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Engineering Mathematics - 1

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Contents
1 Revision of fractions, decimals and
percentages 1
1.1 Fractions 1
1.2 Ratio and proportion 3
1.3 Decimals 4
1.4 Percentages 7
8 Simple equations 57
8.1 Expressions, equations and
identities 57
8.2 Worked problems on simple
equations 57
8.3 Further worked problems on simple
equations 59
8.4 Practical problems involving simple
equations 61
8.5 Further practical problems involving
simple equations 62
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2 Indices and standard form 9
2.1 Indices 9
2.2 Worked problems on indices 9
2.3 Further worked problems on
indices 11
2.4 Standard form 13
2.5 Worked problems on standard
form 13
2.6 Further worked problems on standard
form 14
3 Computer numbering systems 16
3.1 Binary numbers 16
3.2 Conversion of binary to decimal 16
3.3 Conversion of decimal to binary 17
3.4 Conversion of decimal to binary via
octal 18
3.5 Hexadecimal numbers 20
4 Calculations and evaluation of
formulae 24
4.1 Errors and approximations 24
4.2 Use of calculator 26
4.3 Conversion tables and charts 28
4.4 Evaluation of formulae 30
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Part 1 Number and Algebra 1
7 Partial fractions 51
7.1 Introduction to partial fractions 51
7.2 Worked problems on partial fractions
with linear factors 51
7.3 Worked problems on partial fractions
with repeated linear factors 54
7.4 Worked problems on partial fractions
with quadratic factors 55
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Preface xi
Assignment 2 64
9 Simultaneous equations 65
9.1 Introduction to simultaneous
equations 65
9.2 Worked problems on simultaneous
equations in two unknowns 65
9.3 Further worked problems on
simultaneous equations 67
9.4 More difficult worked problems on
simultaneous equations 69
9.5 Practical problems involving
simultaneous equations 70
5 Algebra 34
5.1 Basic operations 34
5.2 Laws of Indices 36
5.3 Brackets and factorisation 38
5.4 Fundamental laws and precedence 40
5.5 Direct and inverse proportionality 42
10 Transposition of formulae 74
10.1 Introduction to transposition of
formulae 74
10.2 Worked problems on transposition of
formulae 74
10.3 Further worked problems on
transposition of formulae 75
10.4 Harder worked problems on
transposition of formulae 77
6 Further algebra 44
6.1 Polynomial division 44
6.2 The factor theorem 46
6.3 The remainder theorem 48
11 Quadratic equations 80
11.1 Introduction to quadratic equations
11.2 Solution of quadratic equations by
factorisation 80
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Assignment 1
33
80

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CONTENTS
Multiple choice questions on chapters 1 to
16 127
11.3 Solution of quadratic equations by
‘completing the square’ 82
11.4 Solution of quadratic equations by
formula 84
11.5 Practical problems involving quadratic
equations 85
11.6 The solution of linear and quadratic
equations simultaneously 87
Part 2 Mensuration 131
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17 Areas of plane figures 131
17.1 Mensuration 131
17.2 Properties of quadrilaterals 131
17.3 Worked problems on areas of plane
figures 132
17.4 Further worked problems on areas of
plane figures 135
17.5 Worked problems on areas of
composite figures 137
17.6 Areas of similar shapes 138
Assignment 3
94
18 The circle and its properties 139
18.1 Introduction 139
18.2 Properties of circles 139
18.3 Arc length and area of a sector 140
18.4 Worked problems on arc length and
sector of a circle 141
18.5 The equation of a circle 143
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13 Exponential functions 95
13.1 The exponential function 95
13.2 Evaluating exponential functions 95
13.3 The power series for e x 96
13.4 Graphs of exponential functions 98
13.5 Napierian logarithms 100
13.6 Evaluating Napierian logarithms 100
13.7 Laws of growth and decay 102
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12 Logarithms 89
12.1 Introduction to logarithms 89
12.2 Laws of logarithms 89
12.3 Indicial equations 92
12.4 Graphs of logarithmic functions 93
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14 Number sequences 106
14.1 Arithmetic progressions 106
14.2 Worked problems on arithmetic
progression 106
14.3 Further worked problems on arithmetic
progressions 107
14.4 Geometric progressions 109
14.5 Worked problems on geometric
progressions 110
14.6 Further worked problems on geometric
progressions 111
14.7 Combinations and permutations 112
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15 The binomial series 114
15.1 Pascal’s triangle 114
15.2 The binomial series 115
15.3 Worked problems on the binomial
series 115
15.4 Further worked problems on the
binomial series 117
15.5 Practical problems involving the
binomial theorem 120
16 Solving equations by iterative
methods 123
16.1 Introduction to iterative methods 123
16.2 The Newton–Raphson method 123
16.3 Worked problems on the
Newton–Raphson method 123
Assignment 4
126
19 Volumes and surface areas of
common solids 145
19.1 Volumes and surface areas of
regular solids 145
19.2 Worked problems on volumes and
surface areas of regular solids 145
19.3 Further worked problems on volumes
and surface areas of regular
solids 147
19.4 Volumes and surface areas of frusta of
pyramids and cones 151
19.5 The frustum and zone of a sphere 155
19.6 Prismoidal rule 157
19.7 Volumes of similar shapes 159
20 Irregular areas and volumes and mean
values of waveforms 161
20.1 Areas of irregular figures 161
20.2 Volumes of irregular solids 163
20.3 The mean or average value of a
waveform 164
Assignment 5
168
Part 3 Trigonometry 171
21 Introduction to trigonometry 171
21.1 Trigonometry 171
21.2 The theorem of Pythagoras 171
21.3 Trigonometric ratios of acute
angles 172

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CONTENTS
26 Compound angles 214
26.1 Compound angle formulae 214
26.2 Conversion of a sin ωt C b cos ωt into
R sinωt C ˛) 216
26.3 Double angles 220
26.4 Changing products of sines and cosines
into sums or differences 221
26.5 Changing sums or differences of sines
and cosines into products 222
Assignment 7
224
Multiple choice questions on chapters 17
to 26 225
Part 4 Graphs 231
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22 Trigonometric waveforms 182
22.1 Graphs of trigonometric functions 182
22.2 Angles of any magnitude 182
22.3 The production of a sine and cosine
wave 185
22.4 Sine and cosine curves 185
22.5 Sinusoidal form A sinωt š ˛ 189
22.6 Waveform harmonics 192
25.7 Worked problems (iv) on trigonometric
equations 212
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21.4 Fractional and surd forms of
trigonometric ratios 174
21.5 Solution of right-angled triangles 175
21.6 Angles of elevation and
depression 176
21.7 Evaluating trigonometric ratios of any
angles 178
21.8 Trigonometric approximations for small
angles 181
Assignment 6
198
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24 Triangles and some practical
applications 199
24.1 Sine and cosine rules 199
24.2 Area of any triangle 199
24.3 Worked problems on the solution of
triangles and their areas 199
24.4 Further worked problems on the
solution of triangles and their
areas 201
24.5 Practical situations involving
trigonometry 203
24.6 Further practical situations involving
trigonometry 205
25 Trigonometric identities and
equations 208
25.1 Trigonometric identities 208
25.2 Worked problems on trigonometric
identities 208
25.3 Trigonometric equations 209
25.4 Worked problems (i) on trigonometric
equations 210
25.5 Worked problems (ii) on trigonometric
equations 211
25.6 Worked problems (iii) on trigonometric
equations 212
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27 Straight line graphs 231
27.1 Introduction to graphs 231
27.2 The straight line graph 231
27.3 Practical problems involving straight
line graphs 237
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23 Cartesian and polar co-ordinates 194
23.1 Introduction 194
23.2 Changing from Cartesian into polar
co-ordinates 194
23.3 Changing from polar into Cartesian
co-ordinates 196
23.4 Use of R ! P and P ! R functions on
calculators 197
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28 Reduction of non-linear laws to linear
form 243
28.1 Determination of law 243
28.2 Determination of law involving
logarithms 246
29 Graphs with logarithmic scales 251
29.1 Logarithmic scales 251
29.2 Graphs of the form y D axn 251
29.3 Graphs of the form y D abx 254
29.4 Graphs of the form y D ae kx 255
30 Graphical solution of equations 258
30.1 Graphical solution of simultaneous
equations 258
30.2 Graphical solution of quadratic
equations 259
30.3 Graphical solution of linear and
quadratic equations simultaneously
263
30.4 Graphical solution of cubic equations
264
31 Functions and their curves 266
31.1 Standard curves 266
31.2 Simple transformations 268
31.3 Periodic functions 273
31.4 Continuous and discontinuous
functions 273
31.5 Even and odd functions 273
31.6 Inverse functions 275
Assignment 8
279

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