← Northern Ireland Further Maths
Northern Ireland · CCEASyllabus
Further Maths syllabus, dot point by dot point
Every dot point in the Northern Ireland 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.
A2 1 Pure Mathematics
Module overview →- How does de Moivre's theorem let you take powers and roots of complex numbers?De Moivre's theorem, the exponential (Euler) form of a complex number, using de Moivre to derive trigonometric identities, and finding the nth roots of a complex number.12 min answer →
- How do you solve first-order and second-order linear differential equations?Solving first-order linear differential equations by the integrating factor, second-order linear equations with constant coefficients via the auxiliary equation, complementary function and particular integral, and modelling with differential equations.13 min answer →
- How do you build a Maclaurin series and evaluate improper integrals, arc lengths and surface areas?Maclaurin series expansions of standard functions, improper integrals with infinite limits or discontinuities, the arc length of a curve, and the area of a surface of revolution.12 min answer →
- What are eigenvalues and eigenvectors, and how do you find them for a 2x2 matrix?Eigenvalues and eigenvectors of a 2x2 matrix, the characteristic equation, invariant lines and lines of invariant points of a transformation, and using eigenvectors to describe the geometry of a matrix.12 min answer →
- What are the hyperbolic functions, and how do you differentiate, integrate and invert them?The definitions of the hyperbolic functions in terms of the exponential function, their identities, derivatives and integrals, and the logarithmic forms of the inverse hyperbolic functions.12 min answer →
- How do polar coordinates describe curves, and how do you find the area they enclose?Polar coordinates and their relationship to Cartesian coordinates, sketching curves given in polar form, and the area enclosed by a polar curve using the half r squared integral.12 min answer →
A2 2 Applied Mathematics
Module overview →- How do you model circular motion and simple harmonic motion in mechanics?Circular motion with angular speed, centripetal acceleration and force, motion in a horizontal and vertical circle, and simple harmonic motion with its defining equation, period and energy.13 min answer →
- How do graph and network algorithms solve route, connection and ordering problems?Graphs and networks, the minimum spanning tree by Kruskal's and Prim's algorithms, the shortest path by Dijkstra's algorithm, and sorting and route-inspection ideas in decision mathematics.13 min answer →
- How do you handle motion with variable acceleration and the flight of a projectile?Kinematics with variable acceleration using calculus, motion in two dimensions with vectors, and projectile motion treating the horizontal and vertical components separately.13 min answer →
- How do you set up and carry out a hypothesis test, and use continuous and Poisson distributions?The Poisson distribution as a model, the normal distribution and standardising, the Central Limit Theorem for the distribution of the sample mean, and hypothesis testing including null and alternative hypotheses, significance levels and conclusions.13 min answer →
- How do momentum and impulse describe collisions, and what does the coefficient of restitution add?Linear momentum and impulse, conservation of momentum in collisions, Newton's experimental law with the coefficient of restitution, and kinetic energy lost in impacts.13 min answer →
AS 1 Pure Mathematics
Module overview →- What are complex numbers, and how do you add, multiply, divide and represent them?The imaginary unit, arithmetic of complex numbers, the complex conjugate, the Argand diagram, modulus and argument, and complex roots of real polynomial equations occurring in conjugate pairs.12 min answer →
- How do you sketch rational functions and find their stationary points and asymptotes?Curve sketching of rational functions, finding vertical and horizontal asymptotes, oblique asymptotes, the range of values a rational function can take, and locating stationary points.12 min answer →
- How does proof by mathematical induction establish a result for all positive integers?The principle of mathematical induction and its use to prove results about summation of series, divisibility and other statements indexed by the positive integers.12 min answer →
- How do matrices encode linear transformations, and how do you invert a 2x2 matrix?Matrix algebra including addition, multiplication and the identity, the determinant and inverse of a 2x2 matrix, and matrices as linear transformations of the plane including rotations, reflections and enlargements.12 min answer →
- How are the roots of a polynomial related to its coefficients?The relationships between the roots and coefficients of quadratic, cubic and quartic equations, and forming new equations whose roots are functions of the original roots.12 min answer →
- How do you sum finite series using standard results and the method of differences?Summation of finite series using the standard results for the sum of r, r squared and r cubed, and the method of differences for telescoping sums.12 min answer →
AS 2 Applied Mathematics
Module overview →- How do Newton's laws relate force to motion, and how do you handle friction and connected bodies?Forces as vectors, resolving and equilibrium, Newton's three laws, the equation of motion F equals ma, friction and the coefficient of friction, and connected particles over pulleys.12 min answer →
- How do the constant-acceleration equations and motion graphs describe motion in a straight line?Kinematics of motion in a straight line with constant acceleration, the suvat equations, vertical motion under gravity, and interpreting displacement-time and velocity-time graphs.12 min answer →
- How do you calculate probabilities and work with discrete random variables and the binomial distribution?Probability of combined events, mutually exclusive and independent events, conditional probability, discrete random variables with their expectation and variance, and the binomial distribution as a model.12 min answer →
- How do you sample a population fairly, summarise data, and measure its centre and spread?Statistical sampling methods, presenting and interpreting data, measures of central tendency (mean, median, mode) and measures of variation (range, interquartile range, variance and standard deviation).12 min answer →