Maths for CAPE® Examinations Volume 2

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Contents

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Language

English

ISBN

9780230465749

Cover

Title Page

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Contents

INTRODUCTION

MODULE 1 COMPLEX NUMBERS AND CALCULUS II

CHAPTER 1 COMPLEX NUMBERS

Complex numbers as an extension to the real numbers

Powers of i

Algebra of complex numbers

Addition of complex numbers

Subtraction of complex numbers

Multiplication of a complex number by a real number

Multiplication of complex numbers

Equality of complex numbers

Conjugate of a complex number

Division of complex numbers

Square root of a complex number

Roots of a polynomial

Quadratic equations

Other polynomials

The Argand diagram

Addition and subtraction on the Argand diagram

Multiplication by i

Modulus (length) of a complex number

Argument of a complex number

Trigonometric or polar form of a complex number

Exponential form of a complex number

De Moivre’s theorem

Locus of a complex number

Circles

Perpendicular bisector of a line segment

Half-line

Straight line

Inequalities

Intersecting loci

Cartesian form of loci

CHAPTER 2 DIFFERENTIATION

Standard differentials

Differentiation of ln x

Differentiation of ex

Chain rule (function of a function rule)

Differentiating exponential functions of the form y = ax

Differentiating logarithms of the form y = loga x

Differentiation of combinations of functions

Differentiation of combinations involving trigonometric functions

Tangents and normals

Gradients of tangents and normals

Equations of tangents and normals

Implicit differentiation

Differentiation of inverse trigonometric functions

Differentiation of y = sin–1x

Differentiation of y = tan–1x

Second derivatives

Parametric differentiation

First derivative of parametric equations

Second derivative of parametric equations

Partial derivatives

First order partial derivatives

Second order partial derivatives

Applications of partial derivatives

CHAPTER 3 PARTIAL FRACTIONS

Rational fractions

Proper fractions: Unrepeated linear factors

Proper fractions: Repeated linear factors

Proper fractions: Unrepeated quadratic factors

Proper fractions: Repeated quadratic factors

Improper fractions

CHAPTER 4 INTEGRATION

Integration by recognition

When the numerator is the differential of the denominator

The form ʃf'(x)[f(x)]n dx, n ǂ –1

The form ʃf'(x)ef(x)dx

Integration by substitution

Integration by parts

Integration using partial fractions

Integration of trigonometric functions

Integrating sin2 x and cos2 x

Integrating sin3 x and cos3 x

Integrating powers of tan x

Integrating products of sines and cosines

Finding integrals using the standard forms

CHAPTER 5 REDUCTION FORMULAE

Reduction formula for ʃsinn x dx

Reduction formula for ʃcosn x dx

Reduction formula for ʃtann x dx

Other reduction formulae

CHAPTER 6 TRAPEZOIDAL RULE (TRAPEZIUM RULE)

The area under a curve

MODULE 1 TESTS

MODULE 2 SEQUENCES, SERIES AND APPROXIMATIONS

CHAPTER 7 SEQUENCES

Types of sequence

Convergent sequences

Divergent sequences

Oscillating sequences

Periodic sequences

Alternating sequences

The terms of a sequence

Finding the general term of a sequence by identifying a pattern

A sequence defined as a recurrence relation

Convergence of a sequence

CHAPTER 8 SERIES

Writing a series in sigma notation (∑)

Sum of a series

Sum of a series in terms of n

Method of differences

Convergence of a series

Tests for convergence of a series

CHAPTER 9 PRINCIPLE OF MATHEMATICAL INDUCTION (PMI): SEQUENCES AND SERIES

PMI and sequences

PMI and series

CHAPTER 10 BINOMIAL THEOREM

Pascal’s triangle

Factorial notation

Combinations

General formula for nCr

Binomial theorem for any positive integer n

The term independent of x in an expansion

Extension of the binomial expansion

Approximations and the binomial expansion

Partial fractions and the binomial expansion

CHAPTER 11 ARITHMETIC AND GEOMETRIC PROGRESSIONS

Arithmetic progressions

Sum of the first n terms of an AP

Proving that a sequence is an AP

Geometric progressions

Sum of the first n terms of a GP (Sn)

Sum to infinity

Proving that a sequence is a GP

Convergence of a geometric series

CHAPTER 12 NUMERICAL TECHNIQUES

The intermediate value theorem (IMVT)

Finding the roots of an equation

Graphical solution of equations

Interval bisection

Linear interpolation

Newton–Raphson method for finding the roots of an equation

CHAPTER 13 POWER SERIES

Power series and functions

Taylor expansion

The Maclaurin expansion

Maclaurin expansions of some common functions

MODULE 2 TESTS

MODULE 3 COUNTING, MATRICES AND DIFFERENTIAL EQUATIONS

CHAPTER 14 PERMUTATIONS AND COMBINATIONS

The counting principles

Multiplication rule

Addition rule

Permutations

Permutations of n distinct objects

Permutation of r out of n distinct objects

Permutations with repeated objects

Permutations with restrictions

Permutations with restrictions and repetition

Combinations

Combinations with repetition

CHAPTER 15 PROBABILITY

Sample space and sample points

Events: mutually exclusive; equally likely

Probability

Rules of probability

Conditional probability

Tree diagrams

Probability and permutations

Probability and combinations

CHAPTER 16 MATRICES

Matrices: elements and order

Square matrices

Equal matrices

Zero matrix

Addition and subtraction of matrices

Multiplication of a matrix by a scalar

Properties of matrix addition

Matrix multiplication

Properties of matrix multiplication

Identity matrix

Multiplication of square matrices

Transpose of a matrix

Properties of the transpose of a matrix

Determinant of a square matrix

Determinant of a 2 × 2 matrix

Determinant of a 3 × 3 matrix

Properties of determinants

Singular and non-singular matrices

Solving equations using determinants (Cramer’s rule)

Using Cramer’s rule to solve three equations in three unknowns

Inverse of a matrix

Inverse of a 2 × 2 matrix

Cofactors of a 3 × 3 matrix

Inverse of a 3 × 3 matrix

Properties of inverses

Systems of linear equations

Row reduction to echelon form

Finding the inverse of a matrix by row reduction

Solving simultaneous equations using row reduction

Systems of linear equations with two unknowns

Intersecting lines

Parallel lines

Lines that coincide

Systems of linear equations with three unknowns

Unique solution

No solutions

Infinite set of solutions

Solution of linear equations in three unknowns: geometrical interpretation

Applications of matrices

CHAPTER 17 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELLING

First order linear differential equations

Practical applications

Second order differential equations

When the roots of the AQE are real and equal

When the roots of the AQE are real and distinct

When the roots of the AQE are complex

Non-homogeneous second order differential equations

When f(x) is a polynomial of degree n

When f(x) is a trigonometric function

When f(x) is an exponential function

Equations reducible to a recognisable form

Mathematical modelling

MODULE 3 TESTS

UNIT 2—MULTIPLE CHOICE TESTS

INDEX

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