Specialist Mathematics Q&A by dot point

Arithmetic and algebra of complex numbers in Cartesian form $z = a + bi$ and polar form $z = r\,\mathrm{cis}\,\theta$, the modulus and argument, conjugates, and representation of complex numbers and their operations on the Argand plane

The vector (cross) product of two three-dimensional vectors, its definition in component form, the geometric meaning of its direction and magnitude, and its applications to finding a normal vector, the area of a parallelogram or triangle, and testing for parallel vectors

De Moivre's theorem $(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)$ for integer $n$, its use in finding powers and the $n$ distinct $n$th roots of a complex number, and the factorisation of polynomials over the complex numbers

The equations and key features of ellipses and hyperbolas, including centre, vertices, axes and asymptotes, and the description of curves by parametric equations together with conversion between parametric and Cartesian forms

Implicit differentiation of relations defined by equations in $x$ and $y$, the second derivative and its use to determine concavity and points of inflection, and the analysis of curves using first and second derivative information

The inverse circular functions $\arcsin$, $\arccos$ and $\arctan$, the domain restrictions needed to define them, their domains, ranges and graphs, and the evaluation of exact values and composite expressions

The principle of mathematical induction and its use to prove propositions about positive integers, including the base step, the inductive assumption and the inductive step, applied to summation formulas, divisibility results and inequalities

Methods of proof including direct proof, proof by contrapositive, proof by contradiction, and the use of a single counterexample to disprove a universal statement, together with the language of quantifiers and implication

Graphs of rational functions including reciprocal functions, the location of vertical, horizontal and oblique asymptotes, and the effect of reciprocal and modulus transformations on the shape and key features of a graph

The transformations $y = \frac{1}{f(x)}$, $y = |f(x)|$ and $y = f(|x|)$ applied to a known graph $y = f(x)$, the effect on intercepts, asymptotes, turning points and symmetry, and the sketching of the resulting curves

Solution of polynomial equations over the complex numbers, the fundamental theorem of algebra, the conjugate root theorem for polynomials with real coefficients, and the full factorisation of real polynomials into linear and irreducible quadratic factors

Description and sketching of subsets of the complex plane defined by conditions on modulus and argument, including circles $|z - z_0| = r$, perpendicular bisectors $|z - a| = |z - b|$, rays $\arg(z - z_0) = \alpha$, and the regions defined by the corresponding inequalities

Vector equations of lines and planes in three dimensions, their parametric and Cartesian forms, the use of a direction vector for a line and a normal vector for a plane, and the determination of intersections and the angle between a line and a plane

Vectors in two and three dimensions in $\mathbf{i}, \mathbf{j}, \mathbf{k}$ form, magnitude and unit vectors, the scalar (dot) product and the angle between vectors, vector projection, and the use of the scalar product to test for perpendicular and parallel vectors

The use of definite integrals to find the arc length of a curve and the surface area of a solid of revolution, in Cartesian form $y = f(x)$ and in parametric form $x = x(t)$, $y = y(t)$, and the setting up of the appropriate integral

Construction and interpretation of approximate confidence intervals for a population mean using the sample mean and standard error, the choice of confidence level and its $z$ value, the effect of sample size on the interval width, and the correct interpretation of a confidence interval

Formulation and solution of first-order differential equations including those solvable by direct integration and by separation of variables, the use of initial conditions to find particular solutions, and the interpretation of solutions in modelling contexts

Differentiation of the inverse circular functions $\arcsin$, $\arccos$ and $\arctan$, the standard derivative results, the use of the chain rule for composite forms, and the related standard antiderivatives

Hypothesis testing for a population mean, the null and alternative hypotheses, one-tailed and two-tailed tests, the test statistic and its $p$ value, the comparison with a significance level, the decision and its interpretation, and the meaning of Type I and Type II errors

Antidifferentiation techniques including integration by substitution, the use of partial fractions, trigonometric identities and inverse-trigonometric standard forms, and the evaluation of definite integrals using these techniques

Application of calculus to rectilinear motion, the relationships between position, velocity and acceleration including the forms $a = \frac{\mathrm{d}v}{\mathrm{d}t} = v\frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x}(\tfrac12 v^2)$, and the use of these to analyse motion with variable acceleration

Linear combinations of independent random variables and their mean and variance, the distribution of the sample mean $\bar{X}$, the construction of confidence intervals for a population mean, and hypothesis testing for the mean using a $p$ value

Newton's laws of motion, the resultant of forces acting on a particle, the resolution of forces into components, the relationship $\mathbf{F} = m\mathbf{a}$, and the analysis of equilibrium and of motion under constant forces including weight, normal reaction and friction

Momentum $p = mv$ and impulse as the change in momentum, the impulse-momentum relationship, and the analysis of connected particles such as bodies linked by a string over a pulley or in contact, which share a common acceleration

Related rates of change problems, the use of the chain rule to connect the rates of change of related variables, the setting up of a relating equation from the geometry or context, and the evaluation of an unknown rate at a given instant

The distribution of the sample mean $\bar{X}$ as a random variable, its mean and standard deviation (the standard error), the effect of sample size, and the central limit theorem giving the approximate normality of $\bar{X}$ for large samples

Slope (direction) fields as a representation of a first-order differential equation, the sketching of solution curves on a slope field, and Euler's method for the numerical approximation of a solution from an initial condition with a chosen step size

Vector functions of a real variable, the differentiation and integration of a position vector $\mathbf{r}(t)$ to obtain velocity and acceleration, the speed as the magnitude of velocity, and the application to motion in two and three dimensions

The use of definite integrals to find the volume of a solid of revolution generated by rotating a region about the $x$-axis or $y$-axis, using the disc and washer (annulus) methods, and the setting up of the appropriate integral