## Mathematics (AS1051)

### Aims

The course is primarily a revision of those topics common to all A-level mathematics syllabuses with emphasis on the solution of problems of a standard slightly above A-level. It also includes a simple introduction to mathematical reasoning and proof.

### Description

The course is primarily a revision of those topics common to all A-level mathematics syllabuses with emphasis on the solution of problems of a standard slightly above A-level. It also includes a simple introduction to mathematical reasoning and proof.

### Syllabus

A simple exposition of the historical development of number systems: the sets of natural numbers, integers, rational numbers and real numbers. 
Laws of indices, positive and negative rational exponents. Permutations and combinations, expansion of (a+x)n for positive integral n. 
Quadratic equations: factorization, completing the square, the formula, relation between roots and coefficients. 
Expansion and factorization of algebraic expressions. 
Polynomials: division, the remainder theorem. 
Manipulation of rational expressions, partial fractions. 
Elementary set theory: equality, inclusion, union, intersection, complementation, the empty set, Venn diagrams. 
Mathematical reasoning: 'paradoxes', 'fallacies' and errors, the nature of mathematical proof, attention to detail and exceptions, direct proofs, proofs by contradiction, counterexamples. 
Mathematical induction and recursion. 
Summation of simple series: arithmetic and geometric progressions. 
Simple inequalities and the modulus. Real-valued functions of one real variable: graphical representation, trigonometric functions, exponential and logarithmic functions. 
Solution of equations: two or three linear simultaneous equations in two or three unknowns, one linear and one quadratic equation, equations of the form ax = b, trigonometric equations. 
Two dimensional coordinate geometry: distance between two points, gradient of a line, equation of a line, Cartesian and parametric equations of simple curves. 
Differentiation: rates of change, differentiation of polynomial, trigonometric, exponential and logarithmic functions, differentiation of products, quotients and composites, maxima and minima, equations of tangents and normals to curves. 
Integration: indefinite and definite integrals of polynomial, trigonometric and exponential functions, areas under curves, the methods of substitution and parts.