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Algebra (1) Use and application of quadratic forms including graphing, completing the square, solution of quadratic equations, simultaneous linear and quadratic equations and inequalities.
Algebra (2) Algebraic manipulation of polynomials, factorisation and use of the Remainder Theorem.
Algebra (3) Location and approximate solutions of equation using simple iterative methods. Numerical integration using Simpsonís Rule.
Algebra (4) Binomial series of any rational n. Rational expressions and functions and their simplification. Introduction to Partial Fractions.
Algebra (5) Sum of finite series using formulae for the sum of the natural numbers, the natural numbers squared and cubed. Introduction to mathematical induction.
Algebra (6) The number and nature of the roots of polynomial equations, real complex and repeated. Relationship between the roots and coefficients of a polynomial equation.
Algebra (7) Further partial fractions.
Algebra (8) Function theory including, image of set of points under transformation, odd and even functions, continuity of functions and identification of asymptotes.
Approximate roots Use of the Intermediate Value Theorem and applications of the Newton-Raphson method.
Binomial expansion Binomial expansions with positive integer powers.
Complex Numbers (1) Introduction to complex numbers, their conjugate, Modulus and Argument. Representation of complex numbers in the Argand Diagram and Loci of functions.
Complex Numbers (2) Knowledge of de Moivreís Theorem and its application to geometry and finding the nth roots of complex numbers.
Differentiation (1) Introduction to the derivative of a function as a limiting process. Applications to geometry including finding equations of tangents and normals as well as determining the stationary values of the function.
Differentiation (2) Differentiation of the logarithm and exponential functions, the basic trigonometric functions. Knowledge of the function of a function Rule, the Product and Quotient Rules and implicit differentiation.
Differentiation (3) Formation of simple differential equations.
Differentiation (4) Finding the derivative of functions from first principles. Logarithmic differentiation.
Functions (1) Functions and composition of functions, with their domain and range. Inverse functions and their graphs.
Functions (2) The natural logarithm and exponential functions and their relationship as inverses of each other. Sketching such functions.
Geometry (1) Equations of straight lines with conditions for them to be parallel or perpendicular.
Geometry (2) The coordinate geometry of circles including the quadratic form for its equation, angle in semicircle is right angled, the perpendicular from the centre to a chord bisects the chord and the perpendicularity of radius and tangent.
Geometry (3) Cartesian and parametric equations of curves and conversion between the two forms
Geometry (4) Loci in Cartesian and parametric form. Derivation of the standard forms of the equations of conics including the identification of focus and directrix.
Graphs Graphing functions, applications to solving equations and knowledge of the graphical impact of simple transformations.
Hyperbolic Functions Definition, integration and differentiation of the six basic hyperbolic functions.
Indices & Surds Laws of indices for all rational exponents and the use and manipulation of surds.
Integration (1) Introduction to indefinite and definite integration for integer powers of x. Approximation as the area under the curve and use of Trapezium Rule.
Integration (2) Integration using exponential and trigonometric functions. Logarithm as the integral of the reciprocal of x.
Integration (3) Integration with applications to finding the volume of revolution and the solution of first order differential equations. Integration by parts, and by using partial fractions.
Integration (4) Further integration resulting in inverse trigonometric forms.
Integration (5) Integration to determine arc length, surface area and volumes of revolution. Reduction formulae.
Laws of Logarithms The properties and Laws of logarithms. Use of logarithms to solve equations.
Logic Proofs Proofs by contradiction.
Matrices Matrices and their properties, including inverse. Application to solving equations which have a unique solution. Also the solution of equations by row reduction.
Polar Coordinates Polar coordinates and their application to geometry with the identification of tangents and points of intersection.
Sequences & Series Sequences and series including Arithmetic series (sum of natural numbers) and Geometric series (finite and infinite). Also the use of recursive formulae.
Series (2) Applications of Maclaurin and Taylor series expansions.
Transformations Transformations in the plane in matrix form. Identification of the fixed points of transformations.
Trigonometry (1) Trigonometry including, Sine and Cosine Rules, area of circle, arc length, area of sector, area of segment. Solution of equations using sine, cosine and tangent formulae together with the Pythagorean result for sine and cosine.
Trigonometry (2) Knowledge of secant, cosecant and cotangent functions and the inverse functions for sine, cosine and tangent and the ability to sketch such functions. Extension of the Pythagorean result to secant and cosecant.
Trigonometry (3) Knowledge of the trigonometric expressions for the sum and difference of angles and their application to the solution of linear equations in sine and cosine.
Trigonometry (4) Solution to trigonometric equations using the tan half angle formula.
Vectors Introduction to vectors both in two and three dimensions.
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