User interface language: English | Español

Section name
Topic 1 - Linear Algebra
Definition of a matrix: the terms element, row, column and order for \(m \times n\) matrices.
Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for \(m \times n\) matrices.
Multiplication of matrices.
Properties of matrix multiplication: associativity, distributivity.
Identity and zero matrices.
Transpose of a matrix including \({A^{\text{T}}}\) notation: \({(AB)^{\text{T}}} = {B^{\text{T}}}{A^{\text{T}}}\) .
Definition and properties of the inverse of a square matrix: \({\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}\) , \({\left( {{A^{\text{T}}}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^{\text{T}}}\) , \({\left( {{A^n}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^n}\) .
Calculation of \({A^{ - 1}}\) .
Use of elementary row operations to find \({A^{ - 1}}\) .
Formulae for the inverse and determinant of a \(2 \times 2\) matrix and the determinant of a \(3 \times 3\) matrix.
Elementary row and column operations for matrices.
Scaling, swapping and pivoting.
Corresponding elementary matrices.
Row reduced echelon form.
Row space, column space and null space.
Row rank and column rank and their equality.
Solutions of \(m\) linear equations in \(n\) unknowns: both augmented matrix method, leading to reduced row echelon form method, and inverse matrix method, when applicable.
The vector space \({\mathbb{R}^n}\) .
Linear combinations of vectors.
Spanning set.
Linear independence of vectors.
Basis and dimension for a vector space.
Linear transformations: \(T(u + v) = T(u) + T(v)\) , \(T(ku) = kT(u)\) .
Composition of linear transformations.
Domain, range, codomain and kernel.
Result and proof that the kernel is a subspace of the domain.
Result and proof that the range is a subspace of the codomain.
Rank-nullity theorem (proof not required).
Result that any linear transformation can be represented by a matrix, and the converse of this result.
Result that the numbers of linearly independent rows and columns are equal, and this is the dimension of the range of the transformation (proof not required).
Application of linear transformations to solutions of system of equations.
Solution of \(A{\text{x}} = {\text{b}}\) .
Geometric transformations represented by \(2 \times 2\) matrices include general rotation, general reflection in \(y = (\tan \alpha )x\) , stretches parallel to axes, shears parallel to axes, and projection onto \(y = (\tan \alpha )x\).
Compositions of the above transformations.
Geometric interpretation of determinant.
Eigenvalues and eigenvectors of \(2 \times 2\) matrices.
Characteristic polynomial of \(2 \times 2\) matrices.
Diagonalization of \(2 \times 2\) matrices (restricted to the case where there are distinct real eigenvalues).
Applications to powers of \(2 \times 2\) matrices.
Topic 2 - Geometry
Similar and congruent triangles.
Euclid’s theorem for proportional segments in a right-angled triangle.
Centres of a triangle: orthocentre, incentre, circumcentre and centroid.
Circle geometry.
Tangents; arcs, chords and secants.
In a cyclic quadrilateral, opposite angles are supplementary, and the converse.
Angle bisector theorem; Apollonius’ circle theorem, Menelaus’ theorem; Ceva’s theorem; Ptolemy’s theorem for cyclic quadrilaterals.
Proofs of these theorems and converses.
Finding equations of loci.
Coordinate geometry of the circle.
Tangents to a circle.
Conic sections.
The parabola, ellipse and hyperbola, including rectangular hyperbola.
Focus–directrix definitions.
Tangents and normals.
Parametric equations.
Parametric differentiation.
Tangents and normals.
The standard parametric equations of the circle, parabola, ellipse, rectangular hyperbola, hyperbola.
The general conic \(a{x^2} + 2bxy + c{y^2} + dx + ey + f = 0\) and the quadratic form \({x^{\text{T}}}Ax = a{x^2} + 2bxy + c{y^2}\) .
Diagonalizing the matrix \(A\) with the rotation matrix \(P\) and reducing the general conic to standard form.
Topic 3 - Statistics and probability
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.
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.
Unbiased estimators and estimates.
Comparison of unbiased estimators based on variances.
\(\overline X \) as an unbiased estimator for \(\mu \) .
\({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .
A linear combination of independent normal random variables is normally distributed. In particular, \(X{\text{ ~ N}}\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ N}}\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) .
The central limit theorem.
Confidence intervals for the mean of a normal population.
Null and alternative hypotheses, \({H_0}\) and \({H_1}\) .
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.
Introduction to bivariate distributions.
Covariance and (population) product moment correlation coefficient \(\rho \).
Proof that \(\rho = 0\) in the case of independence and \(\pm 1\) 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.
The following topics are based on the assumption of bivariate normality.
Use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .
Knowledge of the facts that the regression of \(X\) on \(Y\) (\({\text{E}}(X)|Y = y\)) and \(Y\) on \(X\) (\({\text{E}}(Y)|X = x\)) are linear.
Least-squares estimates of these regression lines (proof not required).
The use of these regression lines to predict the value of one of the variables given the value of the other.
Topic 4 - Sets, relations and groups
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).
Ordered pairs: the Cartesian product of two sets.
Relations: equivalence relations; equivalence classes.
Functions: injections; surjections; bijections.
Composition of functions and inverse functions.
Binary operations.
Operation tables (Cayley tables).
Binary operations: associative, distributive and commutative properties.
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.
The definition of a group \(\left\{ {G, * } \right\}\) .
The operation table of a group is a Latin square, but the converse is false.
Abelian groups.
Example of groups: \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\) and \(\mathbb{C}\) under addition.
Example of groups: integers under addition modulo \(n\).
Example of groups: non-zero integers under multiplication, modulo \(p\), where \(p\) is prime.
Example of groups: symmetries of plane figures, including equilateral triangles and rectangles.
Example of groups: invertible functions under composition of functions.
The order of a group.
The order of a group element.
Cyclic groups.
Proof that all cyclic groups are Abelian.
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.
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.)
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 5 - Calculus
Infinite sequences of real numbers and their convergence or divergence.
Convergence of infinite series.
Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
The \(p\)-series, \(\mathop \sum \nolimits \frac{1}{{{n^p}}}\) .
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.
Continuity and differentiability of a function at a point.
Continuous functions and differentiable functions.
The integral as a limit of a sum; lower and upper Riemann sums.
Fundamental theorem of calculus.
Improper integrals of the type \(\int\limits_a^\infty {f\left( x \right){\text{d}}} x\) .
First-order differential equations.
Geometric interpretation using slope fields, including identification of isoclines.
Numerical solution of \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {x,y} \right)\) using Euler’s method.
Variables separable.
Homogeneous differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right)\) using the substitution \(y = vx\) .
Solution of \(y' + P\left( x \right)y = Q\left( x \right)\), using the integrating factor.
Rolle’s theorem.
Mean value theorem.
Taylor polynomials; the Lagrange form of the error term.
Maclaurin series for \({{\text{e}}^x}\) , \(\sin x\) , \(\cos x\) , \(\ln (1 + x)\) , \({(1 + x)^p}\) , \(P \in \mathbb{Q}\) .
Use of substitution, products, integration and differentiation to obtain other series.
Taylor series developed from differential equations.
The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f(x)}}{{g(x)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}{{g(x)}}\) .
Using l’Hôpital’s rule or the Taylor series.
Topic 6 - Discrete mathematics
Strong induction.
Pigeon-hole principle.
\(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .
The theorem \(\left. a \right|b\) and \(\left. a \right|c \Rightarrow \left. a \right|(bx \pm cy)\) where \(x,y \in \mathbb{Z}\) .
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.
Linear Diophantine equations \(ax + by = c\) .
Modular arithmetic.
The solution of linear congruences.
Solution of simultaneous linear congruences (Chinese remainder theorem).
Representation of integers in different bases.
Fermat’s little theorem.
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 \(e \leqslant 3v - 6\) , \(e \leqslant 2v - 4\) , leading to the results that \({\kappa _5}\) and \({\kappa _{3,3}}\) are not planar.
Walks, trails, paths, circuits, cycles.
Eulerian trails and circuits.
Hamiltonian paths and cycles.
Graph algorithms: Kruskal’s; Dijkstra’s.
Chinese postman problem.
Travelling salesman problem.
Nearest neighbour algorithm for determining an upper bound.
Deleted vertex algorithm for determining a lower bound.
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 \({u_n} = a{u_{n - 1}} + b\) .
Modelling with recurrence relations.