Further Maths Lessons
The compulsory Core Pure content plus the optional module routes, following the A-Level specification. Pick a topic to see its lessons.
Core Pure · Year 1
01
Complex NumbersImaginary numbers, arithmetic, conjugation and quadratic roots
Imaginary and complex numbers
Multiplying complex numbers
Complex conjugation
Roots of quadratic equations
Solving cubic and quartic equations
02
Argand DiagramsModulus, argument, modulus–argument form, loci and regions
Argand diagrams
Modulus and argument
Loci in the Argand diagram
Regions in the Argand diagram
03
SeriesSums of natural numbers, squares and cubes
Sums of natural numbers
Sums of squares and cubes
04
Roots of PolynomialsSymmetric functions of roots and transformations of equations
Roots of a quadratic equation
Roots of a cubic equation
Roots of a quartic equation
Expressions relating to the roots of a polynomial
Linear transformations of roots
05
Volumes of RevolutionRotating curves about the x- and y-axes
Volumes of revolution around the x-axis
Volumes of revolution around the y-axis
Adding and subtracting volumes
Modelling with volumes of revolution
06
MatricesArithmetic, determinants, inverses and solving systems
Introduction to matrices
Matrix multiplication
Determinants
Inverting a 2 × 2 matrix
Inverting a 3 × 3 matrix
Solving systems of equations using matrices
Geometric interpretation of systems of equations
Modelling with matrices
07
Linear TransformationsReflections, rotations, enlargements and 3D transformations
Linear transformations in two dimensions
Reflections and rotations
Enlargements and stretches
Successive transformations
Linear transformations in three dimensions
The inverse of a linear transformation
Invariant points and lines
08
Proof by InductionSeries, divisibility and matrix results
Proof by mathematical induction
Proving divisibility results
Proving statements involving matrices
09
VectorsLines and planes in three dimensions
Equation of a line in three dimensions
Equation of a plane in three dimensions
Scalar product
Calculating angles between lines and planes
Points of intersection
Finding perpendiculars
Modelling with vectors
Core Pure · Year 2
10
Complex NumbersExponential form, De Moivre’s theorem, series and geometric problems
Exponential form of complex numbers
Multiplying and dividing complex numbers
De Moivre’s theorem
Trigonometric identities
Sums of series
nth roots of a complex number
Solving geometric problems
11
SeriesThe method of differences, higher derivatives and Maclaurin series
The method of differences
Higher derivatives
Maclaurin series
Series expansions of compound functions
12
Methods in CalculusImproper integrals, inverse trig calculus and partial fractions
Improper integrals
The mean value of a function
Differentiating inverse trigonometric functions
Integrating with inverse trigonometric functions
Integrating using partial fractions
13
Volumes of RevolutionParametric curves and modelling with volumes
Volumes of revolution around the x-axis
Volumes of revolution around the y-axis
Volumes of revolution of parametrically defined curves
Modelling with volumes of revolution
14
Polar CoordinatesCurves, areas and tangents in polar form
Polar coordinates and equations
Sketching curves
Area enclosed by a polar curve
Tangents to polar curves
15
Hyperbolic FunctionsDefinitions, identities, calculus and inverses
Introduction to hyperbolic functions
Inverse hyperbolic functions
Identities and equations
Differentiating hyperbolic functions
Integrating hyperbolic functions
16
Methods in Differential EquationsFirst- and second-order equations and boundary conditions
First-order differential equations
Second-order homogeneous differential equations
Second-order non-homogeneous differential equations
Using boundary conditions
17
Modelling with Differential EquationsSHM, damped/forced oscillations and coupled systems
Modelling with first-order differential equations
Simple harmonic motion
Damped and forced harmonic motion
Coupled first-order simultaneous differential equations