Mr Kidman's ClassroomKS5 · Edexcel A-level Further Mathematics
Further Mathematics Revision
Study the confirmed Edexcel 9FM0 route: both Core Pure components, Further Mechanics 1 and Decision Mathematics 1, with developed notes, original questions, vocabulary and full mathematical demonstrations.
Further Mathematics topics
24 topics
Complex arithmetic and Argand geometry
Develop complex arithmetic, polynomial conjugate roots, principal arguments and exact Argand loci with boundary conditions.
Complex powers, roots and trigonometry
Derive complex powers, trigonometric identities and finite sums, then construct every root and explain regular-polygon geometry.
Matrices and transformations
Use conformable matrix operations, linear transformations, invariant sets, determinants, inverses and complete three-plane classifications.
Proof by induction
Construct and diagnose induction proofs for sums, divisibility, inequalities and complete matrix-power families.
Polynomial roots and finite sums
Relate polynomial coefficients to roots, transform cubic and quartic root sets and derive finite sums with correct boundary terms.
Maclaurin series
Derive and combine Maclaurin series with exact coefficients, valid ranges, cancellation and justified approximation claims.
Further calculus
Calculate volumes about either axis, evaluate improper integrals and means, and use inverse trigonometric methods and partial fractions.
Further vectors
Describe spatial lines and planes, solve intersections and angles, and justify shortest distances through perpendicular feet and connectors.
Polar coordinates
Convert, sketch and trace polar families, derive exact areas and investigate intersections and Cartesian tangent directions.
Hyperbolic functions
Derive hyperbolic identities and principal inverses from exponentials, then use branch-aware differentiation and integration.
Differential equations and modelling
Solve first-order, second-order and coupled linear differential equations and interpret harmonic, damped and forced models.
Momentum and impulse
Justify momentum balances, calculate signed/vector impulses and solve particle interactions with explicit physical assumptions.
Work, energy and power
Derive and apply work–energy balances, integrate variable resistance and interpret constant-power driving and inclined motion.
Elastic strings, springs and energy
Distinguish taut strings and compressible springs, solve elastic equilibrium and energy problems and track slack or turning stages.
Direct elastic collisions
Solve direct sphere and wall impacts with restitution, derive kinetic-energy losses and establish successive contact order.
Oblique elastic collisions
Resolve smooth oblique contacts into unit normal/tangent components and reconstruct pair and successive-plane impact velocities.
Algorithms, sorting and bin packing
Trace assignments and loops, compare growth, execute both sorts and test three packing heuristics with valid optimum bounds.
Graphs, Eulerian routes and planarity
Distinguish edge/vertex routes, prove graph equivalence or obstructions and execute the Hamiltonian-circle planarity procedure.
Minimum connectors and shortest paths
Construct minimum connectors, execute matrix Prim, Dijkstra and Floyd and recover actual shortest routes with model checks.
Route inspection and travelling salesman
Optimise every-edge inspection walks and compare vertex-visiting tour heuristics, tree bounds and metric shortcuts.
Critical path analysis and scheduling
Build activity-on-arc dependencies, calculate times and floats and prove feasible minimum-worker schedules for a deadline.
Linear programming and graphical solutions
Formulate linear models, solve graphical and integer cases and interpret slack, surplus, feasibility and objective bounds.
Simplex tableaux and optimal solutions
Execute exact simplex pivots, certify maxima/minima and analyse alternate optima, degeneracy and unbounded improvement.
Two-stage simplex and big-M
Find an original feasible basis using two stages or symbolic big-M, restore the real objective and justify final mixed-constraint results.
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