Pure Mathematics 2
Flashcard Bank
Provenance (2026-09-29). This document is a reformatting exercise only. It contains no new
research, no new primary-source reading, and no claim not already stated in the thirteen codex-*.md
files in this same folder — codex-formula-and-method-bank.md and the twelve codex-deep-dive-*.md
files (Proof; Factor and Remainder Theorems; The Circle; Sequences and Arithmetic Series; Geometric
Series; Binomial Expansion; Logarithms and Exponentials; Trigonometric Equations; Stationary Points and
Curve Sketching; Definite Integrals and Areas; The Trapezium Rule; and the Complete Worked Paper, June
2021), all read in full before writing a single card below. Every card's back is a direct reformatting
of a sentence, formula, or worked line already present in one of those files — a term, a formula
statement, a real mark-scheme fact, or a short worked micro-example that already exists in the source
document named. Nothing below states a likelihood, a grade boundary, or a "confirmed in every series
without exception" claim; where a source card carries a "floor, not a rate" qualifier (e.g. "confirmed
in N of 20 series checked"), that qualifier is carried over unchanged rather than dropped. Cards drawn
from a source's own labelled "VERIDIAN-original" or "illustrative" micro-example are marked [worked
example] below to keep that distinction visible; all other cards restate a formula, a definition, or a
real, cited exam finding. Each card names the source file (and, where the source gives one, the section)
it is reformatted from.
WMA12 — CODEX Flashcard Bank
One flashcard per term or question-shape: front, then back. Organised by the eight spec content
areas, in the same order as codex-formula-and-method-bank.md Part 1, followed by a short P1-prerequisite
section and a cross-cutting mark-scheme-convention section.
1. Proof (spec 1.1–1.3)
Source: codex-deep-dive-proof.md, plus codex-formula-and-method-bank.md §1/Part 2 §1.
1.1 — Front: What are the three proof techniques tested at WMA12 spec item 1? Back: Exhaustion (1.2) — domain finite and listable; disproof by counter-example (1.3) — any domain size; general/direct algebraic proof, including proof by contradiction as a credited alternative (1.1, not separately numbered). (codex-deep-dive-proof.md §1.2)
1.2 — Front: Why is disproof structurally cheaper than proof for a "holds for every n" claim? Back: The claim is a conjunction of one sub-statement per domain element. It is true only if every part holds (so proving it needs every part checked — tractable only if the domain is finite and listed), but false the moment one part fails (so disproving it needs exactly one failing case, whatever the domain's size). (codex-deep-dive-proof.md §1.1)
1.3 — Front: Real mark-scheme shape for an exhaustion proof (Oct 2022 Q1; Oct 2023 Q1; 3 marks each)? Back: B1 (complete list) → M1 (calculation attempted across cases) → conclusion mark (B1/A1). (codex-deep-dive-proof.md §1.2, §3.1)
1.4 — Front: Real mark-scheme shape for a counter-example question (June 2022 Q3(i); 2 marks)? Back: M1 (attempt) → A1, dependent, bundling the example, its reason, and the conclusion together into one mark. (codex-deep-dive-proof.md §1.2)
1.5 — Front: Real mark-scheme shape for a general/direct algebraic proof over an infinite domain (June 2019 Q3(i); 3 marks)? Back: M1 (rearrange) → A1 (reach a form whose sign is guaranteed, e.g. a squared term) → A1* (three required closing parts together: correct algebra, the explicit "a square is never negative"-type reasoning, and a reference back to the original claim). (codex-deep-dive-proof.md §1.2, §2.3)
1.6 — Front: Is proof by contradiction its own numbered spec sub-point? Back: No. It is confirmed as a credited alternative route to the direct algebraic proof, on the same real mark scheme (June 2019: "Alt to (i) via contradiction — Proof by contradiction is acceptable"). (codex-deep-dive-proof.md §2.4)
1.7 — Front: What common real error undermines a general/direct proof over an infinite domain? Back: Treating several numerical examples of the claim holding as sufficient evidence — it isn't, for an infinite domain; one verified real series recorded this as the single worst-performing item on its paper. (codex-deep-dive-proof.md §1.2)
1.8 — Front: In an exhaustion proof generated from a relation between two variables (e.g. c=b+2, a+b+c=10), what real trap sits in that first line? Back: The relation's direction is part of the domain definition, not cosmetic — applying it backwards silently generates a different, wrong case set (confirmed real error, Oct 2022: "using c=b−2 instead of c=b+2"). (codex-deep-dive-proof.md §2.1)
1.9 — Front: In an exhaustion proof claiming "a multiple of k" (k>2), what must actually be shown? Back: The actual quotient (or an explicit equivalent, e.g. "=10, a whole number") must be shown — a bare "divisible by 4" assertion with no supporting division is insufficient on a real mark scheme. (codex-formula-and-method-bank.md Part 2 §1)
1.10 — Front: How many failing cases does a counter-example need? Back: Exactly one — more than one wastes time without extra credit; the calculation, the stated reason it fails, and the conclusion must all be shown. (codex-deep-dive-proof.md §2.2)
1.11 — Front: What four real question-shapes has Proof actually taken across the confirmed anchor catalogue (10 question-parts)? Back: (1) Single-technique, exhaustion only. (2) Single-technique, counter-example only. (3) Two independent statements, two techniques (either order). (4) One statement, two complementary domains, where the same expression's truth value flips and the technique flips with it (Oct 2021 Q9: proved true by algebra on a positive domain in (a), disproved by counter-example on the negated domain in (b)). (codex-deep-dive-proof.md §3.2)
1.12 — Front: Which confirmed real anchor is the exception to Proof anchoring Question 1? Back: June 2025 — Proof by exhaustion sits at Q10, the paper's last question, not Q1, confirmed worth 4 marks (the paper's lightest question). (codex-deep-dive-proof.md §3.1, §3.3)
2. Algebra and functions — factor and remainder theorems (spec 2.1)
Source: codex-deep-dive-factor-and-remainder-theorems.md, plus codex-formula-and-method-bank.md §2.
2.1 — Front: State the division identity behind both theorems. Back: f(x) ≡ (divisor) × (quotient) + (remainder) — true for every value of x, not a special few. (codex-formula-and-method-bank.md §2; codex-deep-dive-factor-and-remainder-theorems.md Part 1)
2.2 — Front: State the remainder theorem, with its divisor restriction. Back: Dividing f(x) by (ax−b) leaves remainder f(b/a). Divisor restricted to (ax+b) or (ax−b) only — spec-stated directly: "Only division by (ax + b) or (ax − b) will be required." (codex-formula-and- method-bank.md §2)
2.3 — Front: State the factor theorem. Back: (ax−b) is a factor of f(x) ⟺ f(b/a) = 0 — an "if and only if," since both directions run down the identical identity line. (codex-formula-and-method-bank.md §2; codex-deep-dive-factor-and- remainder-theorems.md Part 1)
2.4 — Front: Why does WMA12 restrict the divisor to a linear expression? Back: deg(remainder) < deg(divisor) always; restricting to a linear divisor collapses "remainder has degree less than 1" to "the remainder is a single number" — the entire reason one substitution pins it down completely. (codex-deep-dive-factor-and-remainder-theorems.md Part 1)
2.5 — Front: Real mark structure for "find an unknown from a stated remainder" (Jan 2024 Q1, 3 marks)? Back: M1 (embed the value and equate to the stated remainder — one mark for both) → M1 dependent (solve the resulting linear equation, tolerant of an arithmetic slip) → A1 (bare numeric value, cao). (codex-formula-and-method-bank.md Part 2 §2)
2.6 — Front: What does the word "hence" require on a "hence factorise" question, and what happens if it's bypassed? Back: The factorisation must be derived via the given factor's division/comparison. Any other correct-looking route (e.g. solving the cubic directly on a calculator) scores zero on that part, even with an identical final answer — confirmed on Oct 2023 Q4(c)(ii). (codex-formula-and-method- bank.md Part 2 §2)
2.7 — Front: For a single "find the unknown" question, which method is confirmed faster and more successful — remainder-theorem substitution, or long division? Back: Remainder-theorem substitution — examiners' own commentary confirms it is measurably faster and safer; candidates who chose division more often gave up partway through (Jan 2024 examiner report). (codex-formula-and-method-bank.md Part 2 §2)
2.8 — Front: Name the real repeated-linear-factor anchor identified in this topic's archive. Back: June 2019 Q6(b): the quotient 9x²−12x+4 = (3x−2)², so the real, full factorisation is f(x) = (x−3)(3x−2)². (codex-deep-dive-factor-and-remainder-theorems.md Part 2, §2a)
2.9 — Front: For a non-monic cubic like 6x³+11x²−x−6, how is the candidate-root list bounded? Back: Any candidate root b/a needs b to divide the constant term (±1,±2,±3,±6) and a to divide the leading coefficient (1,2,3,6) — giving a finite list of 18 values (±1,±2,±3,±6,±½,±⅓,±⅙,±⅔,±3/2), "a bigger list but still a list, not an infinity." (codex-deep-dive-factor-and-remainder-theorems.md Part 2, §2c)
2.10 — Front: When a quadratic quotient's discriminant is negative, what does that mean for the cubic's factorisation? Back: The cubic simply has no third real linear factor — stopping there is correct, not incomplete; forcing a further real split that doesn't exist would itself be the error. (codex-deep-dive-factor- and-remainder-theorems.md Part 2, §2d)
2.11 — Front: Why does pairing one given factor with one given remainder produce exactly enough information to find two unknown coefficients? Back: Each condition (a stated factor, or a stated remainder) contributes exactly one linear equation once the identity is evaluated at that condition's own root — two conditions give two linear equations, enough for two unknowns. (codex-deep-dive-factor-and-remainder-theorems.md Part 4)
2.12 — Front: Give two confirmed real cross-topic sightings of this skill outside spec 2.1 itself. Back: Jan 2021 Q2 (differentiation): factorising the quadratic derivative 3x²−2x−16=(3x−8)(x+2), with the exact fraction 8/3 required, "do not accept 2.67." June 2025 WMA12/01A Q8(b): a factor stated as x−7/2 instead of the required integer-coefficient (2x−7) — "incorrect as integer values were required." (codex-deep-dive-factor-and-remainder-theorems.md Part 6)
3. Coordinate geometry of the circle (spec 3.1)
Source: codex-deep-dive-the-circle.md, plus codex-formula-and-method-bank.md §3.
3.1 — Front: State the centre-radius form of a circle's equation. Back: (x−a)² + (y−b)² = r², centre (a,b), radius r. (codex-formula-and-method-bank.md §3)
3.2 — Front: Convert general form x²+y²+2gx+2fy+c=0 to centre/radius form. Back: Centre (−g,−f), radius = √(g²+f²−c) (complete the square on the x- and y-terms separately). (codex-formula-and-method-bank.md §3)
3.3 — Front: State the circle's non-degeneracy condition, and name the real anchor confirming it is exam-tested. Back: g²+f²−c > 0. Confirmed real-tested on Oct 2021 Q6(i)(b), 2 marks: for x²+y²+10x−12y=k, "state the possible range in values for k" → method line k+(±5)²+(±6)²>0, answer k>−61. (codex-deep-dive-the- circle.md §2)
3.4 — Front: State the angle-in-a-semicircle property (forward direction). Back: Any triangle inscribed with one side as a diameter has a right angle at the third vertex. (codex-formula-and-method-bank.md §3)
3.5 — Front: Give the real converse method for proving a segment is a diameter, from three given points (June 2022 Q3(ii), 5 marks). Back: Three independently credited routes: (1) show the gradients of the two other sides multiply to −1, so the angle between them is 90°, so by the converse of the semicircle property the segment is a diameter; (2) the same conclusion via Pythagoras on the three side lengths; (3) find the segment's midpoint and show it is equidistant from all three points (so it is the circle's own centre). (codex-deep-dive-the-circle.md §5)
3.6 — Front: State the chord-bisection property. Back: The perpendicular from the centre to a chord bisects that chord. (codex-formula-and-method- bank.md §3)
3.7 — Front: How is the perpendicular bisector of a chord used as the primary method for finding an unknown centre (Jan 2025 Q6, 8 marks)? Back: Given two points A, B on a circle with centre P and M the midpoint of AB, the line through M and P is the perpendicular bisector of chord AB by the spec property — finding that line directly is the entire method, worth 4 of the question's 8 marks, with no need to state the property explicitly. (codex-deep-dive-the-circle.md §3)
3.8 — Front: State the radius-perpendicular-to-tangent property. Back: The tangent at any point on a circle is perpendicular to the radius drawn to that point. (codex-formula-and-method-bank.md §3)
3.9 — Front: Name three genuinely different real methods for establishing tangency. Back: (1) Perpendicular-gradient of the radius to the point of contact. (2) Discriminant of the line substituted into the circle's equation, set to zero. (3) Perpendicular distance from a point to the line, d = |ax+by+c|/√(a²+b²), set equal to the radius. (codex-deep-dive-the-circle.md §4)
3.10 — Front: Describe the real trap in Jan 2025 Q6 where the semicircle property is misapplied. Back: Candidates "wrongly assum[ed] that ABP was a right-angled triangle" — misapplying the angle-in-a-semicircle property to an angle at the centre (P), when the property only ever concerns an angle at a point on the circumference subtended by a diameter. (codex-deep-dive-the-circle.md §3, §5)
3.11 — Front: Give two real methods for testing whether two circles neither touch nor intersect (June 2024 Q7, 6 marks). Back: (1) Compare the distance between the two centres against the sum of the two radii, both computed as decimals, with the inequality stated explicitly. (2) Subtract the two circle equations to get a straight line (the radical axis), substitute it into either circle equation, and show the resulting quadratic's discriminant is negative. (codex-deep-dive-the-circle.md §6)
3.12 — Front: Name the diameter/radius confusion trap, in both its directions, with real anchors. Back: Treating a diameter's length as though it were already the radius, forgetting to halve (June 2025 Q2; Oct 2020 Q4). Treating an already-correct radius as though it were a diameter, halving when there was nothing to halve (June 2023 Q3(a)). Treating a chord's length as the radius directly, with no distance-to-radius relationship applied at all (Jan 2025 Q6(c)). (codex-deep-dive-the-circle.md §7)
3.13 — Front: How is a circle's equation built from a centre and one point on the circumference, versus from two diameter endpoints? Back: Centre + point: the radius is the distance from the centre to that point. Diameter endpoints: the centre is the midpoint of the two endpoints; the radius is the distance from that midpoint to either endpoint. (codex-deep-dive-the-circle.md §2)
3.14 — Front: In an axis-tangency construction (centre on the x-axis, circle touches the y-axis), how is the radius found (June 2025 01a Q9)? Back: The radius equals the centre's own x-coordinate: for a centre (a,0) touching the y-axis, the perpendicular distance from the centre to the y-axis — which equals the radius, by the radius-⊥-tangent property applied to the axis itself — is simply |a|. (codex-deep-dive-the-circle.md §2)
4. Sequences and series — arithmetic (spec 4.1–4.3)
Source: codex-deep-dive-sequences-and-arithmetic-series.md, plus codex-formula-and-method-bank.md
§4.
4.1 — Front: State the recurrence notation for a general sequence, and the arithmetic special case. Back: General: u_{n+1} = f(u_n) — a sequence generated from a rule, not a direct formula. Arithmetic: the case where f is "add a fixed constant d," i.e. u_{n+1} = u_n + d, u_1 = a. (codex-formula-and- method-bank.md §4; codex-deep-dive-sequences-and-arithmetic-series.md Part 1)
4.2 — Front: State the arithmetic nth-term formula, and its booklet status. Back: u_n = a + (n−1)d — GIVEN in the P2 formula booklet. (codex-formula-and-method-bank.md §4)
4.3 — Front: State the arithmetic sum formula, and its booklet status. Back: S_n = ½n(a+l) = ½n[2a+(n−1)d] — GIVEN in the P2 formula booklet. (codex-formula-and-method- bank.md §4)
4.4 — Front: Describe the method for proving the arithmetic sum formula (spec explicitly requires this proof be known). Back: Write S_n showing ≥3 terms including the first and last (the last as a+(n−1)d); write the same sum reversed; add the two versions term-by-term, showing ≥2 matching pairs; show every pair sums to 2a+(n−1)d; reach 2S_n = n{2a+(n−1)d}; divide by 2. (codex-formula-and-method-bank.md §4, Part 2 §4)
4.5 — Front: Give the real mark-scheme skeleton for this proof (Jan 2020 Q8(i), 3 marks). Back: B1 (expression for S_n, ≥3 terms, first and last correct) → M1 (reverse and add, ≥2 matching pairs) → A1* (reach the given result via the intermediate line 2S_n = n{2a+(n−1)d}). (codex-deep-dive-sequences-and-arithmetic-series.md Part 1)
4.6 — Front: How does this arithmetic proof's mark scheme differ structurally from the parallel geometric proof's, even though both cost 3 marks? Back: The arithmetic proof gives an independent B1 for writing S_n correctly, before any method is applied. The geometric proof (Oct 2020 Q8(a)) bundles writing S_n and multiplying by r into a single M1 — no standalone mark for the opening line alone. A candidate who writes a correct S_n then stalls has banked one mark on the arithmetic proof and nothing on the geometric one. (codex-deep-dive-sequences-and-arithmetic-series.md Part 1)
4.7 — Front: Give working definitions for increasing, decreasing, and periodic sequences (spec 4.3 has no spec-given definition). Back: Increasing: u_{n+1} > u_n for all n. Decreasing: u_{n+1} < u_n for all n. Periodic with period k: u_{n+k} = u_n for all n, for some fixed positive integer k (the smallest such k is the period). (codex-formula-and-method-bank.md §4)
4.8 — Front: Prove: an arithmetic sequence is periodic if and only if d = 0. Back: If periodic with period k≥1: u_{n+k}=u_n, so a+(n+k−1)d = a+(n−1)d, which rearranges to kd=0; since k≥1, d=0. Conversely a constant sequence (d=0) trivially satisfies u_{n+k}=u_n for every k. (codex-deep-dive-sequences-and-arithmetic-series.md Part 4)
4.9 — Front: Given two terms u_p and u_q of an arithmetic sequence, state the closed form for d and a. Back: d = (u_q − u_p)/(q − p); a = u_p − (p−1)d. (codex-deep-dive-sequences-and-arithmetic-series.md Part 2)
4.10 — Front: Name a confirmed real error recurring across at least two series (June 2019; Oct 2020) on this topic. Back: Reaching for the wrong series type's formula mid-question — arithmetic/geometric formula confusion, a mechanical (not conceptual) error. (codex-deep-dive-sequences-and-arithmetic-series.md Part 1)
4.11 — Front: State the sum of the first n natural numbers, and how it relates to the arithmetic sum formula. Back: ½n(n+1) — the a=1, d=1 special case of the arithmetic sum formula, not a separately printed or separately memorised formula. (codex-formula-and-method-bank.md §4)
5. Geometric series (spec 4.4)
Source: codex-deep-dive-geometric-series.md, plus codex-formula-and-method-bank.md §4.
5.1 — Front: State the geometric nth-term formula, and its booklet status. Back: u_n = ar^(n−1) — GIVEN in the P2 formula booklet. (codex-formula-and-method-bank.md §4)
5.2 — Front: State the geometric sum formula, and its booklet status. Back: S_n = a(1−r^n)/(1−r), r≠1 — GIVEN in the P2 formula booklet. (codex-formula-and-method- bank.md §4)
5.3 — Front: State the sum-to-infinity formula, with its condition. Back: S_∞ = a/(1−r), valid only for |r| < 1 — GIVEN in the booklet, printed with this condition attached. (codex-formula-and-method-bank.md §4)
5.4 — Front: Describe the method for proving the geometric sum formula (spec explicitly requires this proof be known). Back: Write S_n; form rS_n by multiplying every term through by r; subtract one from the other so all but the first and last terms cancel; factorise and solve for S_n. (codex-deep-dive-geometric-series.md §1.3)
5.5 — Front: Give the real mark-scheme skeleton for this proof (Oct 2020 Q8(a), 3 marks). Back: M1 (writes ≥3 correct terms and multiplies the sequence by r) → A1 (obtains S_n − rS_n = a − ar^n, both sides unfactorised) → A1* (factorises both sides and divides by 1−r to reach the printed answer). (codex-deep-dive-geometric-series.md §1.3)
5.6 — Front: Name two confirmed real proof-specific errors on this exact question (Oct 2020 Q8(a)). Back: Subtracting the wrong way round (producing S_n−rS_n = ar^n−a, or = a+ar^n). Importing the arithmetic proof's reverse-and-add move wholesale — the examiner records candidates who "wrote the terms with increasing powers of r and then wrote them as decreasing powers of r and then tried to fudge the answer." (codex-deep-dive-geometric-series.md §1.3)
5.7 — Front: Given two non-consecutive terms of a geometric sequence, how are a and r found? Back: Divide the two term-equations to eliminate a (it appears as a multiplicative factor in both), giving r raised to the gap between the terms; solve for r, then substitute back for a. Confirmed real question shape: Oct 2020 Q8(b)/(c), a gap of 3. (codex-deep-dive-geometric-series.md §2.1)
5.8 — Front: What is the caveat when the gap between two given terms is even, rather than odd? Back: An even power (e.g. r²) gives two solutions for r (±), both genuinely real and both needing consideration, unless a further stated condition (e.g. "all terms positive") eliminates one — an odd power gives a unique real root. (codex-deep-dive-geometric-series.md §2.1)
5.9 — Front: What real error recurs when summing a series via sigma notation (Jan 2024 Q5(i))? Back: One substituted term is "not sufficient evidence" of a genuine geometric sequence — a second consecutive ratio, or ≥3 written-out terms, must be checked before asserting the value of r. (codex-deep-dive-geometric-series.md §2.2; codex-formula-and-method-bank.md Part 2 §4)
5.10 — Front: What real error recurs when summing the terms of a series after an exclusion (Jan 2024)? Back: Candidates "often used a=6 instead of a=1.5" — taking the sequence's original first term rather than the first term of the sum actually being asked for, after the excluded terms are removed. (codex-deep-dive-geometric-series.md §2.3)
5.11 — Front: Describe the two confirmed traps in a mixed arithmetic-then-geometric transition (Jan 2022 Q(c)/(d)). Back: Double-counting the shared "overlap" term in both sums. Wrongly restarting the geometric part from the whole sequence's original first term, rather than from where the arithmetic part actually left off. (codex-deep-dive-geometric-series.md §2.4)
5.12 — Front: Why does the geometric series formula break at r=1 and at r=−1? Back: r=1: the derivation divides by 1−r, illegal here; the series a+a+a+… sums directly as S_n=na and never converges (a≠0). r=−1 (the boundary of |r|<1): r^n alternates −1,1,−1,1,…, so partial sums visit two values endlessly, settling on neither — this is why the spec's inequality is strict, |r|<1, not |r|≤1. (codex-deep-dive-geometric-series.md §1.5)
6. Binomial expansion (spec 4.5)
Source: codex-deep-dive-binomial-expansion.md, plus codex-formula-and-method-bank.md §4.
6.1 — Front: State the binomial expansion formula, and its restriction at P2. Back: (a+b)^n = a^n + C(n,1)a^(n−1)b + … + C(n,r)a^(n−r)b^r + … + b^n, C(n,r) = n!/(r!(n−r)!) — GIVEN in the booklet; n restricted to a positive integer only at P2 (spec 4.5's own restriction). (codex-formula-and-method-bank.md §4)
6.2 — Front: What two formulas does the P2 booklet's binomial page actually print, and which is the trap? Back: The combinatorial (a+b)^n form (for positive integer n), and directly beneath it the general real-index series (1+x)^n = 1+nx+[n(n−1)/2!]x²+…, valid for real n and |x|<1 — genuine P4 content, never required at P2. Real examiner reports confirm candidates reach for the second, harder form anyway. (codex-formula-and-method-bank.md §4)
6.3 — Front: Why do the two booklet formulas agree, term by term, for a positive integer n? Back: The general series' r-th coefficient, n(n−1)…(n−r+1)/r!, equals n!/((n−r)!r!) = C(n,r) for r≤n. The series then terminates rather than running forever because at r=n+1 the descending product contains the factor (n−n)=0, forcing that coefficient — and every one after it — to zero. (codex-deep-dive-binomial-expansion.md Part 2)
6.4 — Front: Name the real "un-raised bracket" error confirmed on Jan 2024 Q2. Back: Examiners recorded candidates "using invisible brackets that were not recovered in further work" — the entire bracketed term (its numeral and its sign) must be raised to the power r, not just the variable. (codex-deep-dive-binomial-expansion.md Part 3)
6.5 — Front: Name the real "isolate the answer" rule confirmed on Jan 2024 Q2. Back: "The term or coefficient must be identified if they have more than one term" — the final mark is commonly withheld from a correct value left buried inside a full expansion or an unannotated Pascal's- triangle row. (codex-deep-dive-binomial-expansion.md Part 6)
6.6 — Front: Give the real mark-scheme skeleton for a single-coefficient binomial question (Jan 2024 Q2, 3 marks, flat/no dependency). Back: M1 (combines the correct binomial coefficient with both bracket terms, each correctly raised) → A1 (correct unsimplified coefficient) → A1 (final, isolated numeric coefficient). (codex-deep-dive-binomial-expansion.md Part 6)
6.7 — Front: What is the real, harder variant confirmed on June 2025 01R Q7, and what fared better? Back: (1+kx)^n with both n and k unknown, recovered simultaneously. The examiner report states candidates who reached for the general series form "made little progress," while those using the combinatorial form fared better. (codex-deep-dive-binomial-expansion.md Part 5)
6.8 — Front: [worked example] Give a fast method for the two-unknown case (n and k), avoiding solving the harder general-series arithmetic. Back: Divide the square of the x-coefficient by the x²-coefficient to eliminate k outright: (nk)²/[n(n−1)k²/2] = 2n/(n−1), a function of n alone — turning a two-unknown problem into a one-unknown equation before any value of k is found. (codex-deep-dive-binomial-expansion.md Part 5 — reasoned directly from the mechanism, not itself sourced to an examiner report, as flagged in that document)
6.9 — Front: What is the confirmed real floor for how often this topic appears? Back: Confirmed present in 2 series / 3 papers of the 20 checked (Jan 2024; June 2025 main paper; June 2025's 01R regional variant), carrying 3–7 marks where found — a floor, not a rarity signal, since spec 4.5 is compulsory every series. (codex-deep-dive-binomial-expansion.md, opening summary)
7. Exponentials and logarithms (spec 5.1–5.3)
Source: codex-deep-dive-logarithms-and-exponentials.md, plus codex-formula-and-method-bank.md §5.
7.1 — Front: State the properties of the exponential graph y=a^x, and why the restrictions a>0, a≠1 exist. Back: Passes through (0,1) for every valid a; a^x>0 for all real x (no x-intercept, ever); increasing if a>1, decreasing if 0<a<1; asymptotic to y=0. If a<0, a^x has gaps in its domain (e.g. (−4)^0.5 has no real value); if a=1, 1^x=1 identically, a horizontal line, not a growth/decay curve. (codex-deep-dive-logarithms-and-exponentials.md §2.1)
7.2 — Front: Define log_a b. Back: log_a b = c means, by definition, that a^c = b (for a>0, a≠1, b>0). The condition b>0 falls directly out of a^c always being positive — there is no power of a producing zero or a negative number. (codex-deep-dive-logarithms-and-exponentials.md §2.2)
7.3 — Front: State the addition, subtraction, and power laws of logarithms. Back: log_a(xy) ≡ log_a x + log_a y (addition). log_a(x/y) ≡ log_a x − log_a y (subtraction). log_a(x^k) ≡ k log_a x (power). (codex-formula-and-method-bank.md §5)
7.4 — Front: State the reciprocal and identity laws of logarithms. Back: log_a(1/x) ≡ −log_a x (reciprocal — since log_a1=0, because a^0=1). log_a a = 1 (identity — immediate from the definition, since a^1=a). (codex-formula-and-method-bank.md §5)
7.5 — Front: State the change-of-base formula, and its booklet/spec status. Back: log_a x = log_b x / log_b a — the only log formula GIVEN in the P2 booklet; spec 5.3 names it by name, "students may use the change of base formula." (codex-formula-and-method-bank.md §5)
7.6 — Front: Which of the five log laws are re-listed in the spec's own P2 "must-know, not in the booklet" callout, and which two are merely implied? Back: Addition, subtraction, and power are the three explicitly re-listed. The reciprocal and identity laws are still spec-5.2 content, just not separately re-flagged there — both are trivially derivable from the other three plus the definition. (codex-formula-and-method-bank.md §5)
7.7 — Front: Name the real error that confuses log subtraction with division. Back: Treating log x − log y as log x / log y — log subtraction is not division of the log expressions themselves. (codex-deep-dive-logarithms-and-exponentials.md §3.3, §4)
7.8 — Front: Name the real error that expands a log of a sum incorrectly. Back: Expanding log(x+y) as log x + log y — a log is never distributed across a sum inside its own argument. (codex-deep-dive-logarithms-and-exponentials.md §3.3, §4)
7.9 — Front: Name the real error confusing (log_a y)² with log_a(y²) (confirmed Jan 2023 Q4(ii), "one of the most challenging parts on the paper"). Back: Squaring the whole logged number is not the same operation as applying the power law to square the argument — they are structurally different expressions. (codex-deep-dive-logarithms-and-exponentials.md §2.3, §3.5, §4)
7.10 — Front: When a question specifies the answer as a single combined log form (e.g. "x=log_a k"), what real mark-scheme distinction applies? Back: Reaching a correct but un-combined exact expression (e.g. log_a p − q) earns only an earlier accuracy mark — the combining step (rewrite the constant via log_a a=1, then apply the addition/ subtraction law) is a genuinely separate, later accuracy mark (confirmed Jan 2021 Q3(i)). (codex-deep-dive-logarithms-and-exponentials.md §3.2)
7.11 — Front: What real mark-scheme fact is confirmed identically on four independent series (Jan 2021 Q3(ii); Jun 2023 Q5; Jun 2024 Q3(i); Jun 2025 Q6(i))? Back: Forming the log-free equation, after combining log terms, is its own standalone accuracy mark — distinct from both the combining step before it and the solving step after it. (codex-deep-dive-logarithms-and-exponentials.md §3.3)
7.12 — Front: What is the real rule for rejecting an invalid root on a combine-then-solve log question? Back: Any root that would make an original log's argument ≤0 must be explicitly rejected in the final line, not silently dropped — every anchor checked states this as part of the final mark's own criterion. (codex-deep-dive-logarithms-and-exponentials.md §3.3, §3.4)
7.13 — Front: [worked example] Solve log_2(x+1) + log_2(x−1) = 3. Back: Combine: log_2[(x+1)(x−1)] = 3, so x²−1=8, x²=9, x=±3. The original equation needs x−1>0, i.e. x>1; reject x=−3; accept x=3. (codex-deep-dive-logarithms-and-exponentials.md §3.3)
7.14 — Front: Is e or ln examinable as a distinguished base at WMA12 (P2)? Back: No confirmed instance anywhere in the spec's P2 content, the booklet page, or any examiner report reviewed. A calculator's ln button is a computational tool for change-of-base, not evidence that e itself is examinable content at P2. (codex-deep-dive-logarithms-and-exponentials.md §2.1)
7.15 — Front: [worked example] Solve 3^x = 40 to 3 s.f. Back: Take logs of both sides: x log 3 = log 40, so x = log₃40 = 3.36 (3 s.f.). (codex-deep-dive-logarithms-and-exponentials.md §3.1)
8. Trigonometry (spec 6.1–6.2)
Source: codex-deep-dive-trigonometric-equations.md, plus codex-formula-and-method-bank.md §6.
8.1 — Front: State the tangent identity, with its restriction. Back: tan θ ≡ sin θ / cos θ — MEMORIZE, undefined where cos θ = 0. (codex-formula-and-method-bank.md §6)
8.2 — Front: State the Pythagorean identity. Back: sin²θ + cos²θ ≡ 1 — MEMORIZE, not printed anywhere on the P2 booklet page. (codex-formula-and-method-bank.md §6)
8.3 — Front: Describe the method for collapsing a mixed sin/cos/tan equation to one function, and name the trap to avoid. Back: Use tan θ = sin θ/cos θ (multiplying every term through by cos θ as its own separately-scored step) where a lone tan θ sits with sin θ/cos θ terms, or sin²θ+cos²θ=1 where a squared term meets a linear one. Never divide out a common factor of sin θ, cos θ, or tan θ — this silently discards an entire solution branch. (codex-formula-and-method-bank.md §7)
8.4 — Front: Describe the method for finding all solutions of a trig equation in a given interval. Back: Find the calculator's principal value, then use the quadrant-sign (CAST) pattern to generate every further solution inside the stated interval — not just a second value by rote. (codex-formula-and-method-bank.md §7)
8.5 — Front: Describe the dominant real WMA12 trig-equation shape, confirmed across eight series. Back: A fixed two-part "show that … (hence) …" structure: part (a) shows a given equation reduces to a printed target (3–4 marks); part (b), "hence solve," gives a compound-angle or related version to actually solve (3–4 more marks) — the algebra of (a) does almost all of (b)'s work. (codex-deep-dive-trigonometric-equations.md §3)
8.6 — Front: Name the three different things "hence" can mean on this exact question type. Back: (1) Scale θ to a compound angle and re-run the identical algebra (the most common, confirmed in six series). (2) Solve the equation just shown to hold, whose constants (a,b,c) the candidate had to find themselves (June 2021 Q8(ii)). (3) Use the just-proven identity to simplify a different-but-related equation, then solve it (Jan 2025 Q10). (codex-deep-dive-trigonometric-equations.md §3)
8.7 — Front: Describe the real sub-case where a mixed trig equation collapses to a cubic, not a quadratic, via a factor-theorem link (Jan 2020 Q3; June 2019 Q6). Back: An earlier part factorises a cubic f(x) via the factor theorem; a later part substitutes tan θ (or cos θ) for x in the identical cubic, so the factorisation already found is reused directly — each linear factor gives one candidate value for the trig ratio. (codex-deep-dive-trigonometric-equations.md §4)
8.8 — Front: [worked example] Why does a cubic in tan θ give exactly one θ per root, per 180°- wide interval, rather than two? Back: Because tan has period 180°, and the interval is exactly one period wide — unlike a quadratic in sin θ or cos θ (period 360°), which usually doubles each root into a pair of θ-values. (codex-deep-dive-trigonometric-equations.md §4)
8.9 — Front: Describe the real sub-case of a compound-angle equation given directly as one function equal to a multiple of another (Jan 2022 Q7(i); June 2021 Q8(i)). Back: No Pythagorean identity is needed — isolate tan(·) directly by dividing both sides by cos(·): e.g. 3 sin(2x−15°) = cos(2x−15°) ⟹ tan(2x−15°) = ⅓. (codex-deep-dive-trigonometric-equations.md §5)
8.10 — Front: What real fact governs the number of solutions of a squared compound-angle equation (Oct 2020 Q7(ii): 3cos²(2x+10°)=1, giving 2 solutions, not the taught tan² example's 4)? Back: The solution count is set by how the bracket's range compares to the function's own period, not a fixed count to memorise — here the bracket sweeps only 180° (one period of cos²), so it produces two final values, not four. (codex-deep-dive-trigonometric-equations.md §6.1)
8.11 — Front: On that same real anchor (Oct 2020 Q7(ii)), what does the real mark scheme say about the "±" sign when taking a square root? Back: The mark scheme states explicitly "the '±' is not required" for the method mark here — a real, quotable contrast with the taught tan² anchor, where dropping the ± sign is the single most costly documented error there. (codex-deep-dive-trigonometric-equations.md §6.1)
8.12 — Front: Describe the real method for a squared linear combination of sin θ and cos θ (Oct 2021 Q10(ii): (2sinθ−cosθ)²=1) — not a compound angle at all. Back: Expand, then substitute the Pythagorean identity (1=sin²θ+cos²θ) so the matching cos²θ terms cancel exactly; what remains factorises — carrying the same divide-out-a-common-factor hazard as the taught tan θ = sinθ/cosθ case, reached this time by squaring and substituting rather than by direct substitution. (codex-deep-dive-trigonometric-equations.md §6.2)
8.13 — Front: Describe the real sub-type of proving a further trig identity (not solving an equation), confirmed on Oct 2020 Q7(i). Back: tan θ + 1/tan θ ≡ 1/(sin θ cos θ), θ ≠ nπ/2 — combine tan θ + cot θ over the common denominator sin θ cos θ; the numerator sin²θ+cos²θ collapses to 1 via the Pythagorean identity. There is no interval to search and no numeric answer — the domain restriction (θ≠nπ/2) is part of the required answer, not a check applied afterward. (codex-deep-dive-trigonometric-equations.md §7.1)
8.14 — Front: [worked example] Given 0°<θ<90° and cos θ = ⅓, find the exact value of tan θ (confirmed real, June 2024 01R Q7(ii)). Back: sin θ = √(1−1/9) = 2√2/3 (since θ is acute), so tan θ = (2√2/3)/(1/3) = 2√2. (codex-deep-dive-trigonometric-equations.md §7.2)
8.15 — Front: Name the real rule for when radians are demanded "in the form kπ." Back: The final answer must be left as the exact standard-angle fraction of π — a correctly-rounded decimal does not satisfy this requirement, a distinct rule from the general "don't round early" caution. (codex-formula-and-method-bank.md §7)
8.16 — Front: For a squared compound-angle equation such as tan²(2x+π/4)=k, what is the confirmed real trap? Back: Take both square roots first, then run the full compound-angle, range-scaled method on each sign separately — dropping either sign is a confirmed way of losing the last two accuracy marks, even when the candidate correctly considered ±√k but then failed to find all four resulting roots. (codex-formula-and-method-bank.md §7)
9. Differentiation — stationary points and curve sketching (spec 7.1)
Source: codex-deep-dive-stationary-points-and-curve-sketching.md, plus codex-formula-and-method- bank.md §7/§8.
9.1 — Front: Describe the full process for finding and stating a stationary point. Back: Differentiate to dy/dx; set it to zero; solve for x; substitute each x back into the original y=f(x) — never into dy/dx — for the y-coordinate. (codex-deep-dive-stationary-points-and-curve-sketching.md §1.1)
9.2 — Front: State the second-derivative test, and its one blind spot. Back: d²y/dx²>0 ⟹ minimum; <0 ⟹ maximum; =0 decides nothing (the test declining to answer, not a third category) — fallback: test the sign of dy/dx just below and above the point (negative-to- positive = minimum; positive-to-negative = maximum; no change = inflection). (codex-deep-dive-stationary-points-and-curve-sketching.md §1.2)
9.3 — Front: Why does comparing a curve's y-values either side of a stationary point NOT earn the classification mark, even though the fallback sign test does (confirmed real, Jan 2021 Q2)? Back: The sign-of-dy/dx fallback is genuine calculus; comparing y-values either side never uses calculus at all, so a real mark scheme does not credit it as "further calculus." (codex-deep-dive-stationary-points-and-curve-sketching.md §1.2)
9.4 — Front: What is the single most commonly missed step on this topic's best-evidenced anchor (Jan 2021 Q2)? Back: Roughly half of candidates who correctly reached the second derivative never went on to substitute the already-found x-value into it. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.1)
9.5 — Front: State the exact-answer rule for a stationary point's coordinate, extended to a surd root (x²−4x+1=0 ⟹ x=2±√3). Back: Both roots must be kept as 2+√3 and 2−√3 — never 3.73 and 0.27, however precisely rounded; "marks will normally be lost if the candidate resorts to using rounded decimals." (codex-deep-dive-stationary-points-and-curve-sketching.md §2.2)
9.6 — Front: Describe the method for "prove no turning points" via the discriminant (confirmed real, Oct 2019 Q10). Back: Set dy/dx=0, rearrange to a 3-term quadratic in x, show B²−4AC<0 (or complete the square to the same effect), then state explicitly that this means no real solutions exist, hence no stationary points. (codex-formula-and-method-bank.md Part 2 §8; codex-deep-dive-stationary-points-and-curve- sketching.md §1.4, §2.3)
9.7 — Front: State the rule for reporting increasing/decreasing intervals, including the integer-coefficient factor trap. Back: State the interval with a consistent inequality direction throughout — reporting the wrong side of a critical-value inequality is a confirmed, common error. Where a factor is required in integer-coefficient form (e.g. (2x−7)), the mathematically-identical non-integer form ((x−7/2)) is marked wrong if the question specifies integer coefficients. (codex-formula-and-method-bank.md Part 2 §8)
9.8 — Front: Describe the three-stage mechanism for real-world optimisation via constraint elimination (spec 7.1's "practical problem" clause). Back: (i) A given constraint (fixed volume/capacity) lets one variable be written in terms of the other. (ii) Substituting into the quantity to be optimised gives a single-variable function, of the shape px²+q/x on this evidence. (iii) The ordinary stationary-point procedure applies — except setting its derivative to zero now produces x³=k (a cube root), not a quadratic. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.5)
9.9 — Front: Name the confirmed cross-series error specific to this optimisation sub-case (Jan 2025 Q3; June 2025 Q9, both independently). Back: "Taking a square root rather than a cube root" when solving x³=k — a root-extraction mistake, distinct from the exact-answer/rounding trap, and specific to this sub-case because it is the only WMA12 stationary-point pattern confirmed to produce a cubic equation in x. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.5)
9.10 — Front: State the confirmed real mark-scheme shortcut for classifying a minimum when the second derivative has the shape 2p+2q/x³ (p,q>0) (June 2025 Q9's mark scheme). Back: This expression is a sum of two positive terms for any x>0, so it is manifestly positive for every positive x — full credit for that deduction alone, stated as an accepted alternative to computing the numeric value at the specific x found. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.5)
9.11 — Front: Name three confirmed real setup-stage (geometry, not calculus) errors specific to 3D optimisation questions (June 2025 Q9; Jan 2025 Q3). Back: Treating an explicitly open prism as closed (adding an area that shouldn't exist). Taking both rectangles to have the same area, double-counting one dimension. Not dividing by 2 when finding a triangular cross-section's area — each feeding a wrong constant into the eventual function, not a calculus error at all. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.5)
9.12 — Front: State the synthesis point about exact vs. rounded answers across this whole topic. Back: The polynomial sub-cases (find-and-classify, no-turning-points, increasing/decreasing) require an exact fraction or surd, no rounding at any precision. Both confirmed real optimisation anchors explicitly instruct rounding instead (x to 3 s.f., or the optimised value to 1 d.p.), because the cube-root answer has no exact form worth demanding — the real skill is reading which instruction the specific question gives. (codex-deep-dive-stationary-points-and-curve-sketching.md §2.6)
9.13 — Front: What structural mark-scheme feature is confirmed across all five real anchors for this topic (pure algebra, a "prove" question, and practical optimisation alike)? Back: Every anchor splits "form/differentiate correctly" (an independent M1) from "the dependent step that actually uses the specific value found" (a dependent dM1 or A1*) into separate marks. (codex-deep-dive-stationary-points-and-curve-sketching.md Part 3)
10. Integration — definite integrals and areas (spec 8.1–8.2)
Source: codex-deep-dive-definite-integrals-and-areas.md, plus codex-formula-and-method-bank.md §8.
10.1 — Front: State how a definite integral is evaluated, mechanically. Back: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′(x) = f(x). Find an antiderivative term by term (raise the power by one, divide by the new power); no "+c" is carried, since any constant cancels in the subtraction. No formula for this is printed in the booklet — 100% memorise-and-apply. (codex-deep-dive-definite-integrals-and-areas.md §1.1; codex-formula-and-method-bank.md §8)
10.2 — Front: State the three cases for when a definite integral equals an area. Back: If f(x)≥0 throughout [a,b]: the integral is the area exactly. If f(x)≤0 throughout: the area is |∫ₐᵇ f(x) dx|. If f(x) changes sign inside (a,b): split at every sign change, take |·| of the integral over each single-sign piece, then add — a single unsplit integral is neither the true area nor an honest signed total. (codex-deep-dive-definite-integrals-and-areas.md §1.2)
10.3 — Front: State the area-between-boundaries formula, and how the limits a, b are found. Back: area = ∫ₐᵇ (upper − lower) dx. Limits come from where the two boundaries meet (solve by setting the two expressions equal) — except where a boundary is an axis, which supplies its own limit directly, with no equation to solve. (codex-deep-dive-definite-integrals-and-areas.md §1.3; codex-formula-and-method-bank.md §8)
10.4 — Front: State the explicit spec restriction on this topic. Back: "∫x dy will not be required" — every region is sliced into vertical strips and integrated with respect to x; re-expressing a boundary as x in terms of y is out of scope. (codex-deep-dive-definite-integrals-and-areas.md §1.4; codex-formula-and-method-bank.md §8)
10.5 — Front: Describe the real mark-scheme rule for an axis-supplied limit, versus a solved limit (confirmed Jan 2020 Q4). Back: The axis-given limit is its own standalone accuracy mark, with no method mark attached — separate from the other limit, which usually does need solving. Confusing a function's value at that axis with the x-limit itself is a confirmed real error. (codex-deep-dive-definite-integrals-and-areas.md Part 3)
10.6 — Front: Name the confirmed cross-series trap of dropping the line boundary entirely (Jan
2020; Jan 2023; June 2025 — three separate examiner reports).
Back: Integrating only the curve and ignoring the line boundary produces a specific, named wrong
answer — already in the lesson's trap-taxonomy as line-boundary-ignored-entirely.
(codex-deep-dive-definite-integrals-and-areas.md Part 3)
10.7 — Front: What efficiency finding does the June 2025 Q3 examiner report confirm about combining versus separately integrating two expressions? Back: Combining (upper − lower) into a single expression before integrating once is faster than integrating each function separately and subtracting — the separate route carries more terms through the working and creates more chances for a sign slip at the end. (codex-deep-dive-definite-integrals-and-areas.md Part 3)
10.8 — Front: Describe the real multi-region decomposition strategy confirmed on Jan 2023 Q9. Back: A second region can be found not by writing a fresh integral, but by subtracting an already-found region from a named elementary shape (a triangle or rectangle) — or by finding one larger bounded region and subtracting a smaller one. Multiple strategies are independently credited. (codex-deep-dive-definite-integrals-and-areas.md §2.6, Part 3)
10.9 — Front: What real mark-scheme structural feature is confirmed across all three of this topic's anchors (different curve types, different series)? Back: All three split "find the antiderivative" from "apply the limits and evaluate" into separate marks, never one combined mark. (codex-deep-dive-definite-integrals-and-areas.md Part 3)
10.10 — Front: Is a pure two-curve (no line) area question confirmed real-tested for WMA12? Back: Spec 8.2 names the case explicitly, so it is real, examinable content — but no confirmed real WMA12 anchor for a pure two-curve region has been found in the reviewed archive; this is an honest gap in what has been checked, not a claim the case is untested. (codex-deep-dive-definite-integrals-and-areas.md §2.5)
11. The trapezium rule (spec 8.3)
Source: codex-deep-dive-the-trapezium-rule.md, plus codex-formula-and-method-bank.md §8.
11.1 — Front: State the trapezium rule formula in full, with its booklet status. Back: ∫ₐᵇ y dx ≈ ½h{(y₀+yₙ) + 2(y₁+y₂+…+y_{n−1})}, h = (b−a)/n — GIVEN in full, including the h formula, on the P1/P2 booklet page. (codex-formula-and-method-bank.md §8)
11.2 — Front: Name the confirmed recurring real error about n and the strip count (at least four series). Back: n is the number of strips (gaps between consecutive ordinates), not the number of ordinates themselves — a confirmed error miscounts this by one, taking h from the point count instead of the interval count. (codex-formula-and-method-bank.md Part 2 §9; codex-deep-dive-the-trapezium-rule.md Part 3)
11.3 — Front: Name the confirmed real "bracket closed early" error (Jan 2023). Back: Closing the bracket after the first pair of terms and adding the rest outside it — this forfeits the accuracy mark even though the arithmetic that follows is otherwise correct. (codex-deep-dive-the-trapezium-rule.md Part 3)
11.4 — Front: What should be watched for specifically when ordinates are read from a labelled graph rather than a table or formula? Back: Zero-valued endpoints — a confirmed source of difficulty distinct from, and harder than, reading the same information from a table. (codex-formula-and-method-bank.md Part 2 §9)
11.5 — Front: State the real mark-scheme rule for justifying an over- or underestimate (confirmed Oct 2023 Q6(c), a single B1). Back: State the direction (over/under) and the mechanism (how the curve's own curvature relates to the straight chords used to approximate it) in the same sentence — this is one combined mark, not two separately-awardable ones, on every verified instance checked. (codex-formula-and-method-bank.md Part 2 §9)
11.6 — Front: What does NOT earn the over/underestimate mark, even though the direction is stated correctly? Back: Merely restating the conclusion — "it's an underestimate because it's smaller than the true area" — with no reasoning tied to the curve's shape relative to the chords. (codex-formula-and-method-bank.md Part 2 §9)
11.7 — Front: Describe the confirmed real trap of recomputing instead of reusing an earlier part's trapezium-rule answer (June 2025 Q5(b)). Back: Candidates who recomputed the trapezium rule in part (b) instead of using their part (a) answer scored nothing on that part, even when the recomputation was itself correct — those marks are for the step part (b) actually asks for, not a repetition of part (a)'s. (codex-deep-dive-the-trapezium-rule.md §2.6)
11.8 — Front: What does the spec's own guidance sentence permit asking about, beyond a single application of the rule? Back: Asking for more strips to sharpen the estimate, and asking for a sense of how far off an estimate is — in every confirmed real anchor, the "how far off" question has meant the qualitative direction (over/underestimate from concavity), not a numeric error estimate. (codex-deep-dive-the-trapezium-rule.md §1.2)
11.9 — Front: Describe the real format of the rule embedded inside another topic's own machinery (confirmed: "via logarithms," Jan 2025; a real-life water-flow context, Oct 2023 Q6). Back: An extra step, outside the trapezium rule itself, has to run before the ordinate table can even be built (e.g. rearranging a logarithmic relationship to isolate y) — the trapezium arithmetic that follows is unchanged once that step is done. (codex-deep-dive-the-trapezium-rule.md §2.5)
11.10 — Front: What is the confirmed real floor for how often this topic appears? Back: 6 of the 20 series checked carry a confirmed, individually-cited trapezium-rule question (Oct 2020; Jan 2023; June 2024; Oct 2023 Q6; Jan 2025; June 2025 Q5). (codex-deep-dive-the-trapezium-rule.md Part 3)
12. P1-prerequisite formulae, examinable within P2
Source: codex-formula-and-method-bank.md, "P1-prerequisite formulae" section.
12.1 — Front: Which P1-level formulae are NOT printed in the booklet at all, and must already be memorised before a P2 candidate opens this paper? Back: The quadratic formula; the sine rule; the area-of-a-triangle formula (½ab sin C); arc length (s=rθ); area of a sector (½r²θ); the differentiation power rule; the integration power rule. (codex-formula-and-method-bank.md, P1-prerequisite section)
12.2 — Front: Which single P1-level formula genuinely IS printed in the booklet? Back: The cosine rule, a² = b² + c² − 2bc cos A — confirmed by direct reading of the P1 booklet page. (codex-formula-and-method-bank.md, P1-prerequisite section)
12.3 — Front: State the differentiation power rule. Back: d/dx(x^n) = n·x^(n−1) — MEMORIZE, a P1-prerequisite formula, directly examinable within P2 under the spec's own carry-forward rule. (codex-formula-and-method-bank.md §7)
12.4 — Front: State the integration power rule. Back: ∫x^n dx = 1/(n+1)·x^(n+1) + c, n≠−1 — MEMORIZE, same P1-prerequisite carry-forward basis as the differentiation power rule. (codex-formula-and-method-bank.md §8)
13. Cross-cutting mark-scheme conventions
Source: codex-formula-and-method-bank.md, general conventions cited throughout Part 2; codex- complete-worked-paper.md, mark-by-mark tables (June 2021).
13.1 — Front: What does "cao" mean on a mark scheme, and where does it typically sit in a method chain? Back: "Correct answer only" — the final, bare numeric or algebraic value, with no credit for a correct method alone; seen in this bank e.g. as the closing A1 of the factor/remainder single-unknown skeleton (card 2.5). (codex-complete-worked-paper.md, Q1 mark table)
13.2 — Front: What does a "dependent" method mark (dM1) require, that an independent M1 does not? Back: A dependent mark is only earned if a specific earlier method mark has already been earned — e.g. substituting a found x-value into a second derivative is dependent on having correctly solved for that x-value first (card 9.13). (codex-deep-dive-stationary-points-and-curve-sketching.md Part 3)
13.3 — Front: What does an asterisked accuracy mark (A1*) signal on a "show that" question? Back: The target result is printed on the question paper; the mark requires an intermediate, genuinely algebraic step to be visible between the working and that printed result — jumping straight to the printed answer is not sufficient (confirmed, e.g., Q7(a) of the June 2021 complete worked paper). (codex-complete-worked-paper.md, Q7 mark table)
13.4 — Front: What real mark-scheme allowance exists for a candidate who consistently (but wrongly) uses the other series type's formula throughout a question (June 2021 Q1(b))? Back: If the wrong-series working is internally consistent and correctly executed on its own terms, the scheme can still credit the middle "solves the equation" method mark (e.g. M0 A0 M1 A0), rather than scoring the whole part zero — a genuinely generous piece of real mark-scheme design. (codex-complete-worked-paper.md, Q1)
13.5 — Front: What is the standing rule, confirmed across every topic in this bank, for when a calculator-only answer is given with no method shown on a "show that" or "using algebra" question? Back: It scores zero, even when the numeric answer is correct — confirmed independently across sum-to-infinity (card 5.9), definite integration (card 10.1), and the logarithmic-equation question in the June 2021 complete worked paper (Q7(c): "a calculator-only answer with no working scores nothing"). (codex-formula-and-method-bank.md Part 2 §4, §9; codex-complete-worked-paper.md Q7)
Coverage note
This bank draws on all thirteen codex-*.md files in this folder. It does not cover every worked
micro-example or every named trap in those files — where a source document's own micro-example is a
close numerical variant of another card already in this bank, only one representative card was made,
per this exercise's reformatting (not padding) discipline. Anyone wanting the full worked derivation
behind any card here should follow that card's cited source file and section.