England · OCRSyllabus
Further Maths syllabus, dot point by dot point
Every dot point in the England Further Mathssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Core Pure: complex numbers
Module overview →- How do you add, multiply and divide complex numbers, and how are they represented on the Argand diagram?The arithmetic of complex numbers, the complex conjugate and division, the Argand diagram, and solving quadratic, cubic and quartic equations with complex roots that occur in conjugate pairs.11 min answer →
- How does de Moivre's theorem give powers of complex numbers and let you derive trigonometric identities?De Moivre's theorem for integer and rational powers, using it to find powers of complex numbers, and applying it with the binomial theorem to derive multiple-angle identities and to express powers of sine and cosine.12 min answer →
- How do you write a complex number in modulus-argument and exponential form, and how do these forms make multiplication and division easy?The modulus and argument of a complex number, modulus-argument form, the exponential form re^(i theta), and the multiplication and division rules in which moduli multiply or divide and arguments add or subtract.11 min answer →
- How do you find the nth roots of a complex number, and how do conditions on z describe loci on the Argand diagram?The nth roots of unity and of a general complex number, their geometric arrangement as a regular polygon, and loci on the Argand diagram defined by modulus and argument conditions (circles, perpendicular bisectors, half-lines and regions).12 min answer →
Core Pure: further calculus
Module overview →- How do you solve first and second order differential equations, and how do they model simple harmonic motion and damping?First order linear differential equations by the integrating factor, second order linear constant-coefficient equations via the auxiliary equation and particular integral, and applications to simple harmonic motion and damped systems.12 min answer →
- How do you evaluate an integral with an infinite limit or an unbounded integrand, and how do you decide whether it converges?Improper integrals with an infinite limit of integration or an integrand that is unbounded at an endpoint, evaluated as a limit, and deciding whether such an integral converges or diverges.11 min answer →
- How do you build the Maclaurin series of a function, and how is it used to approximate functions?The Maclaurin series of a function, the standard series for e^x, ln(1+x), sin x and cos x, finding a series by repeated differentiation or by combining known series, and using a truncated series to approximate values.11 min answer →
- How do you find the volume generated when a region is rotated about an axis?Volumes of revolution about the x-axis and y-axis, volumes generated by the region between two curves, and parametric and improper cases, using integration of pi y squared or pi x squared.11 min answer →
Core Pure: further vectors and planes
Module overview →- How do you find angles between lines and planes and the shortest distances in three dimensions?Angles between two lines, between a line and a plane, and between two planes, and the shortest distance from a point to a line or plane and between two skew lines.12 min answer →
- How do you write the equation of a line in three dimensions, and how do you find where two lines meet?The vector and Cartesian equations of a straight line in three dimensions, the direction vector, and finding the intersection of two lines or showing that they are parallel or skew.11 min answer →
- How do you write the equation of a plane, and how do you find where a line meets a plane or where two planes meet?The vector, scalar product and Cartesian equations of a plane, the normal vector, and the intersection of a line with a plane and of two planes.11 min answer →
- How do the scalar and vector products work, and what geometric information do they give?The scalar (dot) product and its use for angles and perpendicularity, the vector (cross) product and its use for a perpendicular direction and areas, and the modulus of the vector product as an area.11 min answer →
Core Pure: matrices and transformations
Module overview →- How do you find the inverse of a 2x2 and 3x3 matrix, and when does an inverse exist?The inverse of a 2x2 matrix, the existence condition (non-zero determinant), the inverse of a 3x3 matrix via the adjugate or row reduction, and the inverse of a product.12 min answer →
- How do matrices represent geometric transformations of the plane and of space, and how do you compose and reverse them?Matrices as linear transformations in two and three dimensions (rotations, reflections, enlargements, stretches and shears), composition by multiplication, invariant points and lines, and the determinant as an area or volume scale factor.12 min answer →
- How do you add, multiply and take the determinant of matrices, and what does the determinant tell you geometrically?Matrix addition, subtraction, scalar multiplication and multiplication, the zero and identity matrices, non-commutativity, and the determinant of a 2x2 and 3x3 matrix as an area or volume scale factor.11 min answer →
- How do you use matrices to solve a system of linear equations, and how do you interpret the geometry when the determinant is zero?Writing a system of linear equations as a matrix equation, solving by the inverse matrix, and the geometric interpretation of consistent, inconsistent and dependent systems in two and three unknowns.11 min answer →
Core Pure: polar coordinates and hyperbolic functions
Module overview →- How do you find the area enclosed by a polar curve?The area enclosed by a polar curve using the formula one half the integral of r squared with respect to theta, including areas between two curves and the area of one loop.11 min answer →
- How do you differentiate and integrate hyperbolic functions, and how do they help integrate certain algebraic functions?Differentiation and integration of hyperbolic and inverse hyperbolic functions, and using hyperbolic substitutions to integrate functions involving the square root of x squared plus or minus a squared.12 min answer →
- What are the hyperbolic functions, what identities do they satisfy, and what do their inverses look like?The hyperbolic functions defined from exponentials, their graphs and properties, the key identities, and the logarithmic forms of the inverse hyperbolic functions.11 min answer →
- How do polar coordinates describe points and curves, and how do you convert between polar and Cartesian form?Polar coordinates, conversion between polar and Cartesian form, and sketching polar curves r = f(theta) including circles, lines, cardioids and spirals.11 min answer →
Core Pure: series and proof
Module overview →- How does the method of differences sum a series whose terms telescope?The method of differences, expressing a general term as a difference of consecutive terms (often via partial fractions), summing by cancellation, and finding the sum to infinity where it exists.11 min answer →
- How does proof by mathematical induction work, and what must each step contain to be rigorous?Proof by mathematical induction for summation formulae, divisibility results, recurrence relations and powers of matrices, with a correctly stated base case, inductive hypothesis, inductive step and conclusion.12 min answer →
- How are the roots of a polynomial related to its coefficients, and how do you use these relationships?The relationships between the roots and coefficients of quadratic, cubic and quartic equations, symmetric functions of the roots, and forming a new equation whose roots are a given function of the original roots.11 min answer →
- How do you use the standard summation formulae for sums of powers to evaluate more complicated series?The standard results for the sum of r, r squared and r cubed, using them to sum polynomial expressions in r, splitting sums by linearity, and adjusting limits.11 min answer →
Further Mechanics (optional)
Module overview →- How do you find the centre of mass of a system of particles, a lamina or a composite body?The centre of mass of a system of particles, of a uniform lamina (by symmetry or integration), and of a composite body, and the equilibrium of a suspended or tilting body.11 min answer →
- How do you analyse motion in a horizontal or vertical circle, and what provides the centripetal force?Angular speed, the centripetal acceleration and force, motion in a horizontal circle (including the conical pendulum and banked tracks), and motion in a vertical circle with the conditions for maintaining contact or tension.12 min answer →
- How do momentum and impulse describe collisions, and what does the coefficient of restitution tell you?Linear momentum and impulse, conservation of momentum, Newton's experimental law and the coefficient of restitution, and direct and oblique collisions including the impulse during impact.12 min answer →
- How do work, energy and power relate, and how do elastic strings and springs store energy?Work done by a force, kinetic and gravitational potential energy, the work-energy principle, power as the rate of working, Hooke's law for elastic strings and springs, and elastic potential energy.12 min answer →
Further Statistics (optional)
Module overview →- How do you test whether data fit a distribution or whether two factors are independent, and what non-parametric tests are available?The chi-squared goodness-of-fit test and contingency table test for independence, degrees of freedom, and non-parametric tests including the sign test and Wilcoxon signed-rank test.12 min answer →
- How do you work with a continuous random variable defined by a probability density function?Continuous random variables, the probability density function and cumulative distribution function, finding probabilities by integration, and the expectation and variance of a continuous variable.12 min answer →
- How do you describe a discrete random variable and find its expectation and variance?Discrete random variables, the probability distribution, expectation and variance, and the effect of a linear transformation aX + b on the mean and variance.11 min answer →
- How do the Poisson and geometric distributions model counts and waiting times, and when does each apply?The Poisson distribution and its conditions, mean and variance, the sum of independent Poisson variables, the geometric distribution and its mean, and the Poisson approximation to the binomial.12 min answer →