Topic Brief: Quadratic Inequalities Graphically

Provenance: compiled 2026-09-29. Source is research/veridian/WMA11-verified-facts.md §4.1 and its own "Lesson-audit log" entry for quadratic-inequalities-graphically.ts (line ~896) — no fresh PDF extraction was done for this document; every quote below is copied from that facts bank's own verbatim transcriptions, which it states were checked against the real PDFs with both pdftotext -layout and -raw. This is a narrow expansion of the single-line entry ("Quadratic/reciprocal inequalities, graphically | 1.7–1.9 | 2 of 5 | Jan 2019 Q4, Oct 2021 Q3(i)+(ii)") already in signal-frequency-and-boundaries.md §1 of this same folder, and it deliberately splits the quadratic-specific finding out from that row's reciprocal one — see below for why. Cross-checked against forge-exam-technique-guide.md §2.7 and codex-formula-and-method-bank.md §1.1.6/§2.1.3 to confirm this brief adds a per-topic layer neither already covers, rather than repeating it. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Quadratic Inequalities, Interpreted and Represented Graphically (spec 1.7–1.9)

#Frequency: seen in 2 of 5 series checked

The facts bank fully mined five WMA11 examiner reports question-by-question: Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025. The specific skill this brief covers — reading or stating the system of inequalities that defines a region bounded by a line and a quadratic curve, shown in a sketch — is confirmed in 2 of those 5: Jan 2019 Q4 and Oct 2021 Q3(ii).

This is narrower than the frequency table's own "2 of 5" row, which lumps this skill together with a different question on the same Oct 2021 paper, Q3(i) ("solve 3/x > 4"). That is a reciprocal inequality, spec 1.11 territory, solved by a different method (multiplying through by x²) and carrying no line-meets-curve region at all. The facts bank itself flags conflating these two parts as a real error a prior pass made and later corrected, so this brief keeps them apart: the "2 of 5" here is Jan 2019 and Oct 2021 only, and Oct 2021's contribution is its Q3(ii), not Q3(i).

A third data point exists outside the 5-series base: June 2023 Q1 was spot-checked (not fully mined) and found to test a quadratic inequality not pre-arranged with one side already zero — a rearrangement step this course already teaches, just not yet drilled against a non-pre-arranged example. Because June 2023 was read for one targeted question only, it isn't folded into the "2 of 5" count, but it corroborates that this spec area keeps recurring beyond the two fully-documented sightings.

#The confirmed appearances

Jan 2019 Q4 (3 marks). Line $l_1$: $2y=x$; curve $C$: $y=2x-\tfrac18x^2$; unshaded region $R$ bounded by $l_1$, $C$ and a third boundary parallel to the y-axis. Task: state the inequalities defining $R$. Mark scheme: B1 for sighting either $2y\leq x$ or $y\geq 2x-\tfrac18x^2$ straight off the sketch, no method required; M1 for solving where $C$ meets the x-axis ($x=16$) to get the third boundary; A1 for the complete, correctly-strict system, $2y\leq x,\ y\geq 2x-\tfrac18x^2,\ x<16$. The examiner report calls it "a new topic within the qualification" and adds that "only a minority of candidates gained full marks."

Oct 2021 Q3(ii) (3 marks, the second of two independent parts on that question). Curve $C$: $y=2x^2-50$; line $l$ meets $C$ on the negative x-axis; shaded region $R$ in quadrants 2 and 3. Same task shape: state the inequalities defining $R$. Mark scheme: B1 for the line's equation; M1 for two of the three inequalities, with an explicit note that a region-label answer (e.g. "$R<2x^2-50$") scores M0, not just a lost accuracy mark; A1 for the complete system. The facts bank's own correction (2026-09-14) confirms this is the same real question type as Jan 2019 Q4 — a genuinely repeated pattern, not two unrelated findings.

#Mark-tariff pattern: 3 marks, every time it's been checked

Both confirmed sightings are worth exactly 3 marks, and both sit as one self-contained sub-item — the whole of Jan 2019 Q4, or one part of Oct 2021's two-part Q3 — rather than as the tail end of a longer question. Both follow the identical B1 (sight, no method) → M1 (solve for the missing numeric boundary) → A1 (complete, strict system) shape. On this evidence — thin, at N=2, but internally consistent — expect a short, standalone 3-mark item rather than a long multi-step one if this topic appears again.

#The recurring context: a sketch, a line, a quadratic curve, an unlabelled region

Both real appearances use the same setup: a printed sketch (both papers call it "Figure 1") showing a line and a quadratic curve enclosing a region labelled $R$, with the question asking for the inequalities that define $R$ — never asking the student to solve a single one-variable inequality in isolation. Both answers combine two inequalities in $y$ (one from the line, one from the curve) with one inequality in $x$ found by solving where the curve crosses an axis. That third, x-only boundary is the one numeric method mark in both cases; the other two inequalities are read straight off the picture for no working at all.

#Confirmed trap set (both series, verbatim-sourced)

• Wrong-strictness boundary: "Stating y < 16 was a frequent error" in place of the correct $x<16$ — using the wrong variable's inequality symbol on the numeric boundary (Jan 2019 Q4).
• Region-label answers: writing the answer in terms of $R$ itself (e.g. "$R<2x^2-50$") instead of the actual variable — this "scores M0" on Oct 2021 Q3(ii), a lost method mark, and separately loses the equivalent accuracy mark on Jan 2019 Q4, where the examiner report notes some candidates "solved the equation correctly but stated no inequalities."
• Stopping after the algebra, before the inequalities: candidates who correctly solved for the numeric boundary but never converted that into the required inequality statements at all (Jan 2019 Q4).
• No clear entry point: on Oct 2021 Q3(ii) specifically, the examiner report says "little idea of how to approach the question" was common, and that "far fewer achieved the final mark for the inequality in x" than picked up the first two.

#If this topic appears: what to expect

Expect a sketch of a line and a quadratic curve enclosing a labelled region, and a request to state the inequalities that define it — not a bare "solve this inequality" instruction. Budget it as a short, 3-mark item. Pick up the two "sight" inequalities straight from the diagram for free, find the numeric x-boundary by solving where the curve meets an axis, and state the final answer in the actual variable, never in terms of the region's own label — that single habit is what both confirmed series show costing marks. If the diagram instead shows only a curve on its own with no line and no shaded region, that's outside what either real anchor here has tested; the confirmed pattern is specifically line-meets-quadratic, not a quadratic inequality solved in isolation.