 CAPE Training, CXC Training
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RATIONALE
Mathematics is one of the oldest and most universal means of creating, communicating, connecting and applying structural and quantitative ideas. The discipline of Mathematics allows the formulation and solution of realworld problems as well as the creation of new mathematical ideas, both as an intellectual end in itself, as well as a means to increase the success and generality of mathematical applications. This success can be measured by the quantum leap that occurs in the progress made in other traditional disciplines once mathematics is introduced to describe and analyse the problems studied. It is therefore essential that as many persons as possible be taught not only to be able to use mathematics, but also to understand it.
AIMS
The syllabus aims to:
 provide understanding of mathematical concepts and structures, their development and the relationships between them;
 enable the development of skills in the use of mathematical and information, communication and technology (ICT) tools;
 develop an appreciation of the idea of mathematical proof, the internal logical coherence of Mathematics, and its consequent universal applicability;
 develop the ability to make connections between distinct concepts in Mathematics, and between mathematical ideas and those pertaining to other disciplines;
 develop a spirit of mathematical curiosity and creativity, as well as a sense of enjoyment;
 enable the analysis, abstraction and generalisation of mathematical ideas;
 develop in students the skills of recognising essential aspects of concrete, realworld problems, formulating these problems into relevant and solvable mathematical problems and mathematical modelling;
 develop the ability of students to carry out independent or group work on tasks involving mathematical modelling;
 integrate ICT tools and skills;
 provide students with access to more advanced courses in Mathematics and its applications at tertiary institutions.
Course Content

(A) Complex Numbers: UNIT2: COMPLEX NUMBERS, ANALYSIS AND MATRICES MODULE 1: COMPLEX NUMBERS AND CALCULUS II
 recognise the need to use complex numbers to find the roots of the general quadratic equationax2+ bx + c = 0,whenb2 4ac < 0;
 use the concept that complex roots of equations with constant coefficients occur in conjugate pairs;
 write the roots of the equation in that case and relate the sums and products toa, b and c;
 calculate the square root of a complex number;
 express complex numbers in the formaï€«bi wherea, bare real numbers, and identify the real and imaginary parts;
 add, subtract, multiply and divide complex numbers in the formaï€«bi, whereaandbare real numbers;
 ind the principal value of the argument of a nonzero complex number
 find the modulus and conjugate of a given complex number;
 interpret modulus and argument of complex numbers on the Argand diagram;
 represent complex numbers, their sums, differences and products on an Argand diagram;
 find the set of all pointsz(locus of z) on the Argand Diagram such that zsatisfies given properties
 apply De Moivreâ€™s theorem for integral values of n;
 use eix = cos x + i sin x, for real x.

(B) Differentiation II
 find the derivative of ef (x),
 find the derivative of ln f (x) (to include functions of xâ€“ polynomials or trigonometric);
 apply the chain rule to obtain gradients and equations of tangents and normals to curves given by their parametric equations;
 use the concept of implicit differentiation, with the assumption that one of the variables is a function of the other;
 differentiate any combinations of polynomials, trigonometric, exponential and logarithmic functions;
 differentiate inverse trigonometric functions;
 obtain second derivatives,
 find the first and second partial derivatives of u = f (x, y).

(C) Integration II
 express a rational function (proper and improper) in partial fractions
 express an improper rational function as a sum of a polynomial and partial fractions;
 integrate rational functions in Specific Objectives 1 and 2 above;
 integrate trigonometric functions using appropriate trigonometric identities;
 integrate exponential functions and logarithmic functions;
 find integrals
 use substitutions to integrate functions (the substitution will be given in all but the most simple cases)
 use integration by parts forcombinations of functions;
 integrate inverse trigonometric functions
 derive and use reduction formulae to obtain integrals
 use the trapezium rule as an approximation method for evaluating the area under the graph of the function.

UNIT 2 MODULE 2: SEQUENCES, SERIES AND APPROXIMATIONS
 define the concept of a sequence {an} of termsanas a function from the positive integers to the real numbers;
 write a specific term from the formula for thenthterm, or from a recurrence relation;
 describe the behaviour of convergent and divergent sequences, through simple examples;
 apply mathematical induction to establish properties of sequences.
 use the summation
 define a series, as the sum of the terms of a sequence;
 identify the mth term of a series, in the summation notation;
 apply mathematical induction to establish properties of series;
 find the sum to infinity of a convergent series;
 apply the method of differences to appropriate series, and find their sums;
 use the Maclaurin theorem for the expansion of series;
 use the Taylor theorem for the expansion of series.

(C) The Binomial Theorem

(D) Roots of Equations
 test for the existence of a root of f (x) = 0 where f is continuous using the Intermediate Value Theorem;
 use interval bisection to find an approximation for a root in a given interval;
 use linear interpolation to find an approximation for a root in a given interval;
 explain, in geometrical terms, the working of the NewtonRaphson method
 use the NewtonRaphson method to find successive approximations to the roots off(x) = 0, where f is differentiable;
 use a given iteration to determine a root of an equation to a specified degree of accuracy

UNIT 2MODULE 3: COUNTING, MATRICES AND DIFFERENTIAL EQUATIONS On completion of this Module, students should: 1.develop the ability to analyse and solve simple problems dealing with choices and arrangements; 2.developan understanding of the algebra of matrices; 3.develop the ability to analyse and solve systems of linear equations; 4.developskills to model some realworld phenomena by means of differential equations, and solve these; 5.develop the ability to use concepts to model and solve realworld problems.
 (A) Counting: state the principles of counting;
 find the number of ways of arrangingndistinct objects;
 find the number of ways of arrangingnobjects some of which are identical;
 find the number of ways of choosingrdistinct objects from a set ofndistinct objects;
 identifyasamplespace;
 identify the numbers of possibleoutcomes in a given samplespace;
 use Venn diagrams to illustrate the principles of counting;
 use possibility space diagram to identify a sample space
 define and calculate P(A), the probability of an eventAoccurring as the number of possible ways in which Acan occur divided by the total number of possibleways in which all equally likely outcomes, including A, occur
 use the fact
 demonstrate and use the property that the total probability for all possible outcomes in the sample space is 1;
 use the property that P(‘A) = 1 â€“ P(A) is the probability that eventAdoes not occur;
 use the property P(A B) = P (A) + P (B) â€“ P(A B) for event A and B;
 use the property P(A B) = 0 or P (A B) = P (A) + P (B), where AandBare mutually exclusive events;
 use the property P(A B) = P (A) Ã— P (B), where AandBare independent events;
 use the property P(AB)
 use a tree diagram to list all possible outcomes for conditional probability.

(B) Matrices and Systems of Linear Equations
 operate with conformable matrices, carry out simple operations and manipulate matrices using their properties;
 evaluate determinants
 reduce a system of linear equations to echelon form;
 rowreduce the augmented matrix of an n n system of linear equations,n = 2, 3;
 determine whether the system is consistent, and if so, how many solutions it has;
 find all solutions of a consistent system;
 invert a nonsingular 3 x 3 matrix;
 solve a 3×3 system of linear equations, having a nonsingular coefficient matrix, by using its inverse

(C) Differential Equations and Modeling
 solve first order linear differential equations, given that k is a real constant or a function of x, and f is a function;
 solve first order linear differential equations given boundary conditions;
 solve second order ordinary differential equations with constant coefficients of the form
 solve second order ordinary differential equation given boundary conditions;
 use substitution to reduce a second order ordinary differential equation to a suitable form.