Factorising, completing the square, vertex form, discriminant, sketching
Linear inequalities, quadratic inequalities, set notation
Sketching polynomial equations and interpreting algebraic solutions graphically, including transformations and key features.
Factor theorem, remainder theorem, finding factors and roots of polynomials
Polynomial long division, algebraic division, manipulating and simplifying polynomial expressions, sketching polynomial curves
Using binomial theorem for positive integer n to expand expressions and find specific terms or coefficients.
Transformations of functions (translations, stretches, reflections), successive transformations, graph transformations
Composition of functions f(g(x)), inverse functions, domain and range, function notation and definitions
Decomposing rational functions into partial fractions, including linear and repeated linear factors.
Extending binomial theorem to (1+x)^n for any rational n, including series expansions and approximations for small x.
Using partial fractions with binomial expansion to find series in ascending powers of x
Operations with complex numbers: addition, subtraction, multiplication, division, conjugates, solving equations with complex roots
Argand diagram, modulus and argument, polar/exponential form, loci in the complex plane
Counting principles, arrangements, permutations, factorial problems
Combinations, choosing/selecting, binomial coefficients, nCr problems
Measures of location (mean, median, mode) and spread (range, variance, standard deviation), coding data, and identifying outliers.
Probability notation, addition law, inclusion-exclusion principle, and mutually exclusive events.
Using tree diagrams to calculate probabilities for multi-stage experiments and conditional probability.
Applying the principle of inclusion-exclusion for probabilities of unions of events, using Venn diagrams.
Independent events and their properties, calculating probabilities using P(A∩B) = P(A)P(B).
Conditional probability using P(A|B) notation, Venn diagrams, two-way tables, and Bayes' theorem applications.
Discrete probability distributions, expectation E(X), variance Var(X), and the effect of linear coding.
Discrete uniform distribution where all outcomes are equally likely, calculating expectation and variance.
Binomial distribution B(n,p) as a model for repeated independent trials, calculating probabilities, mean np and variance np(1-p).
Geometric distribution for number of trials until first success, including expectation and variance.
Hypergeometric distribution for sampling without replacement from finite populations.
Negative binomial distribution for number of trials until r successes.
Modelling real-world situations with probability distributions and conducting hypothesis tests.
Hypothesis testing using binomial distributions, including one-tailed and two-tailed tests, critical values and p-values.
Graphical representations of data including cumulative frequency curves, box plots, and histograms with unequal class widths.
Continuous random variables, probability density functions (PDFs), and cumulative distribution functions (CDFs).
Continuous uniform distribution on interval [a,b], calculating probabilities using areas and finding expectation.
Geometric probability for continuous random variables in spatial contexts.
Normal distribution N(μ,σ²), using tables or calculators, finding probabilities and inverse normal calculations.
Approximating binomial B(n,p) to normal distribution when n is large, including continuity correction.
Sign change, bisection, decimal search, change of sign to locate roots
Iterative methods, rearranging to x=g(x), staircase/cobweb diagrams, convergence
Newton-Raphson method for finding approximate solutions to equations using tangent line iterations x_{n+1} = x_n - f(x_n)/f'(x_n).