Further MathsQ&A by dot point

Arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem, nth roots, complex roots of polynomials and loci.

First order linear equations by integrating factor, second order constant-coefficient equations using the auxiliary equation, complementary function and particular integral, and modelling damped and forced oscillations and coupled systems.

Summing series of powers of integers, relationships between roots and coefficients of polynomials, transforming equations with new roots, and the method of differences.

Improper integrals, volumes of revolution, the mean value of a function, integration using partial fractions, and the derivation of standard inverse trig and hyperbolic integrals.

Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections, and shortest distances including between skew lines.

Definitions of sinh, cosh and tanh from exponentials, hyperbolic identities, logarithmic forms of the inverse functions, and differentiation and integration of hyperbolic functions.

Matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving linear systems.

Polar coordinates and curves, conversion to and from Cartesian form, sketching cardioids and spirals, tangents parallel and perpendicular to the initial line, and areas enclosed by polar curves.

The structure of proof by induction, applied to summation formulae, divisibility results, recurrence relations and powers of matrices, with rigorous base case, inductive step and conclusion.

Angular speed, acceleration towards the centre, motion in a horizontal circle, the conical pendulum, and motion in a vertical circle with energy conservation.

Newton's experimental law of restitution, direct and oblique impact of smooth spheres, impact with a fixed surface, and kinetic energy lost in a collision.

Momentum and impulse in one and two dimensions, the impulse-momentum principle, conservation of momentum, and impulse as the area under a force-time graph.

Work done by a force, kinetic and potential energy, the work-energy principle, the conservation of mechanical energy, and power as the rate of doing work.

The parabola, ellipse and hyperbola in Cartesian and parametric form, foci and directrices, tangents and normals, and the rectangular hyperbola.

Solving equations numerically by interval bisection, linear interpolation and the Newton-Raphson method, and approximating definite integrals using Simpson's rule and the mid-ordinate rule.

The t-substitution for trigonometric integrals and equations, summing series of sines and cosines, the general solution of trigonometric equations, and inverse trigonometric functions.

Maclaurin and Taylor series of standard functions, finding series solutions of differential equations, and using series to approximate functions and limits.

Goodness of fit tests, contingency tables and tests for independence using the chi-squared statistic, expected frequencies, degrees of freedom, and Yates' correction.

Discrete random variables and probability distributions, expectation and variance, the effect of linear coding, and expectation and variance of functions of a discrete variable.

The geometric distribution as a model for the trial of the first success, the negative binomial distribution for the rth success, and their means and variances.

The Poisson distribution as a model for random events, its mean and variance, the binomial distribution, the additive property of Poisson variables, and the Poisson approximation to the binomial.