Further
Mathematics

Relationships between roots and coefficients. Symmetric functions of roots for quadratics, cubics, and quartics.

Vertical, horizontal and oblique asymptotes. Sketching technique, range by discriminant, five curve transformations

Method of differences, standard results for Σr, Σr², Σr³, and partial fraction decomposition of series.

Matrix algebra, determinants, inverses, and solving systems of linear equations including singular cases.

Polar curves, conversion between Cartesian and polar forms, area under a polar curve, and arc length.

Vector and Cartesian equations of lines and planes, angles, distances, and intersections in three-dimensional space.

Methods of proof including contradiction and counterexample, and further integration techniques.

Standard results, method of differences, partial fractions route, and sums to infinity.

Summation proofs, divisibility, matrix power proofs, and recurrence relation proofs — all four induction types.

Operations, determinants, inverses (2×2 and 3×3), linear systems, and linear transformations with Argand diagrams.

Characteristic equation, finding eigenvectors, diagonalisation M = PDP⁻¹, and Cayley-Hamilton theorem.

Modulus-argument form, de Moivre's theorem, roots of unity, z+z⁻¹ method, and loci in the Argand diagram.

Maclaurin series, improper integrals, reduction formulae, arc length, and surface of revolution.

Range, maximum height, time of flight, trajectory equation, and projectile on inclined planes.

Moments, couples, centre of mass, tilting and sliding conditions, and ladder problems.

Horizontal and vertical circular motion, conical pendulum, conditions for complete circles, and loss of contact.

Natural length, extensions, elastic potential energy, energy conservation with strings and springs.

Newton's 2nd law with F as a function of time, velocity, or displacement — solving differential equations of motion.

Newton's experimental law, coefficient of restitution, direct and oblique impacts of smooth spheres and surfaces.

PDF, CDF, expectation, variance, median, mode, and functions of a CRV including the Jacobian method.

Hypothesis tests and confidence intervals for means — z-test, t-test, two-sample tests, and paired tests.

Goodness-of-fit tests (uniform, Poisson, binomial, normal) and contingency table tests of association.

Sign test, Wilcoxon signed-rank test, and Mann-Whitney U-test — ranking, tied values, and critical regions.

G(t) = E[tˣ], moments via derivatives, standard PGFs, sums of independent variables, compound distributions.