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Note for Applied Mathematics-1 - M-1 By issu issu

  • Applied Mathematics-1 - M-1
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alljntuworld.in 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 TU W 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 ld 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 or 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 JN Assignment 1 33 80

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alljntuworld.in vi 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 ld 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 W 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 or 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 TU 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 JN 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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alljntuworld.in 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 or 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 ld 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 TU 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 JN 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 W 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 vii 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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alljntuworld.in viii CONTENTS 38.3 Worked problems on probability 327 38.4 Further worked problems on probability 329 38.5 Permutations and combinations 331 Part 5 Vectors 281 32 Vectors 281 32.1 Introduction 281 32.2 Vector addition 281 32.3 Resolution of vectors 283 32.4 Vector subtraction 284 39 The binomial and Poisson distribution 333 39.1 The binomial distribution 333 39.2 The Poisson distribution 336 ld 33 Combination of waveforms 287 33.1 Combination of two periodic functions 287 33.2 Plotting periodic functions 287 33.3 Determining resultant phasors by calculation 288 Assignment 10 339 40 The normal distribution 340 40.1 Introduction to the normal distribution 340 40.2 Testing for a normal distribution 344 or Part 6 Complex Numbers 291 41 Linear correlation 347 41.1 Introduction to linear correlation 347 41.2 The product-moment formula for determining the linear correlation coefficient 347 41.3 The significance of a coefficient of correlation 348 41.4 Worked problems on linear correlation 348 W 34 Complex numbers 291 34.1 Cartesian complex numbers 291 34.2 The Argand diagram 292 34.3 Addition and subtraction of complex numbers 292 34.4 Multiplication and division of complex numbers 293 34.5 Complex equations 295 34.6 The polar form of a complex number 296 34.7 Multiplication and division in polar form 298 34.8 Applications of complex numbers 299 TU 35 De Moivre’s theorem 303 35.1 Introduction 303 35.2 Powers of complex numbers 303 35.3 Roots of complex numbers 304 Assignment 9 306 Part 7 Statistics 307 36 Presentation of statistical data 307 36.1 Some statistical terminology 307 36.2 Presentation of ungrouped data 308 36.3 Presentation of grouped data 312 JN 37 Measures of central tendency and dispersion 319 37.1 Measures of central tendency 319 37.2 Mean, median and mode for discrete data 319 37.3 Mean, median and mode for grouped data 320 37.4 Standard deviation 322 37.5 Quartiles, deciles and percentiles 324 38 Probability 326 38.1 Introduction to probability 38.2 Laws of probability 326 326 42 Linear regression 351 42.1 Introduction to linear regression 351 42.2 The least-squares regression lines 351 42.3 Worked problems on linear regression 352 43 Sampling and estimation theories 356 43.1 Introduction 356 43.2 Sampling distributions 356 43.3 The sampling distribution of the means 356 43.4 The estimation of population parameters based on a large sample size 359 43.5 Estimating the mean of a population based on a small sample size 364 Assignment 11 368 Multiple choice questions on chapters 27 to 43 369 Part 8 Differential Calculus 375 44 Introduction to differentiation 375 44.1 Introduction to calculus 375 44.2 Functional notation 375 44.3 The gradient of a curve 376 44.4 Differentiation from first principles 377

Lecture Notes