Technique Masterclasses

Pure Mathematics 3

Flashcard Bank

Provenance (2026-09-29): this document is a reformatting pass, not new research. Every flashcard below is a front/back restatement of a fact, method, or worked micro-example that is already stated in one of the thirteen codex-*.md files in this folder — codex-formula-and-method-bank.md, codex-complete-worked-paper.md, and the ten codex-deep-dive-*.md files (compound-and-double-angle-formulae, differentiation-rules, exponential-growth-and-decay, exponentials-and-logarithmic-graphs, functions-composition-and-inverses, harmonic-form, integration-by-recognition, modulus-and-graph-transformations, numerical-methods, rational-expressions-and-algebraic-division, reciprocal-and-inverse-trigonometric-functions). No new factual claim, number, formula, or example is introduced here — every card's back is traceable to a real sentence already in one of those files, cited by file (and section, where the source numbers its own sections) at the end of the card. Where a source file itself flags a fact as unconfirmed, self-generated/illustrative rather than exam-sourced, or a genuine open gap, this bank preserves that flag rather than presenting the item as settled. Exam-board wording is never reproduced at length; any quoted fragment is under fifteen words, exactly as it appears (quoted or paraphrased) in the source codex file. Not affiliated with or endorsed by Pearson Edexcel.

WMA13 (Pure Mathematics 3) — CODEX Flashcard Bank

How to use this bank

One card = one atomic fact, method skeleton, or worked micro-example. Front is the term, formula, or question; Back is the answer, ending with the source file (and section) it was reformatted from. Two labels recur throughout, both meaning exactly what the source files mean by them:

114 cards total, organised by spec topic (1.1–1.4 Algebra and functions, 2.1–2.5 Trigonometry, 3.1–3.3 Exponentials and logarithms, 4.1–4.4 Differentiation, 5.1–5.2 Integration, 6.1–6.2 Numerical methods), plus a short section of additive worked micro-examples drawn from the one full past paper (October 2020) reproduced in codex-complete-worked-paper.md. A closing section lists every item the source files themselves flag as a genuine open gap — these are not phrased as cards, because they are not settled facts to test recall on.

Section 1 — Algebra and Functions (spec 1.1–1.4)

1.1 Algebraic division and rational expressions

FC1.1.1 — Front: What real, checkable constraint does the spec place on divisors for algebraic-division/rational-expression questions on WMA13? Back: Denominators tested are restricted to linear or quadratic — verbatim from the spec's own guidance column. (codex-formula-and-method-bank.md §1.1; codex-deep-dive-rational-expressions-and-algebraic-division.md §1)

FC1.1.2 — Front: What is the stopping condition that decides when polynomial division is "finished"? Back: From the division algorithm identity f(x) ≡ D(x)Q(x) + R(x): division is not finished until deg R < deg D. (codex-deep-dive-rational-expressions-and-algebraic-division.md §3)

FC1.1.3 — Front: Dividing by a linear divisor leaves a remainder that is still degree 1 (e.g. "x − 2"). Is the division finished? Back: No. Against a linear divisor (deg D = 1), deg R < 1 forces R to have no x-term at all — a linear leftover is not a valid final remainder. Real, confirmed error, Oct 2024 Q9(c): candidates who stopped after the first division step and reported (x−2) as the remainder had "terminate[d] the division early." (codex-deep-dive-rational-expressions-and-algebraic-division.md §3.1)

FC1.1.4 — Front: Algebraic division "Pattern A" (real anchor Oct 2020 Q9(a)) — linear factor genuinely cancels. State the method. Back: Divide to reach the remainder over the given quadratic denominator → factorise that denominator → cancel the common factor between the remainder and the factorised denominator → the value that survives is the constant to be shown. (codex-formula-and-method-bank.md §2.1)

FC1.1.5 — Front: Algebraic division "Pattern B" (real anchor Jan 2025 Q4(a)) — quadratic divisor with no real factor, e.g. x²+4. Why doesn't Pattern A's cancel step work, and what's the method instead? Back: x²+4 has a negative discriminant, so it has no real linear factor — there is nothing to cancel. Method: set up the identity, compare coefficients to form at least two equations, and solve — or divide fully through and read the constant off directly. A method that teaches "factorise and cancel" as the universal fix is wrong whenever the divisor's discriminant is negative. (codex-formula-and-method-bank.md §2.1)

FC1.1.6 — Front: State the three-step decision procedure that covers every real algebraic-division sub-case on WMA13. Back: (1) Is the divisor linear or quadratic? If linear, the remainder must end as a bare constant. (2) If quadratic, does it factorise over the reals (discriminant ≥ 0)? If yes, a correctly finished division may still admit cancellation. (3) If the discriminant is negative, no factorising will produce a cancellation — use coefficient-matching or substitution instead. (codex-deep-dive-rational-expressions-and-algebraic-division.md §3, synthesis)

FC1.1.7 — Front: When can a rational expression P(x)/Q(x) actually be simplified? Back: Only once P and Q are fully factorised, and only a factor multiplying the whole of each side may be cancelled — never a term that merely looks the same inside an unfactorised sum. (codex-deep-dive-rational-expressions-and-algebraic-division.md §2)

1.2 Functions — composition and inverses

FC1.2.1 — Front: What does "fg" mean, and what is the defining inverse-function identity? Back: fg means "do g first, then f." If f⁻¹ exists, f⁻¹f(x) = ff⁻¹(x) = x — a definition, not a formula to look up. (codex-formula-and-method-bank.md §1.2)

FC1.2.2 — Front: When does f⁻¹ exist, and what is the domain of f⁻¹? Back: f⁻¹ exists only where f is one-one. The domain of f⁻¹ equals the range of f. (codex-formula-and-method-bank.md §1.2)

FC1.2.3 — Front: Real mark-scheme trap (Jan 2023 Q1(a)): a range answer of "x ≤ 9" is numerically correct. Why is it rejected? Back: A range is a set of outputs (y or f(x) notation); a domain is a set of inputs (x notation). The scheme accepts y ≤ 9 or f ≤ 9 but explicitly rejects x ≤ 9. (codex-formula-and-method-bank.md §1.2)

FC1.2.4 — Front: Mirror-image trap to FC1.2.3 (Jan 2024 Q4(c)): what domain-notation error did the real examiner report name? Back: The domain of g⁻¹ was sometimes stated as "g⁻¹(x) ≥ 3" (output notation) instead of "x ≥ 3" (input notation) — the same input-for-domains/output-for-ranges rule, caught from the other side. (codex-deep-dive-functions-composition-and-inverses.md §3.3)

FC1.2.5 — Front: What is the general condition for a Möbius function f(x) = (ax+b)/(cx+d) to be its own inverse? Back: a + d = 0 (the two "diagonal" coefficients are negatives of each other). Checked against every real WMA13 linear-fractional function located in this course's research: none of the five is self-inverse. (codex-deep-dive-functions-composition-and-inverses.md §2)

FC1.2.6 — Front: How do you find where y = f(x) meets y = f⁻¹(x) without ever constructing f⁻¹? Back: Because f and f⁻¹ are reflections in y = x, for a strictly monotonic f the only points where the two graphs can meet are points already on y = x — solve f(x) = x directly. Real anchor Jun 2023 Q4(c): the literal route f(x) = f⁻¹(x) leads to an intractable quartic; f(x) = x reaches the identical exact answer directly. (codex-deep-dive-functions-composition-and-inverses.md §3.8)

FC1.2.7 — Front: Solving gg(x) = k (self-composition, real anchor Oct 2024 Q6(c)) — what's the named real trap? Back: "Undoing" g only once (solving g(x) = k directly) instead of twice — a plausible-looking wrong answer from a genuine misunderstanding of what "gg" asks. The real mark scheme scores this 0,0,0 unless further work is shown. (codex-deep-dive-functions-composition-and-inverses.md §3.4)

FC1.2.8 — Front: For a composed algebraic argument like fg(a²−5) = g(59) (real anchor Jun 2026 Q2(d)), what must happen before an equation can even be formed? Back: Evaluate the numerical side first, then substitute the algebraic argument into the inner function and use the question's own stated constraint (e.g. a > 0) to collapse a square root to a clean expression before an equation can be formed. (codex-formula-and-method-bank.md §2.2)

1.3 The modulus function

FC1.3.1 — Front: How is |x| defined? Back: Piecewise: |x| = x for x ≥ 0; |x| = −x for x < 0. A definition, not a formula to memorise separately. (codex-formula-and-method-bank.md §1.3)

FC1.3.2 — Front: What does y = |f(x)| do to the graph of y = f(x)? Back: Reflects in the x-axis any portion of y = f(x) lying below it; everything above the x-axis stays fixed. (codex-formula-and-method-bank.md §1.3)

FC1.3.3 — Front: What does y = f(|x|) do to the graph of y = f(x)? Back: Discards the original x < 0 branch entirely and replaces it with a mirror image of the x ≥ 0 branch, reflected in the y-axis — f(|x|) is therefore always an even function. (codex-formula-and-method-bank.md §1.3; codex-deep-dive-modulus-and-graph-transformations.md §1)

FC1.3.4 — Front: Real trap (Jan 2024 Q1(c)): P(−4,−3) lies on y = f(x). Find its image under y = |f(x)|, and name the common wrong answer. Back: Correct: (−4, 3) — x is untouched, y = |−3| = 3. Common real error: negating both coordinates to give (4,3), mistaking an output-only modulus reflection for a full point reflection through the origin. (codex-deep-dive-modulus-and-graph-transformations.md §3.1)

FC1.3.5 — Front: Method skeleton for solving "|linear| > linear" (real anchor Jan 2022 Q7(b)). Back: Solve one case's equation (split into two linear cases, or square both sides) → both critical values normally fall from a single accuracy mark → select the outside/inside region the inequality's direction requires → state both bounds together. (codex-formula-and-method-bank.md §2.3)

FC1.3.6 — Front: Can a whole branch of a modulus inequality be empty (not only a modulus equation)? Back: Yes. Real anchor Jun 2024 Q1(b): one case's own region contradicted its own case restriction entirely, so that whole branch contributed nothing to the solution set — a different failure shape from a single rejected root. (codex-deep-dive-modulus-and-graph-transformations.md §3.2)

FC1.3.7 — Front: What's the real, nameable difference between sketching y = |x| − 1 and y = |x − 1|? Back: y = |x| − 1 has vertex (0,−1) — an output translation (constant outside the bars). y = |x − 1| has vertex (1,0) — an input translation (constant inside the bars, attached to x). Real anchor Jan 2024 Q8(b): candidates sketched the wrong one. (codex-deep-dive-modulus-and-graph-transformations.md §3.3)

1.4 Combinations of transformations

FC1.4.1 — Front: State the four building-block transformations of y = f(x) and their effects. Back: y = af(x): vertical stretch, scale factor a, from the x-axis. y = f(x)+a: vertical translation, a units up. y = f(x+a): horizontal translation, a units in the negative x-direction. y = f(ax): horizontal stretch, scale factor 1/a, from the y-axis. (codex-formula-and-method-bank.md §1.4)

FC1.4.2 — Front: What combined single-bracket transformation does the spec explicitly exclude? Back: "The graph of y = f(ax+b) will not be required" — a combined transformation must be built from the four listed types applied in sequence (e.g. y=2f(3x), y=f(−x)+1), never asked for as one fused rule. (codex-formula-and-method-bank.md §1.4)

FC1.4.3 — Front: Real method skeleton (Summer 2026 Q3(c)): under y = f(ax) + b, how does a point (p,q) on y=f(x) map? Back: (p,q) maps to (p/a, q+b) — apply the transformation's input/output rule directly to the point the question stem itself names, not a point computed in an earlier part. (codex-formula-and-method-bank.md §2.4)

FC1.4.4 — Front: Real, named trap when transforming a named point: what mistake costs marks even with a correctly-executed transformation? Back: Transforming a different, more recently computed point instead of the point the question stem itself names — the failure is which of two candidate numbers gets picked up, not the arithmetic. Real anchor Summer 2026 Q3(c): scored B0dB0 outright for a correctly-executed transformation applied to the wrong point. (codex-formula-and-method-bank.md §2.4)

Section 2 — Trigonometry (spec 2.1–2.5)

2.1 Reciprocal trig functions

FC2.1.1 — Front: State the three reciprocal trig definitions. Back: sec θ = 1/cos θ; cosec θ = 1/sin θ; cot θ = 1/tan θ = cos θ/sin θ. MEMORISE — not printed as such, though implied by other booklet entries. (codex-formula-and-method-bank.md §2.1)

FC2.1.2 — Front: Where are sec θ, cosec θ, and cot θ undefined? Back: sec θ is undefined wherever cos θ = 0 (odd multiples of 90°/π/2); cosec θ and cot θ are both undefined wherever sin θ = 0 (integer multiples of 180°/π) — each function's asymptotes sit exactly where the original ratio it reciprocates hits zero. (codex-formula-and-method-bank.md §2.1)

FC2.1.3 — Front: What boundedness fact follows directly from the reciprocal definitions? Back: Since |cos θ|, |sin θ| ≤ 1 always, |sec θ|, |cosec θ| ≥ 1 always. cot θ carries no such floor, because it reciprocates tan θ, which is already unbounded. (codex-deep-dive-reciprocal-and-inverse-trigonometric-functions.md §2)

FC2.1.4 — Front: Named real error (Jan 2022 Q2 examiner report): what's the single most confused pairing on this spec item? Back: Writing cosec θ as the reciprocal of cosine rather than sine — the two three-letter reciprocal names share a first syllable. (codex-formula-and-method-bank.md §2.1)

FC2.1.5 — Front: Real, scored-zero error: what's wrong with writing 3 cosec θ as 1/(3 sin θ)? Back: 3 cosec θ = 3/sin θ, not 1/(3 sin θ). Scored B0 on the real Jan 2022 Q2(a) mark scheme even though the underlying identity is "known" — the coefficient was applied to the wrong side of the reciprocal. (codex-formula-and-method-bank.md §2.1)

FC2.1.6 — Front: State the two Pythagorean-derived reciprocal identities and how each is derived. GIVEN or MEMORISE? Back: sec²θ ≡ 1+tan²θ and cosec²θ ≡ 1+cot²θ — both follow by dividing cos²θ+sin²θ≡1 through by cos²θ or sin²θ respectively. MEMORISE — on the spec's own "will not appear in the booklet" list. (codex-formula-and-method-bank.md §2.2)

Inverse trig — restricted domains

FC2.1.7 — Front: arcsin x — restricted domain of sin θ, domain of arcsin, and range of arcsin? Back: sin θ restricted to −π/2 ≤ θ ≤ π/2; domain of arcsin is −1 ≤ x ≤ 1; range of arcsin is −π/2 ≤ arcsin x ≤ π/2. (codex-formula-and-method-bank.md §2.1)

FC2.1.8 — Front: arccos x — restricted domain of cos θ, domain of arccos, and range of arccos? Back: cos θ restricted to 0 ≤ θ ≤ π; domain of arccos is −1 ≤ x ≤ 1; range of arccos is 0 ≤ arccos x ≤ π. (codex-formula-and-method-bank.md §2.1)

FC2.1.9 — Front: arctan x — restricted domain of tan θ, domain of arctan, and range of arctan? Why is the range open? Back: tan θ restricted to −π/2 < θ < π/2 (open); domain of arctan is all real x; range is the open interval −π/2 < arctan x < π/2 — open because tan θ itself is undefined at ±π/2, so those values are never reached. (codex-formula-and-method-bank.md §2.1)

FC2.1.10 — Front: A value outside [−1,1] is handed to arcsin or arccos. Is the result a large angle? Back: No — it's undefined outright. sin θ and cos θ never produce a value outside [−1,1] in the first place, so there is no angle to find. (codex-deep-dive-reciprocal-and-inverse-trigonometric-functions.md §2)

FC2.1.11 — Front: Confirmed real boundary (Oct 2020 Q8(ii) examiner report): does WMA13 ever require differentiating arcsin, arccos or arctan? Back: No. A candidate route rewriting an equation via arcsin and differentiating it directly is named as "beyond the specification so should not be encouraged." Spec 2.1 requires knowing these functions' definitions, graphs and domains — never their derivatives. (codex-deep-dive-differentiation-rules.md §5; codex-deep-dive-reciprocal-and-inverse-trigonometric-functions.md §4(v))

2.3 Double angle, compound angle, and related identities

FC2.3.1 — Front: State the three GIVEN compound-angle formulae. Back: sin(A±B) ≡ sin A cos B ± cos A sin B; cos(A±B) ≡ cos A cos B ∓ sin A sin B; tan(A±B) ≡ (tan A ± tan B)/(1 ∓ tan A tan B). (codex-formula-and-method-bank.md §2.3)

FC2.3.2 — Front: What single substitution generates sin2A, all three forms of cos2A, and tan2A from the compound-angle formulae? Back: Setting B = A in each of the three compound-angle formulae — one substitution, not three separate derivations. cos2A's three forms come from substituting cos²A+sin²A≡1 into the first form two different ways. (codex-formula-and-method-bank.md §2.3; codex-deep-dive-compound-and-double-angle-formulae.md §2.2)

FC2.3.3 — Front: State sin2A, the three equivalent forms of cos2A, and tan2A. GIVEN or MEMORISE? Back: sin2A ≡ 2 sin A cos A; cos2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A; tan2A ≡ 2tan A/(1 − tan²A). MEMORISE — verbatim from the spec's "will not appear in the booklet" list. (codex-formula-and-method-bank.md §2.3)

FC2.3.4 — Front: Decision rule for which form of cos2A to substitute into an equation. Back: Equation already in cos θ → reach for 2cos²θ − 1. Equation already in sin θ → reach for 1 − 2sin²θ. Reserve cos²θ − sin²θ for when a difference of squares is being factorised. (codex-deep-dive-compound-and-double-angle-formulae.md §2.3)

FC2.3.5 — Front: A fraction contains the block 1 + cos2θ or 1 − cos2θ. What do these simplify to? Back: 1 + cos2θ ≡ 2cos²θ; 1 − cos2θ ≡ 2sin²θ — the identical substitution, read as "what does adding/subtracting 1 do to this form." Example: sin2θ/(1+cos2θ) = 2sinθcosθ/2cos²θ = tanθ. (codex-deep-dive-compound-and-double-angle-formulae.md §2.3)

FC2.3.6 — Front: Is the half-angle tangent substitution t = tan(θ/2) examinable on WMA13? Back: No — explicitly excluded: "the t = tan(½θ) formulae will NOT be required." (codex-formula-and-method-bank.md §2.3)

FC2.3.7 — Front: Is sin3x ≡ 3sinx − 4sin³x examinable, even though it is not named in the spec's guidance column? Back: Yes — real, assessed content. Confirmed Oct 2020 Q5(a), a genuine 4-mark "show that." "Not named in the guidance column" is not the same as "safely excludable." (codex-formula-and-method-bank.md §2.3)

FC2.3.8 — Front: How is sin3x ≡ 3sinx − 4sin³x proved (real anchor Oct 2020 Q5(a))? Back: Write sin3x = sin(2x+x) = sin2x·cosx + cos2x·sinx. Substitute sin2x≡2sinxcosx and cos2x≡1−2sin²x (this form, since the target is entirely in sinx): = 2sinxcos²x + sinx − 2sin³x. Eliminate the remaining cos²x via cos²x≡1−sin²x: = 2sinx(1−sin²x)+sinx−2sin³x = 3sinx − 4sin³x. (codex-complete-worked-paper.md Q5)

FC2.3.9 — Front: The factor (sum-to-product) formulae, e.g. sinA+sinB ≡ 2sin((A+B)/2)cos((A−B)/2), are printed in the booklet. Are they confirmed as directly examined? Back: No — flagged gap. They are genuinely on the sheet, but no question in the series reviewed for this course visibly requires them. Treat "on the sheet" and "confirmed as directly tested" as separate facts. (codex-formula-and-method-bank.md §2.3)

FC2.3.10 — Front: Once a double-angle substitution collapses an equation to a 3-term quadratic in one ratio, how many solutions does each root contribute? Back: Each root with |k| < 1 contributes an ordinary reflected pair over one period; each root at exactly |k| = 1 contributes exactly one solution; any root with |k| > 1 is rejected outright. If the substituted variable is 2θ rather than θ, double the interval first, then halve every result and re-check against the original interval. (codex-deep-dive-compound-and-double-angle-formulae.md §4)

FC2.3.11 — Front: Real trap (Jan 2022 Q2): a candidate correctly solves for 2θ but reports the doubled value instead of dividing by 2. What number results, and what should it be? Back: Reporting 48.6° instead of dividing by 2 to recover 24.3°. (codex-deep-dive-compound-and-double-angle-formulae.md §3.C)

FC2.3.12 — Front: Real trap (Jan 2022 Q2): the question asks for the "smallest positive" solution, but a candidate correctly finds a second, larger one too. What happens? Back: Failing to reject the extra solution costs the final mark despite doing strictly more correct mathematics than asked — the mirror image of the far more common "missing a second solution" trap. (codex-deep-dive-compound-and-double-angle-formulae.md §3.C)

FC2.3.13 — Front: Bracket-loss trap (Oct 2020 Q1): what happens if you write "2×2cos²x−1 = 7cosx" with no bracket around (2cos²x−1)? Back: Scores nothing (M0) unless the identity was stated correctly first — the un-bracketed line is algebraically a different (wrong) equation. (codex-formula-and-method-bank.md §2.5; codex-complete-worked-paper.md Q1)

2.4 Harmonic form

FC2.4.1 — Front: Is there a boxed formula for R and α in a cosθ + b sinθ ≡ R cos(θ∓α)? Back: No — genuinely no formula to look up or memorise. R and α are derived every time by expanding R cos(θ∓α) with the compound-angle formulae and matching coefficients against a cosθ + b sinθ. The "formula" is the method, not a fact in closed form. (codex-formula-and-method-bank.md §2.3)

FC2.4.2 — Front: Real, mark-scheme-confirmed condition on the construction marks for R and α (Oct 2020 Q7(a))? Back: R must be left exact (a surd, not a rounded decimal) and α must be given in the unit the question names. A numerically correct α = awrt 75.96° scored A0 when the question asked for radians to 3 d.p. (codex-formula-and-method-bank.md §2.3)

FC2.4.3 — Front: State the four target-form/sign-condition pairs for a cosθ + b sinθ (first-quadrant α). Back: R cos(θ−α): a>0, b>0. R cos(θ+α): a>0, b<0. R sin(θ−α): b>0, a<0. R sin(θ+α): b>0, a>0. (codex-deep-dive-harmonic-form.md §2)

FC2.4.4 — Front: What sign combination (first-quadrant α) is not reachable by any of the four harmonic-form target forms? Back: a<0 and b<0 together — no located WMA13 question tests this combination; the evidence simply runs out here. (codex-deep-dive-harmonic-form.md §2)

FC2.4.5 — Front: "Minimise/maximise a fraction k/(a + R cos(...))" (positive k) — what's the key reasoning step, and the confirmed real point of confusion? Back: Minimising the fraction requires maximising the wave (the denominator's cosine term), not minimising it — a real, confirmed point of confusion ("the denominator had to be a maximum," Oct 2020 Q7(b) examiner report). No calculus is needed once R is known. (codex-formula-and-method-bank.md §2.3; codex-deep-dive-harmonic-form.md §4)

FC2.4.6 — Front: When the harmonic-form argument is a half-angle (e.g. θ = ½t), what extra step is needed to recover t? Back: An extra doubling step, on top of the usual "convert the shifted angle back" step — confirmed by Oct 2020 Q7(c), the only located WMA13 question testing this. (codex-formula-and-method-bank.md §2.3; codex-deep-dive-harmonic-form.md §3)

FC2.4.7 — Front: Method skeleton for "solve for t when H = [value]" where the harmonic-form argument is a half-angle. Back: Use the R and α from the construction part to reach cos(βt±α)=k → evaluate k → find the full method to make t the subject (look for 2×(arccos k ± α) if the argument was ½t) → state the final t, checking for genuine extra solutions inside the domain. (codex-formula-and-method-bank.md §2.5)

2.5 Reciprocal-identity-embedded-in-an-equation pattern

FC2.5.1 — Front: Method skeleton for "show 8cosθ=3cosecθ can be written sin2θ=k; hence find the smallest positive solution" (real anchor Jan 2022 Q2). Back: Part (a): state/use cosecθ=1/sinθ (refused if a coefficient is folded in) → use sin2θ=2sinθcosθ to reach sin2θ=k → state k exactly. Part (b): find any value satisfying sin2θ=k → report only the value the question names. (codex-formula-and-method-bank.md §2.5)

Section 3 — Exponentials and Logarithms (spec 3.1–3.3)

FC3.1 — Front: What are e^x and ln x, and what identity connects a^x and e^x? Back: e^x and ln x are mutual inverses; ln x's domain (x>0) follows directly. GIVEN, printed in the P3 section: e^(x ln a) = a^x — lets a^x be rewritten in base e for differentiation or integration. (codex-formula-and-method-bank.md §3.1–3.2)

FC3.2 — Front: Key shape properties of y = e^x. Back: Defined for every real x, strictly positive throughout, strictly increasing everywhere, no turning point. As x→−∞, e^x→0⁺ (never reaching it, asymptote y=0); as x→+∞, e^x→∞. e^0=1. (codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.1)

FC3.3 — Front: Why does ln x exist as a genuine function with no domain restriction needed first (unlike inverting a quadratic)? Back: e^x is strictly increasing on all of ℝ, so it is already one-one everywhere. Domain of ln x is {x: x>0} (the range of e^x); range of ln x is all of ℝ. y=ln x is the reflection of y=e^x in y=x. (codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.3)

FC3.4 — Front: Method for solving e^(ax+b) = p. Back: Apply ln to both sides; the left side collapses to ax+b, leaving ax+b = ln p. Requires p > 0 (e^(ax+b) is always positive). (codex-formula-and-method-bank.md §3.1; codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.4)

FC3.5 — Front: Method for solving ln(ax+b) = q. Back: Apply e^(·) to both sides; the left side collapses to ax+b, leaving ax+b = e^q. (codex-formula-and-method-bank.md §3.1; codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.5)

FC3.6 — Front: A model y = c + de^(−kt) (d,k>0) is asked to reach a target below c. What is the correct reasoning, and what phrasing does the real mark scheme reject? Back: The model has a genuine floor at y=c, so a target below the floor requires ln(negative or zero) — a real, correctly-reasoned "no solution." Accepted: "the log of a negative number cannot be found." Rejected as ambiguous: "logs can't be negative." Confirmed Jan 2024 Q5(d). (codex-formula-and-method-bank.md §3.1)

FC3.7 — Front: Log-linearisation of y = ax^n: which axes get logged, and what do the gradient and intercept give? Back: log y = n log x + log a. Plot log y against log x (both axes logged) — gradient n, intercept log a. (codex-formula-and-method-bank.md §3.3)

FC3.8 — Front: Log-linearisation of y = kb^x: which axes get logged, and what do the gradient and intercept give? Back: log y = x log b + log k. Plot log y against unlogged x — gradient log b, intercept log k. (codex-formula-and-method-bank.md §3.3)

FC3.9 — Front: General diagnostic for deciding which axes to log in a linearisation question. Back: Ask: is the unknown a fixed power applied to x (log both axes — y=ax^n), or is x itself the varying exponent (log only y — y=kb^x)? (codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.6)

FC3.10 — Front: Base-dependence rule: which quantities change if you switch the base of the logs in a log-linear fit? Back: The gradient of a y=ax^n (power-law) fit is n, and is base-invariant. The intercept of either line type, and the gradient of a y=kb^x (semi-log) fit, DO change with the base — un-log using the same base the graph was plotted in, never base 10 by default habit. (codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.6)

FC3.11 — Front: Given a fitted linear log-equation directly (e.g. log₆T = 4 − 2log₆x), how do you eliminate the logs entirely to state the original relationship? Back: Exponentiate both sides and split the exponent as a sum: log_c y = P + Q log_c x ⇒ y = c^P · x^Q. A full derivation using the product and power laws in reverse — distinct from reading two separate numbers off a graph, and real, confirmed as where marks are most often lost (Jun 2023 Q2). (codex-formula-and-method-bank.md §3.3; codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.6)

FC3.12 — Front: For y = kb^x, what does the constant b mean in words, and how is the percentage computed? Back: b is the multiplier applied once per unit of x. b>1 → percentage increase of (b−1)×100% per unit; 0<b<1 → percentage decrease of (1−b)×100% per unit. Real mark scheme (Jun 2026 Q5(b)) requires a direction word, the percentage to 2 s.f., and a per-year unit, all three together. (codex-formula-and-method-bank.md §3.3; codex-deep-dive-exponentials-and-logarithmic-graphs.md §3.6)

FC3.13 — Front: State the GIVEN change-of-base formula (carried forward from P2). Back: log_a x = log_b x / log_b a. (codex-formula-and-method-bank.md §3.3)

FC3.14 — Front: A numerically correct modelling answer is still wrong. What is the single most consistently lost mark across this whole spec area? Back: Missing the unit and/or a direction word ("£", "tonnes", "m²", "loss," "increasing") — confirmed across four-plus series with near-identical wording each time. (codex-formula-and-method-bank.md §3.1)

Section 4 — Differentiation (spec 4.1–4.4)

FC4.1 — Front: State the five standard derivatives of spec 4.1, marking GIVEN vs MEMORISE. Back: d/dx[e^(kx)]=ke^(kx) (MEMORISE); d/dx[ln kx]=1/x, no k (MEMORISE); d/dx[sin kx]=k cos kx (MEMORISE); d/dx[cos kx]=−k sin kx (MEMORISE); d/dx[tan kx]=k sec²kx (GIVEN). (codex-formula-and-method-bank.md §4.1; codex-deep-dive-differentiation-rules.md §2)

FC4.2 — Front: Why does d/dx[ln kx] carry no k, even though k sits inside the log? Back: ln(kx) = ln k + ln x by the log-of-a-product law, and ln k is a constant once k is fixed — its derivative is 0, so k never survives into the result. (codex-formula-and-method-bank.md §4.1)

FC4.3 — Front: State the product rule, the chain rule, and the GIVEN quotient rule. Back: Product (MEMORISE): (uv)' = u'v + uv'. Chain (MEMORISE): dy/dx = dy/du · du/dx. Quotient (GIVEN, printed): (u/v)' = (u'v−uv')/v². (codex-formula-and-method-bank.md §4.2)

FC4.4 — Front: Is the quotient rule a genuinely separate, fourth rule? Back: No — it is the product rule applied to u·v⁻¹, with the chain rule supplying d/dx[v⁻¹] = −v⁻²v'; both routes reach the identical result. (codex-formula-and-method-bank.md §4.2)

FC4.5 — Front: State the four further GIVEN trig derivatives. Back: sec x → sec x tan x; cot x → −csc²x; csc x → −csc x cot x; tan kx → k sec²kx — all printed verbatim; all rebuildable on the spot from sin kx/cos kx and the quotient rule. (codex-formula-and-method-bank.md §4.2)

FC4.6 — Front: State the relationship dy/dx = 1/(dx/dy), and when it is used. Back: MEMORISE the relationship (a direct consequence of the chain rule applied to g(y(x))=x). Valid wherever dx/dy ≠ 0. Used whenever differentiating with respect to the other variable is structurally easier (e.g. x = sin3y). (codex-formula-and-method-bank.md §4.3)

FC4.7 — Front: Real, repeatedly-confirmed error using dy/dx = 1/(dx/dy): what happens if the final answer is left in terms of y? Back: A genuinely incomplete answer if the question asks for the result in terms of x — confirmed across multiple series (Oct 2020 Q8(ii)). (codex-formula-and-method-bank.md §4.3)

FC4.8 — Front: The invalid shortcut (Oct 2024 Q2): on x=f(y), a candidate inverts dy/dx=1/(dx/dy) first, then sets the inverted fraction "equal to zero" and solves via the denominator. Why does this score zero, even if it lands on the right number? Back: A fraction is never made equal to zero by zeroing its denominator — that operation finds where the fraction is undefined, not where it equals zero. Correct method: find dx/dy → set THAT (before inverting) equal to zero → solve for y → only then invert if needed. The turning point of x=f(y) is exactly where dx/dy=0, which is where the inverted dy/dx is undefined. (codex-formula-and-method-bank.md §4.3)

FC4.9 — Front: State d/dx[a^x], and how it is rebuilt on demand if forgotten. Back: d/dx[a^x] = a^x ln a — MEMORISE, not printed anywhere in the P3 section. Rebuild: rewrite a^x as e^(x ln a) (the given identity), then apply e^(kx)→ke^(kx) with k=ln a: d/dx[e^(x ln a)] = (ln a)e^(x ln a) = a^x ln a. (codex-formula-and-method-bank.md §4.4; codex-deep-dive-exponential-growth-and-decay.md §2)

FC4.10 — Front: Real, confirmed error distinct from a sign slip (Oct 2024 Q4(c)): what happens if you apply the power rule to N=100×2.24^t? Back: Differentiating as though t were a fixed index rather than the varying exponent (e.g. producing 100t×2.24^(t−1)) scores nothing. Always name which symbol is fixed (the base) and which is varying (the exponent) before choosing a rule — if the variable is in the exponent, the rule is always a^x→a^x ln a, never nx^(n−1). (codex-formula-and-method-bank.md §4.4; codex-deep-dive-exponential-growth-and-decay.md §2)

FC4.11 — Front: Method skeleton for a product of a polynomial and a root/reciprocal-root factor, e.g. (2x+3)(4x−1)^(−1/2) — what's the discriminating step? Back: Apply the product rule with the chain rule folded into v' — the differentiation itself is the accessible half. The genuinely discriminating step is combining the two unsimplified terms over a common denominator (the shared root), then factorising the numerator. Real statistic (Oct 2020): only about a third of candidates scored the combining mark, and only a quarter scored all four. (codex-formula-and-method-bank.md §4.2)

FC4.12 — Front: Method skeleton for a sum of two differently-rated exponential terms (e.g. H=32+40e^(−0.2t)−20e^(−0.9t)), finding a maximum. Back: Differentiate term-by-term (each its own chain rule) → set the derivative to zero → rearrange to isolate one exponential term per side → divide one side by the other to collapse the two rate constants into a single exponential term → only then take logs → solve. Real, exact cost of skipping the algebra for a calculator solver (Oct 2024 Q8(c)): only 2 of 5 marks, because the equation-solving marks are chained to method the working never showed. (codex-formula-and-method-bank.md §4.4)

FC4.13 — Front: Real anchor Jun 2023 Q7(a): given two data points (not a stated formula), how do you find A and k in N=Ae^(kt)? Back: The initial-value convention gives A directly (e^(k·0)=1 regardless of k) — no equation needed, just substitute t=0. The second data point then gives one equation in k alone (e.g. 10000=2500e^(8k) ⇒ k = ln4/8). A distinct technique from both log-linear-graph methods, because there is no graph and no given linear log-equation. (codex-deep-dive-exponential-growth-and-decay.md §3.1)

FC4.14 — Front: Real anchor Jun 2023 Q7(c): two separate models cross (60000e^(−0.6t) = 2500e^(kt)). How does this differ from the two-exponential-term-maximum technique? Back: There, one model's own derivative is set to zero; here, two entirely separate models' own values are set equal to each other (growth on one side, decay on the other). Same underlying move — collapse to one exponential term, then take logs. (codex-deep-dive-exponential-growth-and-decay.md §3.2)

FC4.15 — Front: What compound sub-case pairs "convert a log-linear relationship into N=ab^t form" with "then differentiate for a rate," and how many real series show it? Back: Three real series (Jan 2022 Q8, Oct 2024 Q4, Jun 2026 Q5) pair, in order: convert a stated linear log₁₀ relationship (by algebra or graph) into N=ab^t form, then differentiate that converted form for a rate. A candidate drilled only on differentiating an already-given exponential model has not covered this recurring pattern. (codex-deep-dive-exponential-growth-and-decay.md §3.3)

FC4.16 — Front: Two structurally different real shapes test spec 4.4's "is the predicted range of values sensible" clause. What are they? Back: Shape A — hard mathematical impossibility (a target requires ln of a negative/zero number; answer is "not possible," Jan 2024 Q5(d)). Shape B — a plausibility check by direct substitution at one specific stated t, with the answer able to be "yes" (Jan 2024 Q3(c)). (codex-deep-dive-exponential-growth-and-decay.md §4)

Section 5 — Integration (spec 5.1–5.2)

FC5.1 — Front: State the four standard MEMORISE integrals of spec 5.1. Back: ∫cos kx dx = (1/k)sin kx + c; ∫sin kx dx = −(1/k)cos kx + c; ∫e^(kx) dx = (1/k)e^(kx) + c; ∫(1/x) dx = ln|x| + c (x≠0). Also ∫a^x dx = a^x/ln a + c (MEMORISE, the reverse of d/dx[a^x]=a^x ln a). (codex-formula-and-method-bank.md §5.1)

FC5.2 — Front: State the three GIVEN integrals printed in the P3 integration table. Back: ∫sec²kx dx = (1/k)tan kx + c; ∫tan x dx = ln|sec x| + c; ∫cot x dx = ln|sin x| + c. (codex-formula-and-method-bank.md §5.1)

FC5.3 — Front: State the two integration-by-recognition formulae. GIVEN or MEMORISE? Back: ∫f'(x)/f(x) dx = ln|f(x)| + c; ∫f'(x)[f(x)]ⁿ dx = [f(x)]^(n+1)/(n+1) + c. Both are the chain rule run backwards. MEMORISE — verbatim from the spec's content list. (codex-formula-and-method-bank.md §5.2)

FC5.4 — Front: Why does the log form ∫f'(x)/f(x) dx = ln|f(x)|+c exist as a separate formula from the power form? Back: The ordinary power rule for integration, ∫uⁿ du = u^(n+1)/(n+1)+c, divides by zero at n=−1. ln|u| is the patch for exactly that one missing case — the two "formulae" are one power-rule family for [g(x)]ⁿg'(x), split at the value of n where the formula breaks. (codex-deep-dive-integration-by-recognition.md §1.2)

FC5.5 — Front: Real, well-confirmed error across two series (Oct 2020 Q9(c), Jun 2023 Q9(c)): what happens if the modulus is dropped in the log form? Back: Ends up trying to evaluate ln of a negative number instead of recognising the domain issue. (codex-formula-and-method-bank.md §5.2)

FC5.6 — Front: Why doesn't ∫sin²x dx fit the power-form recognition pattern directly? Back: The power-form pattern needs f'(x) present as a genuine multiplying factor alongside [f(x)]ⁿ. sin²x is [sin x]² with no accompanying cos x factor — nothing plays the role of f'(x). Fix: convert via the double-angle identities first (spec 2.3), then integrate two ordinary spec-5.1 terms with no recognition step left. (codex-formula-and-method-bank.md §5.2; codex-deep-dive-integration-by-recognition.md §1.4)

FC5.7 — Front: For a definite power-form or log-form integral, where does the dependent mark (dM1) actually belong? Back: dM1 is reserved for the substitute-the-limits step — never used to separate "spot the pattern" from "get the constant right" within the antiderivative itself; those two live inside one M1/A1 pair. Confirmed identically across three real questions (Oct 2021 Q5, Jan 2022 Q3, Jun 2023 Q9(c)). (codex-formula-and-method-bank.md §5.2; codex-deep-dive-integration-by-recognition.md §3.2)

FC5.8 — Front: Is ∫cosec x dx printed in the formula booklet, and is it examinable on WMA13? Back: It IS printed — but under the "Pure Mathematics P4" heading, not P3. The P3 section's own opening line only reaches backward into P1/P2, never forward into P4. Whether ∫cosec x dx has ever been directly required on a real WMA13 paper is a genuine open question. (codex-formula-and-method-bank.md §1.5)

FC5.9 — Front: June 2023 Q9(c) was originally described (in this course's own earlier research) as testing ∫cosec x dx. What does it actually test? Back: It tests ∫cosec x cot x dx, credited via the P3-printed derivative cosec x → −cosec x cot x read in reverse — a materially different, entirely P3-resident fact from the harder ∫cosec x dx. (codex-formula-and-method-bank.md §1.5; codex-deep-dive-integration-by-recognition.md §3.5)

FC5.10 — Front: Real, zero-credit trap on Jun 2023 Q9(c): what happens if a numerically correct final value is given with no antiderivative line shown? Back: Scores M0A0 outright — "answer only with no integration shown" earns nothing regardless of correctness. (codex-deep-dive-integration-by-recognition.md §3.2, §3.5)

Section 6 — Numerical Methods (spec 6.1–6.2)

FC6.1 — Front: State the root-location test (spec 6.1). Back: If f is continuous on [a,b] and f(a), f(b) have opposite signs, at least one root of f(x)=0 lies in (a,b) — in effect the Intermediate Value Theorem, though the spec never names it that. (codex-formula-and-method-bank.md §6.1)

FC6.2 — Front: What is the single most mark-scheme-enforced word on the entire WMA13 paper, and where does it belong? Back: "Continuity"/"continuous," stated explicitly as part of the root-location reasoning — not just "sign change" alone. Confirmed verbatim (Oct 2023 Q1(a)): three components required — both function values correct, a reason naming continuity and stating the sign change, and a "hence root" conclusion. The single most repeated specific loss across five-plus series. (codex-formula-and-method-bank.md §6.1)

FC6.3 — Front: State the iteration relationship (spec 6.2) and what convergence means. Back: x_(n+1) = g(x_n), converging (where it does) to a root of x=g(x), i.e. a root of the original f(x)=0 once rearranged into that form. The spec's own note: "leads will be given" — a candidate is never asked to invent a convergent rearrangement from a blank page. (codex-formula-and-method-bank.md §6.2)

FC6.4 — Front: Two real, confirmed secondary errors during iteration, beyond an arithmetic slip. Back: (1) Giving an iterate to fewer decimal places than asked, or rounding prematurely mid-chain. (2) Embedding the iteration via a calculator's ANS key with the substitution never written down — treated as insufficient method evidence even when every printed value is correct. (codex-formula-and-method-bank.md §6.2)

FC6.5 — Front: How is a claimed rounded root value confirmed by a sign-change test (real anchor Jan 2021 Q6(d))? Back: Test an interval exactly half a unit wide either side of the candidate value, in the last decimal place — e.g. to confirm 2.581 to 3 dp, test 2.5815 and 2.5805 (interval width 0.001). (codex-deep-dive-numerical-methods.md §2.3)

FC6.6 — Front: Can the sign-change/continuity test (spec 6.1) be applied to something other than f(x) itself? Back: Yes — real anchor Jan 2021 Q6(d) applies the identical test to f'(x)=0 (a stationary point) rather than f(x)=0 (a root). The spec's "changes of sign of f(x)" wording does not restrict which function is being tested. (codex-deep-dive-numerical-methods.md §2.2)

FC6.7 — Front: After an iteration converges to a decimal root, is that decimal always the finished answer? Back: Not always — real anchor Oct 2022 Q5(e): after converging to t=6.315 years, a further, separately-marked step converts the fractional part into whole months. (codex-deep-dive-numerical-methods.md §2.6)

FC6.8 — Front: Does "find the value the iteration converges to" always mean the same as "find x₅"? Back: No — some real mark schemes ask for and mark a specific indexed iterate by name (e.g. Jan 2020 Q7(b)(ii) asks for x₅ CAO; Jan 2021 Q6(c) asks for x₆ specifically) rather than accepting any sufficiently-converged value. (codex-deep-dive-numerical-methods.md §2.8)

FC6.9 — Front: What is the real mark-scheme shape for the sign-change/continuity step and for iteration, reinforced across a widened archive of anchors? Back: The sign-change/continuity mark is a lenient M1 (one value correct to 1 s.f./1 d.p. suffices) followed by a single dependent A1 bundling both values, the continuity-naming reason, and a conclusion — never two independent marks. Iteration awards the first produced value an M1 A1 pair, with each subsequent value its own separate, dependent accuracy mark. (codex-deep-dive-numerical-methods.md §4)

Section 7 — Worked-Paper Micro-Examples (October 2020, the one full series independently re-verified end to end)

These seven cards use genuinely additive worked material from codex-complete-worked-paper.md — the one real WMA13 series this course's research has verified question-by-question against a primary PDF — where a fact isn't already captured in Sections 1–6 above.

FC-WP1 — Front: Real anchor Oct 2020 Q3(b): given f'(x) = (4x−8)/(4x−1)^(3/2) for f(x)=(2x+3)/√(4x−1), x>¼, how do you find the range of f? Back: Set the numerator of f'(x) to zero (4x−8=0 ⇒ x=2, inside the domain) → substitute back into f to find f(2)=√7 → check end behaviour (f→+∞ as x→¼⁺ and as x→∞) to confirm x=2 is a minimum → state the range in function notation: f(x) ≥ √7. Written as "x ≥ √7" is not accepted — the inequality describes the output of f. (codex-complete-worked-paper.md Q3)

FC-WP2 — Front: Real anchor Oct 2020 Q4(c): f(x)=21−2|2−x| is a "tent" shape (peak (2,21)). For what range of k does f(x)=k have exactly two roots? Back: 17 ≤ k < 21 — above the peak (k>21), zero intersections; exactly at the peak (k=21), one; between the peak and the rising piece's start (17≤k<21), the line crosses both pieces, giving two; below that (k<17), only the falling piece reaches that low, giving one. (codex-complete-worked-paper.md Q4)

FC-WP3 — Front: Real anchor Oct 2020 Q6(b): to justify α=1.134 (3 dp) as a root of h(x)=x²+5e^(x−1)−7, how narrow was the actual sign-change interval? Back: A 0.001-wide interval (h(1.1335)<0, h(1.1345)>0) — a real sign-change interval is often far narrower than the requested decimal precision might suggest at first attempt. (codex-complete-worked-paper.md Q6)

FC-WP4 — Front: Real anchor Oct 2020 Q8(ii): show dy/dx = eˣ/f(x) for x=ln(sin y), 0<y<π/2, having already found dx/dy = cos y/sin y. Back: Take the reciprocal to form dy/dx = sin y/cos y, then express cos y in terms of eˣ via sin²y+cos²y≡1 (sin y=eˣ, and cos y>0 on this domain, so cos y=√(1−e^(2x))). Combining gives dy/dx = eˣ/√(1−e^(2x)) — stopping at eˣ/cos y, never converting cos y back to x, is the real, confirmed way this question's marks are lost. (codex-complete-worked-paper.md Q8)

FC-WP5 — Front: Real anchor Oct 2020 Q9(b): given g(x)=x²+2+5/(x−4), find the tangent at x=2. Back: g'(x)=2x−5/(x−4)² (chain rule on 5(x−4)^(−1)). At x=2: gradient = 4−5/4 = 11/4; g(2)=4+2−2.5=3.5. Tangent: y−3.5=(11/4)(x−2) ⇒ y=2.75x−2. (codex-complete-worked-paper.md Q9)

FC-WP6 — Front: Real anchor Oct 2020 Q9(c): the area under g(x)=x²+2+5/(x−4) from 0 to 2 evaluates to a+b ln2. Find a and b. Back: Antiderivative: x³/3+2x+5ln|x−4|+c. Evaluating from 0 to 2 gives (8/3+4+5ln2) − 5ln4; since ln4=2ln2, this is 8/3+4+5ln2−10ln2 = 20/3−5ln2, so a=20/3, b=−5. Dropping the modulus and trying ln(−2)/ln(−4) directly cannot reach this — both are undefined. (codex-complete-worked-paper.md Q9)

FC-WP7 — Front: Real anchor Oct 2020 Q4(d): the vertex of y=f(x) is (2,21). Under y=af(x−b), the transformed vertex is (6,3). Find a and b. Back: Under y=af(x−b) (horizontal shift b, vertical scale a), the peak (2,21) maps to (2+b, 21a). Matching to (6,3): 2+b=6 ⇒ b=4; 21a=3 ⇒ a=1/7. (codex-complete-worked-paper.md Q4)

Genuine gaps and open items — flagged, not turned into cards

These are not phrased as flashcards because they are not settled facts. Each is carried over, unresolved, from the source codex file that flagged it.