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 TOPIC 1

Topic 1 – Core: Algebra

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TOPIC 3

Topic 3 – Core: Circular functions and trigonometry

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TOPIC 5

Topic 5 – Core: Statistics and probability

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 TOPIC 7

 Topic 7 – Option: Statistics and probability

  • Topic 7.1

    • Cumulative distribution functions for both discrete and continuous distributions.

    • Geometric distribution.

    • Negative binomial distribution.

    • Probability generating functions for discrete random variables.

    • Using probability generating functions to find mean, variance and the distribution of the sum of n independent random variables.

  • Topic 7.2

    • Linear transformation of a single random variable.

    • Mean of linear combinations of n random variables.

    • Variance of linear combinations of n independent random variables.

    • Expectation of the product of independent random variables.

  • Topic 7.3

    • Unbiased estimators and estimates.

    • Comparison of unbiased estimators based on variances.

    • X¯ as an unbiased estimator for μ .

    • S2 as an unbiased estimator for σ2 .

  • Topic 7.4

    • A linear combination of independent normal random variables is normally distributed. In particular, X ~ N(μ,σ2)⇒X¯ ~ N(μ,σ2n) .

    • The central limit theorem.

  • Topic 7.5

    • Confidence intervals for the mean of a normal population.

  • Topic 7.6

    • Null and alternative hypotheses, H0 and H1 .

    • Significance level.

    • Critical regions, critical values, p-values, one-tailed and two-tailed tests.

    • Type I and II errors, including calculations of their probabilities.

    • Testing hypotheses for the mean of a normal population.

  • Topic 7.7

    • Introduction to bivariate distributions.

    • Covariance and (population) product moment correlation coefficient ρ.

    • Proof that ρ=0 in the case of independence and ±1 in the case of a linear relationship between X and Y.

    • Definition of the (sample) product moment correlation coefficient R in terms of n paired observations on X and Y.

    • Its application to the estimation of ρ.Informal interpretation of r, the observed value of R. Scatter diagrams.

    • Topics based on the assumption of bivariate normality: use of the t-statistic to test the null hypothesis ρ=0 .

    • Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of X on Y (E(X)|Y=y) and Y on X (E(Y)|X=x) are linear.

    • Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).

    • Topics based on the assumption of bivariate normalitT

 TOPIC 9

Topic 9 – Option: Calculus

  • Topic 9.1

    • Infinite sequences of real numbers and their convergence or divergence.

  • Topic 9.2

    • Convergence of infinite series.

    • Tests for convergence: comparison test; limit comparison test; ratio test; integral test.

    • The p-series, ∑⁡1np .Series that converge absolutely.

    • Series that converge conditionally.

    • Alternating series.

    • Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

  • Topic 9.3

    • Continuity and differentiability of a function at a point.

    • Continuous functions and differentiable functions.

  • Topic 9.4

    • The integral as a limit of a sum; lower and upper Riemann sums.

    • Fundamental theorem of calculus.

    • Improper integrals of the type ∫a∞f(x)dx .

  • Topic 9.5

    • First-order differential equations.

    • Geometric interpretation using slope fields, including identification of isoclines.

    • Numerical solution of dydx=f(x,y) using Euler’s method.

    • Variables separable.

    • Homogeneous differential equation dydx=f(yx) using the substitution y=vx .Solution of y′+P(x)y=Q(x), using the integrating factor.

  • Topic 9.6

    • Rolle’s theorem.

    • Mean value theorem.

    • Taylor polynomials; the Lagrange form of the error term.

    • Maclaurin series for ex , sinx , cos⁡x , ln⁡(1+x) , (1+x)p , P∈Q .

    • Use of substitution, products, integration and differentiation to obtain other series.

    • Taylor series developed from differential equations.

  • Topic 9.7

    • The evaluation of limits of the form limx→a⁡f(x)g(x) and limx→∞⁡f(x)g(x) .

    • Using l’Hôpital’s rule or the Taylor series.

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TOPIC 2

Topic 2 – Core: Functions and equations

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TOPIC 4

Topic 4 – Core: Vectors

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 TOPIC 6 

Topic 6 – Core: Calculus

 TOPIC 8 

 Topic 8 – Option: Sets, relations and groups

  • Topic 8.1

    • Finite and infinite sets.

    • Subsets.

    • Operations on sets: union; intersection; complement; set difference; symmetric difference.

    • De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

  • Topic 8.2

    • Ordered pairs: the Cartesian product of two sets.

    • Relations: equivalence relations; equivalence classes.

  • Topic 8.3

    • Functions: injections; surjections; bijections.

    • Composition of functions and inverse functions.

  • Topic 8.4

    • Binary operations.

    • Operation tables (Cayley tables).

  • Topic 8.5

    • Binary operations: associative, distributive and commutative properties.

  • Topic 8.6

    • 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.

  • Topic 8.7

    • The definition of a group {G,∗} .

    • The operation table of a group is a Latin square, but the converse is false.

    • Abelian groups.

  • Topic 8.8

    • Example of groups: R, Q, Z and C under addition.

    • Example of groups: integers under addition modulo n.

    • Example of groups: non-zero integers under multiplication, modulo p, where p is prime.

    • Symmetries of plane figures, including equilateral triangles and rectangles.

    • Invertible functions under composition of functions.

  • Topic 8.9

    • The order of a group.

    • The order of a group element.

    • Cyclic groups.

    • Generators.

    • Proof that all cyclic groups are Abelian.

  • Topic 8.10

    • Permutations under composition of permutations.

    • Cycle notation for permutations.

    • Result that every permutation can be written as a composition of disjoint cycles.

    • The order of a combination of cycles.

  • Topic 8.11

    • Subgroups, proper subgroups.

    • Use and proof of subgroup tests.

    • Definition and examples of left and right cosets of a subgroup of a group.

    • 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.)

  • Topic 8.12

    • Definition of a group homomorphism.

    • Definition of the kernel of a homomorphism.

    • Proof that the kernel and range of a homomorphism are subgroups.

    • Proof of homomorphism properties for identities and inverses.

    • Isomorphism of groups.

    • The order of an element is unchanged by an isomorphism.

TOPIC 10

Topic 10 – Option: Discrete mathematics

  • Topic 10.1

    • Strong induction.

    • Pigeon-hole principle.

  • Topic 10.2

    • a|b⇒b=na for some n∈Z .

    • The theorem a|b and a|c⇒a|(bx±cy) where x,y∈Z .

    • Division and Euclidean algorithms.

    • The greatest common divisor, gcd(a,b), and the least common multiple, lcm(a,b), of integers a and b.

    • Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.

  • Topic 10.3

    • Linear Diophantine equations ax+by=c .

  • Topic 10.4

    • Modular arithmetic.

    • The solution of linear congruences.

    • Solution of simultaneous linear congruences (Chinese remainder theorem).

  • Topic 10.5

    • Representation of integers in different bases.

  • Topic 10.6

    • Fermat’s little theorem.

  • Topic 10.7

    • Graphs, vertices, edges, faces.

    • Adjacent vertices, adjacent edges.

    • Degree of a vertex, degree sequence.

    • Handshaking lemma.

    • Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.

    • Subgraphs; complements of graphs

    • .Euler’s relation: v–e+f=2 ; theorems for planar graphs including e⩽3v–6 , e⩽2v–4 , leading to the results that κ5 and κ3,3 are not planar.

  • Topic 10.8

    • Walks, trails, paths, circuits, cycles.

    • Eulerian trails and circuits.

    • Hamiltonian paths and cycles.

  • Topic 10.9

    • Graph algorithms: Kruskal’s; Dijkstra’s.

  • Topic 10.10

    • Chinese postman problem.

    • Travelling salesman problem.

    • Nearest-neighbour algorithm for determining an upper bound.

    • Deleted vertex algorithm for determining a lower bound.

  • Topic 10.11

    • Recurrence relations.

    • Initial conditions, recursive definition of a sequence.

    • Solution of first- and second-degree linear homogeneous recurrence relations with constant coefficients.

    • The first-degree linear recurrence relation un=aun–1+b .

    • Modelling with recurrence relations

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