directly in entry 11 below, applied here to a constant reached algebraically rather than read straight off a given equation.

A multiplicative rate constant. Summer 2026 Q5(b): H=abtH=ab^t models a growing count of households; interpreting bb needs three separate components stated together — direction, the percentage to 2 s.f., and the time period ("per year"/"each year") — and the scheme's own reject list shows how specifically each of the three is checked. Explicitly refused, despite each getting the general idea right: "The amount it is rising," "The rate by which it increases," "The percentage increase each year," "The annual rate of increase," "It is changing by 19% per year," and — a distinct numerical conflation, not just a missing component — "The annual increase is 119%" (mistaking the new-value multiplier, b×100b\times100, for the increase itself, (b−1)×100(b-1)\times100).

The two tasks share the instruction "interpret...in words" and nothing else: one constant is a starting quantity, interpreted with a single fact; the other is a growth rate, interpreted only when direction, size, and period are all present together. Treating every "interpret the constant" question as needing the same one-line answer is exactly the conflation these two real questions, four series apart, are each individually evidence against.

11. "Initial" — a spec-defined technical term, not a general word for "given"

This one is not mark-scheme wording but the specification's own glossary note, and it is worth stating with its primary-source citation rather than only as an assumed convention: guidance for spec item 4.4 tells students to be familiar with terms such as "'initial', meaning 'when t=0t=0'" (spec p.22, item 4.4 guidance). It is a precise substitution instruction, not a synonym for "the number given in the question" in general — a model's initial value is specifically its value at t=0t=0, which matters once a question states several numerical facts about a model and asks which one is "initial" versus which is a later, given data point.

What this document does not cover, and why

Two candidate entries were investigated and left out rather than padded in. First, whether "any correct method" phrasing in the general marking guidance genuinely licenses solving a "hence" part by an unrelated route (entry 2's open question) — no examiner-report sentence naming this was found in the reviewed series, so the entry above states the mark-scheme-structure evidence only and does not claim more. Second, whether a domain must be written in one specific notational register (a yy/f(x)f(x)-form for a range, an xx-form for a domain) as a scored distinction in its own right — this claim appears in this course's own battle card, but could not be independently re-verified against WMA13-verified-facts.md or the primary lesson file in the time available for this pass, so it is not repeated here as a sourced fact. Where the real material ran out, this document stops rather than filling the gap with an invented example.

Sources