PURE MATHEMATICS Unit 2 FOR CAPE® EXAMINATIONS

DIPCHAND BAHALL

CAPE® is a registered trade mark of the Caribbean Examinations Council (CXC). Pure Mathematics for CAPE® Examinations Unit 2 is an independent publication and has not been authorised, sponsored, or otherwise approved by CXC.

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ISBN: 978-0-230-46574-9 AER

Text © Dipchand Bahall 2013Design and illustration © Macmillan Publishers Limited 2013

First published in 2013

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Contents

INTRODUCTION ix

MODULE 1 COMPLEX NUMBERS AND CALCULUS II

CHAPTER 1 COMPLEX NUMBERS 2

Complex numbers as an extension to the real numbers 3

Powers of i 4

Algebra of complex numbers 5

Addition of complex numbers 5

Subtraction of complex numbers 5

Multiplication of a complex number by a real number 5

Multiplication of complex numbers 5

Equality of complex numbers 6

Conjugate of a complex number 7

Division of complex numbers 8

Square root of a complex number 9

Roots of a polynomial 11

Quadratic equations 11

Other polynomials 13

The Argand diagram 15

Addition and subtraction on the Argand diagram 15

Multiplication by i 16

Modulus (length) of a complex number 16

Argument of a complex number 17

Trigonometric or polar form of a complex number 19

Exponential form of a complex number 21

De Moivre’s theorem 22

Locus of a complex number 27

Circles 27

Perpendicular bisector of a line segment 28

Half-line 29

Straight line 30

Inequalities 31

Intersecting loci 33

Cartesian form of loci 35

CHAPTER 2 DIFFERENTIATION 41

Standard differentials 42

Differentiation of ln x 42

Differentiation of ex 43iii

iv

Chain rule (function of a function rule) 43

Differentiating exponential functions of the form y = ax 46

Differentiating logarithms of the form y = loga x 47

Differentiation of combinations of functions 50

Differentiation of combinations involving

trigonometric functions 51

Tangents and normals 54

Gradients of tangents and normals 54

Equations of tangents and normals 56

Implicit differentiation 58

Differentiation of inverse trigonometric functions 62

Differentiation of y = sin−1x 62

Differentiation of y = tan−1x 63

Second derivatives 65

Parametric differentiation 67

First derivative of parametric equations 67

Second derivative of parametric equations 70

Partial derivatives 72

First order partial derivatives 72

Second order partial derivatives 73

Applications of partial derivatives 74

CHAPTER 3 PARTIAL FRACTIONS 84

Rational fractions 85

Proper fractions: Unrepeated linear factors 85

Proper fractions: Repeated linear factors 88

Proper fractions: Unrepeated quadratic factors 91

Proper fractions: Repeated quadratic factors 93

Improper fractions 94

CHAPTER 4 INTEGRATION 99

Integration by recognition 100

When the numerator is the differential of the denominator 104

The form ∫f ′(x)[ f(x)]n dx, n ≠ −1 105

The form ∫f ′(x)ef(x)dx 106

Integration by substitution 108

Integration by parts 112

Integration using partial fractions 116

Integration of trigonometric functions 121

Integrating sin2 x and cos2 x 123

Integrating sin3 x and cos3 x 123

Integrating powers of tan x 125

v

Integrating products of sines and cosines 125

Finding integrals using the standard forms

∫ 1 ________ √

_______

a2 − x2 dx = sin−1 ( x __ a ) + c

and ∫ 1 ______ a2 + x2

dx = 1 __ a tan−1 ( x __ a ) + c 126

CHAPTER 5 REDUCTION FORMULAE 136

Reduction formula for ∫sinn x dx 137

Reduction formula for ∫cosn x dx 138

Reduction formula for ∫tann x dx 139

Other reduction formulae 139

CHAPTER 6 TRAPEZOIDAL RULE (TRAPEZIUM RULE) 145

The area under a curve 146

MODULE 1 TESTS 153

MODULE 2 SEQUENCES, SERIES AND APPROXIMATIONS

CHAPTER 7 SEQUENCES 156

Types of sequence 157

Convergent sequences 157

Divergent sequences 157

Oscillating sequences 157

Periodic sequences 158

Alternating sequences 158

The terms of a sequence 158

Finding the general term of a sequence by identifying

a pattern 160

A sequence defined as a recurrence relation 161

Convergence of a sequence 162

CHAPTER 8 SERIES 167

Writing a series in sigma notation (Σ) 168

Sum of a series 169

Sum of a series in terms of n 172

Method of differences 176

Convergence of a series 180

Tests for convergence of a series 181

CHAPTER 9 PRINCIPLE OF MATHEMATICAL INDUCTION (PMI):

SEQUENCES AND SERIES 185

PMI and sequences 186

PMI and series 190

vi

CHAPTER 10 BINOMIAL THEOREM 196

Pascal’s triangle 197

Factorial notation 197

Combinations 199

General formula for nCr 199

Binomial theorem for any positive integer n 200

The term independent of x in an expansion 203

Extension of the binomial expansion 205

Approximations and the binomial expansion 208

Partial fractions and the binomial expansion 209

CHAPTER 11 ARITHMETIC AND GEOMETRIC PROGRESSIONS 215

Arithmetic progressions 216

Sum of the first n terms of an AP 218

Proving that a sequence is an AP 220

Geometric progressions 224

Sum of the first n terms of a GP (Sn) 227

Sum to infinity 229

Proving that a sequence is a GP 230

Convergence of a geometric series 231

CHAPTER 12 NUMERICAL TECHNIQUES 240

The intermediate value theorem (IMVT) 241

Finding the roots of an equation 241

Graphical solution of equations 242

Interval bisection 242

Linear interpolation 243

Newton–Raphson method for finding the roots of an

equation 247

CHAPTER 13 POWER SERIES 255

Power series and functions 256

Taylor expansion 256

The Maclaurin expansion 259

Maclaurin expansions of some common functions 265

MODULE 2 TESTS 267

MODULE 3 COUNTING, MATRICES AND

DIFFERENTIAL EQUATIONS

CHAPTER 14 PERMUTATIONS AND COMBINATIONS 270

The counting principles 271

Multiplication rule 271

Addition rule 271

vii

Permutations 272

Permutations of n distinct objects 272

Permutation of r out of n distinct objects 274

Permutations with repeated objects 275

Permutations with restrictions 277

Permutations with restrictions and repetition 281

Combinations 285

Combinations with repetition 287

CHAPTER 15 PROBABILITY 294

Sample space and sample points 295

Events: mutually exclusive; equally likely 296

Probability 296

Rules of probability 298

Conditional probability 299

Tree diagrams 302

Probability and permutations 307

Probability and combinations 311

CHAPTER 16 MATRICES 321

Matrices: elements and order 322

Square matrices 322

Equal matrices 323

Zero matrix 323

Addition and subtraction of matrices 323

Multiplication of a matrix by a scalar 324

Properties of matrix addition 324

Matrix multiplication 325

Properties of matrix multiplication 328

Identity matrix 328

Multiplication of square matrices 329

Transpose of a matrix 330

Properties of the transpose of a matrix 330

Determinant of a square matrix 331

Determinant of a 2 × 2 matrix 331

Determinant of a 3 × 3 matrix 331

Properties of determinants 332

Singular and non-singular matrices 334

Solving equations using determinants (Cramer’s rule) 335

Using Cramer’s rule to solve three equations in three

unknowns 337

Inverse of a matrix 339

Inverse of a 2 × 2 matrix 339

viii

Cofactors of a 3 × 3 matrix 339

Inverse of a 3 × 3 matrix 341

Properties of inverses 343

Systems of linear equations 343

Row reduction to echelon form 346

Finding the inverse of a matrix by row reduction 348

Solving simultaneous equations using row reduction 352

Systems of linear equations with two unknowns 355

Intersecting lines 355

Parallel lines 356

Lines that coincide 357

Systems of linear equations with three unknowns 358

Unique solution 358

No solutions 361

Infinite set of solutions 362

Solution of linear equations in three unknowns: geometrical

interpretation 365

Applications of matrices 367

CHAPTER 17 DIFFERENTIAL EQUATIONS AND

MATHEMATICAL MODELLING 380

First order linear differential equations 381

Practical applications 385

Second order differential equations 388

When the roots of the AQE are real and equal 388

When the roots of the AQE are real and distinct 389

When the roots of the AQE are complex 390

Non-homogeneous second order differential equations 392

When f(x) is a polynomial of degree n 393

When f(x) is a trigonometric function 398

When f(x) is an exponential function 403

Equations reducible to a recognisable form 405

Mathematical modelling 417

MODULE 3 TESTS 421

UNIT 2—MULTIPLE CHOICE TESTS 424

INDEX 441

Answers are available online at

www.macmillan-caribbean.com/resources

ix

Introduction

These two volumes provide students with an understanding of pure mathematics

at the CAPE level taken from both a theoretical and an application aspect and

encourage the learning of mathematics. They provide the medium through

which a student can find problems applied to different disciplines. The concepts

are developed step by step; they start from the basics (for those who did not do

additional mathematics) and move to the more advanced content areas, thereby

satisfying the needs of the syllabus. Examination questions all seem to have answers

that are considered ‘nice’ whole numbers or small fractions that are easy to work

with; not all real-world problems have such answers and these books have avoided

that to some extent. Expect any kind of numbers for your answers; there are no

strange or weird numbers.

The objectives are outlined at the beginning of each chapter, followed by the

keywords and terms that a student should be familiar with for a better understanding

of the subject. Every student should have a section of their work book for the

language of the subject. I have met many students who do not understand terms such

as ‘root’ and ‘factor’. A dictionary developed in class from topic to topic may assist the

students in understanding the terms involved. Each objective is fulfilled throughout

the chapters with examples clearly explained. Mathematical modelling is a concept

that is developed throughout, with each chapter containing the relevant modelling

questions.

The exercises at the end of each section are graded in difficulty and have adequate

problems so that a student can move on once they feel comfortable with the concepts.

Additionally, review exercises give the student a feel for solving problems that are

varied in content. There are three multiple choice papers at the end of each Unit,

and at the end of each module there are tests based on that module. For additional

practice the student can go to the relevant past papers and solve the problems given.

After going through the questions in each chapter, a student should be able to do past

paper questions from different examining boards for further practice.

A checklist at the end of each chapter enables the student to note easily what is

understood and to what extent. A student can identify areas that need work with

proper use of this checklist. Furthermore, each chapter is summarised as far as

possible as a diagram. Students can use this to revise the content that was covered in

the chapter.

The text provides all the material that is needed for the CAPE syllabus so that

teachers will not have to search for additional material. Both new and experienced

teachers will benefit from the text since it goes through the syllabus chapter by

chapter and objective to objective. All objectives in the syllabus are dealt with

in detail and both students and teachers can work through the text comfortably

knowing that the content of the syllabus will be covered.