Pure Mathematics 4

Vocabulary Precision List

Provenance: written 2026-09-29, 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 June 2022, October 2022, January 2023, June 2023, October 2023, June 2024 and January 2026). Before writing anything below, this pass read in full: the eleven lesson files in content/veridian/curriculum/wma14/ (including each topic's own trap-taxonomy), content/veridian/question-bank/wma14.ts (no vocabulary/glossary content found there — checked directly, grep for "vocabulary", "glossary", "command word" returns nothing), and every existing technique-hub document for this paper (codex-formula-and-method-bank.md, forge-exam-technique-guide.md, forge-battle-card.md, forge-exam-day-protocol-extended.md, forge-forty-eight-hour-checklist.md, forge-spaced-repetition-schedule.md, signal-frequency-and-boundaries.md, and the eleven codex-deep-dive-*.md / signal-topic-brief-*.md files) to confirm no dedicated command-word/grading-vocabulary glossary already exists for this paper — several of those documents quote isolated abbreviations (isw, cao, awrt, ag) in passing inside a worked example, but none assembles the mark-scheme's own defined terms into one reference, and none treats command-word phrasing ("hence" vs "hence, or otherwise") or paired technical terms (skew vs perpendicular; position vector vs direction vector) as vocabulary in their own right. This document is that missing layer. No new primary Pearson document was fetched for it — every claim below traces through the verified-facts file's own independently-checked citations, re-quoted here at the word/phrase level rather than restated at the worked-example level. Not affiliated with or endorsed by Pearson Edexcel.

#WMA14 · FORGE — Vocabulary Precision List

Pure Mathematics 4's mark schemes are not vague about language. A handful of words and phrases carry a specific, checkable meaning that differs — sometimes sharply — from how the same word gets used in ordinary classroom speech, and the examiner reports name the exact moment a candidate's answer was marked down for blurring one of these distinctions. This document collects those distinctions as vocabulary entries: what the term precisely means, what it is commonly confused with, and the real citation that shows the confusion costing marks. It does not re-teach the mathematics — the eleven curriculum lessons and the other technique-hub documents already do that — and it does not repeat a full worked example where one already exists elsewhere in this course's WMA14 material; where an entry below touches ground FORGE or CODEX already covers, it is kept to the vocabulary point itself and points to the fuller treatment rather than reproducing it.

A note on scope, honestly stated. This paper's own evidence base is what it is: four full Examiners' Reports and a handful of mark schemes across eleven series, all reviewed in WMA14-verified-facts.md. Every entry below is traceable to a specific series and question inside that file. Where the real material runs out on a given term, the entry says so rather than inventing a second example to round the list out.

#1. The mark scheme's own grading vocabulary

Every WMA14 mark scheme reviewed opens with an identical "General Marking Guidance" section, verified word-for-word stable from January 2021 through June 2024. These are not stylistic flourishes — each abbreviation below gates whether a specific mark can be awarded, and mixing two of them up (treating an "isw" situation as a "cao" one, say) is the difference between a mark surviving a slip and being lost to it.

Term	What it precisely means	Commonly blurred with
M	"Method marks are awarded for 'knowing a method and attempting to apply it'."	Treated as automatic for showing any working, when in fact a method mark requires the right method, attempted.
A	Accuracy mark — "can only be awarded if the relevant method (M) marks have been earned."	Assumed available for a correct-looking final number reached by an uncredited method — it isn't.
B	"Unconditional accuracy marks (independent of M marks)."	Confused with A marks — a B mark can be earned even when no method mark exists for that step, which is the opposite of how A marks work.
cao	Correct answer only — the default status of an A mark unless stated otherwise.	cso (below) — cao tolerates earlier slips that don't affect this specific mark; cso does not.
cso	Correct solution only: "there must be no errors in this part of the question to obtain this mark."	cao — cso is the stricter of the two, and the two are not interchangeable even though both start "correct s/answer only."
ft	Follow through — a wrong earlier value can still earn a later mark explicitly flagged "ft," carried forward consistently.	Assumed to apply generally after any error; it only applies to marks the scheme itself marks "ft."
isw	Ignore subsequent working — applies after a correct answer has already been written down; anything wrong written afterward does not retract it.	Used to excuse an error that occurs before the correct answer is reached — isw only protects what comes after.
awrt	Answers which round to — a numerical tolerance instruction.	The "exact answer" requirement (§3) — awrt and "exact" are opposite instructions about the same numeral.
oe	"Or equivalent (and appropriate)" — an alternative but mathematically equivalent final form is accepted.	condone (§2) — oe accepts a different correct form; condone permits a specific slip inside an otherwise-standard method.
bod	Benefit of doubt.	—
SC	Special case — a named, scheme-specific partial-credit route, distinct from the general M/A/B ladder.	—
dep / dM1	Dependent — a mark that can only be awarded if a specific earlier mark was already earned.	indep — the two sit at opposite ends of the same dependency question.
indep	Independent — not gated on any other specific mark.	dep/dM1 (above).
dp / sf	Decimal places / significant figures — a stated precision instruction.	"Exact" (§3) — dp/sf specify a rounding tolerance; "exact" forbids rounding altogether.
ag	Answer given — "the answer is printed on the paper."	An ordinary final-answer mark. On an ag line every prior step must be genuinely correct — see §4.

Source: WMA14-verified-facts.md §4, "Mark-scheme conventions — verbatim, and confirmed stable across at least three years of series," itself quoting the Summer 2024 and January 2021 mark schemes' "General Marking Guidance" sections word-for-word.

#2. "Condone" vs "oe" — a permitted slip is not an accepted alternative form

"Oe" is in the mark scheme's own quoted abbreviations list (§1). "Condone" is not — it never appears in that formal list — but it recurs constantly across real per-question scheme lines with a consistent, distinct job: naming one specific slip that will not cost the mark, inside a method that is otherwise being marked exactly as written. Confirmed real instances: a scalar-product angle mark that will "condone slips in dot product/moduli" (Jan 2026 Q9(b)); a cross-product alternative method that will "condone sign slips for their cross product" (Jan 2026 Q9, Alt(c)); and two integration-by- parts method marks that each explicitly "condone missing dx" (Jun 2022 Q8(b), both the first and second application).

The precision that costs marks when this is blurred: "condone" only forgives the named slip. It is not a general licence to skip steps, and it is not the same move as "oe," which instead accepts a genuinely different final form of an answer (e.g. 1/3 or −1/3 for a cosine, per Jan 2026 Q9(b)'s own A1). A candidate who reads "condone missing dx" as permission to drop other required structure, or who reads "oe" as permission for an uncorrected arithmetic slip, is applying the wrong one of two superficially similar-sounding mark-scheme licences.

#3. "Exact" vs "awrt" vs a bare decimal

The general marking guidance states the rule directly: where an exact answer is asked for, or surd working is clearly required, examiners' reports "have emphasised that... marks will normally be lost" if a candidate resorts to a rounded decimal (WMA14-verified-facts.md §4). This is the standing rule. Two real instances sharpen exactly where it bites:

January 2021, Q1 (binomial approximation to a surd): "a very small minority of candidates unfortunately gave a decimal approximation" where the exact surd form was required.
January 2026, Q7(d): the question does not merely say "find the exact area" — it prints, twice, "Solutions relying on calculator technology are not acceptable." This is a question-specific strengthening of the general exact-answer rule into an explicit method ban: a numerically correct decimal reached by any calculator route scores nothing here, not merely "loses the exactness mark."

The vocabulary distinction worth holding separately: "exact" forbids rounding at all; "awrt" (§1) is the opposite instruction, explicitly inviting a rounded numeral within a stated tolerance; and "3 s.f." /"1 d.p." (dp/sf, §1) name how much rounding is acceptable when rounding is in fact allowed. Reading an "exact" instruction as though it were an "awrt" one — and reaching for the calculator — is a different error from reaching an exact-looking but wrong surd; the mark scheme treats them differently, and only the first is what this section documents.

#4. "Show that" / "ag" vs "Hence" vs "Hence, or otherwise"

These three instructions look similar — each tells you to use something already established — but they license three different amounts of freedom, and conflating them costs marks in different ways.

"Show that" (an ag line). The target answer is printed on the paper. The October 2021 examiner's report on Q10(b) is explicit about what this is actually testing: candidates "forgot... to add a statement that a/b was fully simplified" — a required stated step, not merely consistent working — and the January 2021 report on a separable-differential-equation "show that" item records the specific failure mode directly at stake: some candidates, working toward a printed target, "adjust" earlier working once they can see where they are supposed to land, rather than genuinely re-deriving it. An ag mark requires every preceding line to be independently correct — it is not awarded for merely arriving at the printed string.

"Hence." A bare "hence" ties the next step to the specific result just derived. The real January 2026 Q3(b) mark scheme is precise about this: the method mark for the limiting height is earned specifically as "Their a / Their b² from an answer to (a)" — i.e., built from the exact closed form h = at²/(1+bt)² just derived, not from any other correct route to the same numerical limit (differentiating h to hunt for a stationary point is independently confirmed to be a dead end for this exact question, since h is strictly increasing for all finite t > 0). "Hence" here is a routing instruction, not just a sequencing one.

"Hence, or otherwise." Adding "or otherwise" genuinely widens the accepted method. The real January 2021 Q4(b) mark scheme instructs candidates to "Hence, or otherwise, state the range of g," and its M1 is correspondingly written to accept either of two different routes to the same two boundary values: "attempting g(2) for their g" or "attempting their a/their c" — not a single prescribed chain from part (a) at all. The precise distinction: "hence" without qualification binds you to the previous result's specific form; "hence, or otherwise" releases that binding and rewards any valid route to the same answer.

A fourth real variant sharpens this further: January 2026 Q7(d) reads "Hence, using algebraic integration, find the exact area of S" — "hence" here is qualified by its own named method restriction (algebraic integration only), stacked on top of the ordinary hence-routing requirement and the exact-answer rule of §3 simultaneously. Three separate instructions are compressed into one sentence, and a candidate who satisfies only one of the three (reaches the exact number, say, by calculator) has not actually answered the question as set.

#5. "Assumption" — generous at the start, strict at the finish (proof by contradiction)

Proof by contradiction (spec 1.1) hinges on the word "assumption," and the mark scheme treats it with two different levels of strictness depending on where in the proof it appears.

At the setup. The June 2022 mark scheme for Q9 is explicit that no formal wording is required to earn the method mark for stating the assumption: "No need for explicit statement of assumption" — setting up a suitable algebraic equation is enough, and the scheme even models an acceptable phrasing as loosely as "Assume n² − 2 is divisible be 4 for all n."

At the contradiction. The same scheme is equally explicit that the final mark requires something stricter: examiners will "be generous with the technicality" of how the assumption was worded, but only if "a clear assumption... is contradicted, and all working must have been correct." The January 2024 report on Q8 documents exactly what goes wrong when this second, stricter bar is missed: candidates "simply commented that the equation formed had no solutions, giving no explanation as to why" — an assertion standing in for the derivation the mark actually requires.

A second, distinct vocabulary point sits inside the same topic: "fully simplified." The October 2021 report on the √2-irrationality proof (Q10(b)) records that most candidates who reached a/b as a ratio of integers still lost the final mark because they never wrote "a statement that a/b was fully simplified" — i.e., that a and b share no common factor. The mathematics was usually done correctly; the stated coprimality claim, in words, is itself a scored step, distinct from having silently assumed it.

#6. "Skew" is two negatives, and each negative has its own precise standard

"Skew" (spec 7.6) means not-parallel and not-intersecting simultaneously. The January 2021 Q8 examiner's report shows candidates substituting the wrong second condition for "not-intersecting": having shown two lines don't meet, some candidates "went on to show that they were not perpendicular" — a real property, but irrelevant to skewness, which needs not-parallel, not not-perpendicular. The same report states plainly that many candidates "did not fully understand the meaning of skew" at all, stopping at finding the two parameter values with no follow-up conclusion about either required negative.

Each of the two negatives, in turn, has its own precise bar, confirmed from a different series:

"Not parallel" is not established merely by two direction vectors looking different component-wise. The October 2022 Q9 mark scheme's own required standard is that a candidate must show the direction vectors are not scalar multiples of one another — concluding only that "[direction vector 1] ≠ [direction vector 2]... was insufficient" per that scheme, because non-identical vectors could still be parallel (one a scalar multiple of the other).
"Not intersecting" is not established merely by reaching a contradiction in the working — it requires the candidate to draw the conclusion in words. The same Q9 report records that a high proportion of candidates who reached a pair of conflicting values "failed to then conclude that this implied the lines did not intersect" — the algebra was often right; the sentence stating what the algebra proved was the missing, separately-marked step.

So "skew" is not one word to define but three: the compound term itself, plus the precise content of each of its two negated halves — and the real error pattern shows candidates substituting a wrong condition for one half (§6, Jan 2021) or leaving the correct condition's conclusion unstated (§6, Oct 2022), which are two different failure shapes, not one.

#7. Scalar product vs vector (cross) product — and which vertex an angle name fixes

"Scalar product" names an operation whose result is a single number. The January 2021 Q2(a) examiner's report records candidates producing "the vector (12,−10,24)" — i.e. performing a cross product — where the scalar (dot) product value was required: a confusion about what kind of object the named operation returns, not an arithmetic slip inside the right operation.

The same report, on the same question, documents a second and independent precision failure: the angle requested was ∠ABC (vertex B), and the report notes it was "unclear whether candidates were finding angle BAC rather than the requested angle ABC" — i.e. using the pair BA and CA (fixing vertex A) instead of BA and BC (fixing vertex B). The middle letter of a three-letter angle name is the vertex; using the wrong pair of vectors for the angle formula produces a real, differently-valued angle, not a rounding variant of the right one. October 2022 Q3(b) and January 2024 Q6(c) each independently confirm a closely related error — candidates substituting a position vector where a direction vector was needed (§8) — making mis-selection of the vectors fed into the scalar-product angle formula the single most consistently attested precision failure in this whole document, across three separate series (Jan 2021, Oct 2022, Jan 2024).

A related pair, confirmed independently at October 2022 Q3(b): "acute" and "obtuse" are not interchangeable just because one is 180° minus the other. That report notes candidates who found the acute angle "then subtracted from 180°" as a fix — a workable rescue, but evidence that the original angle computed was between the wrong ray-pair in the first place, not simply mislabelled.

#8. Position vector vs direction vector — and the "general point" is neither of the intuitive answers

A position vector locates a point relative to the origin (spec 7.4); a direction vector encodes a line's orientation and is what the scalar-product angle formula and the parallel/skew tests (§6) actually need. January 2024 Q6(c) documents candidates substituting the wrong one directly: incorrect direction vectors were "fairly common," the report notes, with candidates reaching instead for a position vector — often the point of intersection already found in an earlier part of the same question — where the line's actual direction vector was required.

A third, more subtle term enters foot-of-perpendicular problems (spec 7.4–7.5): the line's "general point," c + λd. This is neither the fixed point c nor the fixed direction vector d — it is itself a position vector, but one that varies with λ. The October 2021 Q7(a) examiner's report records the dominant real error as dotting this general point directly against d, rather than first forming the vector AX (= X − a, the displacement from the fixed external point to the general point) and dotting that against d instead — a setup independently confirmed this session to expand to a different, solvable-but-wrong equation for λ, not an unsolvable one, which is why it slips past a candidate rather than visibly failing. The report is direct about the cost: the wrong dot product "immediately resulted in 0 marks" for that part. The general point and the direction vector are both needed, but never dotted directly against each other — AX has to be assembled first.

#9. "General solution" vs "particular solution" — a spec-named pair, and what this paper's own citations actually test

Spec 6.4's own guidance text names both terms directly: "General and particular solutions will be required." A general solution is the family obtained by integrating a separated differential equation, carrying an unresolved constant; a particular solution fixes that constant using a given boundary condition. Stated plainly rather than assumed: every separable-differential-equation citation reviewed in this facts bank — January 2021 Q10, October 2021 Q2, October 2021 Q9, January 2026 Q3 — tests the transition to a particular solution via a stated boundary condition; none of the four is a bare "state the general solution" prompt with no boundary condition attached. This is a real, checkable asymmetry in what has actually been evidenced, not a claim about which is more likely to appear — the spec itself requires both, and this facts bank's own citations happen to all land on the same half of the pair.

#10. "Estimate" / "approximate" — approximating what, exactly

A binomial expansion evaluated at a substituted x-value approximates whatever expression the original series was built from at that x — not automatically the quantity the question asks for. The October 2024 Q1 examiner's report documents this precisely: candidates built (8−3x)⁻¹ᐟ³'s expansion, correctly substituted x = 2/3, and then submitted that series value directly as their estimate for ∛6 — but "the expansion approximated the reciprocal of what they were asked to find," since (8−3(2/3))⁻¹ᐟ³ = 6⁻¹ᐟ³, the reciprocal of 6¹ᐟ³. Every step of the method was executed correctly; the final word "estimate" was applied to the wrong quantity. The report's own generalising instruction: substitute the value into the original expression, not just the series, to check what is actually being approximated before reporting it as the answer.

#11. "Misread" — a formally defined term with its own capped consequence

"Misread" is not a synonym for "made an error." The general marking guidance defines a specific, numeric consequence for it: for a misreading that doesn't alter a question's character or materially simplify it, examiners "deduct two from any A or B marks gained" in the affected part — a fixed penalty, distinct from the ordinary consequence of a wrong method (which simply fails to earn the relevant M and dependent A marks). The October 2024 Q1 mark scheme shows this cap applied to a specific real case: misreading (8−3x)⁻¹ᐟ³ with the wrong sign of the index "can score a maximum of B1M1A0A0 M0A0" — the method-shaped marks (B1, M1) survive because the shape of the working is unaffected by the sign, while the marks that depend on the misread quantity's actual value are capped at zero. A genuine misinterpretation of what a question is asking for (as in §10, where the wrong quantity was reported entirely) is a different failure from a misread in this formal sense, and the two are not scored the same way.

#12. Flagged — where this list stops rather than pads

Consistent with this course's standing no-fabrication discipline: the entries above are every precision-costing vocabulary pair this pass could ground in a real, named citation inside WMA14-verified-facts.md. Several plausible candidate entries were considered and dropped because the facts bank does not evidence them for this paper specifically:

A formal "state" vs "prove"/"justify" command-word split, of the kind published in some exam boards' general command-word glossaries, is not independently evidenced anywhere in the four WMA14 examiner reports reviewed — the reports discuss specific missing justifications, but never contrast "state" against "prove" as command words in the abstract. Asserting a general rule for those two words on this paper would go beyond what has actually been checked, so no such entry appears above.
"Accept" and "allow," which recur in mark-scheme prose alongside "condone" (§2) and "oe" (§1), are not distinguished from one another anywhere in the material reviewed with enough consistency to support a precise, separate definition of each — they appear to function similarly in the lines quoted in WMA14-verified-facts.md, and this document does not manufacture a distinction the source material does not draw.
No examiner's report exists yet for the January 2026 series (confirmed directly in WMA14-verified-facts.md §5.1 and §5.7 — both the report URL pattern used for every other series fetched and 404s for January 2026). Several January 2026 items above (§4's "hence" examples, §7's scalar-product citation, §9's boundary condition) are therefore drawn from real, verified mark-scheme mechanics rather than from a documented candidate misconception for that series — each such instance says so at the point it is used, rather than presenting mark-scheme wording as if it carried examiner-report weight.

#Sources

All citations above trace to research/veridian/WMA14-verified-facts.md, itself built from primary Pearson documents (fetched from qualifications.pearson.com directly, or from a third-party mirror whose cover-page metadata — publication code, "© Pearson Education Ltd" — was independently checked against Pearson's own convention before use):

General Marking Guidance — Summer 2024 mark scheme (wma14-01-rms-20240815.pdf) and January 2021 mark scheme, cross-checked word-for-word (§1, §3, §11).
January 2021: full Examiner's Report (Q2(a) scalar/vector product and angle-vertex error, §7; Q4(b) "hence, or otherwise," §4; Q8 skew, §6) and mark schemes (Q1 binomial exact-answer, §3; Q9 volume of revolution, referenced not repeated).
October 2021: full Examiner's Report (Q7(a) foot-of-perpendicular general point vs AX, §8; Q10(b) fully-simplified statement, §5) and mark scheme.
June 2022: mark scheme (Q8(b) "condone missing dx," §2; Q9 proof-by-contradiction assumption wording, §5).
October 2022: full Examiner's Report (Q3(b) acute/obtuse and direction vector, §7; Q9 skew's two negatives, §6) and mark scheme.
October 2024: mark scheme and Examiner's Report (Q1 misread cap and reciprocal-estimate error, §10, §11).
January 2024: full Examiner's Report (Q6(c) position vs direction vector, §8; Q8 proof-by- contradiction justification, §5).
January 2026: mark scheme only, no examiner's report yet published (Q3 "hence," §4; Q7 "hence, using algebraic integration" and calculator ban, §3, §4; Q9 "condone" instances, §2).

This document adds no new primary-source fetch of its own; it re-reads citations already verified in WMA14-verified-facts.md at the vocabulary level.