Advanced Higher Mathematics

Splitting a fraction with a complicated denominator into simpler fractions, including a repeated or irreducible quadratic factor.

Odd and even functions, sketching rational functions with asymptotes, and the full set of graph features and transformations this course examines.

Adding, multiplying, dividing, and finding the square root of complex numbers, and plotting them on the Argand diagram.

Writing a complex number in polar form, and using De Moivre's theorem to find powers, roots, and multiple-angle trig identities.

Sigma notation, arithmetic and geometric series, and the Maclaurin series.

The structure of an inductive proof, and how to write one out properly.

Adding, multiplying matrices, finding a 2×2 or 3×3 determinant, and the key properties matrix algebra follows.

Finding the inverse of a 2×2 or 3×3 matrix, and using a matrix inverse to solve simultaneous equations directly.

The product rule, quotient rule, chain rule, logarithmic differentiation, and differentiating inverse trig, exponential, and log functions.

Related rates of change, optimisation, and curve sketching for rational functions with asymptotes.

Integration by substitution, integration by parts, and the standard results this course expects to be recognised directly.

Integrating a fraction by splitting it first, and finding the volume of a solid of revolution about either axis.

Solving a differential equation by separating the variables, and solving a linear differential equation using an integrating factor.

The equation of a line and plane in several forms, the vector product and scalar triple product, and finding angles and intersections.