Further MathsQ&A by dot point

The arithmetic of complex numbers, the complex conjugate and division, the Argand diagram, and solving quadratic, cubic and quartic equations with complex roots that occur in conjugate pairs.

De Moivre's theorem for integer and rational powers, using it to find powers of complex numbers, and applying it with the binomial theorem to derive multiple-angle identities and to express powers of sine and cosine.

The modulus and argument of a complex number, modulus-argument form, the exponential form re^(i theta), and the multiplication and division rules in which moduli multiply or divide and arguments add or subtract.

The nth roots of unity and of a general complex number, their geometric arrangement as a regular polygon, and loci on the Argand diagram defined by modulus and argument conditions (circles, perpendicular bisectors, half-lines and regions).

First order linear differential equations by the integrating factor, second order linear constant-coefficient equations via the auxiliary equation and particular integral, and applications to simple harmonic motion and damped systems.

Improper integrals with an infinite limit of integration or an integrand that is unbounded at an endpoint, evaluated as a limit, and deciding whether such an integral converges or diverges.

The Maclaurin series of a function, the standard series for e^x, ln(1+x), sin x and cos x, finding a series by repeated differentiation or by combining known series, and using a truncated series to approximate values.

Volumes of revolution about the x-axis and y-axis, volumes generated by the region between two curves, and parametric and improper cases, using integration of pi y squared or pi x squared.

Angles between two lines, between a line and a plane, and between two planes, and the shortest distance from a point to a line or plane and between two skew lines.

The vector and Cartesian equations of a straight line in three dimensions, the direction vector, and finding the intersection of two lines or showing that they are parallel or skew.

The vector, scalar product and Cartesian equations of a plane, the normal vector, and the intersection of a line with a plane and of two planes.

The scalar (dot) product and its use for angles and perpendicularity, the vector (cross) product and its use for a perpendicular direction and areas, and the modulus of the vector product as an area.

The inverse of a 2x2 matrix, the existence condition (non-zero determinant), the inverse of a 3x3 matrix via the adjugate or row reduction, and the inverse of a product.

Matrices as linear transformations in two and three dimensions (rotations, reflections, enlargements, stretches and shears), composition by multiplication, invariant points and lines, and the determinant as an area or volume scale factor.

Matrix addition, subtraction, scalar multiplication and multiplication, the zero and identity matrices, non-commutativity, and the determinant of a 2x2 and 3x3 matrix as an area or volume scale factor.

Writing a system of linear equations as a matrix equation, solving by the inverse matrix, and the geometric interpretation of consistent, inconsistent and dependent systems in two and three unknowns.

The area enclosed by a polar curve using the formula one half the integral of r squared with respect to theta, including areas between two curves and the area of one loop.

Differentiation and integration of hyperbolic and inverse hyperbolic functions, and using hyperbolic substitutions to integrate functions involving the square root of x squared plus or minus a squared.

The hyperbolic functions defined from exponentials, their graphs and properties, the key identities, and the logarithmic forms of the inverse hyperbolic functions.

Polar coordinates, conversion between polar and Cartesian form, and sketching polar curves r = f(theta) including circles, lines, cardioids and spirals.

The method of differences, expressing a general term as a difference of consecutive terms (often via partial fractions), summing by cancellation, and finding the sum to infinity where it exists.

Proof by mathematical induction for summation formulae, divisibility results, recurrence relations and powers of matrices, with a correctly stated base case, inductive hypothesis, inductive step and conclusion.

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

The standard results for the sum of r, r squared and r cubed, using them to sum polynomial expressions in r, splitting sums by linearity, and adjusting limits.

The centre of mass of a system of particles, of a uniform lamina (by symmetry or integration), and of a composite body, and the equilibrium of a suspended or tilting body.

Angular speed, the centripetal acceleration and force, motion in a horizontal circle (including the conical pendulum and banked tracks), and motion in a vertical circle with the conditions for maintaining contact or tension.

Linear momentum and impulse, conservation of momentum, Newton's experimental law and the coefficient of restitution, and direct and oblique collisions including the impulse during impact.

Work done by a force, kinetic and gravitational potential energy, the work-energy principle, power as the rate of working, Hooke's law for elastic strings and springs, and elastic potential energy.

The chi-squared goodness-of-fit test and contingency table test for independence, degrees of freedom, and non-parametric tests including the sign test and Wilcoxon signed-rank test.

Continuous random variables, the probability density function and cumulative distribution function, finding probabilities by integration, and the expectation and variance of a continuous variable.

Discrete random variables, the probability distribution, expectation and variance, and the effect of a linear transformation aX + b on the mean and variance.

The Poisson distribution and its conditions, mean and variance, the sum of independent Poisson variables, the geometric distribution and its mean, and the Poisson approximation to the binomial.