## Mathematics (AS1051)

### General note on past papers

Past exam papers are published for illustrative purposes only. They can be used as a study aid but do not provide a definitive guide to either the format of the next exam, the topics that will be examined or the style of questions that will be set. Students should not expect their own exam to be directly comparable with previous papers. Remember that a degree requires an amount of self-study, reading around topics, and lateral thinking - particularly at the higher level modules and for higher marks.

### 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. [1]
Laws of indices, positive and negative rational exponents. Permutations and combinations, expansion of (a+x)n for positive integral n. [2]
Quadratic equations: factorization, completing the square, the formula, relation between roots and coefficients. [1]
Expansion and factorization of algebraic expressions. [2]
Polynomials: division, the remainder theorem. [2]
Manipulation of rational expressions, partial fractions. [2]
Elementary set theory: equality, inclusion, union, intersection, complementation, the empty set, Venn diagrams. [2]
Mathematical reasoning: 'paradoxes', 'fallacies' and errors, the nature of mathematical proof, attention to detail and exceptions, direct proofs, proofs by contradiction, counterexamples. [4]
Mathematical induction and recursion. [2]
Summation of simple series: arithmetic and geometric progressions. [2]
Simple inequalities and the modulus. Real-valued functions of one real variable: graphical representation, trigonometric functions, exponential and logarithmic functions. [4]
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. [3]
Two dimensional coordinate geometry: distance between two points, gradient of a line, equation of a line, Cartesian and parametric equations of simple curves. [3]
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. [5]
Integration: indefinite and definite integrals of polynomial, trigonometric and exponential functions, areas under curves, the methods of substitution and parts. [5]