Technique Masterclasses

Pure Mathematics 1

Formula and Method Bank

Provenance: compiled 2026-09-28. Primary sourcing is research/veridian/WMA11-verified-facts.md and content/veridian/technique-hub/wma11/forge-exam-technique-guide.md (both read in full before this document was started). This pass additionally re-verified the two load-bearing primary sources directly and independently, rather than trusting either document's own transcription on trust: the specification PDF (international-a-level-maths-spec.pdf, Specification Issue 3, April 2019, "Unit P1: Pure Mathematics 1", pp. 12–16, local copies at ~/Downloads/international-a-level-maths-spec.pdf and ~/Documents/SOL/Old/EDgine summaires + getting started guides/Spec/international-a-level-maths-spec.pdf) and the exam formula booklet (IAL-Mathematics-Formula-Book.pdf, Issue 2, January 2021, local copy at ~/Downloads/IAL-Mathematics-Formula-Book.pdf), both converted with pdftotext -layout and cross-checked against pdftotext -raw where a formula's fraction layout was ambiguous in one mode (the completing-the-square identity in §1.1.3 and the reciprocal-inequality example in §1.1.6 were both re-extracted in -raw mode specifically to resolve column-mangled fractions before being quoted here). Mark-scheme method conventions in Part 2 are drawn from the verified-facts bank's own §3–§4, itself checked against real Jan 2023 / Jan 2024 / Oct 2021 mark-scheme PDFs, and were not independently re-fetched a third time in this pass except where noted inline. Not affiliated with or endorsed by Pearson Edexcel. Every formula below is tagged against the spec's own explicit "will not appear in the booklet" list (P1.2) and the real formula booklet's own P1 section — the two are genuinely different lists, and conflating them is flagged explicitly in §0.2 below, not glossed over.

WMA11 — Formula & Identity Masterlist and Worked-Method Reference

Pearson Edexcel IAL Mathematics, Unit 1: Pure Mathematics P1 — the CODEX-equivalent layer

0. What this document is, and how it differs from the other two WMA11 documents

0.1 Scope. This is a content/knowledge-reference layer, not a technique layer and not a re-teach. The FORGE guide at content/veridian/technique-hub/wma11/forge-exam-technique-guide.md already covers how to write a solution that captures the marks on offer — the mark-coding system, the self-mark checklist, the exam-day protocol. The 16 curriculum lesson files in content/veridian/curriculum/wma11/ already carry full worked-chain and marked-solution exercises for every topic with strong evidence behind it. This document sits underneath both: Part 1 is a masterlist of every real formula, identity and condition the WMA11 spec actually requires, organised by spec topic, with each one flagged for whether it is given in the real exam formula booklet, must be memorised, or is assumed prior knowledge the spec never itemises at all — a three-way distinction that turns out to matter and that no other WMA11 document in this course states this explicitly. Part 2 is a bank of short worked-method snippets — the standard recipe for a common question shape, three to six lines long, not a full exam question — cross-referenced to the curriculum lesson that already carries the full worked-chain treatment of the same skill, so nothing here duplicates that lesson's own exercises.

0.2 The distinction that genuinely matters, stated once, up front. WMA11's specification (P1.2, "Notation and formulae") makes an explicit, narrow claim: "Formulae that students are expected to know are given below and will not appear in the booklet." That sentence introduces a specific, short, named list — verified directly against the primary PDF this pass: the quadratic formula, the sine rule, the area-of-a-triangle formula, arc length, sector area, the differentiation rule for xⁿ, and the integration rule for xⁿ. Nothing else in the P1 spec carries that flag. Separately, the real exam formula booklet's own "Pure Mathematics P1" section (verified directly against the booklet PDF this pass) contains exactly two entries — sphere/cone mensuration and the cosine rule — and nothing else. These are two different lists, and a formula can fail to appear on either one without being "forgettable." Laws of indices, surd manipulation, the discriminant conditions, completing the square, the straight-line equation forms, the parallel/perpendicular gradient conditions, polynomial factorisation, and the graph-transformation rules are all genuine, examined P1 spec content — but none of them is in the booklet, and none of them is on the spec's own short "will not appear in the booklet" callout either. They sit in a third category: assumed prior knowledge (largely IGCSE-level) that the spec's content list simply expects without flagging it as a formula to memorise, because from the spec's own point of view it was never "in the booklet" to begin with — there was never a booklet entry to be missing. Every entry in Part 1 below is tagged [BOOKLET], [MEMORISE] or [ASSUMED] on exactly this basis, checked against both primary sources directly, not inferred.

0.3 Scope exclusions, confirmed directly from the spec. P1 does not include: proof (proof by exhaustion / counterexample is P2 1.1–1.3), the factor theorem or remainder theorem (P2 2.1), circle geometry (P2 3.1), sequences and series (P2 4), exponentials and logarithms (P2 5), the chain rule (explicitly excluded, verbatim, spec 4.1: "Knowledge of the chain rule is not required"), and n = −1 in integration (explicitly excluded, spec 5.2 — meaning ∫x⁻¹ dx = ln|x| + c, which needs logarithms, is not valid P1 content). A lesson-builder or reviser importing formulas from a general "Pure Maths" reference risks pulling in P2/P3 content that cannot legally appear on this paper; every formula below was checked against the P1 unit-content section specifically (spec pp. 12–16), not the qualification as a whole.

PART 1 — Formula & Identity Masterlist

Organised by the spec's own five numbered content strands (P1.3): Algebra and functions, Coordinate geometry, Trigonometry, Differentiation, Integration.

1.1 Algebra and functions (spec strand 1)

1.1.1 Laws of indices (spec 1.1) — [ASSUMED], not in the booklet, not on the P1.2 memorise-list; stated directly in the spec's own guidance column and treated as prior knowledge to be applied, not a formula newly introduced at P1.

1.1.2 Surd manipulation (spec 1.2) — [ASSUMED], not in the booklet, not on the memorise-list; universal algebraic identities the spec expects rather than introduces.

1.1.3 The quadratic, its discriminant, and completing the square (spec 1.3, 1.4, 1.5)

1.1.4 Simultaneous equations (spec 1.6) — no dedicated identity; the technique is substitution (solve one equation for one variable, substitute into the other, reduce to a single-variable polynomial). Condition worth flagging: where the resulting equation is quadratic, both roots typically correspond to genuine solution pairs and both x AND y must be found and correctly paired for full marks — stopping at the x-coordinate, or pairing x- and y-values incorrectly against a stated constraint (e.g. "x > y"), is a real, confirmed, repeatable mark loss (Jan 2023 Q7(d): "A small number forgot to substitute their value for x to find the y coordinate"; June 2023 Q2: "The final A1 mark was sometimes lost because candidates did not realise the requirement that x > y" — both facts bank §4.1).

1.1.5 Linear inequalities (spec 1.7–1.9) — [ASSUMED], standard algebraic rearrangement; the one condition that must never be dropped: multiplying or dividing an inequality by a negative number reverses its direction. No WMA11-specific formula; the technique is treated as prior knowledge.

1.1.6 Quadratic and reciprocal inequalities (spec 1.7–1.9)

1.1.7 Polynomial factorisation, degree n ≥ 3 (spec 1.10) — [ASSUMED]; P1 explicitly excludes the factor theorem and remainder theorem (those are P2 2.1 — confirmed via the facts bank's scope note and independently checked against the spec's P2 content list, which introduces factor/remainder theorem for the first time at P2). At P1, cubic-and-above factorisation relies on extracting a common factor and then factorising the remaining quadratic by inspection — the spec's own worked example, verbatim: "Factorisation of polynomials of degree n ≥ 3, e.g. x³ + 4x² + 3x" (spec P1.3, confirmed this pass) — which factorises as x(x² + 4x + 3) = x(x+1)(x+3), a common-factor extraction followed by an ordinary quadratic factorisation, not a factor-theorem search for a root by trial substitution.

1.1.8 Graphs of functions (spec 1.11) — [ASSUMED].

1.1.9 Graph transformations (spec 1.12) — [ASSUMED], applied to quadratics, cubics, reciprocals, sine, cosine and tangent.

1.2 Coordinate geometry in the (x, y) plane (spec strand 2)

1.2.1 Straight-line forms (spec 2.1) — [ASSUMED], not in the booklet, not on the P1.2 memorise-list (coordinate geometry has no entry there at all — confirmed this pass, the spec's P1.2 box covers only quadratics, trigonometry, differentiation and integration).

1.2.2 Parallel and perpendicular conditions (spec 2.2) — [ASSUMED], universal geometric facts, not spec-boxed, not booklet-given.

1.3 Trigonometry (spec strand 3)

1.3.1 Sine rule, cosine rule, and triangle area (spec 3.1)

1.3.2 Radians, arc length, and sector area (spec 3.2)

1.3.3 Sine, cosine and tangent graphs (spec 3.3) — [ASSUMED].

1.4 Differentiation (spec strand 4)

1.4.1 The derivative and second-order derivatives (spec 4.1) — [ASSUMED] for the definition; the power rule itself is [MEMORISE].

1.4.2 Differentiation of xⁿ (spec 4.2)

1.4.3 Tangents and normals (spec 4.3) — [ASSUMED], built directly from 1.4.2 and 1.2.1–1.2.2 combined; no separate spec-boxed formula.

1.5 Integration (spec strand 5)

1.5.1 Indefinite integration as the reverse of differentiation (spec 5.1) — [ASSUMED] as a concept, the rule itself [MEMORISE].

1.5.2 Integration of xⁿ (spec 5.2)

PART 2 — Worked-Method Reference

Short, standard-recipe snippets for common WMA11 question shapes, organised to mirror Part 1's spec strands. Each one names the real mark-scheme convention it is grounded in and cross-references the curriculum lesson carrying the full worked-chain/marked-solution treatment of the same skill — these snippets deliberately stop short of that depth so as not to duplicate it.

2.1 Algebra and functions

2.1.1 Solving a three-term quadratic — pick a route, then quote it.

  1. Check the discriminant sign first if a route isn't obvious (Δ = b² − 4ac): a perfect square ⟹ integer factorisation is findable; otherwise go straight to the formula rather than hunting for factors that don't exist.
  2. Factorisation: seek (mx+p)(nx+q) with mn = a, pq = c (sign-matched) — write the bracket pair down before expanding to check it.
  3. Formula: write x = (−b ± √(b²−4ac))/2a in full, in letters, before substituting — the mark-scheme convention (verified verbatim) is that quoting the formula first protects the method mark against a substitution slip; substituting straight into unlabelled numbers risks losing the mark to any arithmetic error, since the method then has to be inferred.
  4. Completing the square: only when the question specifically asks for turning-point/range/proof information alongside the roots — it is the slowest route to the roots alone. → Full worked derivation and three-way method-comparison: quadratic-functions.ts.

2.1.2 Recognising a "hidden" quadratic before applying the discriminant or solving it. Three real, recurring disguises, confirmed across independent series (facts bank §4.1):

  1. Line meets curve: set the two y-expressions equal; the resulting equation's discriminant counts the intersections (positive: two meetings; zero: tangent; negative: none).
  2. Substitution (e.g. p = 3ˣ turns an exponential equation into a quadratic in p): solve for the substituted variable, then go back and solve for the original variable — stopping at p is a real, twice-independently-confirmed mark loss ("a significant number of responses did not go any further than finding solutions to the quadratic in p").
  3. Unknown in a denominator, cleared by multiplying through: the quadratic only appears after the multiplication step. Before writing b²−4ac, be able to point at one equation "(something) = 0" with exactly three terms and name a, b, c. → Full worked chains: discriminant-with-a-parameter.ts, hidden-quadratic-substitution.ts.

2.1.3 Quadratic/linear inequalities: critical values, then a sketch, never a region label.

  1. Rearrange to "(expression) [inequality] 0" if not already there.
  2. Solve "(expression) = 0" for the critical value(s).
  3. Sketch (mentally or on paper) the shape — a line for a linear inequality, a parabola (arms up if the x² coefficient is positive) for a quadratic one — and read off which side/region satisfies the original inequality.
  4. State the answer in the actual variable — never in terms of a region label like "R." Confirmed as a real M0, not just a lost accuracy mark: "Do not allow inequalities in terms of R... This scores M0" (Oct 2021 Q3(ii)). Reciprocal case (a/x compared to b): multiply through by x² (never x alone) to avoid a sign-of-x case split — this is the spec's own named method (§1.1.6 above), not a trick, and is the examiner-confirmed higher-success route versus a naive one-sided rearrangement that silently assumes x > 0. → Full worked treatment: quadratic-inequalities-graphically.ts.

2.1.4 Simultaneous equations by substitution — line meets curve.

  1. Rearrange the linear equation for one variable if not already isolated.
  2. Substitute directly into the other equation — do not rearrange both into y = … first and set them equal if a direct numeric substitution is available; the mark scheme rewards the shorter route and a real examiner report confirms the "equate-then-rearrange" habit produces more algebra errors in practice ("a lot of students made this more complicated by equating the expressions for y... More errors in the algebra were found when taking this approach", Jan 2023 Q7(c)).
  3. Solve the resulting single-variable equation for all valid values.
  4. Substitute back to find the paired second-variable value for each — stopping at the first variable is a real, repeated, confirmed mark loss.
  5. If a constraint is stated in the question's prose (e.g. "x > y") rather than in either equation, apply it after solving, to select between multiple valid solution pairs — don't assume the algebra alone settles which pair is "the" answer. → Full worked chains: simultaneous-equations-by-substitution.ts.

2.1.5 Sketching a cubic from factored form.

  1. Identify each root and its multiplicity from the factorised form.
  2. A single (non-repeated) root: the curve crosses the x-axis there.
  3. A repeated (squared) factor: the curve touches the axis there — smoothly, as a proper turning point, never a sharp cusp (a real, confirmed, examiner-documented drawing error: "a few candidates were inclined to feature a cusp at the minimum point", Oct 2021 Q6(a)).
  4. Use the sign of the leading coefficient to fix the far-left/far-right direction of the curve's arms.
  5. Label the y-intercept explicitly (read off directly from the constant term, or by evaluating at x = 0) — a real, confirmed, otherwise-avoidable mark loss on sketches that are shape-correct but leave this unlabelled. → Full worked treatment: cubic-graph-sketching-from-factored-form.ts.

2.1.6 Describing a graph transformation in words.

  1. Identify which of the four forms applies: y = af(x), y = f(x)+a, y = f(x+a), y = f(ax).
  2. State the transformation type using the required word "translation" (not "shift," "move," or "slide") for a translation, "stretch" (with the correct scale factor and axis) for a stretch.
  3. For y = f(x+a): the translation is by (−a, 0) — the sign flips relative to how it appears inside the bracket; this is the single most common sign-direction error in this topic.
  4. For y = f(ax): the stretch is horizontal, scale factor 1/a — inverted relative to a's own size, and a common, confirmed real confusion is describing this as a vertical stretch instead. → Full worked treatment: graph-transformations-in-words.ts.

2.1.7 Surds — rationalising and exact-value discipline.

  1. Denominator is a bare surd (1/√a): multiply top and bottom by √a.
  2. Denominator is a binomial with a surd (1/(a+√b)): multiply top and bottom by the conjugate (a−√b); the denominator becomes a² − b, surd-free.
  3. Keep every intermediate value exact (as a surd or fraction) — convert to a decimal only at the very final step, and only if the question explicitly permits it; per the general marking principles, resorting to a rounded decimal where an exact value or surd working is clearly required normally costs the mark even if the decimal itself is numerically correct to the stated precision. → Full worked treatment: surds-manipulation-and-exact-answers.ts.

2.2 Coordinate geometry

2.2.1 Proving two lines are perpendicular.

  1. Find both gradients (via Δy/Δx, or by rearranging a general-form equation ax+by+c=0 into y = mx + … first).
  2. Show m₁ × m₂ = −1 (or use the length/Pythagoras route, or the vector dot-product route — all three are equally credited).
  3. Write an explicit closing sentence connecting the arithmetic to the geometric claim — "therefore PQ ⊥ QR, so angle PQR = 90°" or equivalent. The calculation alone, with no such sentence, is a real, confirmed, documented way to lose the final mark even when every number is correct: "a small number of candidates simply left it at that without any reference to the fact that it indicated the two lines were perpendicular." → Full worked treatment (including the length/Pythagoras and vector alternative routes, and the "find a fourth vertex" extension): perpendicularity-proofs-and-coordinate-geometry.ts.

2.2.2 Finding a line's equation from a point and a parallel/perpendicular condition.

  1. Find the gradient of the given line (rearrange to y = mx + c form if given in general form).
  2. Apply the condition: same gradient for parallel, negative reciprocal for perpendicular.
  3. Substitute the given point and the new gradient into y − y₁ = m(x − x₁), then simplify — leaving the answer in the form the question specifies (general form ax+by+c=0, or gradient-intercept form) rather than defaulting to whichever form the working happened to produce.

2.3 Trigonometry

2.3.1 Sine rule with the ambiguous case.

  1. Set up a/sin A = b/sin B = c/sin C with the known sides/angles correctly paired (a side and the angle opposite it belong together).
  2. Solve for the unknown angle; a calculator returns the acute solution by default.
  3. Check whether an obtuse solution is also valid: compute 180° − (the acute angle found), and check the resulting angle sum and side-length consistency.
  4. Report both candidate values where both are valid — the mark scheme credits either one for the first accuracy mark, but a specific final question (e.g. "find the obtuse angle") requires identifying the correct one specifically, cao (correct answer only, not follow-through-able). → Full worked treatment: sine-and-cosine-rules-ambiguous-case.ts.

2.3.2 Composite sector/arc-length/perimeter problems.

  1. Convert any given degree angle to radians before applying s = rθ or A = ½r²θ.
  2. Identify whether the angle needed is the one drawn/labelled, or its 2π-complement (major vs minor arc/sector) — do not assume the labelled angle is the one the question is asking about.
  3. Decompose the composite shape into its named regions explicitly (e.g. "sector + two triangles," or "sector minus triangle") before computing any areas — do not treat the whole figure as a single sector by default.
  4. For a perimeter: sum every bounding edge separately (arc length(s), any straight radii, any chord/base lengths) — a common, confirmed real error is doubling one length that shouldn't be doubled, or omitting a radius.
  5. Do not drop the ½ from the sector-area formula — confirmed as the single most common error on this question type. → Full worked treatment: sector-area-and-arc-length.ts.

2.3.3 Sketching or interpreting a trig graph.

  1. State the period as the length of one repeat (2π for sin/cos, π for tan) — not as an interval or domain restriction; these are different concepts and conflating them is a real, heavily-documented (>50% of candidates scoring zero on a direct period-of-tan-x question) conceptual error.
  2. Read amplitude, phase shift and period changes directly from the function's form using the transformation rules in §2.1.6, applied to a trig base function.
  3. Label a scale on both axes of any sketch — an unscaled sketch is a real, confirmed, avoidable mark loss even when the shape is otherwise correct. → Full worked treatment: trig-graphs-period-and-domain.ts.

2.4 Differentiation

2.4.1 Differentiating term-by-term, including negative/fractional indices.

  1. Rewrite every root and fraction as a power of x first (√x = x^(1/2), 1/x² = x⁻², etc.) — nothing can be differentiated by the power rule until it is in xⁿ form.
  2. Differentiate each term separately, writing the index shift as its own visible line (xⁿ → nxⁿ⁻¹) — the method mark is earned the instant any one term is correctly power-shifted, so collapsing three terms into one mental jump risks losing a mark that a single visible line would have banked regardless of what happens to the other terms.
  3. Do not append "+c" — that is an integration habit, and a real, confirmed, examiner-documented error is differentiating correctly and then adding a constant anyway out of procedural muscle memory.
  4. If an exact gradient is required at a specific x-value, keep the value exact (as a fraction or surd) throughout the substitution — resorting to a decimal early is a real, confirmed mark loss even when differentiation itself was correct. → Full worked treatment: differentiating-negative-and-fractional-indices.ts.

2.4.2 Tangent and normal at a point, including the zero-gradient case.

  1. Differentiate, then evaluate f'(a) at the given x-value.
  2. Tangent: y − f(a) = f'(a)(x − a). Normal: y − f(a) = −1/f'(a) · (x − a).
  3. If f'(a) = 0: don't stop. The tangent is simply y = f(a) (a horizontal line) and the normal is x = a (a vertical line) — a zero gradient is a valid, simpler outcome, not a dead end; a real, confirmed, examiner-documented finding is candidates reaching f'(a) = 0 correctly and then failing to write down the (now-trivial) tangent equation at all.
  4. The y-value alone is not the answer — it must be presented as an equation.

2.4.3 A "hidden" higher-degree equation produced by setting a derivative equal to a value.

  1. Differentiate, and set f'(x) equal to the stated gradient value.
  2. Rearrange to "(polynomial) = 0" — check whether the result is a disguised quadratic (commonly in x², when only even powers of x are present).
  3. If a common factor can be taken out before solving, take it out explicitly and show that step — a real, confirmed mark loss is presenting a factorised form that does not actually multiply back out to the unsimplified line above it (e.g. writing 15x⁴+12x²−3 as if it factorised directly to (5x²−1)(x²+1), silently dropping the factor of 3 that must be divided through first).
  4. Solve the resulting quadratic (in x² or whichever substituted variable), then convert back to x — including taking the square root of both sign branches where the substituted variable was x², unless a stated domain restriction (e.g. "x > 0" in the original question) rules one branch out. Both directions of this error are real and independently confirmed: keeping only one root when both are needed (Oct 2021 Q2), and keeping both roots when the question's own domain restriction rules one out (Jan 2025 Q5(c)) — the correct behaviour depends on reading the question's stated domain, not on a default rule applied automatically either way. → Full worked treatment: hidden-higher-degree-equations-from-differentiation.ts.

2.5 Integration

2.5.1 Splitting a fraction before integrating.

  1. Confirm the denominator is a single term (e.g. 2x², 3√x) — this method applies to a fixed denominator with a shifted numerator, not the reverse.
  2. Divide every term of the numerator by the denominator separately, simplifying each resulting index.
  3. Integrate each resulting power-of-x term separately, add one constant of integration at the end (not one per term).
  4. Do not attempt this split on a shifted-denominator expression like 6/(x−2) — that is not a valid algebraic identity (6/(x−2) ≠ 6/x − 3, a real, confirmed WMA11 error); a shifted denominator can only be handled by cross-multiplying or genuine algebraic manipulation of the whole expression, not by "pulling the shift out." → Full worked treatment: splitting-fractions-before-integrating.ts.

2.5.2 Two-stage integration: f''(x) → f'(x) → f(x), with two separate constants.

  1. Before integrating anything, sort the given conditions by which function each belongs to: a gradient condition f'(a) = m belongs to f'(x); a point (p, q) on the curve belongs to f(x).
  2. Integrate f''(x) once to reach f'(x) + c₁.
  3. Use the gradient condition immediately to solve for c₁ — do not defer it.
  4. Integrate the now-complete f'(x) a second time to reach f(x) + c₂.
  5. Use the point condition now, substituting into f(x), to solve for c₂.
  6. Check both conditions against the finished f(x) in one line each before moving on. → Full worked treatment: integrate-twice-two-constants.ts.

Flagged gaps and what remains unverified