Pure Mathematics 2
Topic Brief: The Circle
Provenance (2026-09-29): written against research/veridian/WMA12-verified-facts.md — §2 (spec wording), §5 (the archive's real scope: 20 series with a question paper + mark scheme, 17 with an examiner report), §6.3 and §7 (the topic's own prose summary), and the two independent lesson-audit passes that re-extracted primary sources directly (dated 2026-09-14), plus one further citation found by grepping the Proof topic's own audit section for "circle" rather than trusting §6.3's own list of anchors. content/veridian/curriculum/wma12/the-circle.ts was read in full (935 lines) so nothing below repeats its worked exemplars, mechanism blocks, or trap-taxonomy prose. signal-frequency-and-boundaries.md and codex-formula-and-method-bank.md (both existing technique-hub documents) were read for their own circle content so this brief adds to them rather than restating them — one factual correction against the former is noted below. content/veridian/question-bank/wma12.ts was grepped for circle content: none present (239 lines, no matches).
SIGNAL Topic Master Brief: The Circle (WMA12, spec 3.1)
What this is
A revision-facing "what to expect" briefing for this one topic — narrower than signal-frequency-and-boundaries.md's whole-paper table, and pattern-facing rather than method-teaching (that job belongs to the-circle.ts itself; nothing here re-derives a formula or a worked example).
Every real series confirmed testing this topic
Seen in 4 of 20 series checked (the archive's own full scope, per the facts bank §5 — a floor on individually-cited series, not a completed topic-tag of all 20; never a percentage or likelihood).
| Series & Q | Setup | Marks | Question no. |
|---|---|---|---|
| Jan 2021, Q9 | Centre-radius form with an unresolved constant ; point on the circle gives a "show that" quadratic in ; tangent meets the x-axis; area of triangle | 10 (2+3+5) | Q9 |
| Jan 2023, Q6 | True general form ; centre/radius; tangent-and-normal intersection point; equation of the normal | 10 (3+4+3) | Q6 |
| Jun 2022, Q3 | A second, unrelated circle-geometry part sitting inside a question whose part (i) is a Proof-topic counter-example — found only by re-reading the Proof audit, not §6.3 | not stated in facts bank | Q3 |
| Jun 2025, Q9 | Examiners flag it explicitly as unexpected; the radius turns out numerically equal to the centre's x-coordinate | not itemised by part in facts bank | Q9 |
(Total question count per series is not fixed on this paper — confirmed 9 to 11 across the archive, per the facts bank §1 — so "Q9" is not necessarily a paper's final question; no source read for this pass confirms the exact total for these four series specifically.)
Both Jan 2021 and Jan 2023 total exactly 10 marks across three lettered parts. Jun 2022 Q3's circle component and Jun 2025 Q9's per-part breakdown are not itemised in the facts bank at the depth the two January anchors are — flagged rather than estimated. One correction against an existing technique-hub document: signal-frequency-and-boundaries.md cites the June 2025 anchor as "Q2"; the facts bank itself (§6.3, twice) and the-circle.ts's own sourcing note both say Q9 — this brief follows the facts bank as the primary source.
The real mark-tariff pattern
Where marks are itemised (Jan 2021, Jan 2023), the shape is consistent: a short, independent (no-method-required) mark for stating the centre; a dependent mark for the radius; then the bulk of the question — 5–7 of the 10 marks — sitting in a tangent, normal, or area calculation that follows from that centre and radius. In both cases the circle question is a full 10-mark, three-part question rather than a short standalone item. Both are numbered Q9 and Q6 respectively; since this paper's own question count is confirmed to vary between 9 and 11 (facts bank §1), neither number by itself confirms "early" or "late" placement without knowing that series' total.
The recurring context and angle
- The centre/radius step is graded as two separate marks with different rules, confirmed independently on both fully-itemised anchors: the centre is credited free of any shown method (B1), the radius only once a correct method is shown (A1 or A1ft) — true whether the equation started in centre-radius form with an unresolved constant (Jan 2021) or in genuine general form (Jan 2023).
- A circle equation carrying an unresolved constant across multiple parts, resolved to a specific value later in the same question, is a confirmed real shape (Jan 2021) — not a one-off invention. The re-audit found the actual mark-loss point on that anchor was not completing the square (there was none to complete) but re-substituting the newly solved constant back into an earlier general expression — a distinct trap from the general-form sign-flip error confirmed on Jan 2023.
- The tangent-gradient step is graded as one combined method mark, not two, on both fully-itemised anchors — "find the radius gradient, then take its negative reciprocal" is never split into separate marks in either real scheme, regardless of which direction the algebra runs.
- Examiners explicitly frame at least one recent circle question as deliberately non-routine: Jun 2025 Q9's report calls it "unexpected" and states the question turned on noticing the radius equalled the centre's x-coordinate — a constrained-parameter framing rather than a direct plug-into-the-formula task. This is the most recent of the four confirmed anchors (as of this pass).
- A circle question has also appeared as a second, structurally unrelated component bolted onto a different topic's question (Jun 2022 Q3, alongside a Proof-topic counter-example) — the facts bank does not itemise what that circle part actually tested, so this brief can confirm the pairing exists but not its content.
If this topic appears, here's what to expect
- A full 10-mark, three-part question is the confirmed shape whenever the circle is the lead role of a question — not a short 3–4 mark item.
- Expect the centre and radius to be marked separately, with the centre markable even from an otherwise-incomplete method.
- If the circle's equation carries an unresolved constant into a later part, expect the actual mark-losing step to be re-substituting the solved value back into the earlier general expression — not the algebra of solving for it.
- Expect "find the tangent gradient" to be one method mark covering both the radius-gradient step and the perpendicular-reciprocal step, not two.
- Do not assume the circle question is routine: the most recent confirmed anchor (Jun 2025) was examiner-flagged as deliberately unexpected in its framing.
What remains genuinely unverified for this topic
The facts bank's own §7 summary asserts this topic verified "across at least 5 series" and separately claims "right-angle-in-a-semicircle applications" have been verified — but no named series, question, or quote for a semicircle or perpendicular-chord-bisection question (spec properties (i) and (ii)) appears anywhere else in the document; every itemised anchor above tests only property (iii), radius-tangent perpendicularity. That broader claim should be treated as unconfirmed until a specific anchor is found, not repeated as settled. Six series flagged in the facts bank's §8 as grepped but not read start-to-finish (Oct 2019, Oct 2021, Oct 2024, Jun 2023, Jun 2024 + regional, Jan 2025) could raise the "4 of 20" count or surface a genuine semicircle/chord anchor on a fuller re-read; nothing above is extrapolated past what the sources checked for this pass actually state.