Further Mathematics

Work through the topics in order, or jump to the one you need.

  1. Vectors Column and i–j notation, adding, subtracting and scaling vectors, position vectors, magnitude, unit vectors, the scalar (dot) product, the angle between vectors and the perpendicular test, with WAEC worked examples.
  2. Statics & Equilibrium Types of force, free-body diagrams, the resultant of several forces, resolving a force into components, the conditions for equilibrium, Lami's theorem, the triangle of forces and moments, with WAEC worked examples.
  3. Kinematics (SUVAT) Uniformly accelerated motion in a straight line, the four SUVAT equations, choosing the right equation, motion under gravity with a sign convention, and velocity–time graphs, with WAEC worked examples.
  4. Momentum & Impulse Linear momentum, impulse as change in momentum, the impulse–force relationship, conservation of momentum in collisions and explosions, recoil, and kinetic energy in collisions, with WAEC worked examples.
  5. Projectile Motion Why horizontal and vertical motion are independent, resolving the launch velocity, the formulas for time of flight, greatest height and range, the 45° angle for maximum range, and horizontal projection, with WAEC worked examples.
  6. Work, Energy & Power Work done by a force, kinetic and potential energy, the work–energy theorem, conservation of mechanical energy, energy in projectile motion, power and efficiency, with WAEC worked examples.
  7. Statistics Frequency tables, histograms, frequency polygons and ogives, the mean by coding, median and mode by formula, mean deviation, variance and standard deviation, Spearman's rank correlation and the least-squares regression line, with WAEC worked examples.
  8. Sets What makes a collection a set, set notation and set-builder form, types of sets, subsets and power sets, union, intersection, complement and De Morgan's laws, and two- and three-set Venn diagram problems in the WAEC style.
  9. Binary Operations How to evaluate a binary operation, test it for closure, commutativity, associativity and distributivity, find the identity element and inverses by algebra, and read all of these from an operation (Cayley) table, with WAEC worked examples.
  10. Mathematical Induction The domino idea, P(n), the base case, the inductive hypothesis and the inductive step, with full proofs for sums of series and divisibility, how to lay out the proof for full marks, and common errors.
  11. Differentiation From the gradient of a straight line to the gradient of a curve, the limit definition dy/dx = lim (f(x + h) − f(x))/h, five-step worked examples for x², 3x² + 2x, x³ and 1/x, the power rule, rates of change, and tangents.