A-Level Further Maths — Diagnostic Guide

Coverage Map

Pure Mathematics

Diagnostic File	Topics Covered	Source File
diag-complex-numbers.md	Argand diagrams, modulus-argument form, De Moivre's theorem, roots of unity, complex transformations	pure-mathematics/01-complex-numbers.md
diag-matrices.md	Operations, 3x3 inverses, eigenvalues/eigenvectors, geometric transformations, systems	pure-mathematics/02-matrices.md
diag-further-algebra.md	Roots of polynomials, partial fractions, series summation, mathematical induction	pure-mathematics/03-further-algebra.md
diag-further-calculus.md	Improper integrals, volumes of revolution, parametric differentiation, arc length	pure-mathematics/04-further-calculus.md
diag-polar-coordinates.md	Conversion, sketching, area enclosed, tangent at a point, conics in polar form	pure-mathematics/05-polar-coordinates.md
diag-hyperbolic-functions.md	Definitions, identities, calculus, inverse hyperbolic functions, Osborn's rule	pure-mathematics/06-hyperbolic-functions.md
diag-differential-equations.md	First order separable, second order linear, particular integrals, forming DEs	pure-mathematics/07-differential-equations.md
diag-maclaurin-taylor-series.md	Standard expansions, radius of convergence, differentiation/integration of series	pure-mathematics/08-maclaurin-and-taylor-series.md
diag-vectors-3d.md	Scalar product, vector product, equations of lines/planes, volumes	pure-mathematics/09-vectors-in-3d.md

Further Mechanics

Diagnostic File	Topics Covered	Source File
diag-projectile-motion.md	(Covered through integration tests)	further-mechanics/01-projectile-motion.md
diag-circular-motion.md	(Covered through integration tests)	further-mechanics/02-circular-motion.md
diag-collisions.md	(Covered through integration tests)	further-mechanics/03-centres-of-mass-and-elastic-collisions.md

Further Statistics

Diagnostic File	Topics Covered	Source File
diag-poisson-geometric.md	(Covered through integration tests)	further-statistics/01-poisson-and-geometric-distributions.md
diag-exponential-continuous.md	(Covered through integration tests)	further-statistics/02-exponential-and-continuous-random-variables.md
diag-chi-squared.md	(Covered through integration tests)	further-statistics/03-chi-squared-tests.md

Grading Rubric

PASS Criteria

• Correctly solve at least 2 out of 3 Unit Tests with full working
• Correctly solve at least 2 out of 3 Integration Tests
• Correct use of mathematical notation and LaTeX
• Clear logical flow in proofs and derivations

PARTIAL Criteria

• Correctly solve 1--2 Unit Tests and 1 Integration Test
• Shows understanding of concepts but with calculation errors
• Partially correct proofs with gaps in reasoning
• Struggles with multi-step problems

FAIL Indicators

• Cannot compute eigenvalues or matrix inverses
• Confuses trigonometric and hyperbolic identities
• Cannot separate variables in first-order DEs
• Cannot identify convergence of series
• Cannot convert between Cartesian and polar forms

Prerequisite Chains

Further Algebra (polynomials, series)
  ├── Complex Numbers (roots of unity, polynomial factorisation)
  └── Matrices (characteristic polynomials for eigenvalues)

Complex Numbers
  ├── Polar Coordinates (Argand diagrams, transformations)
  └── Hyperbolic Functions (Euler's formula connections)

Further Calculus
  ├── Differential Equations (integration techniques)
  └── Maclaurin Series (series for integration)

Polar Coordinates
  └── Vectors in 3D (position vectors, direction)

Differential Equations
  └── Further Mechanics (modelling physical systems)

Maclaurin Series
  └── Hyperbolic Functions (series definitions)

Recommended order of diagnostic completion:

1. diag-further-algebra -- foundational algebraic techniques
2. diag-complex-numbers -- essential for matrices and DEs
3. diag-matrices -- requires algebra skills
4. diag-further-calculus -- requires integration skills
5. diag-differential-equations -- builds on calculus
6. diag-maclaurin-taylor-series -- requires calculus
7. diag-polar-coordinates -- requires trigonometry
8. diag-hyperbolic-functions -- parallels trigonometry
9. diag-vectors-3d -- requires algebra and geometry

Timing Recommendations

Diagnostic	Recommended Time	Notes
diag-further-algebra	40 minutes	Induction proof takes time
diag-complex-numbers	45 minutes	De Moivre and roots of unity
diag-matrices	45 minutes	3x3 inverses and eigenvalues
diag-further-calculus	40 minutes	Volumes and arc length
diag-differential-equations	40 minutes	Particular integrals can be complex
diag-maclaurin-taylor-series	35 minutes	Series manipulation
diag-polar-coordinates	35 minutes	Area calculations
diag-hyperbolic-functions	30 minutes	Parallels trigonometry
diag-vectors-3d	35 minutes	Equations of planes

Total recommended time: approximately 6 hours (spread across 3--4 sessions).

How to Use These Diagnostics

1. Complete each diagnostic without notes, showing full algebraic working.
2. Check solutions immediately, comparing both method and final answer.
3. If you score FAIL, review the corresponding source file before retrying.
4. Integration Tests combine skills across pure maths topics -- these mirror A-Level exam style.
5. Practise writing proofs with clear logical steps.
6. Pay special attention to convergence conditions for series and Maclaurin expansions.