Unit 2 · Pure Mathematics 3 (Paper 3, A Level)
The Complete Cambridge International AS & A Level Mathematics (9709) Guide · Cambridge International AS & A Level Mathematics (9709)Unit 2 · Pure Mathematics 3 (Paper 3, A Level)
- 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. Answers become defensible when the chosen identity, iteration, branch, or initial condition is visible.
- 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 partial fractions, logarithmic equations, and trigonometric identities, iteration values and vector component or scalar-product form, separated differential equations and Cartesian or polar complex form
- Key skills Algebra: Modulus, Polynomial Division, Factor and Remainder Theorems, Partial Fractions, Binomial Series, Logarithmic and Exponential Functions: Laws of Logarithms, e and ln, Linear Form, Trigonometry: sec, cosec and cot, Compound-Angle and Double-Angle Formulae, the Form R sin(θ ± α), Differentiation: Exponential, Logarithmic and Trigonometric Functions, Products and Quotients, Parametric and Implicit, Integration: Standard Forms, Partial Fractions, Integration by Parts and by Substitution, Numerical Solution of Equations: Locating Roots and Iterative Formulae, Vectors: Position Vectors, Equations of Lines, Scalar Product and Intersections, Differential Equations: Forming and Solving by Separation of Variables, Complex Numbers: Cartesian and Polar Forms, Roots of Equations, Argand Diagrams and Loci
- How to study for Unit 2 This page turns partial fractions, logarithmic equations, and trigonometric identities, iteration values and vector component or scalar-product form, separated differential equations and Cartesian or polar complex form into one route: Record the domain, interval, initial value, or quadrant before carrying out the transformation..
- The organizing decision Choose the transformation, numerical method, vector relation, initial condition, or complex-number form that preserves every stated restriction.
What Cambridge International AS & A Level Mathematics (9709) Unit 2 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Algebra: Modulus, Polynomial Division, Factor and Remainder Theorems, Partial Fractions, Binomial Series
partial fractions; logarithmic form; double-angle identity02Logarithmic and Exponential Functions: Laws of Logarithms, e and ln, Linear Form
double-angle identity; Newton iteration; scalar product03Trigonometry: sec, cosec and cot, Compound-Angle and Double-Angle Formulae, the Form R sin(θ ± α)
scalar product; differential equation; initial condition04Differentiation: Exponential, Logarithmic and Trigonometric Functions, Products and Quotients, Parametric and Implicit
initial condition; complex argument; polar form05Integration: Standard Forms, Partial Fractions, Integration by Parts and by Substitution
polar form; partial fractions; logarithmic form06Numerical Solution of Equations: Locating Roots and Iterative Formulae
logarithmic form; double-angle identity; Newton iteration07Vectors: Position Vectors, Equations of Lines, Scalar Product and Intersections
Newton iteration; scalar product; differential equation08Differential Equations: Forming and Solving by Separation of Variables
differential equation; initial condition; complex argument09Complex Numbers: Cartesian and Polar Forms, Roots of Equations, Argand Diagrams and Loci
complex argument; polar form; partial fractionsCambridge International does not publish a per-unit weighting for this exam.
Official unit name and weighting: Cambridge International syllabus.
Advanced pure methods
Connect the published share to the unit model
Answers become defensible when the chosen identity, iteration, branch, or initial condition is visible.
For advanced pure methods, the following pages use newly written examples and data. The advanced pure methods 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 advanced pure methods, identify the domain, units, accuracy instruction, and event boundary. Preserve exact values in advanced pure methods until a final approximation is requested.
The decision that organizes this unit
Define the system and choose the route before calculating
Choose the transformation, numerical method, vector relation, initial condition, or complex-number form that preserves every stated restriction.
Record the domain, interval, initial value, or quadrant before carrying out the transformation.
Mechanism route and repair branches
Relationships to preserve
- Equivalent algebraic and trigonometric forms can reveal a solvable equation or integrable expression.
- A numerical iteration needs a located root and visible successive updates.
- Initial conditions fix constants in differential equations, while the quadrant fixes the complex argument.
Representations to read
- partial fractions, logarithmic equations, and trigonometric identities
- iteration values and vector component or scalar-product form
- separated differential equations and Cartesian or polar complex form
Branches to reject
- dropping a domain restriction during an algebraic transformation
- reporting a numerical root without bracketing or showing the update
- using an initial value or reference angle without carrying it into the final model
| Key concept | Why it's hard | What scores |
|---|---|---|
| Exact transformation | Equivalent forms have different useful domains | Preserves restrictions through every rewrite |
| Numerical method | A decimal needs a location and convergence argument | Displays the update and successive values |
| Model solution | Constants and arguments depend on conditions | Applies the initial value or quadrant |
How Cambridge International AS & A Level Mathematics (9709) assesses Pure Mathematics 3 (Paper 3, A Level)
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 |
|---|---|---|
| Algebra: Modulus, Polynomial Division, Factor and Remainder Theorems, Partial Fractions, Binomial Series | partial fractions; logarithmic form; double-angle identity | dropping a domain restriction during an algebraic transformation |
| Logarithmic and Exponential Functions: Laws of Logarithms, e and ln, Linear Form | double-angle identity; Newton iteration; scalar product | reporting a numerical root without bracketing or showing the update |
| Trigonometry: sec, cosec and cot, Compound-Angle and Double-Angle Formulae, the Form R sin(θ ± α) | scalar product; differential equation; initial condition | using an initial value or reference angle without carrying it into the final model |
| Differentiation: Exponential, Logarithmic and Trigonometric Functions, Products and Quotients, Parametric and Implicit | initial condition; complex argument; polar form | dropping a domain restriction during an algebraic transformation |
| Integration: Standard Forms, Partial Fractions, Integration by Parts and by Substitution | polar form; partial fractions; logarithmic form | reporting a numerical root without bracketing or showing the update |
| Numerical Solution of Equations: Locating Roots and Iterative Formulae | logarithmic form; double-angle identity; Newton iteration | using an initial value or reference angle without carrying it into the final model |
| Vectors: Position Vectors, Equations of Lines, Scalar Product and Intersections | Newton iteration; scalar product; differential equation | dropping a domain restriction during an algebraic transformation |
| Differential Equations: Forming and Solving by Separation of Variables | differential equation; initial condition; complex argument | reporting a numerical root without bracketing or showing the update |
| Complex Numbers: Cartesian and Polar Forms, Roots of Equations, Argand Diagrams and Loci | complex argument; polar form; partial fractions | using an initial value or reference angle without carrying it into the final model |
Algebraic forms and exponential models
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Algebraic forms and exponential models
Answer both parts.
(a) Express (5x + 1)/((x − 1)(x + 2)) in partial fractions. (b) Solve 3e⁰·⁴ᵗ = 20, giving t to 3 significant figures.
[5]Show all necessary working.
A. Use A/(x − 1) + B/(x + 2), giving A = 2 and B = 3; then t = ln(20/3)/0.4.
✅ Correct! 😊 Correct. Equating numerators gives the stated partial fractions, and logarithms isolate the exponent after dividing by 3.
B. Use A/(x − 1) + B/(x + 2), giving A = 3 and B = 2; then t = ln(20) − ln(3)/0.4.
❌ Not quite 😭 The partial-fraction constants are interchanged, and the logarithmic numerator is ambiguous because the entire difference must be divided by 0.4.
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C. Use 5/(x − 1) + 1/(x + 2); then t = ln(20/3 − 0.4).
❌ Not quite 😭 The original numerator coefficients do not directly become partial-fraction numerators, and logarithms do not turn a product in the exponent into this subtraction.
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D. Use 2/(x − 1) + 3/(x + 2); then t = e²⁰⁄³/0.4.
❌ Not quite 😭 The decomposition is correct, but exponentiating again does not undo the exponential equation. Taking natural logarithms is required.
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Full explanation
- Step 1Write (5x + 1)/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2).
- Step 2Then 5x + 1 = A(x + 2) + B(x − 1); x = 1 gives A = 2 and x = −2 gives B = 3.
- Step 3From 3e⁰·⁴ᵗ = 20, obtain e⁰·⁴ᵗ = 20/3 and hence 0.4t = ln(20/3).
- Step 4Therefore t = ln(20/3)/0.4 ≈ 4.74.
Practice 2: Differential equation with an initial condition
A function y satisfies a differential equation and an initial condition.
Given that dy/dx = 3x²y and y = 2 when x = 0, solve the differential equation, expressing y in terms of x.
[6]Show all necessary working.
A. Integrate dy/y = 3x² dx to get ln y = x³ + C, then use y(0) = 2 to obtain y = 2eˣ³.
✅ Correct! 😊 Correct. Separation gives a logarithm in y, and the initial value fixes the multiplicative constant as 2.
B. Integrate dy = 3x²y dx to get y = x³y + C, then set C = 2.
❌ Not quite 😭 This integrates while y still depends on x and therefore does not separate the variables. The resulting equation is not a valid antiderivative relation.
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C. Obtain ln y = x³ + 2, so y = eˣ³ + 2.
❌ Not quite 😭 The initial value does not make the integration constant 2 in the logarithmic equation. It gives C = ln 2, producing a multiplicative factor.
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D. Obtain ln y = 3x³ + C, so y = 2e³ˣ³.
❌ Not quite 😭 This integrates 3x² incorrectly. Its antiderivative is x³, not 3x³.
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Full explanation
- Step 1Separate the variables: dy/y = 3x² dx.
- Step 2Integrate to obtain ln y = x³ + C.
- Step 3Using y = 2 at x = 0 gives C = ln 2.
- Step 4Exponentiating gives y = eˣ³⁺ˡⁿ ² = 2eˣ³.
Key terms for Unit 2: Pure Mathematics 3 (Paper 3, A Level)
Models, uses, and boundaries
- The sequence carries a stated domain through transformation and verification
- Decompose the rational expression into partial fractions and solve the exponential model by taking logarithms. The sequence carries a stated domain through transformation and verification.
- The partial-fraction decomposition and exponential model show the two algebraic forms used in the example
- Use a double-angle identity to rewrite the integrand before evaluating the definite integral. The partial-fraction decomposition and exponential model show the two algebraic forms used in the example.
- The double-angle identity converts the integrand before the definite integral is evaluated
- Apply Newton's update after locating the root, and use the scalar product divided by both vector magnitudes to find the angle. The double-angle identity converts the integrand before the definite integral is evaluated.
- One Newton update is shown beside the scalar-product condition for perpendicular vectors
- Separate and integrate the differential equation, apply the initial condition, and use modulus with the correct quadrant for polar form. One Newton update is shown beside the scalar-product condition for perpendicular vectors.
- The separated differential equation and polar form record the two final mathematical models
- Carry the stated domain through a transformation and verify the result in the original relationship. The separated differential equation and polar form record the two final mathematical models.
Cambridge International AS & A Level Mathematics (9709) Unit 2 FAQ
How much of Cambridge International AS & A Level Mathematics (9709) does Unit 2 carry?
Cambridge International does not publish a per-unit weighting for this exam.
What is the first move on a Pure Mathematics 3 (Paper 3, A Level) problem?
Record the domain, interval, initial value, or quadrant before carrying out the transformation.
Which relationships should I preserve?
Equivalent algebraic and trigonometric forms can reveal a solvable equation or integrable expression. A numerical iteration needs a located root and visible successive updates. Initial conditions fix constants in differential equations, while the quadrant fixes the complex argument.
Which representations should I practice?
Practice moving among partial fractions, logarithmic equations, and trigonometric identities, iteration values and vector component or scalar-product form, separated differential equations and Cartesian or polar complex form.
What error should I check before submitting an answer?
Check for dropping a domain restriction during an algebraic transformation; reporting a numerical root without bracketing or showing the update; using an initial value or reference angle without carrying it into the final model.
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.
- Algebraic forms and exponential models
- Advanced trigonometry and calculus
- Root finding and vector geometry
- Differential equations and complex numbers
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.01Pure Mathematics 1 (Paper 1)
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 2
Start with the organizing decision
Before solving, restate the decision in operational terms: Choose the transformation, numerical method, vector relation, initial condition, or complex-number form that preserves every stated restriction.. Your first written move should be to Record the domain, interval, initial value, or quadrant before carrying out the transformation..
Practice the same idea in several representations
Rotate through partial fractions, logarithmic equations, and trigonometric identities, iteration values and vector component or scalar-product form, separated differential equations and Cartesian or polar complex form. Use each representation to practice Algebra: Modulus, Polynomial Division, Factor and Remainder Theorems, Partial Fractions, Binomial Series, Logarithmic and Exponential Functions: Laws of Logarithms, e and ln, Linear Form, Trigonometry: sec, cosec and cot, Compound-Angle and Double-Angle Formulae, the Form R sin(θ ± α), Differentiation: Exponential, Logarithmic and Trigonometric Functions, Products and Quotients, Parametric and Implicit, Integration: Standard Forms, Partial Fractions, Integration by Parts and by Substitution, Numerical Solution of Equations: Locating Roots and Iterative Formulae, Vectors: Position Vectors, Equations of Lines, Scalar Product and Intersections, Differential Equations: Forming and Solving by Separation of Variables, Complex Numbers: Cartesian and Polar Forms, Roots of Equations, Argand Diagrams and Loci, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for dropping a domain restriction during an algebraic transformation; reporting a numerical root without bracketing or showing the update; using an initial value or reference angle without carrying it into the final model. 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.