Unit 1 · Pure Mathematics 1 (Paper 1)
The Complete Cambridge International AS & A Level Mathematics (9709) Guide · Cambridge International AS & A Level Mathematics (9709)Unit 1 · Pure Mathematics 1 (Paper 1)
- The Complete Cambridge International AS & A Level Mathematics (9709) Guide
- Cambridge International AS & A Level Mathematics (9709)
- 5 sections
Cambridge International does not publish a per-unit weighting for this exam. Exam structure (official): AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%); A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. A restriction, interval, or constant can decide whether an otherwise correct method earns full credit.
- How Cambridge International AS & A Level Mathematics (9709) assesses this Cambridge International publishes no per-unit percentage for Cambridge International A Level Mathematics; Pure Mathematics 1 accounts for 60% of the AS Level total and 30% of the A Level total. · AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%) | A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%) · show the model with factorized and completed-square algebraic forms, coordinate diagrams and trigonometric graphs, derivative sign information and definite-integral notation
- Key skills Quadratics: Completing the Square, the Discriminant, Equations and Inequalities, Functions: Domain and Range, Composite and Inverse Functions, Graph Transformations, Coordinate Geometry: Straight Lines, Circles and Intersections, Circular Measure: Radians, Arc Length and Sector Area, Trigonometry: Graphs, Exact Values, Identities and Equations, Series: Binomial Expansion, Arithmetic and Geometric Progressions, Differentiation: Chain Rule, Tangents and Normals, Rates of Change, Stationary Points, Integration: Reverse Differentiation, Definite Integrals, Areas and Volumes of Revolution
- How to study for Unit 1 This page turns factorized and completed-square algebraic forms, coordinate diagrams and trigonometric graphs, derivative sign information and definite-integral notation into one route: Write the domain, interval, units, and exact form that the question requires..
- The organizing decision Decide which algebraic, geometric, trigonometric, sequence, derivative, or integral structure controls the requested result before substituting values.
What Cambridge International AS & A Level Mathematics (9709) Unit 1 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Quadratics: Completing the Square, the Discriminant, Equations and Inequalities
discriminant; restricted domain; inverse function02Functions: Domain and Range, Composite and Inverse Functions, Graph Transformations
inverse function; tangent gradient; radian measure03Coordinate Geometry: Straight Lines, Circles and Intersections
radian measure; geometric progression; stationary point04Circular Measure: Radians, Arc Length and Sector Area
stationary point; definite integral; discriminant05Trigonometry: Graphs, Exact Values, Identities and Equations
discriminant; restricted domain; inverse function06Series: Binomial Expansion, Arithmetic and Geometric Progressions
inverse function; tangent gradient; radian measure07Differentiation: Chain Rule, Tangents and Normals, Rates of Change, Stationary Points
radian measure; geometric progression; stationary point08Integration: Reverse Differentiation, Definite Integrals, Areas and Volumes of Revolution
stationary point; definite integral; discriminantCambridge International does not publish a per-unit weighting for this exam.
Official unit name and weighting: Cambridge International syllabus.
Algebra, geometry, and first calculus
Connect the published share to the unit model
A restriction, interval, or constant can decide whether an otherwise correct method earns full credit.
For algebra, geometry, and first calculus, the following pages use newly written examples and data. The algebra, geometry, and first calculus scoring comparisons are learning aids based on the published syllabus and the general requirement to show necessary working, not official mark schemes.
Before work on algebra, geometry, and first calculus, identify the domain, units, accuracy instruction, and event boundary. Preserve exact values in algebra, geometry, and first calculus until a final approximation is requested.
The decision that organizes this unit
Define the system and choose the route before calculating
Decide which algebraic, geometric, trigonometric, sequence, derivative, or integral structure controls the requested result before substituting values.
Write the domain, interval, units, and exact form that the question requires.
Mechanism route and repair branches
Relationships to preserve
- A restricted domain can turn a many-to-one rule into an invertible function.
- A radius is perpendicular to the tangent at the point of contact.
- Differentiation locates and classifies local behavior, while definite integration measures signed accumulation or area when the sign is known.
Representations to read
- factorized and completed-square algebraic forms
- coordinate diagrams and trigonometric graphs
- derivative sign information and definite-integral notation
Branches to reject
- keeping both inverse branches after a domain restriction selects one
- mixing degrees and radians or using the wrong geometric distance
- giving stationary inputs without classification or using an integral without checking the curve's sign
| Key concept | Why it's hard | What scores |
|---|---|---|
| Quadratic and function structure | Multiple branches can satisfy the algebra | States domains, roots, ranges, and checks |
| Geometry and trigonometry | Radians and perpendicular conditions are easy to mix | Names the relation before substitution |
| Calculus foundations | A coordinate alone does not classify a point | Shows derivative or integral evidence |
How Cambridge International AS & A Level Mathematics (9709) assesses Pure Mathematics 1 (Paper 1)
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Quadratics: Completing the Square, the Discriminant, Equations and Inequalities | discriminant; restricted domain; inverse function | keeping both inverse branches after a domain restriction selects one |
| Functions: Domain and Range, Composite and Inverse Functions, Graph Transformations | inverse function; tangent gradient; radian measure | mixing degrees and radians or using the wrong geometric distance |
| Coordinate Geometry: Straight Lines, Circles and Intersections | radian measure; geometric progression; stationary point | giving stationary inputs without classification or using an integral without checking the curve's sign |
| Circular Measure: Radians, Arc Length and Sector Area | stationary point; definite integral; discriminant | keeping both inverse branches after a domain restriction selects one |
| Trigonometry: Graphs, Exact Values, Identities and Equations | discriminant; restricted domain; inverse function | mixing degrees and radians or using the wrong geometric distance |
| Series: Binomial Expansion, Arithmetic and Geometric Progressions | inverse function; tangent gradient; radian measure | giving stationary inputs without classification or using an integral without checking the curve's sign |
| Differentiation: Chain Rule, Tangents and Normals, Rates of Change, Stationary Points | radian measure; geometric progression; stationary point | keeping both inverse branches after a domain restriction selects one |
| Integration: Reverse Differentiation, Definite Integrals, Areas and Volumes of Revolution | stationary point; definite integral; discriminant | mixing degrees and radians or using the wrong geometric distance |
Quadratics and inverse functions
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Quadratics and inverse functions
The curve has equation y = x² − 5x + 6.
Find the zeros of the curve. The domain is then restricted to x ≥ 2.5. Find the inverse branch and hence find x when y = 6.
[5]Show all necessary working.
A. x = 2.5 + √(y + 0.25), so x = 5 when y = 6.
✅ Correct! 😊 Correct. Completing the square and using x ≥ 2.5 selects the positive square-root branch, which gives x = 5.
B. x = 2.5 ± √(y + 0.25), so x = 0 or x = 5 when y = 6.
❌ Not quite 😭 Both values solve the unrestricted quadratic, but x = 0 lies outside the stated domain. Keeping both signs fails to define the required inverse function.
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C. x = 2.5 − √(y + 0.25), so x = 0 when y = 6.
❌ Not quite 😭 This selects the left-hand branch of the parabola. It contradicts the restriction x ≥ 2.5.
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D. x = √(y − 6), so x = 0 when y = 6.
❌ Not quite 😭 This rearrangement does not follow from completing the square and loses the linear term. Substitution into the original equation exposes the error.
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Full explanation
- Step 1Factor x² − 5x + 6 = (x − 2)(x − 3), so the zeros are 2 and 3.
- Step 2Complete the square: y = (x − 2.5)² − 0.25, hence (x − 2.5)² = y + 0.25.
- Step 3Since x ≥ 2.5, take the positive branch: x = 2.5 + √(y + 0.25).
- Step 4At y = 6, x = 2.5 + √6.25 = 5, which satisfies the original equation and the restriction.
Practice 2: Stationary points
The curve C has equation y = x³ − 3x² + 2.
Find the coordinates of the stationary points of C and determine the nature of each stationary point.
[6]Show all necessary working.
A. Solve 3x² − 6x = 0, find the corresponding y-values, then use 6x − 6 to classify each point.
✅ Correct! 😊 Correct. The first derivative locates stationary inputs, while the second derivative distinguishes the maximum from the minimum.
B. Solve x³ − 3x² + 2 = 0 and classify each root using its sign.
❌ Not quite 😭 This finds x-intercepts rather than stationary points. Stationarity is determined by a zero gradient, not a zero function value.
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C. Solve 6x − 6 = 0 and state that x = 1 is the only stationary point.
❌ Not quite 😭 Setting the second derivative to zero finds a possible inflection input, not the stationary inputs required here.
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D. Solve 3x² − 6x = 0 and report x = 0 and x = 2 without finding y or classifying them.
❌ Not quite 😭 The stationary inputs are correct, but the question requires coordinates and the nature of each point, so the response is incomplete.
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Full explanation
- Step 1Differentiate: dy/dx = 3x² − 6x = 3x(x − 2).
- Step 2Set dy/dx = 0 to obtain x = 0 and x = 2; substitution gives (0, 2) and (2, −2).
- Step 3The second derivative is 6x − 6.
- Step 4It is negative at x = 0 and positive at x = 2, so (0, 2) is a maximum and (2, −2) is a minimum.
Key terms for Unit 1: Pure Mathematics 1 (Paper 1)
Models, uses, and boundaries
- The sequence links structural recognition to a chosen method and a final mathematical check
- Factor the quadratic and use the positive square-root branch selected by the stated domain. The sequence links structural recognition to a chosen method and a final mathematical check.
- A factorised quadratic and the positive square-root branch give the roots and restricted inverse
- Use the perpendicular tangent gradient, then calculate arc length and sector area with the angle in radians. A factorised quadratic and the positive square-root branch give the roots and restricted inverse.
- The tangent gradient, arc-length rule, and sector-area rule give the displayed geometric results
- Find every trigonometric solution in the interval and check that the geometric series has a ratio whose magnitude is below one. The tangent gradient, arc-length rule, and sector-area rule give the displayed geometric results.
- The two interval solutions sit beside the convergent geometric-series total
- Factor the derivative to locate stationary inputs, use the second derivative to classify them, and evaluate the definite integral for the stated area. The two interval solutions sit beside the convergent geometric-series total.
- The factorised derivative identifies stationary inputs, while the definite integral gives the stated area
- Move from the mathematical structure to a suitable method, then check the result against the original conditions. The factorised derivative identifies stationary inputs, while the definite integral gives the stated area.
Cambridge International AS & A Level Mathematics (9709) Unit 1 FAQ
How much of Cambridge International AS & A Level Mathematics (9709) does Unit 1 carry?
Cambridge International does not publish a per-unit weighting for this exam.
What is the first move on a Pure Mathematics 1 (Paper 1) problem?
Write the domain, interval, units, and exact form that the question requires.
Which relationships should I preserve?
A restricted domain can turn a many-to-one rule into an invertible function. A radius is perpendicular to the tangent at the point of contact. Differentiation locates and classifies local behavior, while definite integration measures signed accumulation or area when the sign is known.
Which representations should I practice?
Practice moving among factorized and completed-square algebraic forms, coordinate diagrams and trigonometric graphs, derivative sign information and definite-integral notation.
What error should I check before submitting an answer?
Check for keeping both inverse branches after a domain restriction selects one; mixing degrees and radians or using the wrong geometric distance; giving stationary inputs without classification or using an integral without checking the curve's sign.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Quadratics and inverse functions
- Circles and radians
- Trigonometry and progressions
- Differentiation and integration
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related Cambridge International AS & A Level Mathematics (9709) unit guides
Cambridge International AS & A Level Mathematics (9709) Exam Guide & Review
The whole exam and its official unit sequence.02Pure Mathematics 3 (Paper 3, A Level)
Cambridge International does not publish a per-unit weighting for this exam.03Mechanics (Paper 4)
Cambridge International does not publish a per-unit weighting for this exam.04Probability & Statistics 1 (Paper 5)
Cambridge International does not publish a per-unit weighting for this exam.05Probability & Statistics 2 (Paper 6, A Level)
Cambridge International does not publish a per-unit weighting for this exam.How to study Cambridge International AS & A Level Mathematics (9709) Unit 1
Start with the organizing decision
Before solving, restate the decision in operational terms: Decide which algebraic, geometric, trigonometric, sequence, derivative, or integral structure controls the requested result before substituting values.. Your first written move should be to Write the domain, interval, units, and exact form that the question requires..
Practice the same idea in several representations
Rotate through factorized and completed-square algebraic forms, coordinate diagrams and trigonometric graphs, derivative sign information and definite-integral notation. Use each representation to practice Quadratics: Completing the Square, the Discriminant, Equations and Inequalities, Functions: Domain and Range, Composite and Inverse Functions, Graph Transformations, Coordinate Geometry: Straight Lines, Circles and Intersections, Circular Measure: Radians, Arc Length and Sector Area, Trigonometry: Graphs, Exact Values, Identities and Equations, Series: Binomial Expansion, Arithmetic and Geometric Progressions, Differentiation: Chain Rule, Tangents and Normals, Rates of Change, Stationary Points, Integration: Reverse Differentiation, Definite Integrals, Areas and Volumes of Revolution, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for keeping both inverse branches after a domain restriction selects one; mixing degrees and radians or using the wrong geometric distance; giving stationary inputs without classification or using an integral without checking the curve's sign. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official Cambridge International syllabus for the 2026-2027 administration.
Where to go next
Return to the Cambridge International AS & A Level Mathematics (9709) to see every unit and plan your revision.