A-level Maths → Further Maths

You can already

Nothing in A-level Maths — a negative discriminant just means 'no real roots'.

Now you'll

The imaginary unit $i = \sqrt{-1}$, the Argand diagram, modulus–argument form, de Moivre's theorem, and roots of unity.

You can already

Nothing in A-level Maths.

Now you'll

Matrix algebra, determinants and inverses, solving systems of linear equations, and matrices as transformations.

You can already

Deduction, exhaustion, contradiction and counterexample.

Now you'll

Proof by mathematical induction — for series, divisibility and matrix results.

You can already

Differentiate and integrate standard functions; find areas.

Now you'll

Improper integrals, the mean value of a function, volumes of revolution, and further integration techniques.

You can already

3D vectors, magnitude and position vectors.

Now you'll

The scalar (dot) product, and the equations of lines and planes with the distances between them (Core Pure); the vector/cross product comes in the optional Further Pure papers.

You can already

Arithmetic, geometric and binomial series.

Now you'll

The method of differences, Maclaurin series, and summing $\sum r$, $\sum r^2$ and $\sum r^3$.

You can already

Nothing in A-level Maths.

Now you'll

$\sinh x$, $\cosh x$ and $\tanh x$, their calculus, and the inverse hyperbolic functions.

You can already

Nothing in A-level Maths — everything is Cartesian $(x, y)$.

Now you'll

Curves written in polar form $r = f(\theta)$ and the area enclosed by a polar curve.

You can already

Simple first-order separable equations.

Now you'll

First- and second-order differential equations, modelling with them, and simple harmonic motion.

You can already

A-level Maths fixes your applied content (Statistics and Mechanics).

Now you'll

You choose your optional content — from Further Pure, Further Statistics, Further Mechanics and Decision maths.