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1.1 Triangles: medians; altitudes; angle bisectors; perpendicular bisectors of sides. Concurrency: orthocentre; incentre; circumcentre; centroid, Euler line.
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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.
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Option A: Statistics and probability
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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.
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A2 Cumulative distribution functions. Discrete distributions: uniform, Bernoulli, binomial, negative binomial, Poisson, geometric, hypergeometric. Continuous distributions: uniform, exponential, normal.
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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.
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A4 Finding confidence intervals for the mean of a population. Finding confidence intervals for the proportion of successes in a population.
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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.
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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.
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B2 Ordered pairs: the Cartesian product of two sets. Relations; equivalence relations; equivalence classes.
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B3 Functions: injections; surjections; bijections. Composition of functions and inverse functions.
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B4 Binary operations. Operation tables (Cayley tables).
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B5. Binary operations with associative, distributive and commutative properties.
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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.
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B7 The axioms of a group {G, *}. Abelian groups.
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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.
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B9 Finite and infinite groups. The order of a group element and the order of a group.
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B10 Cyclic groups. Proof that all cyclic groups are Abelian.
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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.)
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B12 Isomorphism of groups. Proof of isomorphism properties for identities and inverses.
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C1 Infinite sequences of real numbers. Limit theorems as n approaches infinity. Limit of a sequence. Improper integrals. The integral as a limit of a sum; lower sum and upper sum. (30)
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C3 Series that converge absolutely. Series that converge conditionally. Alternating series.
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C4 Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.
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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.
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Option D: Discrete mathematics
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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.
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D2 Representation of integers in different bases.
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D3 Linear diophantine equations ax + by = c.
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D4 Modular arithmetic. Linear congruences. Chinese remainder theorem.
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D5 Fermat’s little theorem.
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D6 Graphs, vertices, edges. Adjacent vertices, adjacent edges. Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs, trees, weighted graphs. Subgraphs; complements of graphs. Graph isomorphism.
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D7 Walks, trails, paths, circuits, cycles. Hamiltonian paths and cycles; Eulerian trails and circuits.
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D8 Adjacency matrix. Cost adjacency matrix.
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D9 Graph algorithms: Prim’s; Kruskal’s; Dijkstra’s.
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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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