A-Level Further Maths -- Diagnostic Guide

• 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

• 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

• 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

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

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

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.

This topic explores fundamental concepts that shape our understanding of the world.

The key principles covered in this topic are linked in the sub-pages above. Focus on understanding the definitions, applying the formulas or frameworks, and evaluating strengths and limitations of each approach.

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

• Confusing terminology or concepts that appear similar but have distinct meanings.
• Overlooking key assumptions or boundary conditions that limit applicability.

• Diagnostics
• Complex Numbers — Diagnostic Tests
• Differential Equations — Diagnostic Tests