1.2 Euclid’s theorem for proportional segments in a right-angled triangle. Proportional division of a line segment (internal and external); the harmonic ratio; proportional segments in right-angled triangles.
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.
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.
