Further MathsQ&A by dot point

De Moivre's theorem, the exponential (Euler) form of a complex number, using de Moivre to derive trigonometric identities, and finding the nth roots of a complex number.

Solving first-order linear differential equations by the integrating factor, second-order linear equations with constant coefficients via the auxiliary equation, complementary function and particular integral, and modelling with differential equations.

Maclaurin series expansions of standard functions, improper integrals with infinite limits or discontinuities, the arc length of a curve, and the area of a surface of revolution.

Eigenvalues and eigenvectors of a 2x2 matrix, the characteristic equation, invariant lines and lines of invariant points of a transformation, and using eigenvectors to describe the geometry of a matrix.

The definitions of the hyperbolic functions in terms of the exponential function, their identities, derivatives and integrals, and the logarithmic forms of the inverse hyperbolic functions.

Polar coordinates and their relationship to Cartesian coordinates, sketching curves given in polar form, and the area enclosed by a polar curve using the half r squared integral.

Circular motion with angular speed, centripetal acceleration and force, motion in a horizontal and vertical circle, and simple harmonic motion with its defining equation, period and energy.

Graphs and networks, the minimum spanning tree by Kruskal's and Prim's algorithms, the shortest path by Dijkstra's algorithm, and sorting and route-inspection ideas in decision mathematics.

Kinematics with variable acceleration using calculus, motion in two dimensions with vectors, and projectile motion treating the horizontal and vertical components separately.

The Poisson distribution as a model, the normal distribution and standardising, the Central Limit Theorem for the distribution of the sample mean, and hypothesis testing including null and alternative hypotheses, significance levels and conclusions.

Linear momentum and impulse, conservation of momentum in collisions, Newton's experimental law with the coefficient of restitution, and kinetic energy lost in impacts.

The imaginary unit, arithmetic of complex numbers, the complex conjugate, the Argand diagram, modulus and argument, and complex roots of real polynomial equations occurring in conjugate pairs.

Curve sketching of rational functions, finding vertical and horizontal asymptotes, oblique asymptotes, the range of values a rational function can take, and locating stationary points.

The principle of mathematical induction and its use to prove results about summation of series, divisibility and other statements indexed by the positive integers.

Matrix algebra including addition, multiplication and the identity, the determinant and inverse of a 2x2 matrix, and matrices as linear transformations of the plane including rotations, reflections and enlargements.

The relationships between the roots and coefficients of quadratic, cubic and quartic equations, and forming new equations whose roots are functions of the original roots.

Summation of finite series using the standard results for the sum of r, r squared and r cubed, and the method of differences for telescoping sums.

Forces as vectors, resolving and equilibrium, Newton's three laws, the equation of motion F equals ma, friction and the coefficient of friction, and connected particles over pulleys.

Kinematics of motion in a straight line with constant acceleration, the suvat equations, vertical motion under gravity, and interpreting displacement-time and velocity-time graphs.

Probability of combined events, mutually exclusive and independent events, conditional probability, discrete random variables with their expectation and variance, and the binomial distribution as a model.

Statistical sampling methods, presenting and interpreting data, measures of central tendency (mean, median, mode) and measures of variation (range, interquartile range, variance and standard deviation).