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Further Mathematics: Cambridge 2027 syllabus mapping

Cambridge International AS & A Level 9231 · Examination year 2027

AI-assisted conceptual placements, not official Cambridge endorsement or independent teacher approval. A linked practice question does not establish complete teaching, practical competence or exam readiness.

Current question placements
80 / 80
Direct conceptual matches
63
Partial matches
7
Enrichment
10
Placements needing review
0
Inventory entries without linked practice
6

Inventory scope: individual learning outcomes. Source inventory: recorded in full at this scope. Independent teacher approval: pending.

Official references

Practice by syllabus reference

both · Paper 1: Further Pure Mathematics 1; compulsory AS/A (28 entries)
Further Mathematics 9231: both · Paper 1: Further Pure Mathematics 1; compulsory AS/A
ReferenceConceptLinked practice
1.1 Roots of polynomial equations; candidate bullet 1 (bullet ordinal, not an official subcode)Use degree-two/three/four root-coefficient relations
1.1 Roots of polynomial equations; candidate bullet 2 (bullet ordinal, not an official subcode)Form equations for transformed roots
1.2 Rational functions and graphs; candidate bullet 1 (bullet ordinal, not an official subcode)Sketch quadratic-over-quadratic rational graphs and ranges
1.2 Rational functions and graphs; candidate bullet 2 (bullet ordinal, not an official subcode)Relate y=f(x) to squared, reciprocal and absolute-value graphs
1.3 Summation of series; candidate bullet 1 (bullet ordinal, not an official subcode)Use standard sums of first powers, squares and cubes
1.3 Summation of series; candidate bullet 2 (bullet ordinal, not an official subcode)Sum finite series by differences
1.3 Summation of series; candidate bullet 3 (bullet ordinal, not an official subcode)Use partial sums to establish convergence and infinite sums
1.4 Matrices; candidate bullet 1 (bullet ordinal, not an official subcode)Add/subtract/multiply matrices with at most three rows/columnsNo linked practice
1.4 Matrices; candidate bullet 2 (bullet ordinal, not an official subcode)Find determinants and inverses of 2x2/3x3 matrices
1.4 Matrices; candidate bullet 3 (bullet ordinal, not an official subcode)Invert matrix products in reversed order
1.4 Matrices; candidate bullet 4, inverse subitem (bullet ordinal, not an official subcode)Relate a transformation to its inverse
1.4 Matrices; candidate bullet 4, composition subitem (bullet ordinal, not an official subcode)Compose plane transformations in matrix order
1.4 Matrices; candidate bullet 4, area subitem (bullet ordinal, not an official subcode)Use determinant as signed area scale
1.4 Matrices; candidate bullet 4, construction subitem (bullet ordinal, not an official subcode)Construct a matrix for a transformation sequence
1.4 Matrices; candidate bullet 5 (bullet ordinal, not an official subcode)Find invariant points and lines
1.5 Polar coordinates; candidate bullet 1 (bullet ordinal, not an official subcode)Convert polar and Cartesian curve equations with r>=0
1.5 Polar coordinates; candidate bullet 2 (bullet ordinal, not an official subcode)Sketch polar curves with significant features
1.5 Polar coordinates; candidate bullet 3 (bullet ordinal, not an official subcode)Integrate r^2/2 for polar sector areas
1.6 Vectors; candidate bullet 1 (bullet ordinal, not an official subcode)Convert Cartesian/normal/parametric plane equations
1.6 Vectors; candidate bullet 2 (bullet ordinal, not an official subcode)Compute and interpret vector cross products
1.6 Vectors; candidate bullet 3, intersection subitem (bullet ordinal, not an official subcode)Classify and intersect a line and plane
1.6 Vectors; candidate bullet 3, foot subitem (bullet ordinal, not an official subcode)Project a point perpendicularly to a plane
1.6 Vectors; candidate bullet 3, angle subitem (bullet ordinal, not an official subcode)Find line-plane and plane-plane angles
1.6 Vectors; candidate bullet 3, planes subitem (bullet ordinal, not an official subcode)Find the line of intersection of planes
1.6 Vectors; candidate bullet 3, distance subitem (bullet ordinal, not an official subcode)Find shortest distances between skew lines
1.6 Vectors; candidate bullet 3, perpendicular subitem (bullet ordinal, not an official subcode)Find a common perpendicular to skew linesNo linked practice
1.7 Proof by induction; candidate bullet 1 (bullet ordinal, not an official subcode)Prove a stated result by induction
1.7 Proof by induction; candidate bullet 2 (bullet ordinal, not an official subcode)Conjecture from limited trials and prove inductively
A · Paper 2: Further Pure Mathematics 2; A only (30 entries)
Further Mathematics 9231: A · Paper 2: Further Pure Mathematics 2; A only
ReferenceConceptLinked practice
2.1 Hyperbolic functions; candidate bullet 1 (bullet ordinal, not an official subcode)Use six hyperbolic exponential definitions
2.1 Hyperbolic functions; candidate bullet 2 (bullet ordinal, not an official subcode)Sketch hyperbolic graphs
2.1 Hyperbolic functions; candidate bullet 3 (bullet ordinal, not an official subcode)Prove and apply hyperbolic identities
2.1 Hyperbolic functions; candidate bullet 4 (bullet ordinal, not an official subcode)Derive and apply logarithmic inverse-hyperbolic forms
2.2 Matrices; candidate bullet 1 (bullet ordinal, not an official subcode)Translate three linear equations into matrix form
2.2 Matrices; candidate bullet 2 (bullet ordinal, not an official subcode)Solve/classify consistent or inconsistent three-variable systems geometrically
2.2 Matrices; candidate bullet 3 (bullet ordinal, not an official subcode)Use characteristic/eigenvalue/eigenvector definitions
2.2 Matrices; candidate bullet 4 (bullet ordinal, not an official subcode)Find distinct real eigenvalues/eigenvectors in 2x2/3x3 cases
2.2 Matrices; candidate bullet 5 (bullet ordinal, not an official subcode)Diagonalise with an eigenbasis and use matrix powers
2.2 Matrices; candidate bullet 6 (bullet ordinal, not an official subcode)Apply a matrix's characteristic equation
2.3 Differentiation; candidate bullet 1 (bullet ordinal, not an official subcode)Differentiate hyperbolic and specified inverse circular/hyperbolic functions
2.3 Differentiation; candidate bullet 2 (bullet ordinal, not an official subcode)Find second derivatives of implicit/parametric curves
2.3 Differentiation; candidate bullet 3 (bullet ordinal, not an official subcode)Derive/use initial Maclaurin terms
2.4 Integration; candidate bullet 1 (bullet ordinal, not an official subcode)Integrate hyperbolic/associated inverse-function forms by substitution
2.4 Integration; candidate bullet 2 (bullet ordinal, not an official subcode)Derive and apply definite-integral reduction formulae
2.4 Integration; candidate bullet 3 (bullet ordinal, not an official subcode)Use rectangle sums for area bounds, inequalities and limits
2.4 Integration; candidate bullet 4, length subitem (bullet ordinal, not an official subcode)Find Cartesian, parametric or polar arc lengths
2.4 Integration; candidate bullet 4, surface subitem (bullet ordinal, not an official subcode)Find Cartesian/parametric surfaces of revolution
2.5 Complex numbers; candidate bullet 1 (bullet ordinal, not an official subcode)Interpret integer de Moivre powers geometricallyNo linked practice
2.5 Complex numbers; candidate bullet 2 (bullet ordinal, not an official subcode)Prove positive-integer de Moivre by induction
2.5 Complex numbers; candidate bullet 3, ratios subitem (bullet ordinal, not an official subcode)Express multiple-angle ratios through fundamental powers
2.5 Complex numbers; candidate bullet 3, powers subitem (bullet ordinal, not an official subcode)Express trigonometric powers through multiple angles
2.5 Complex numbers; candidate bullet 3, series subitem (bullet ordinal, not an official subcode)Sum trigonometric series with complex numbers
2.5 Complex numbers; candidate bullet 3, unity subitem (bullet ordinal, not an official subcode)Find and use roots of unity
2.6 Differential equations; candidate bullet 1 (bullet ordinal, not an official subcode)Solve first-order linear equations with integrating factors
2.6 Differential equations; candidate bullet 2 (bullet ordinal, not an official subcode)Distinguish complementary function and particular integral
2.6 Differential equations; candidate bullet 3 (bullet ordinal, not an official subcode)Find constant-coefficient complementary functions for all root types
2.6 Differential equations; candidate bullet 4 (bullet ordinal, not an official subcode)Find suitable particular integrals, including given resonant trials
2.6 Differential equations; candidate bullet 5 (bullet ordinal, not an official subcode)Reduce equations using a given substitution
2.6 Differential equations; candidate bullet 6 (bullet ordinal, not an official subcode)Use initial conditions and interpret a differential-equation model
both · Paper 3: Further Mechanics; AS 1+3; compulsory A (19 entries)
Further Mathematics 9231: both · Paper 3: Further Mechanics; AS 1+3; compulsory A
ReferenceConceptLinked practice
3.1 Motion of a projectile; candidate bullet 1 (bullet ordinal, not an official subcode)Model projectiles with constant gravity and limitations
3.1 Motion of a projectile; candidate bullet 2 (bullet ordinal, not an official subcode)Resolve projectile velocity, range and height
3.1 Motion of a projectile; candidate bullet 3 (bullet ordinal, not an official subcode)Derive Cartesian trajectories and solve inverse launch problems
3.2 Equilibrium of a rigid body; candidate bullet 1 (bullet ordinal, not an official subcode)Calculate coplanar moments
3.2 Equilibrium of a rigid body; candidate bullet 2 (bullet ordinal, not an official subcode)Locate symmetric uniform centres of mass
3.2 Equilibrium of a rigid body; candidate bullet 3 (bullet ordinal, not an official subcode)Use supplied centres of mass of simple shapesNo linked practice
3.2 Equilibrium of a rigid body; candidate bullet 4 (bullet ordinal, not an official subcode)Find composite centres of mass
3.2 Equilibrium of a rigid body; candidate bullet 5 (bullet ordinal, not an official subcode)Balance coplanar forces and moments
3.2 Equilibrium of a rigid body; candidate bullet 6 (bullet ordinal, not an official subcode)Solve rigid-body equilibrium including toppling/sliding
3.3 Circular motion; candidate bullet 1 (bullet ordinal, not an official subcode)Relate angular and linear speeds
3.3 Circular motion; candidate bullet 2 (bullet ordinal, not an official subcode)Use radial acceleration r*omega^2 or v^2/r
3.3 Circular motion; candidate bullet 3 (bullet ordinal, not an official subcode)Solve uniform horizontal-circle force problems
3.3 Circular motion; candidate bullet 4 (bullet ordinal, not an official subcode)Use energy/contact conditions in vertical circles
3.4 Hooke's law; candidate bullet 1 (bullet ordinal, not an official subcode)Model elastic force and modulus of elasticity
3.4 Hooke's law; candidate bullet 2 (bullet ordinal, not an official subcode)Calculate elastic stored energy
3.4 Hooke's law; candidate bullet 3 (bullet ordinal, not an official subcode)Combine elastic forces with work/energy
3.5 Linear motion under a variable force; candidate bullet 1 (bullet ordinal, not an official subcode)Form and solve separable variable-force motion equations
3.6 Momentum; candidate bullet 1 (bullet ordinal, not an official subcode)Use restitution and elastic/inelastic impact meanings
3.6 Momentum; candidate bullet 2 (bullet ordinal, not an official subcode)Solve direct/oblique smooth-sphere or wall impacts
both · Paper 4: Further Probability & Statistics; AS 1+4; compulsory A (21 entries)
Further Mathematics 9231: both · Paper 4: Further Probability & Statistics; AS 1+4; compulsory A
ReferenceConceptLinked practice
4.1 Continuous random variables; candidate bullet 1 (bullet ordinal, not an official subcode)Use piecewise probability densities
4.1 Continuous random variables; candidate bullet 2 (bullet ordinal, not an official subcode)Compute expectation of functions by integration
4.1 Continuous random variables; candidate bullet 3 (bullet ordinal, not an official subcode)Relate density/CDF and evaluate percentiles
4.1 Continuous random variables; candidate bullet 4 (bullet ordinal, not an official subcode)Transform distributions through CDFs
4.2 Inference using normal and t-distributions; candidate bullet 1 (bullet ordinal, not an official subcode)Test one normal mean with small sample and unknown variance
4.2 Inference using normal and t-distributions; candidate bullet 2 (bullet ordinal, not an official subcode)Pool two sample variances
4.2 Inference using normal and t-distributions; candidate bullet 3, two subitem (bullet ordinal, not an official subcode)Test a difference of means with two-sample t
4.2 Inference using normal and t-distributions; candidate bullet 3, paired subitem (bullet ordinal, not an official subcode)Test a mean paired difference with t
4.2 Inference using normal and t-distributions; candidate bullet 3, normal subitem (bullet ordinal, not an official subcode)Test differences using a suitable normal distributionNo linked practice
4.2 Inference using normal and t-distributions; candidate bullet 4 (bullet ordinal, not an official subcode)Construct small-sample t intervals for a mean
4.2 Inference using normal and t-distributions; candidate bullet 5 (bullet ordinal, not an official subcode)Construct intervals for mean differences
4.3 χ²-tests; candidate bullet 1 (bullet ordinal, not an official subcode)Fit theoretical expected frequencies
4.3 χ²-tests; candidate bullet 2 (bullet ordinal, not an official subcode)Perform goodness-of-fit tests with suitable degrees of freedom
4.3 χ²-tests; candidate bullet 3 (bullet ordinal, not an official subcode)Test contingency-table independence
4.4 Non-parametric tests; candidate bullet 1 (bullet ordinal, not an official subcode)Choose when a nonparametric test is appropriate
4.4 Non-parametric tests; candidate bullet 2 (bullet ordinal, not an official subcode)Understand sign, signed-rank and rank-sum bases
4.4 Non-parametric tests; candidate bullet 3 (bullet ordinal, not an official subcode)Test one population median with sign/signed-rank methodsNo linked practice
4.4 Non-parametric tests; candidate bullet 4 (bullet ordinal, not an official subcode)Compare populations with paired sign/rank or rank-sum methods
4.5 Probability generating functions; candidate bullet 1 (bullet ordinal, not an official subcode)Construct and use probability generating functions
4.5 Probability generating functions; candidate bullet 2 (bullet ordinal, not an official subcode)Derive mean/variance from PGF derivatives
4.5 Probability generating functions; candidate bullet 3 (bullet ordinal, not an official subcode)Multiply PGFs for independent sums

Remaining academic requirements

  • Conceptual alignment only, not Cambridge certification, teacher approval, exam-entry advice or complete teaching coverage. Every placement has teacherApproved:false. The 120 pre-existing questions retain their IDs and current English editorial revisions.
  • Inventory enumerates every candidate-content bullet in all numbered subject-content sections and splits independently assessable nested lists. Internal b-number identifiers are editorial locators, never official subcodes. Worked examples and notes constrain the outcomes rather than create separate official codes. pdfPage is one-based actual PDF page.
  • Read subject-content PDF pages 16-28 and prior-knowledge page 15, routes on 10-13, assessment on 29-30, change log on 49, and the one-page update. The version-3 front note says July 2025 while the change log/update say August 2025; this date discrepancy is retained, not silently resolved. Version-3 corrects a page-19 typo; earlier page-16/23/25 corrections are in this PDF.
  • Route constraints: AS requires Further Pure 1 plus either Further Mechanics or Further Probability & Statistics. All four components are compulsory for A level; Further Pure 2 is A-only. Assumed 9709 Pure 1/3 and the applicable Mechanics/Statistics 1/2 prerequisites are not automatically new 9231 outcomes.
  • Distinct real eigenvalues are the stated eigenvalue-finding scope; the defective repeated-eigenvalue example remains enrichment. Polar conventions require r>=0. Resonant particular-integral trials beyond listed simple forms are normally supplied; a current unsupplied-trial question remains partial.
  • Further vectors/planes/common perpendiculars, polar areas and reduction formulae still need broader multi-stage practice. Further Statistics additions do not yet supply a full two-independent-sample Wilcoxon rank-sum calculation, fitted-parameter/grouped chi-squared case or known-variance normal difference test. A signed-rank example is not complete non-parametric-test coverage.
  • No current mapped or partial question directly evidences these 6 inventory items: caie27-9231-1.4-b1 (Add/subtract/multiply matrices with at most three rows/columns); caie27-9231-1.6-b3-perpendicular (Find a common perpendicular to skew lines); caie27-9231-2.5-b1 (Interpret integer de Moivre powers geometrically); caie27-9231-3.2-b3 (Use supplied centres of mass of simple shapes); caie27-9231-4.2-b3-normal (Test differences using a suitable normal distribution); caie27-9231-4.4-b3 (Test one population median with sign/signed-rank methods).
  • A represented outcome means at least one limited task, not mastery or complete coverage of all its forms, examples and modelling contexts. Partial placements must not be counted as full-outcome coverage.
  • Teacher review is still required for indicative mark allocations, diagrams/sketch quality, alternative methods, data/model limitations and complete timed structured papers. Short self-marking answers do not replace teaching, externally assessed work or MF19/non-graphical-calculator exam preparation. No new question is teacher approved.

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