Ordinary Level !Mathematics

L. HARWOOD CLARKE

SIXTH EDITION prepared by F. G. J. NORTON Haad of the Mathematics Department Rugby School

HEINEMANN EDUCATIONAL BOOKS LONDON

Contents

ARITHMETIC

Formulae

1 Numerals and Number Bases Numerals Number bases Binary numbers Octal numbers Addition, subtraction in bases other than ten Multiplication, division Decimal fractions, decimals Binary fractions, bicimals Modular arithmetic Division

2 Approximations Estimates of calculations Significant figures Decimal places ~tandard-form Mani~ulation of numbers in standard form Degree of accuracy

3 Aids for Calculations Abacus Slide rule Multiplication Reading intermediate values Division Squares and square roots Reciprocals Use of trigonometric scales

4 Logarithms Numbers in standard form Negative characteristics Antilogarithms Multiplication Division Powers Roots Worked examples

vii

viii CONTENTS

5 Areas and Volumes The triangle The area of a triangle in terms of the sides The parallelogram The rectangle The box or cuboid The cube The area of walls of a room Solid with uniform cross section The prism The tetrahedron Units Region between two rectangles S ~ a c e between two boxes ensi it^; relative density Circumference and area of a circle Area of an ellipse Surface area of a right circular cylinder Volume of a right circular cylinder Volume of a circular cone Surface area of a cone Sphere Area of a ring Volume of material in a pipe Volume of a spherical shell Discharge of water from a pipe Areas of similar figures Volumes of similar solids Worked examples

<!I 6 Ratio and Percentage Ratio

4 Proportional parts .l !

Percentage

1 Percentage gain and loss Worked examples

j ' 7 Simple and Componnd Interest

Simple interest To find the principal, given the amount Compound interest

. 3. 1 Formula for compound interest 1 Periodical borrowing or repayment , j 8 Rates and Taxes, Stocks and Shares ,i Rates 4, Income tax

Taxable income Value-added tax, duty Stocks and shares Preference and ordinary shares Difference between stock and share Nominal value and market value The yield Worked example

CONTENTS ix

74 76 77 77 78 78 79 80

ALGEBRA

9 Factors Common factors Difference of two squares Trinomials Grouping Sum and difference of two cubes Expressions containing five terms Factor theorem Remainder theorem

10 Formulae, Expressions ' Constructing formulae Simplifying expressions Lowest common multiple Simplification of fractions Changing the subject of a formula

l1 Linear Equations and Inequalities Equations Linear equations Inequalities Problems leading to linear equations Problems involving inequalities Simultaneous equations Problems leading to simultaneous equations Identities

12 Quadratic Equations and Inequalities Quadratic equations Inequalities Problems leading to quadratic equations Quadratic equations by completing the square and by

formula

Simultaneous equations, one linear, one quadratic Harder problems leading to quadratic equations

13 Graphs Linear functions Quadratic functions Intersecting graphs Travel graphs

14 Graphical Ioequalities; Linear Programming Inequalities Description of region Boundaries of region Shading of region Drawing straight line graphs Linear programming Variations on this problem Cost lines, profit lines Transport problem

15 Indices and Logarithms Indices Negative and fractional indices Logarithms Formulae connecting logarithms Change of base Worked examples

16 Variation Direct proportion Other cases of direct proportion Inverse proportion Joint variation Variation as the sum of two parts Worked examples

17 Arithmetic and Geometric Progressions Arithmetic progressions The nth term The arithmetic mean The sum of an A.P. Geometric progressions The nth term The geometric mean The sum of a G.P. Worked examples

Notation; summary

18 Set Theory Definition of a set Intersection and union Empty set Subsets Venn diagrams Equal sets Number of elements in a set: cardinal number The universal set Complement of a set

19 Binary Operations Binary operations Closure Commutative Associative Identity element Inverse element Distributive law

20 Relations, Mappings, lbctions Relations Domain, codomain; image, range Into, onto relations One-one; one-many Non-algebraic relations Symmetric Transitive Reflexive Reflexive, symmetric, transitive: equivalence relations Functions One-one, many-one mappings Inverse functions Restricted domain Composite functions

21 Groups Problem: to solve a*x = b, for all operations * Group: definition Modulo arithmetic Group of rotations

xii CONTENTS

STATISTICS AND PROBABILITY

22 Averages; Mean, Mode, Median Average Mean Median Mode Frequency Use of working zero Scaling Weighted mean Moving averages

23 Representation of Data Collection of data Representation of data Misrepresentation of data Pie chart: circular diagrams Bar chart Histogram Cumulative frequency curve

24 Probability Use of symmetry Definition Importance of equiprobable events Theoretical probability, experimental probability Addition of probabilities Harder example on experimental probability

25 Conditional Probability Independent events Mutually exclusive events Multiplication Law Tree diagram

GEOMETRY

Some definitions and theorems

26 Axioms, Parallel Lines, Triangles and Polygons Measurement of angle The straight line Axioms Definitions Vertically opposite angles

CONTENTS sfi

Parallel lines 263 Corresponding, alternate, and adjacent angles 264 The triangle 267 Exterior angle 268 The polygon . 269

27 Congruent Trisngles; Triangles and Qnadrilaterals; Inequalities 273 Congruent triangles 273 The isosceles triangle 275 Parallelogram 278 Rhombus 278 Rectangle Square Inequalities

28 Ratio: Similar Figures, especially Triangles Ratio Internal and external division Similar figures Similar triangles The angle bisector theorems Areas of similar figures Pythagoras' theorem: proof using similar triangles

29 The Circle Definitions Theorems about chords Angle in a segment Four theorems on these angles Tangent Alternate segment Alternate segment theorem Internal and external contact of tangents Theorems Intersecting chord theorem Converses

3 0 M Definition Some plane loci Some loci in space Two important loci Constant angle locus

31 Constrnctions Section A Section B

X ~ V CONTENTS

32 Symmetry Symmetry about an axis Rotational symmetry Use of symmetry in solving geometrical problems The isosceles triangle The circle

33 Matrices Debition Order of a matrix Zero matrix, unit matrix Multiplication of matrices Square matrices Associative law Transpose of a matrix The inverse matrix Determinant of a matrix 3 Singular matrix 3 Matrix method for solving simultaneous equations 3

34 Geometrical Applications of Matrices Reflection Rotation Enlargement shear- Finding the matrix to describe a transformation Mapping described by a singular matrix Change of area Inverse mappings

35 Vectors Vector: definition Displacement vectors Magnitude and direction of a vector Addition of vectors Subtraction of vectors Scalar multiples of a vector Transformation of a vector by a matrix Translation

36 Vector Geometry Parallel vectors Position vector of a point A relative to a point B Midpoint theorem The diagonals of a parallelogram bisect each other

CONTENTS xv

TRIGONOMETRY

Formulae

37 Ratios of an Acute Angle Sine Cosine Tangent The three ratios Three other ratios Use of tables Given one ratio, to 6nd the others Complementary angles The isosceles triangle The ratios of 45" The ratios of 30" and 60" The ratios of 0" and 90"

38 Solution of a Right-angled Triangle Given one side and one angle Given two sides The isosceles triangle Angles of elevation and depression Height of an inaccessible object Gradient Bearings Projections

3!J The General Angle Projections Obtuse angles Signs of the ratios Magnitudes of the ratios General angle formulae

40 Graphs The sine curve The cosine curve The tangent curve Graphical applications

41 Problems in Three Dimensions The plane Skew lines Angle between a line and a plane Line of greatest slope

X< CONTENT

Angle between two planes Problems

42 The Circle and the Sphere Length of circular arc Area of circular sector Area of segment Radian measure Latitude and longitude The nautical mile

43 She a0a Cc&% Fomolae The sine formula Given two angles and a side The cosine formula The area of a triangle

c-us Notation, Fomolae

44 The Derived Function The gradient of a straight line The gradient of a curve The gradient of y = xZ at any point The derived function The derived function of m, of axZ, of axa + bx + c The derived function of X', of l/x, of l/xZ General rule Notation

45 Applications of the Derived Function Maxima and minima Velocity Acceleration Rate of change

46 Integration The inverse of differentiation Notation Velocity and acceleration Area of a region Area of the region under a velocity-time graph Area of the region under an acceleration-time graph The definite integral Solid of revolution

Revision Exercise Index

Index

Acceleration 480,487 Ansk

acute 263 obtuse 263 reflex 263 - - - - -

to measure 262 Angk bisector theorem 288 Angle in a segment 297,298.313 Angk in a semicirck 299 Angle in the altemate segment 302 Angle of deoression 405

of elevation 405 Angles

adjacent 261.264 alternate 261, 264 mrrespnding 261,264

Area of triangle 39.41, 388,455 Associative Law 198,216 Axioms, Euclid's 262

Bar chart 234 Bearinesm Bidmah 11 Binary numbers 7 Binary oprati0110 183,197 Butter6eld. William 340

Cabbage, time to cook 238 .Cirde

defiotion 295 properties of 295 symmetry of 311

Circular hagrams 234 Closed set, closure 184, 197

' Codomain 204 ; Commutative Law 198,221 Compound Interest 3, 39

I Complement of a set 183, 190

A volume 47 Conversa of a theorem 292, 293,

l Cosine curve 423 deiinition 389, 390 formula 387,451

Cumulative frequency curve 238

Decimal fractions 10 recurring 10

Decimal places 19 Degrec of accuracy 23 Degree, definition 262 Density 44

relative 44 Determinant 355,366 Distributive Law 202 Domain 204 Duty 76

Emtian fractions 249 Ellipse

area 46 definition 46

Empty set 183, 186 Equal sets 188 Equivalence relation 207 Estimates of calculation 18 Eudid's Axioms 262 Exponents 35

Factor theorem 94 Factors 87 Formulae

changhg subject of 102 construction of 97

Frequency 225 Frustum 49 Function 208

compsite 212 derived 463,466 inverse 210

Golden ratio 284 Gradient of road 407

of nwe 464 Graphs

linear 134 P ~ P Y 204 quadratic 135 travel 140

Group, definition 216

H.C.F. 189 Histogram 235 Horizon, distance of 39

510 INDEX

Identities in algebra 118 in trigonometry 420

Identity element 184, 200 matrix 346

Image 184, 204, 362, 367, 376 Indices 158 Inequalities 108, 123

graphical 145 Inflexion, points of 475 Inverse

element 201 function 210 matrix 354

Interest compound 3,66 simple 3, 66

Isosceles triangle definition 261, 267 finding third side 404,448 propetties of 275,341 symmetry of 341

Isomorphism 218, 220

Kite definition 275 symmetry of 342

Kkin four-group 220,222 Knot 442

Latitude 441 ~- ~ ~

Line of greatest slope 429 Linear programming 1 SO

cost line 153 profit line 154

Logarithms changing base of 85, 162 formulae 85, 160 theory 159 use of 33

L.C.M. 108. 189 Longitude h1 Mapping 184,208 Mass of an electron 20 Matrix

delinitinn 345 dominance 359 electrical 360 geometrical 361 inverse 354 route 347 singular 355, 366 transpose 353

Maxima 474

Mean (arithmetic) 176.223 geometric 176 proportional 327

Median 223, 239 Median of a triangle 314 Mediator 31 1 Minima 474 Misrepresentation of data 234 Mode 224 Modular arithmetic 12, 198, 203, 216 Moving average 229

Nautical mile 442 Null set 183, 186 Number bases 5 Numbers

binary 7 cardinal 183. 190 counting 5 natural 5 octal 7

Numerals 5

Ogive 238

Papy graphs 204 Parabola, definition 311 Parallelogram 278, 381 Parallel lines, definition 263 Percentages 57 Percentiles 239 Pie chart W4 Plane, definition 428 Polygon 268 Prism, deiinition 43 Probability

definition 242, 247 experimental 247 theoretical 247

Progressions arithmetic 174 geometric 176

Projections 389,390,415 Proper subset 186 Proportion

direct 165 inverse 167

Quadratic equation 81, 122 Quartiles 239

Radian, defined 439 Range 204 Rates 73 Rational number 183, 191

Reciprocal 30 Reflection 291, 331, 332, 361, 361 Reflexive relation 207 Relations 194. 191.204.207 Remainder theoreA 95 Rhombus 278 Rotation 326,362, 364, 377

group of rotations 271, 219

Sampling 233 Scaling 226 Set, definition 183, 185 Shares 77 Signiiicant figures 18 Similar figures, areas of 47, 289 Similar solids, volumes of 48, 290 Sine, definition of 389. 390

C& 423 definition 389, 390 formula 387, 447 on slide rule 31

Singular matrix 355, 366 Skew lines 428 Solid of revolution 493 Sphere

surface area 47 volume 47 volume of hollow 48

Square, definition 278 Squares and square roots 29 Standard form 20 Straight line, definition 262 Subset 186

proper 186 Symmetric relation 207 Symmetries, group of 220.222 Symmetry

in geometry 306, 312, 313, 326, 331, 334, 337

in nature 337

in probability 242, 247, 257 of the circle 341 of the isosceles triangle 341 rotational 326, 338

Tangent of an angle c u m 424 definition 390 on slide rule 31

Tangent to a circle construction of 321 definition 302 properties of 303,305, 308

Tax Income 74 Value-added 76

Taxable income 74 Transitive relation 247 Transport problem 154 Tree diagram 252 Triangle

equilateral 261, 267 isosceles 261, 267, 275, 341,404 scalene 261.267

Union of sets 183 Universal set 183, 190

Variation 165 joint 169

Vedm . --.-- definition 370 repmenled by a matrix 361, 371 to describe a translation 371

Venn diagram 186, 203,248,249,279

Weighted mean 228 Working zero U 5

Yield of a share 79