1.4 Proof by mathematical induction Forming conjectures to be proved by mathematical induction
1.5 Complex numbers: the number i; the terms real part, imaginary part, conjugate, modulus and argument Cartesian form z = a + ib Modulus-argument form z = r(cosθ + isinθ) The complex plane
1.6 Sums, products and quotients of complex numbers
1.7 De Moivre’s theorem Powers and roots of a complex number
1.8 Conjugate roots of polynomial equations with real coefficients
4.1 Definition of a matrix: the terms element, row, column and order
4.2 Algebra of matrices: equality; addition; subtraction; multiplication by a scalar Multiplication of matrices Identity and zero matrices
4.3 Determinant of a square matrix Calculation of 2 x 2 and 3 x 3 determinants Inverse of a matrix: conditions for its existence
4.4 Solution of systems of linear equations (a maximum of three equations in three unknowns) Conditions for the existence of a unique solution, no solution and an infinity of solutions
7.7 Graphical behaviour of functions: tangents and normals, behaviour for large |x|; asymptotes The significance of the second derivative; distinction between maximum and minimum points Points of inflexion with zero and non-zero gradients
A1 Expectation algebra Linear transformation of a single random variable Mean and variance of linear combinations of two independent random variables Extension to linear combinations of n independent random variables
A2 Cumulative distribution functions Discrete distributions: uniform, Bernoulli, binomial, negative binomial, Poisson, geometric, hypergeometric Continuous distributions: uniform, exponential, normal
A3 Distribution of the sample mean The distribution of linear combinations of independent normal random variables The central limit theorem The approximate normality of the proportion of successes in a large sample
A4 Finding confidence intervals for the mean of a population Finding confidence intervals for the proportion of successes in a population
A5 Significance testing for a mean Significance testing for a proportion Null and alternative hypotheses H0 and H1 Type I and Type II errors Significance levels; critical region, critical values, p-values; one-tailed and two-tailed tests
A6 The chi-squared distribution: degrees of freedom, υ The χ^2 statistic The χ^2 goodness of fit test Contingency tables: the χ^2 test for the independence of two variables
B5 Binary operations with associative, distributive and commutative properties
B6 The identity element e The inverse a^(−1) of an element a Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse Proofs of the uniqueness of the identity and inverse elements
B7 The axioms of a group {G, *} Abelian groups
B8 R, Q, Z, C under addition; matrices of the same order under addition; invertible matrices under multiplication; symmetries of a triangle, rectangle; invertible functions under composition of functions; permutations under composition of permutations
B9 Finite and infinite groups The order of a group element and the order of a group
B10 Cyclic groups Proof that all cyclic groups are Abelian
B11 Subgroups, proper subgroups Use and proof of subgroup tests Lagrange’s theorem Use and proof of the result that the order of a finite group is divisible by the order of any element (Corollary to Lagrange’s theorem)
B12 Isomorphism of groups Proof of isomorphism properties for identities and inverses
C3 Series that converge absolutely Series that converge conditionally Alternating series
C4 Power series: radius of convergence and interval of convergence Determination of the radius of convergence by the ratio test
C5 Taylor polynomials and series, including the error term Maclaurin series for e^x, sinx, cosx, arctanx, ln(1 + x), (1 + x)^p Use of substitution to obtain other series The evaluation of limits using l’Hôpital’s Rule and/or the Taylor series
D1 Division and Euclidean algorithms The greatest common divisor, gcd(a, b), and the least common multiple, lcm(a, b), of integers a and b Relatively prime numbers; prime numbers and the fundamental theorem of arithmetic
D2 Representation of integers in different bases
D3 Linear diophantine equations ax + by = c
D4 Modular arithmetic Linear congruences Chinese remainder theorem
D10 “Chinese postman” problem (“route inspection”) “Travelling salesman” problem Algorithms for determining upper and lower bounds of the travelling salesman problem
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