Operations on the Set of Real Numbers

You will treat absolute value as distance and use its properties; work with powers, nth roots and radicals and rationalise denominators; define the decimal (common) logarithm and use its laws; and use logarithms to model growth and decay, compound interest and the magnitude of earthquakes.

2

Binary Operations

You will define binary operations on a set and test them for closure, commutativity, associativity and distributivity; find identity and inverse elements; build and read Cayley tables; and recognise the structures of a group, a ring and a field, from clock arithmetic and symmetries of a triangle to the arithmetic behind ISBN check digits.

3

Introduction to Logic

You will tell statements from other sentences; combine them with negation, conjunction, disjunction, the conditional and the biconditional; build truth tables and recognise tautologies, contradictions and contingencies; prove equivalences with De Morgan's and the other laws of logic; work with predicates and the quantifiers 'for all' and 'there exists'; and use logic to test arguments, describe sets and design switching circuits.

4

Fundamentals of Trigonometry

You will measure angles in degrees and radians and use arc length and sector area; define the six trigonometric ratios on the unit circle and find exact values for special angles; prove basic identities and reduce any angle to a positive acute angle; sketch and read the graphs of sine, cosine and tangent; and solve any triangle with the sine and cosine rules, with applications to bearings, navigation, angles of elevation and depression, and inclined planes.

5

Linear and Quadratic Equations and Inequalities

You will solve equations and inequalities with absolute values and with linear factors; discuss equations whose coefficients contain a parameter; use the discriminant, the sum and the product of the roots of a quadratic; reduce bi-quadratic, nested, irrational and reciprocal equations to quadratics; solve quadratic inequalities with a sign table and a graph; solve systems in which one or both equations are quadratic; and model real problems with all of these.

6

Polynomial, Rational and Irrational Functions

You will factorise polynomials with the remainder and factor theorems; find the domain and range of polynomial, rational and irrational functions; add, multiply, divide and compose functions and find inverse functions; test functions for parity; split rational functions into partial fractions; and use rational and irrational functions to model real situations.

7

Limits of Polynomial, Rational and Irrational Functions

You will describe neighbourhoods and what it means for a variable to tend to a number; read and prove limits from tables, graphs and the epsilon–delta definition; use the operations on limits and the squeeze theorem; find one-sided limits, infinite limits and limits at infinity; remove indeterminate forms by factorising and by conjugates; test continuity and classify discontinuities; and find the vertical, horizontal and oblique asymptotes of a curve.

8

Differentiation of Polynomial, Rational and Irrational Functions and Its Applications

You will define the derivative as the limit of a rate of change and differentiate from first principles; use the sum, product, quotient and chain rules and find higher-order derivatives; write tangents and normals; interpret derivatives as rates of change, velocity and acceleration; use the mean value theorems of Rolle and Lagrange and L'Hôpital's rule; find where a function increases or decreases, its turning points, concavity and points of inflexion; sketch polynomial, rational and irrational curves; and solve optimisation problems.

Units