Further Maths A-Level Help
The 22 main topics of Further Maths A-Level alongside standard A-Level Maths — complex numbers, matrices, hyperbolic functions, ODEs, polar curves, induction.
Further Maths is sat alongside the standard A-Level Maths. Boards (Edexcel, AQA, OCR, MEI) split the content slightly differently across Core Pure 1/2 and optional papers. Coverage below is the Core Pure content all four boards share.
- Complex numbers (introduction)i = √-1; a + bi form. Argand diagram.
- Complex arithmeticAdd, subtract, multiply, divide a+bi.
- Modulus and argument|z| and arg(z); modulus-argument form.
- Polar form of complex numbersz = r(cosθ + i sinθ) = re^(iθ).
- de Moivre's theorem(cosθ + i sinθ)^n = cos nθ + i sin nθ.
- Roots of unitySolutions of z^n = 1 lie on unit circle.
- Matrices (introduction)Rows and columns; addition; scalar multiplication.
- Matrix arithmeticMultiplication; identity; non-commutativity.
- Determinant and inversedet A; A−1 = adj(A)/det A; 2×2 and 3×3.
- Matrices as transformationsRotations, reflections, enlargements via 2×2 matrices.
- Eigenvalues and eigenvectorsAv = λv; characteristic equation det(A − λI) = 0.
- Proof by inductionBase case + inductive step. Standard FM proof method.
- Hyperbolic functionssinh, cosh, tanh; identities and graphs.
- Hyperbolic identitiescosh²x − sinh²x = 1; addition formulae.
- Differentiating hyperbolicsd(sinh)/dx = cosh; d(cosh)/dx = sinh.
- Integration with partial fractionsDecompose, then integrate term-by-term.
- First-order separable ODEsdy/dx = f(x)g(y); separate, integrate both sides.
- Integrating factorLinear ODE dy/dx + P(x)y = Q(x); IF = e^∫P dx.
- Second-order linear (homogeneous)ay'' + by' + cy = 0; auxiliary equation roots.
- Polar coordinates(r, θ) instead of (x, y); conversion formulae.
- Polar curves and areaArea enclosed = ½∫r²dθ.
- Maclaurin and Taylor seriesf(x) = Σ f^(n)(0) x^n / n!.