← 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 represent, manipulate and find roots of them?

Arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem, nth roots, complex roots of polynomials and loci.
A focused answer to the Edexcel A-Level Further Mathematics complex numbers content, covering arithmetic, the Argand diagram, modulus-argument and exponential form, de Moivre's theorem, nth roots, roots of unity, complex roots of polynomials and loci.
13 min answer →

How do you solve first and second order differential equations, and model oscillations with them?

First order linear equations by integrating factor, second order constant-coefficient equations using the auxiliary equation, complementary function and particular integral, and modelling damped and forced oscillations and coupled systems.
A focused answer to the Edexcel A-Level Further Mathematics differential equations content, covering first order linear equations by integrating factor, second order constant-coefficient equations via the auxiliary equation, the complementary function and particular integral, and modelling simple harmonic, damped and forced systems.
13 min answer →

How do you sum series, relate roots to coefficients, and use the method of differences?

Summing series of powers of integers, relationships between roots and coefficients of polynomials, transforming equations with new roots, and the method of differences.
A focused answer to the Edexcel A-Level Further Mathematics further algebra content, covering standard summation formulae for powers of integers, the relationships between roots and coefficients of polynomials, forming equations with transformed roots, and the method of differences.
12 min answer →

How do you extend integration to improper integrals, volumes, arc lengths and mean values?

Improper integrals, volumes of revolution, the mean value of a function, integration using partial fractions, and the derivation of standard inverse trig and hyperbolic integrals.
A focused answer to the Edexcel A-Level Further Mathematics further calculus content, covering improper integrals evaluated as limits, volumes of revolution about both axes, the mean value of a function, integration by partial fractions, and the standard inverse trig and inverse hyperbolic integral results.
12 min answer →

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

Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections, and shortest distances including between skew lines.
A focused answer to the Edexcel 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, points of intersection, and shortest distances including the distance between two skew lines.
13 min answer →

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

Definitions of sinh, cosh and tanh from exponentials, hyperbolic identities, logarithmic forms of the inverse functions, and differentiation and integration of hyperbolic functions.
A focused answer to the Edexcel A-Level Further Mathematics hyperbolic functions content, covering the definitions of sinh, cosh and tanh from exponentials, the hyperbolic identities, logarithmic forms of the inverse functions, and the differentiation and integration of hyperbolic functions.
12 min answer →

How do matrices encode linear 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 linear systems.
A focused answer to the Edexcel 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 using the inverse to solve systems of linear equations.
13 min answer →

How do polar coordinates describe curves, and how do you find areas they enclose?

Polar coordinates and curves, conversion to and from Cartesian form, sketching cardioids and spirals, tangents parallel and perpendicular to the initial line, and areas enclosed by polar curves.
A focused answer to the Edexcel A-Level Further Mathematics polar coordinates content, covering polar coordinates and curves, conversion between polar and Cartesian form, sketching cardioids and spirals, finding tangents parallel and perpendicular to the initial line, and computing areas enclosed by polar curves.
12 min answer →

How does mathematical induction prove a statement for all positive integers?

The structure of proof by induction, applied to summation formulae, divisibility results, recurrence relations and powers of matrices, with rigorous base case, inductive step and conclusion.
A focused answer to the Edexcel A-Level Further Mathematics proof by induction content, covering the structure of an induction proof and its application to summation formulae, divisibility results, recurrence relations and powers of matrices, with the rigorous base case, inductive step and conclusion examiners reward.
12 min answer →