## 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. [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]

### Recommended books

- Core maths: for advanced level / L. Bostock, S. Chandler (Nelson Thornes, 2000) [City library shelfmark: 510 BOS]
- Further pure mathematics / L. Bostock, S. Chandler, C. Rourke (Cheltenham : Thornes, 1982) [City library shelfmark: 510 BOS]
- Understanding pure mathematics / A.J. Sadler, D.W.S. Thorning (Oxford: Oxford University Press, 1987) [City library shelfmark: 510 SAD]
- A course in pure mathematics / M.M. Gow (Hodder & Stoughton, 1960) [City library shelfmark: 510 GOW]