Complete course covering Pure Mathematics, Statistics, and Mechanics. Track your progress through each strand.
Simplifying surds, laws of indices, rationalising denominators.
Completing the square, discriminant, solving quadratic equations and inequalities.
Solving simultaneous equations by elimination and substitution, including one linear and one quadratic.
Solving linear and quadratic inequalities, representing solutions on a number line and using set notation.
Algebraic division, factor theorem, remainder theorem, factorising cubics.
Decomposing rational expressions into partial fractions with linear and repeated factors.
Graphs and equations involving the modulus function, solving modulus equations and inequalities.
Domain and range, composite functions, inverse functions and their graphs.
Translations, stretches, and reflections of graphs. Effect of transformations on equations.
Equation of a straight line, gradient, midpoint, distance between two points, parallel and perpendicular lines.
Equation of a circle, finding tangents and normals, circle properties and intersection with lines.
Parametric and Cartesian forms, converting between them, sketching parametric curves.
nth term, common difference, sum of arithmetic series, applications.
Common ratio, nth term, sum to n terms, sum to infinity, convergence.
Using sigma notation to express and evaluate series.
Binomial expansion for positive integer powers, binomial coefficients, approximations.
Graphs of sin, cos, tan and their transformations. Exact values of trig ratios.
Fundamental trig identities and using them to simplify expressions and solve equations.
Solving trig equations in given intervals, finding principal and secondary solutions.
Radian measure, converting degrees and radians, arc length and sector area formulae.
Small angle approximations for sin, cos, and tan. Applications in simplification.
Definitions, graphs, and identities involving sec, cosec, and cot.
Addition formulae for sin(A±B), cos(A±B), tan(A±B). Double angle formulae and their applications.
Expressing a sin x + b cos x in the form R sin(x+a) or R cos(x+a). Solving equations and finding maxima/minima.
Constructing proofs of trigonometric identities using known results.
Definition of logarithms, laws of logs, change of base formula.
Using logarithms to solve exponential equations. Graphs of exponential functions.
The natural logarithm and exponential function. Properties of e^x and ln x.
Exponential growth and decay models, using logarithms to linearise data.
First principles, differentiating x^n, gradient functions, rates of change.
Finding equations of tangents and normals to curves at given points.
Finding and classifying stationary points. Second derivative test. Optimisation problems.
Differentiating composite functions using the chain rule.
Differentiating products of two functions using the product rule.
Differentiating quotients of functions using the quotient rule.
Derivatives of sin x, cos x, tan x and their compositions.
Derivatives of e^x, a^x, ln x and their compositions.
Differentiating implicitly defined functions, finding tangents to implicit curves.
Using the chain rule to connect rates of change in applied contexts.
Integration as the reverse of differentiation, integrating x^n, finding constants of integration.
Evaluating definite integrals, area under a curve, area between curves.
Using substitution to integrate composite functions, reversing the chain rule.
Integration by parts formula, choosing u and dv, repeated application.
Integrating sin, cos, tan, sec² and their compositions. Using trig identities to integrate.
Using partial fraction decomposition to integrate rational functions.
Numerical integration using the trapezium rule, estimating accuracy.
Solving first-order differential equations by separation of variables, modelling with DEs.
Vector notation, magnitude, direction, position vectors, vector arithmetic in 2D.
Extending vector operations to three dimensions, distance in 3D.
Using vectors to prove geometric properties, collinearity, ratios on line segments.
Logical argument, algebraic proof, proof of mathematical statements by direct deduction.
Proving statements by checking all possible cases systematically.
Assuming the negation of a statement and deriving a contradiction to prove it true.
Using change of sign to locate roots of equations, interval bisection.
Fixed-point iteration, staircase and cobweb diagrams, convergence.
Using the Newton-Raphson formula to find approximate roots, graphical interpretation.
Simple random, stratified, systematic, quota, and opportunity sampling. Advantages and limitations.
Interpreting and constructing histograms, box plots, cumulative frequency diagrams, and stem-and-leaf.
Mean, median, mode, range, IQR, variance, standard deviation from raw and grouped data.
Identifying outliers using IQR and standard deviation, data cleaning techniques.
Scatter diagrams, correlation coefficients, regression lines, interpolation and extrapolation.
Probability using set notation, Venn diagrams, addition and multiplication rules.
Conditional probability formula, tree diagrams, independence.
Probability distributions, expected value, variance of discrete random variables.
Binomial distribution conditions, calculating probabilities, mean and variance.
Normal distribution properties, standardising, inverse normal, normal approximation to binomial.
Setting up hypotheses, significance levels, critical regions for binomial tests.
Hypothesis tests for the mean of a normal distribution, z-tests, interpreting results.
Constant acceleration equations (suvat), choosing and applying the correct equation.
Using differentiation and integration for variable acceleration, displacement-velocity-acceleration relationships.
Interpreting and sketching motion graphs, finding displacement and acceleration from graphs.
Newton's three laws, weight, normal reaction, tension, force diagrams, F=ma.
Systems of connected particles, pulleys, applying Newton's laws to each particle.
Friction force, coefficient of friction, limiting equilibrium, motion on rough surfaces.
Moment of a force, principle of moments, equilibrium of rigid bodies, tilting.
Resolving forces into components, inclined planes, equilibrium with angled forces.
Horizontal and vertical components of projectile motion, range, maximum height, time of flight.