Topic Brief: Discriminant With a Parameter

Provenance: compiled 2026-09-29. Source is research/veridian/WMA11-verified-facts.md (its own primary-source re-extractions, cross-checked against content/veridian/curriculum/wma11/discriminant-with-a-parameter.ts's audit trail) — no fresh PDF extraction was done for this document. This is a narrow expansion of the single-line entry ("Discriminant with a parameter | 1.4 | 2 of 5 | Jan 2019 Q9, Jan 2023 Q4") already in signal-frequency-and-boundaries.md §1 of this same folder — that document does the whole-paper frequency table; this one is the per-topic drill-down it deliberately doesn't repeat. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Discriminant With a Parameter (spec 1.4)

#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. This exact topic — a parameter inside the discriminant, condition given as a real-roots case — is confirmed in 2 of those 5: Jan 2023 Q4 and Jan 2019 Q9.

Two further series (Oct 2022, Jan 2024) were confirmed as genuine mark schemes but never mined topic-by-topic, and two June series (June 2022, June 2023) were read for one targeted question each, neither of which was this topic. So "2 of 5" is the honest count where absence is informative; across the other four series touched, this topic's presence or absence simply isn't known yet — that is a gap in this research pass, not a claim that the topic only appears twice in nine sittings.

#The two confirmed appearances

Jan 2023 Q4 — kx² + 6kx + 5 = 0, k a non-zero constant, "has no real roots," find the range of k. 4 marks total: M1 (attempt b²−4ac with a=k, b=6k, c=5) → dM1 (form the inequality and solve it to a critical value — one dependent mark, not two) → A1 (upper limit 5/9) → A1 (complete range, 0 < k < 5/9, both bounds strict).

Jan 2019 Q9 — 3/x + 5 = −2x + c, "has no real roots," find the range of c. 7 marks — a single unsplit part, not (a)/(b)/(c). M1 (clear the fraction, multiply through by x) → A1 (correct quadratic, 2x² + (5−c)x + 3 = 0) → M1 (attempt b²−4ac) → dM1 (solve to critical values) → A1 (c = 5 ± 2√6, exact) → M1 (select the inside region) → A1 (final range).

#Mark-tariff pattern: the same topic, two very different lengths

The two confirmed sightings sit at opposite ends of a tariff range — 4 marks and 7 marks — and the gap is structural, not a difference in how hard the discriminant step itself is. Jan 2019 Q9 carries two extra method marks Jan 2023 Q4 never needed: one to derive the quadratic from a fractional equation in the first place, and one to select which side of the critical values actually satisfies the question (Jan 2023 Q4's answer region needed no separate selection mark because the "outer" root, k=0, was already pinned down by the stem's own non-zero condition). On this evidence, don't expect a fixed length for this topic — expect either a short, self-contained 4-ish-mark question where the quadratic is already sitting on the page, or a longer 7-ish-mark question where the discriminant step is the back half of a question that starts somewhere else entirely (here, clearing a fraction).

#The recurring angle: "no real roots," and a quadratic that isn't always already there

Both confirmed appearances use exactly the same real-roots case — "has no real roots," i.e. the strict b²−4ac < 0. Neither uses "two distinct real roots," "at least one real root," or "does not cross the x-axis" in the real record; those alternate framings exist in this course's own teaching material (the lesson's MCQs and chain-drill) as deliberately different practice cases, not because a real paper has been found using them for this topic. Anyone revising from past-paper pattern alone should expect "no real roots" specifically, while still knowing the other framings cold, since the spec point itself (1.4) covers all three cases and a paper is free to ask any of them.

The two appearances also diverge on where the parameter sits: Jan 2023 Q4 puts it in both a and b (k multiplies x² and x), which is what forces the non-zero-leading-coefficient check; Jan 2019 Q9 puts it only in c, inside an equation that isn't a quadratic at all until a fraction is cleared. That second shape is the one genuinely distinct "angle" in the confirmed record — the trap isn't in the discriminant arithmetic, it's in recognising a quadratic has to be built first. The real Jan 2019 examiner report is explicit about this exact failure: candidates who "failed to realise that they needed to form a quadratic in x" before applying the discriminant test at all, and, separately, some who multiplied "all terms except c" through by x, producing an expression linear in c that goes nowhere.

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

• Wrong sign: b²+4ac used in place of b²−4ac (Jan 2023 Q4 examiner report) — scores the method mark nothing, however clean the rest of the working looks.
• Partial squaring: squaring only the letter inside a compound coefficient, e.g. 6k² instead of (6k)²=36k² (Jan 2023 Q4) — condoned for the first method mark only, then costly.
• Dropped parameter boundary: solving the quadratic-in-k correctly but losing the stem's own non-zero condition, most often by dividing an inequality by k instead of factorising (Jan 2023 Q4 — "the most common error seen was the absence of the lower bound 0 < k").
• Boundary treated as non-strict: "0 ≤ k ≤ 5/9" in place of the correct strict range (Jan 2023 Q4).
• Quadratic not recognised/built: not seeing that the discriminant needs a quadratic formed first from a non-quadratic-looking equation, or multiplying only some terms through by x (Jan 2019 Q9).

#If this topic appears: what to expect

Expect the phrase "no real roots" specifically, and expect one of two shapes: a short, ~4-mark question with the quadratic already given and a parameter in a and/or b (check the leading-coefficient exclusion, square the whole compound coefficient, keep the boundary strict); or a longer, ~7-mark question where the discriminant is the second half of the problem and the first half is deriving a genuine three-term quadratic from something that doesn't initially look like one — most confirmed here as a fraction cleared by multiplying through. Either way, the confirmed trap set above, not the length of the question, is what determines whether the marks are actually banked.