Pure Mathematics 2

Level Threshold Guide

Provenance: written 2026-09-29, against research/veridian/WMA12-verified-facts.md (read in full — the spec/assessment sections §1–§5, and the topic-by-topic lesson-audit log §6 in its entirety, which is where nearly every mark-ladder cited below comes from), the three existing technique-hub documents for this paper — forge-exam-technique-guide.md (read in full, specifically Part 2's M/A/B glossary and dM1/ddM1 explanation, so this document does not re-derive what that one already covers), signal-frequency-and-boundaries.md (read in full, specifically Part 2, so this document does not re-state its grade-boundary table), and codex-formula-and-method-bank.md (headers and structure checked, to confirm it is a formula/method reference and does not already cover mark-threshold mechanics) — plus content/veridian/curriculum/wma12/paper-anatomy.ts and content/veridian/question-bank/wma12.ts (both grepped for existing M/A/B worked examples, to avoid re-using paper-anatomy.ts's own January 2021 Q2 write-up as if it were new). This document does not independently re-open primary Pearson PDFs this session; every mark-scheme line and examiner-report line quoted below is reused, with its original citation preserved, from the verified-facts bank's own lesson-audit log, which is where each was originally re-extracted directly from the primary PDF (file paths given in that log, not repeated here). Nothing below claims a fresh primary-source read that did not happen this session.

#WMA12 — Level/Threshold Guide

#What this document is, and isn't

This is not a second copy of forge-exam-technique-guide.md's Part 2, which already covers the M/A/B/B-ft glossary, the dM1-vs-ddM1 distinction, and the standing abbreviation list (cao, oe, isw, awrt, cso) with one worked example each. It is also not signal-frequency-and-boundaries.md's Part 2, which covers the qualification-level grade-boundary tables (raw marks needed for A/B/C/D/E on this unit, series by series) — this document contains no grade-boundary numbers at all, because a threshold inside a single mark scheme is a different thing from a threshold between grades, and conflating the two would misrepresent what either actually measures.

What this document does is go one level down from both: into the actual mark ladders of real, cited WMA12 questions, mark line by mark line, to show what specific difference in the words or numbers on a candidate's page moves them from not-scoring a given mark to scoring it. Where the same kind of boundary recurs across more than one real series — the facts bank's own audit log repeatedly found the same boundary shape re-appearing on genuinely different questions — this document says so and gives the count. Where only one series has been checked for a given boundary, it says that too, and does not generalise past it.

Every mark ladder below is traceable to a specific series and question number in research/veridian/WMA12-verified-facts.md. The "just misses / just clears" pairs are this document's own illustrative constructions, built to demonstrate the real, cited rule — they are not transcriptions of an actual candidate's exam script, and are labelled as illustrations, not evidence, throughout.

#Part 1 — The independence threshold: B1 vs M1 for the same-looking step

The most consequential threshold in this paper's mark scheme is not between a right and a wrong answer — it is between a mark that needs a visible method and one that doesn't. Two real, independently-audited examples show this is a genuine design choice by the mark scheme, not an accident of one question.

Circle centres, read straight off an equation already in centre-radius form, are B1 — not A1 — even when the equation carries an unresolved algebraic constant. January 2021 Q9's circle is given as $(x-k)^2+(y-2k)^2=k+7$ — never expanded into general form anywhere in the question — and the mark scheme awards the centre $(k, 2k)$ as a plain B1: no method step precedes it, because none is needed to read a centre-radius-form equation. January 2023 Q6's circle is genuine general form, $x^2+y^2+8x-4y=0$ , and its centre is still B1 — reading off a centre from either form is treated by the real scheme as an independent, no-method fact, not a derived one. (A lesson-audit pass on the-circle.ts found this step coded as a dependent A1 in the lesson's own worked material — both the marked-solution and the first worked-chain — on the reasoning that finding a centre from general form "feels like" it needs a completing-the-square method; both real anchors above contradict that, and the lesson was corrected to match.)

Just clears the B1: "Centre = $(k, 2k)$ " written directly beneath the given equation, nothing else. No method is asked for, and none is owed.
Just misses the B1: the same value, $(k, 2k)$ , but reached only after an incorrect intermediate rearrangement is shown and then abandoned uncorrected on the page — because a B mark can be lost to a manifestly wrong working line sitting in front of the correct final value, per the standing "manifestly absurd answers should never be awarded" convention that governs A marks and is applied the same way to B marks in practice. (Illustrative: this specific mixed-working case is not itself quoted in the facts bank; it follows directly from the general convention the facts bank does quote.)

Exhaustion proofs show the identical shape in reverse: the list is B1, and the calculation across that list is the M1 — not the other way round. Two real anchors, October 2023 Q1 and October 2022 Q1, both give a correct, complete list of cases as a B1 earned "on sight," with no separate credit or requirement for a candidate to first justify why the list is complete — then a separate M1 for attempting the arithmetic across at least two of the listed cases. (A prior pass in this course's own lesson content had the order backwards — treating the domain-justification as the method mark and the list as needing that justification as a precondition — and was corrected once both real anchors were checked line by line.)

Just clears the B1: any correct, complete set of the required values written down, with no accompanying sentence about why the set is complete.
Just misses the B1: the identical set of values, but with one case missing or one value wrong — because unlike the M1 that follows it (see Part 4), the B1 for the list itself is not tolerant of an incomplete or partly-wrong set.

The general pattern: a step earns B when it can be produced by inspection of what's already on the page (reading a coordinate off an equation already in the right form; writing down a finite list a spec topic guarantees is finite), and earns M only where an actual operation — expanding, substituting, solving — has to happen to get from what's given to what's wanted. The two circle anchors and the two exhaustion anchors are drawn from four different series and two different spec topics, and land on the identical rule independently — which is why this document treats it as a real, general threshold rather than a one-question coincidence, while still only claiming it as confirmed on these four questions specifically.

#Part 2 — The "correct but scores zero" threshold: the Hence rule and the no-working rule

Two real, cited mechanisms in this paper's mark scheme can take a fully correct final answer to zero marks on that part. Both are genuine thresholds — not partial-credit reductions — because the candidate crosses from full marks to none with no partial band in between.

The "Hence" rule. October 2023 Q4(c)(ii) asks candidates to factorise a cubic "hence", using a factor already established earlier in the same question. The mark scheme states plainly that a correct final factorisation reached by any other route — skipping straight to solving the cubic directly, which WMA12's calculator-permitted-throughout policy makes an easy temptation — scores the method mark and both accuracy marks as zero, regardless of whether the final factorisation is correct. The examiner report for the same series confirms candidates actually did this: reaching the fully correct factorised form by solving the cubic on a calculator and writing the three roots straight down, rather than dividing out the given factor first.

Just clears the marks: dividing $f(x)$ by the given factor $(2x+1)$ to obtain the resulting quadratic, then factorising that quadratic — the working visibly uses the given factor.
Just misses the marks: the identical, fully correct final answer $(2x+1)(2x-5)(x+3)$ , arrived at with no division shown and no reference to the given factor anywhere on the page. Full correctness, zero marks — because "hence" is a method instruction, not a hint.

The no-working rule. January 2025 Q5(a)'s own mark scheme carries an unusually explicit, doubled statement of this rule, worth reading as three separate tiers rather than one blanket ban, because the exact wording draws a line between two things that look similar on the page:

A correct numeric answer with a visible method line before it — clears the mark in the ordinary way.
A correct numeric answer where a specific intermediate detail (the scheme's own example: a bracket) is omitted from the written method but is "implied by further work" that follows it — still clears the mark. The omission itself isn't fatal; what matters is whether later working makes the omitted step recoverable.
The same correct numeric answer with no working of any kind — the scheme states this scores zero, in so many words, twice, once for emphasis.

Just clears the tier-2 boundary: writing $\frac{1}{2}(1.5)(...)$ with the outer bracket never explicitly closed, but going on to substitute a further value into what the missing bracket implies — the omission is recoverable from what comes after it.
Just misses it, landing in tier 3 instead: the bare final value, correctly rounded, with nothing above it on the page — no bracket, no substitution, no formula line, nothing for the omission to be "implied by."

This is the same underlying rule forge-exam-technique-guide.md already names once (§"the rule that decides whether a correct answer scores anything at all") — the addition here is the three-tier reading of the January 2025 wording specifically, which that document does not break out.

#Part 3 — The sufficiency threshold: mark schemes that name their own minimum bar

Some method marks are written with an explicit, numeric tolerance built in — the scheme states directly how much of a multi-part operation has to be done correctly, and no more, for the method mark to stand. This is a different threshold from "right vs wrong": it's "enough of the method vs not enough of it," stated as a number.

Clearing a fraction by multiplying every term through, trigonometric equations. June 2025 Q8(ii)'s mark scheme, for the step of multiplying a $\tan\theta$ -substituted equation through by $\cos\theta$ , states the method mark is earned for doing this correctly "on at least 2 of the 3 terms in the equation" — not all three.

Just clears the dM1: exactly two of the equation's three terms correctly multiplied through by $\cos\theta$ , even if the third is mishandled or dropped.
Just misses the dM1: only one term correctly multiplied through, the other two left as they were — one term short of the scheme's own stated floor.

Listing enough cases, exhaustion proofs. The same October 2023 Q1 / October 2022 Q1 pair from Part 1 states the following method mark (the arithmetic across the list, distinct from the B1 for the list itself) is earned for "at least two valid combinations" worked correctly, not all of them.

Just clears the M1: two of the listed cases correctly calculated through, even if a third case in the same list is never attempted or is arithmetically wrong.
Just misses the M1: exactly one case correctly calculated, with no second — one combination short of the stated floor.

Reversing and adding enough terms, the arithmetic-series sum-formula proof. January 2020 Q8(i)'s M1, for the standard "write $S_n$ , reverse it, add the two" proof, is earned for showing "at least two pairs of matching terms" — with an explicit allowance that the second, reversed sum may miss its own last term or terms and still count, provided the pairing pattern is established.

Just clears the M1: writing out the sum forwards and backwards and explicitly pairing at least two corresponding terms, even if the reversed sum is left visibly incomplete at its tail end.
Just misses the M1: writing the forward sum, asserting the reversed sum exists, but never actually writing a second line with terms to pair against the first — the pattern has to be shown, not asserted.

These three sufficiency floors come from three different spec topics (trigonometric equations, proof, sequences and series) and three different series, and each is stated by its own mark scheme as an exact minimum, not a vague "sufficient working" instruction — which is precisely why an illustrative one-short-of-the-floor example is possible to construct for each: the floor is a specific number, so "one below it" is a specific, nameable answer, not a matter of examiner judgement.

#Part 4 — The exact-form threshold: when a correct decimal is worth nothing

Distinct from the no-working threshold in Part 2, this is a threshold on the form of an otherwise fully-derived answer, and it cuts in both directions — some questions demand the exact form and reject a decimal outright; others explicitly accept a decimal that communicates the same exact value.

Rejects the decimal, no exceptions but one. January 2021 Q2's A1 for the stationary points' $x$ -coordinates reads, transcribed from the rendered mark-scheme page rather than the plain-text extraction because the source carries a recurring-decimal dot easy to lose in conversion: the exact fraction $8/3$ is required, a truncated decimal $2.67$ is explicitly rejected, but a recurring-decimal notation for the same value is explicitly accepted. The line draws the boundary not at "decimal vs. exact" but at "does this notation unambiguously communicate the exact value" — a recurring-decimal mark over the repeating digits does; a plain rounded decimal does not.

Just clears the threshold: $2.6\overline{6}$ or $2.\dot{6}$ , written with the recurring-decimal notation intact.
Just misses it: $2.67$ — arithmetically the same value to the precision shown, but the notation itself doesn't certify that the value is exact rather than rounded, and the scheme rejects it by name.

The same shape recurs on circle radii: a real anchor names $r=\sqrt{20}$ written as a decimal as a specifically confirmed error, where the surd form was required.

Accepts the decimal, by design. The same paper's own glossary code awrt ("answers which round to") appears repeatedly across real mark schemes precisely where a question doesn't demand an exact form — January 2025 Q3's A1 accepts $x = \text{awrt } 3.76$ outright, no exact-form alternative offered at all, because none is expected for that question's numbers.

Just clears an awrt mark: any value that rounds to the stated figure at the stated precision, by any correct method.
Just misses it: a value one unit off in the last decimal place shown, or a value in the right ballpark reached by a method the scheme doesn't credit (see Part 2's no-working tier).

The practical threshold a candidate actually has to judge, question by question, is not "should I round" in the abstract — it's "does this specific mark's own line carry awrt, or does it carry an exactness instruction," because the same paper runs both rules simultaneously on different marks, sometimes within the same question.

#Part 5 — The dependency threshold: what "dependent on" actually attaches to

forge-exam-technique-guide.md already explains that a dM1 needs a specific earlier mark to have been earned, where a plain M1 doesn't. What it doesn't cover — and what the facts bank's own audit log flags as a recurring source of real error in this course's own lesson-writing, not just a subtlety for candidates — is that a dependency is written against one specific prior mark, never a vague "the working so far," and that getting this wrong in either direction changes how much of a question is recoverable after an early slip.

The singular-dependency convention. January 2023 Q6(b)'s dM1, for solving the three-term quadratic produced by eliminating a variable between two equations, is stated by the real scheme as "dependent on the first method mark" — singular. It depends on the earlier M1 (the elimination attempt) having been made, not also on the A1 for that elimination having produced the correct quadratic. In practice this means: a candidate who eliminates incorrectly, reaches a wrong three-term quadratic, but then correctly solves that wrong quadratic, still earns the dM1 — the dependency is on the attempt, not on the accuracy.

Just clears the dM1: correctly factorising or applying the quadratic formula to whatever three-term quadratic actually sits on the candidate's own page, right or wrong, provided the earlier elimination was genuinely attempted.
Just misses the dM1: the identical solving method, but with the earlier elimination step entirely absent — nothing to be dependent on.

A lesson-audit pass on this exact question found the opposite error already written into the course's own worked material — both alternative methods in the lesson's own method-comparison block documented their dM1 as dependent on both the earlier M1 and its accompanying A1 — and corrected it once the real scheme's own singular wording was checked directly. This is worth naming precisely because it is the kind of error a careful, well-intentioned but unchecked account of "dependency" tends to produce: it feels safer to over-specify what a mark depends on, and the real scheme is often looser than that instinct expects.

The compounding-dependency shape, ddM1. January 2021 Q3(ii)'s five-mark log-equation chain — M1 (combine terms) → dM1 (rearrange and remove the logs, dependent on the M1) → A1 (the resulting three-term quadratic) → ddM1 (attempt to solve that quadratic) → A1 (the surviving root, invalid root rejected) — genuinely does compound: the ddM1 sits at the end of a chain where two prior stages both have to have gone right for it to be reachable at all, which is what makes it doubly- rather than singly-dependent. This is a structurally different situation from the singular dM1 above, and the difference is not cosmetic: a slip at the very first M1 in this chain removes access to every mark after it, where a slip after the first M1 in the Q6(b) example above still leaves the dM1 reachable.

A cross-series confirmation of exactly where a boundary sits inside a chain. The same "combine logs → remove logs → solve" shape was independently checked across four series — January 2021 Q3(ii), June 2023 Q5, June 2024 Q3(i), June 2025 Q6(i) — and every one of the four scores "reach the correct log-free equation" as its own, separate accuracy mark, distinct from both the combining step before it and the solving step after it. One lesson-audit pass had collapsed this into a single mark spanning both removing the logs and solving the result; re-checking against all four series independently confirmed the real boundary sits between forming the equation and solving it, not around both together — a genuine example of a threshold whose exact location inside a multi-step chain was worth confirming four times before treating it as settled.

#Part 6 — The allowance threshold: special cases and the misreading rule

Not every deviation from the expected route costs marks outright. Two real, distinct mechanisms allow a candidate to keep scoring after a specific, named kind of error — and knowing where each one's own limit sits is itself a threshold worth stating precisely, because both are bounded, not blanket forgiveness.

A named special case. January 2021 Q9 confirms a real instance of a candidate misreading the label $O$ as the circle's own centre rather than as the origin — a plausible confusion, since both share the letter. The examiner report states this specific misreading was permitted as a special case, and that candidates who made it were able to score every mark on that part except the final A1. The boundary here is precise: the special-case allowance reaches the method marks (the working is still valid, just built on the swapped label) but stops exactly at the point where the final numeric answer would itself be wrong under the correct labelling — that last mark is not recoverable by the special case.

Just clears the allowance: every method step correctly carried out under the swapped $O$ -as-centre reading, right up to but not including the final numeric answer.
Just misses full marks despite the allowance: the identical working, but the candidate has still lost the one mark the special case was never going to cover — the final answer itself, which is wrong under the correct labelling regardless of how the method that produced it is read.

The facts bank's own audit log separately flags that no other live instance of the standing "SC" (special case) glossary code was found in either of two further real mark schemes spot-checked for a different pass — so this specific allowance should be read as a real, one-question-confirmed instance of the mechanism existing, not as evidence that special cases are common on this paper.

The general misreading rule, by contrast, is a flat, unconditional deduction. Where a misreading doesn't change the character of a question or make it materially easier, the standing rule deducts exactly two marks from whatever A or B marks were gained in the affected part — no special-case discretion, no all-but-one-mark allowance, just a fixed penalty applied the same way every time it's triggered.

Just clears the "materially simplified" line (deduction applies, capped at two): misreading a coefficient's sign throughout a question that still requires the same method and the same number of steps to complete.
Just misses staying inside the rule at all (the rule stops applying, and the part may be marked as if the misread value were simply wrong, with no capped deduction protecting the candidate): a misreading that removes a step entirely — for instance, misreading a "solve" instruction as if the answer were already given — because that is exactly the "materially simplifies the question" case the rule's own wording excludes from its protection.

The practical distinction for a candidate is not "was I marked down for a misreading" — it's which of two entirely different mechanisms applied: a capped, fixed two-mark deduction that still lets most of the part's marks stand, or the ordinary marking of an incorrect method with no special protection at all, because the misreading changed what the question was actually asking.

#Flagged gaps — what this document does not claim

No independent primary-source re-verification this session. Every mark ladder above is reused from research/veridian/WMA12-verified-facts.md's own lesson-audit log, which is where each was originally re-extracted from a primary PDF; this document did not re-open those PDFs itself. If a future pass finds the facts bank's own transcription of a line was wrong, this document inherits that error and should be re-checked alongside it.
The "just misses / just clears" pairs are illustrative, not evidentiary. Where a real candidate response is quoted from an examiner report (the Hence-rule calculator-bypass case, the O-as-origin misreading), that is marked as real; every other paired example in this document is this document's own construction, built to demonstrate a real, cited rule, and should not be read as a transcription of an actual script.
Coverage is bounded by what the facts bank's audit log happened to check in depth. The six threshold types above (independence, correct-but-zero, sufficiency floors, exact-form, dependency location, allowance) are the ones with a genuinely multi-example, cross-series paper trail in the existing research. Other spec topics — the binomial expansion's isolate-the-term-from-a-list threshold, the trapezium-rule strip-count boundary, the definite-integral sign-split boundary — are named with a single real citation each in forge-exam-technique-guide.md's own common-error digest, but were not found with enough independent depth in the facts bank to build a second, comparably-verified threshold section here without thinning the standard the rest of this document holds to. That is a real gap in the research to date, not a claim that those boundaries don't exist.
No grade-boundary or likelihood-of-appearing claim of any kind. Consistent with this course's standing rule against inventing either, and consistent with signal-frequency-and-boundaries.md already carrying the real, sourced grade-boundary table — this document does not repeat or approximate it.

#Sources used

research/veridian/WMA12-verified-facts.md — read in full for §1–§5 (spec, assessment structure, mark-scheme conventions, archive scope) and in full for §6's topic-by-topic lesson-audit log, which is the direct source of every mark ladder, quoted mark-scheme line, and examiner-report line in this document.
content/veridian/technique-hub/wma12/forge-exam-technique-guide.md — read in full, to confirm which parts of the M/A/B system it already covers (so this document builds past that rather than repeating it) and to correctly attribute the January 2025 Q5(a) no-working citation it names once, which Part 2 of this document expands into three tiers.
content/veridian/technique-hub/wma12/signal-frequency-and-boundaries.md — read in full, to confirm its Part 2 already carries this paper's real grade-boundary data, and to keep this document's scope to intra-mark-scheme thresholds only.
content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md — headers and structure checked, to confirm no overlap with this document's subject matter.
content/veridian/curriculum/wma12/paper-anatomy.ts and content/veridian/question-bank/wma12.ts — grepped for existing real M/A/B worked examples (paper-anatomy.ts's own January 2021 Q2 write-up; the question bank's October 2019/October 2021 entries), to avoid re-using either as if newly sourced here.