← Further Maths syllabus

EnglandFurther Maths

Core pure

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do complex numbers extend the real numbers, and how do you add, multiply, divide and represent them geometrically?

Solving quadratic, cubic and quartic equations with complex roots, arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem and loci.
A focused answer to the AQA A-Level Further Mathematics complex numbers content, covering the arithmetic of complex numbers, the Argand diagram, modulus-argument and exponential form, de Moivre's theorem, complex roots of polynomials, roots of unity and loci.
11 min answer →

How do you solve first and second order differential equations, and how do you model real situations with them?

First order linear differential equations using an integrating factor, second order equations with constant coefficients including the complementary function and particular integral, and modelling with damped and forced systems.
A focused answer to the AQA A-Level Further Mathematics differential equations content, covering first order linear equations using an integrating factor, second order equations with constant coefficients via the auxiliary equation, the complementary function and particular integral, and modelling damped and forced systems.
12 min answer →

How do the roots of a polynomial relate to its coefficients, and how do you sum series and split fractions?

Roots of polynomials and their relationships to coefficients, summation of series using standard results, the method of differences, partial fractions and the Maclaurin series.
A focused answer to the AQA A-Level Further Mathematics further algebra content, covering relationships between roots and coefficients, summation of series with standard results, the method of differences, partial fractions and the Maclaurin series.
11 min answer →

How do you find improper integrals, arc lengths, surface areas and the mean value of a function, and how do you use the Maclaurin series?

Improper integrals, volumes of revolution, mean value of a function, arc length, surface area of revolution, integration using partial fractions and the Maclaurin series of standard functions.
A focused answer to the AQA A-Level Further Mathematics further calculus content, covering improper integrals, volumes of revolution, the mean value of a function, arc length, surface area of revolution, integration with partial fractions and the Maclaurin series of standard functions.
11 min answer →

How do you describe lines and planes in three dimensions, and how do you find angles, intersections and distances between them?

Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections and shortest distances in three dimensions.
A focused answer to the AQA A-Level Further Mathematics further vectors content, covering vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections and shortest distances in three dimensions.
11 min answer →

What are the hyperbolic functions, how are they defined, and how do you differentiate, integrate and invert them?

Definitions of hyperbolic functions in terms of exponentials, their graphs and identities, inverse hyperbolic functions in logarithmic form, and differentiation and integration involving them.
A focused answer to the AQA A-Level Further Mathematics hyperbolic functions content, covering the exponential definitions of sinh, cosh and tanh, their graphs and identities, the logarithmic form of the inverse hyperbolic functions, and differentiation and integration involving them.
10 min answer →

How do matrices represent transformations and systems of equations, and how do you invert them?

Matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving systems of linear equations.
A focused answer to the AQA A-Level Further Mathematics matrices content, covering matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving systems of linear equations.
11 min answer →

How do polar coordinates describe curves, and how do you sketch them and find the area they enclose?

Polar coordinates and the relationship with Cartesian coordinates, sketching polar curves, and finding areas enclosed by polar curves using integration.
A focused answer to the AQA A-Level Further Mathematics polar coordinates content, covering the relationship between polar and Cartesian coordinates, sketching polar curves such as cardioids and spirals, and finding areas enclosed by polar curves using integration.
10 min answer →

How does proof by induction work, and how do you use it for sums, divisibility and matrices?

Proof by mathematical induction applied to summation formulae, divisibility results, recurrence relations and powers of matrices, with a clearly stated base case, inductive step and conclusion.
A focused answer to the AQA A-Level Further Mathematics proof by induction content, covering the structure of an induction proof and its use for summation formulae, divisibility results, recurrence relations and powers of matrices, with a clear base case, inductive step and conclusion.
10 min answer →