MathsQ&A by dot point

Examine the properties of functions, including domain, asymptotes, symmetry and stationary points, to sketch the graph of a rational or other function, and apply differentiation to rates of change and optimisation problems.

Find the Maclaurin series expansion of a function using the standard formula, derive the standard expansions of exponential, logarithmic and trigonometric functions, and use known expansions to build the series of composite or product functions.

Work with arithmetic and geometric sequences and series, using the formulae for the nth term and the sum to n terms, the sum to infinity of a convergent geometric series, and the condition for convergence.

Apply the standard summation formulae for the sum of the first n natural numbers, their squares and their cubes, use sigma notation, and prove statements about series, divisibility and inequalities for all positive integers by mathematical induction.

Use the binomial theorem to expand expressions of the form (a + b) to the power n for a positive integer n, using binomial coefficients, and find a general term or a specific term such as the constant term or the coefficient of a chosen power.

Perform arithmetic with complex numbers in Cartesian form, represent them on an Argand diagram, convert to polar (modulus-argument) form, and use de Moivre's theorem to find powers and the nth roots of a complex number.

Add, subtract and multiply matrices, find the determinant and inverse of 2x2 and 3x3 matrices, and solve systems of linear equations using the inverse matrix and Gaussian elimination, identifying unique, no, and infinitely many solutions.

Construct proofs using direct proof, proof by contradiction and proof by contrapositive, disprove a conjecture by counterexample, and use the Euclidean algorithm to find the highest common factor and express it as a linear combination.

Use the scalar and vector products of vectors in three dimensions, find the equation of a line in three dimensions and the equation of a plane in vector, parametric and Cartesian form, and find angles and intersections between lines and planes.

Differentiate using the chain, product and quotient rules; differentiate exponential, logarithmic, inverse trigonometric, implicit and parametrically defined functions; and use logarithmic differentiation and higher derivatives.

Solve first-order differential equations by separating the variables and by the integrating-factor method for linear equations, find particular solutions from initial conditions, and apply differential equations to growth and decay models.

Integrate using standard results, integration by substitution, integration by parts and integration using partial fractions, and apply integration to find areas and volumes of revolution.

Express a proper rational function as a sum of partial fractions where the denominator factorises into distinct linear factors, repeated linear factors, or an irreducible quadratic factor, and reduce an improper rational function first by algebraic division.

Solve homogeneous and non-homogeneous second-order linear differential equations with constant coefficients using the auxiliary equation, the complementary function and a particular integral, covering distinct real, equal and complex roots.