Engineering Mathematics

Summary of contents
Contents
Preface to the first edition
Preface to the second edition
Preface to the third edition
Preface to the fourth edition
Preface to the eighth edition
How to use this book
	The companion website – www.macmillanihe.com/stroud
	Student review panel
The Personal Tutor
Useful background information
	Symbols used in the text
	Useful mathematical information
		1 Algebraic identities
		2 Trigonometrical identities
		3 Standard curves
		4 Laws of mathematics
PART I Foundation topics
Programme F.1 Arithmetic
	Learning outcomes
	Quiz F.1
	Types of number
		The natural numbers
		Numerals and place value
		Points on a line and order
		The integers
		Brackets
		Addition and subtraction
		Multiplication and division
		Brackets and precedence rules
		Rounding
		Basic laws of arithmetic
			1 Commutativity
			2 Associativity
			3 Distributivity
		Estimating
	Review summary
	Review exercise
	Factors and prime numbers
		Factors
		Prime numbers
		Prime factorization
		Fundamental theorem of arithmetic
		Highest common factor (HCF)
		Lowest common multiple (LCM)
	Review summary
	Review exercise
	Fractions, ratios and percentages
		Division of integers
		Multiplying fractions
		Of
		Equivalent fractions
		Dividing fractions
		Adding and subtracting fractions
		Fractions on a calculator
		Ratios
		Percentages
	Review summary
	Review exercise
	Decimal numbers
		Division of integers
		Rounding
		Significant figures
		Decimal places
		Trailing zeros
		Fractions as decimals
		Decimals as fractions
		Unending decimals
		Unending decimals as fractions
		Rational, irrational and real numbers
	Review summary
	Review exercise
	Powers
		Raising a number to a power
		The laws of powers
		Powers on a calculator
		Fractional powers and roots
		Surds
		Multiplication and division by integer powers of 10
		Precedence rules
		Standard form
		Working in standard form
		Using a calculator
		Preferred standard form
		Checking calculations
		Accuracy
	Review summary
	Review exercise
	Number systems
	Denary (or decimal) system
	Binary system
	Octal system (base 8)
	Duodecimal system (base 12)
	Hexadecimal system (base 16)
	An alternative method
	Review summary
	Review exercise
	Change of base from denary to a new base
		Binary form
		Octal form
		Duodecimal form
		A denary decimal in octal form
		Use of octals as an intermediate step
	Reverse method
	Review summary
	Review exercise
	Can you?
		Checklist F.1
	Test exercise F.1
	Further problems F.1
Programme F.2 Introduction to algebra
	Learning outcomes
	Quiz F.2
	Algebraic expressions
		Symbols other than numerals
		Constants
		Variables
		Rules of algebra
			Commutativity
			Associativity
			Distributivity
		Rules of precedence
		Terms and coefficients
		Collecting like terms
		Similar terms
		Expanding brackets
		Nested brackets
			Review summary
			Review exercise
			Powers and logarithms
		Powers
		Rules of indices
		Logarithms
		Rules of logarithms
		Base 10 and base
		Change of base
		Logarithmic equations
	Review summary
	Review exercise
	Algebraic multiplication and division
		Multiplication
		Division
		Review summary
		Review exercise
		Algebraic fractions
		Addition and subtraction
		Multiplication and division
	Review summary
	Review exercise
	Factorization of algebraic expressions
		Common factors
		Common factors by grouping
		Useful products of two simple factors
		Quadratic expressions as the product of two factors
	Review summary
	Review exercise
	Can you?
		Checklist F.2
	Test exercise F.2
	Further problems F.2
Programme F.3 Expressions and equations
	Learning outcomes
	Quiz F.3
	Expressions and equations
		Evaluating expressions
		Equations
			Conditional equation
			Identity
			Defining equation
			Assigning equation
			Formula
		Evaluating independent variables
		Transposition of formulas
		The evaluation process
	Review summary
	Polynomials
	Polynomial expressions
		Review exercise
		Evaluation of polynomials
		Evaluation of a polynomial by nesting
		Remainder theorem
		Factor theorem
		The general quadratic equation
		Factorization of fourth-order polynomials
	Review summary
	Review exercise
	Can you?
		Checklist F.3
	Test exercise F.3
	Further problems F.3
Programme F.4 Graphs
	Learning outcomes
	Quiz F.4
	Graphs of equations
		Equations
		Ordered pairs of numbers
		Cartesian axes
		Drawing a graph
	Review summary
	Review exercise
	Using a spreadsheet
		Spreadsheets
		Rows and columns
		Text and number entry
		Formulas
		Clearing entries
		Construction of a Cartesian graph
		Displays
	Review summary
	Review exercise
	Inequalities
		Less than or greater than
	Review summary
	Review exercise
	Absolute values
		Modulus
		Graphs
		Inequalities
			Less-than inequalities
			Greater-than inequalities
		Interaction
	Review summary
	Review exercise
	Can you?
		Checklist F.4
	Test exercise F.4
	Further problems F.4
Programme F.5 Linear equations
	Quiz F.5
	Linear equations
		Solution of simple equations
		Simultaneous linear equations with two unknowns
			1 Solution by substitution
			2 Solution by elimination
		Simultaneous linear equations with three unknowns
		Pre-simplification
	Review summary
	Review exercise
	Can you?
		Checklist F.5
	Test exercise F.5
	Further problems F.5
Programme F.6 Polynomial equations
	Learning outcomes
	Quiz F.6
	Polynomial equations
		Quadratic equations
		Cubic equations having at least one simple linear factor
		Fourth-order equations having at least two linear factors
	Review summary
	Review exercise
	Can you?
		Checklist F.6
	Test exercise F.6
	Further problems F.6
Programme F.7 Binomials
	Learning outcomes
	Quiz F.7
	Factorials and combinations
		Factorials
		Combinations
		Three properties of combinatorial coefficients
	Review summary
	Review exercise
	Binomial expansions
		Pascal’s triangle
		Binomial expansions
		The general term of the binomial expansion
	Review summary
	Review exercise
	The Σ (sigma) notation
		General terms
		The sum of the first n natural numbers
		Rules for manipulating sums
		The exponential number e
	Review summary
	Review exercise
	Can you?
		Checklist F.7
	Test exercise F.7
	Further problems F.7
Programme F.8 Partial fractions
	Learning outcomes
	Quiz F.8
	Partial fractions
	Review summary
	Review exercise
	Denominators with repeated and quadratic factors
	Review summary
	Review exercise
	Can you?
		Checklist F.8
	Test exercise F.8
	Further problems F.8
Programme F.9
	Trigonometry
	Learning outcomes
	Quiz F.9
	Angles
		Rotation
		Radians
		Triangles
		Trigonometric ratios
		Reciprocal ratios
		Pythagoras’ theorem
		Special triangles
		Half equilateral
	Review summary
	Review exercise
	Trigonometric identities
		The fundamental identity
		Two more identities
		Identities for compound angles
		Trigonometric formulas
			Sums and differences of angles
			Double angles
			Sums and differences of ratios
			Products of ratios
	Review summary
	Review exercise
	Can you?
		Checklist F.9
	Test exercise F.9
	Further problems F.9
Programme F.10 Functions
	Learning outcomes
	Quiz F.10
	Processing numbers
		Functions are rules but not all rules are functions
		Functions and the arithmetic operations
		Inverses of functions
		Graphs of inverses
		The graph of y = x3
		The graph of y = x1/3
		The graphs of y = x3 and y = x1/3 plotted together
	Review summary
	Review exercise
	Composition
	Function of a function
	Inverses of compositions
	Review summary
	Can you?
		Checklist F.10
	Review exercise
	Test exercise F.10
	Further problems F.10
Programme F.11 Trigonometric and exponential functions
	Learning outcomes
	Quiz F.11
	Introduction
	Trigonometric functions
		Rotation
		The tangent
		Period
		Amplitude
		Phase difference
		Inverse trigonometric functions
		Trigonometric equations
		Equations of the form acosx + bsinx = c
	Review summary
	Review exercise
	Exponential and logarithmic functions
		Exponential functions
		Review summary
		Review exercise
		Indicial equations
	Odd and even functions
		Odd and even parts
		Odd and even parts of the exponential function
		Limits of functions
		The rules of limits
	Review summary
	Review exercise
	Can you?
		Checklist F.11
	Test exercise F.11
	Further problems F.11
Programme F.12 Differentiation
	Learning outcomes
	Quiz F.12
	Gradients
		The gradient of a straight-line
		The gradient of a curve at a given point
		Algebraic determination of the gradient of a curve
		Derivatives of powers of x
			Two straight lines
			Two curves
		Differentiation of polynomials
		Second and higher derivatives and an alternative notation
	Review summary
	Review exercise
	Standard derivatives and rules
		Limiting value of
		Standard derivatives
			Derivative of y = sinx
			Derivative of y =  cosx
			Derivative of y = ex
		Derivative of a product of functions
		Derivative of a quotient of functions
		Derivative of a function of a function
		Derivative of y = ax
	Review summary
	Review exercise
	Newton–Raphson iterative method
		Notation
		Tabular display of results
			First approximations
	Can you?
		Checklist F.12
	Review summary
	Review exercise
	Test exercise F.12
	Further problems F.12
Programme F.13 Integration
	Learning outcomes
	Quiz F.13
	Integration
		Constant of integration
		Standard integrals
	Review summary
	Review exercise
	Integration of polynomial expressions
		Functions of a linear function of x
	Review summary
	Review exercise
	Integration by partial fractions
	Review summary
	Review exercise
	Areas under curves
	Review summary
	Review exercise
	Integration as a summation
		The area between a curve and an intersecting line
	Review summary
	Review exercise
	Can you?
		Checklist F.13
	Test exercise F.13
	Further problems F.13
PART II
Programme 1 Complex numbers 1
	Learning outcomes
	Introduction
	Ideas and symbols
	The symbol j
	Quadratic equations
	Powers of j
		Positive integer powers
		Negative integer powers
	Complex numbers
		Addition and subtraction
		Multiplication
		Division
		Equal complex numbers
	Review exercise
	Graphical representation of a complex number
		Argand diagram
		Graphical addition of complex numbers
	Polar form of a complex number
	Exponential form of a complex number
	Review summary
	Can you?
		Checklist 1
	Test exercise 1
	Further problems 1
Programme 2 Complex numbers 2
	Learning outcomes
	Polar-form calculations
	Review exercise
	Roots of a complex number
	Expansions
	Expansions of sinnθ nd cosnθ
	Expansions for cosnθ and sinnθ
	Loci problems
	Review summary
	Can you?
		Checklist 2
	Test exercise 2
	Further problems 2
Programme 3 Hyperbolic functions
	Learning outcomes
	Introduction
	Graphs of hyperbolic functions
	Review exercise
	Evaluation of hyperbolic functions
	Inverse hyperbolic functions
	Log form of the inverse hyperbolic functions
	Hyperbolic identities
	Relationship between trigonometric and hyperbolic functions
	Review summary
	Can you?
		Checklist 3
	Determinants of the third order
	Test exercise 3
	Further problems 3
Programme 4 Determinants
	Learning outcomes
	Determinants
		Evaluation of a third-order determinant
	Simultaneous equations in three unknowns
	Review exercise
	Consistency of a set of equations
	Properties of determinants
	Review summary
	Can you?
		Checklist 4
	Test exercise 4
	Further problems 4
Programme 5 Matrices
	Learning outcomes
	Matrices – definitions
	Matrix notation
	Equal matrices
	Addition and subtraction of matrices
	Multiplication of matrices
		Scalar multiplication
		Multiplication of two matrices
	Transpose of a matrix
	Special matrices
	Determinant of a square matrix
		Cofactors
		Adjoint of a square matrix
	Inverse of a square matrix
		Product of a square matrix and its inverse
	Solution of a set of linear equations
		Gaussian elimination method for solving a set of linear equations
	Eigenvectors and eigenvalues
		Eigenvalues
		Eigenvectors
	Cayley–Hamilton theorem
		Inverse matrices
		Raising a matrix to a whole number power
	Review summary
	Can you?
		Checklist 5
	Test exercise 5
	Further problems 5
Programme 6 Vectors
	Learning outcomes
	Introduction: scalar and vector quantities
	Vector representation
		Types of vector
		Addition of vectors
		The sum of a number of vectors a+b+c+d + . . .
	Components of a given vector
		Components of a vector in terms of unit vectors
	Vectors in space
	Direction cosines
	Scalar product of two vectors
	Vector product of two vectors
	Angle between two vectors
	Direction ratios
	Review summary
	Can you?
		Checklist 6
	Test exercise 6
	Further problems 6
Programme 7 Differentiation
	Learning outcomes
	Standard derivatives
	Functions of a function
		Products
		Quotients
	Logarithmic differentiation
	Review exercise
	Implicit functions
	Parametric equations
	Review summary
	Can you?
		Checklist 7
	Test exercise 7
	Further problems 7
Programme 8 Differentiation applications
	Learning outcomes
	Differentiation of inverse trigonometric functions
	Review exercise
	Derivatives of inverse hyperbolic functions
	Review exercise
	Maximum and minimum values
	Points of inflexion
	Review summary
	Can you?
		Checklist 8
	Test exercise 8
	Further problems 8
Programme 9 Tangents, normals and curvature
	Learning outcomes
	Equation of a straight line
	Tangents and normals to a curve at a given point
	Curvature
		Centre of curvature
	Review summary
	Can you?
		Checklist 9
	Test exercise 9
	Further problems 9
Programme 10 Sequences
	Learning outcomes
	Functions with integer input
		Sequences
		Graphs of sequences
		Arithmetic sequence
		Geometric sequence
		Harmonic sequence
		Recursive prescriptions
		Other sequences
	Review summary
	Review exercise
	Difference equations
		Solving difference equations
		Second-order, homogeneous equations
		Equal roots of the characteristic equation
	Review summary
	Review exercise
	Limits of sequences
		Infinity
		Limits
		Infinite limits
		Rules of limits
		Multiplication by a constant
		Sums and differences
		Products and quotients
		Indeterminate limits
	Review summary
	Review exercise
	Can you?
		Checklist 10
	Test exercise 10
	Further problems 10
Programme 11 Series 1
	Learning outcomes
	Series
		Arithmetic series
		Arithmetic mean
		Geometric series
		Geometric mean
	Series of powers of the natural numbers
		Sum of natural numbers
		Sum of squares
		Sum of cubes
	Infinite series
	Limiting values
		Convergent and divergent series
		Tests for convergence
			Test 1. A series
			Test 2. The comparison test
			Test 3: D’Alembert’s ratio test for positive terms
		Absolute convergence
	Review summary
	Can you?
		Checklist 11
	Test exercise 11
	Further problems 11
Programme 12 Series 2
	Learning outcomes
	Power series
		Introduction
		Maclaurin’s series
			Series for tan x
		Standard series
	Review summary
		The binomial series
		Taylor’s series
	Limiting values – indeterminate forms
		L’Hȏpital’s rule for finding limiting values
	Approximate values
	Review summary
	Can you?
		Checklist 12
	Test exercise 12
	Further problems 12
Programme 13 Curves and curve fitting
	Learning outcomes
	Introduction
	Standard curves
		Straight line
		Second-degree curves
		Third-degree curves
		Circle
		Ellipse
		Hyperbola
		Logarithmic curves
		Exponential curves
		Hyperbolic curves
		Trigonometrical curves
		Tangent curve
	Asymptotes
		Determination of an asymptote
		Asymptotes parallel to the x- and y-axes
	Systematic curve sketching, given the equation of the curve
		Symmetry
		Intersection with the axes
		Change of origin
		Asymptotes
		Large and small values of x and y
		Stationary points
		Limitations
	Curve fitting
		Straight-line law
		Graphs of the form y = axn, where a and n are constants
		Graphs of the form y = aenx
	Method of least squares
		Fitting a straight-line graph
		Using a spreadsheet
	Correlation
		Correlation
		Measures of correlation
		The Pearson product-moment correlation coefficient
		Spearman’s rank correlation coefficient
	Review summary
	Can you?
		Checklist 13
	Test exercise 13
	Further problems 13
Programme 14 Partial differentiation 1
	Learning outcomes
	Partial differentiation
	Review summary
	Review exercise
	Small increments
	Can you?
		Checklist 14
	Test exercise 14
	Review summary
	Further problems 14
Programme 15 Partial differentiation 2
	Learning outcomes
	Partial differentiation
	Rate-of-change problems
	Change of variables
	Can you?
		Checklist 15
	Review summary
	Test exercise 15
	Further problems 15
Programme 16 Integration 1
	Learning outcomes
	Introduction
		Standard integrals
	Functions of a linear function of x
	Integrals of the forms
	Integration of products – integration by parts
	Integration by partial fractions
	Integration of trigonometric functions
	Review summary
	Can you?
		Checklist 16
	Test exercise 16
	Further problems 16
Programme 17 Integration 2
	Learning outcomes
	Review summary
	Can you?
		Checklist 17
	Test exercise 17
	Further problems 17
Programme 18 Reduction formulas
	Learning outcomes
	Review summary
	Can You?
		Checklist 18
	Test exercise 18
	Further problems 18
Programme 19 Integration applications 1
	Learning outcomes
	Basic applications
		Areas under curves
		Definite integrals
	Parametric equations
	Mean values
	Root mean square (rms) value
	Review summary
	Can you?
		Checklist 19
	Test exercise 19
	Further problems 19
Programme 20 Integration applications 2
	Learning outcomes
	Introduction
	Volume of a solid of revolution
	Centroid of a plane figure
	Centre of gravity of a solid of revolution
	Length of a curve
	Parametric equations
	Surface of revolution
		Parametric equations
	Rules of Pappus
	Review summary
	Can you?
		Checklist 20
	Test exercise 20
	Further problems 20
Programme 21 Integration applications 3
	Learning outcomes
	Moments of inertia
		Radius of gyration
		Parallel axes theorem
		Perpendicular axes theorem (for thin plates)
		Useful standard results
	Second moment of area
		Composite figures
	Centre of pressure
		Pressure at a point P, depth z below the surface of a liquid
		Total thrust on a vertical plate immersed in liquid
		Depth of the centre of pressure
	Can you?
		Checklist 21
	Review summary
	Test exercise 21
	Further problems 21
Programme 22 Approximate integration
	Learning outcomes
	Introduction
	Approximate integration
		Method 1: By series
		Method 2: By Simpson’s rule
	Proof of Simpson’s rule
	Review summary
	Can you?
		Checklist 22
	Test exercise 22
	Further problems 22
Programme 23 Polar coordinate systems
	Learning outcomes
	Introduction to polar coordinates
	Polar curves
	Standard polar curves
	Applications
	Review summary
	Can you?
		Checklist 23
	Test exercise 23
	Further problems 23
Programme 24 Multiple integrals
	Learning outcomes
	Summation in two directions
	Double integrals
	Triple integrals
	Applications
	Review exercise
	Alternative notation
	Determination of areas by multiple integrals
	Determination of volumes by multiple integrals
	Review summary
	Can you?
		Checklist 24
	Test exercise 24
	Further problems 24
Programme 25 First-order differential equations
	Learning outcomes
	Introduction
	Formation of differential equations
	Solution of differential equations
		Method 1: By direct integration
		Method 2: By separating the variables
	Review exercise
		Method 3: Homogeneous equations – by substituting y = vx
	Review exercise
		Method 4: Linear equations – use of integrating factor
	Review exercise
	Bernoulli’s equation
	Review summary
	Can you?
		Checklist 25
	Further problems 25
Programme 26 Second-order differential equations
	Learning outcomes
	Homogeneous equations
	Test exercise 25
	Review exercise
	Inhomogeneous equations
	Particular solution
	Can you?
		Checklist 26
	Review summary
	Test exercise 26
	Further problems 26
Programme 27 Introduction to Laplace transforms
	Learning outcomes
	The Laplace transform
		The inverse Laplace transform
		Table of Laplace transforms
	Review summary
	Review exercise
		Laplace transform of a derivative
		Two properties of Laplace transforms
		Table of Laplace transforms
	Review summary
	Review exercise
		Generating new transforms
		Laplace transforms of higher derivatives
		Table of Laplace transforms
		Linear, constant-coefficient, inhomogeneous differential equations
	Review summary
	Review exercise
	Can you?
		Checklist 27
	Test exercise 27
	Further problems 27
Programme 28 Data handling and statistics
	Learning outcomes
	Introduction
	Arrangement of data
		Tally diagram
		Grouped data
		Grouping with continuous data
		Relative frequency
		Rounding off data
		Class boundaries
	Histograms
	Frequency histogram
		Relative frequency histogram
	Measures of central tendency
		Mean
		Coding for calculating the mean
		Decoding
		Coding with a grouped frequency distribution
		Mode
		Mode with grouped data
		Median
		Median with grouped data
	Measures of dispersion
		Mean deviation
		Range
		Standard deviation
		Alternative formula for the standard deviation
	Distribution curves
		Frequency polygons
		Frequency curves
		Normal distribution curve
	Standardized normal curve
	Review summary
	Can you?
		Checklist 28
	Test exercise 28
	Further problems 28
Programme 29 Probability
	Learning outcomes
	Probability
		Random experiments
		Events
		Sequences of random experiments
		Combining events
	Events and probabilities
		Probability
		Assigning probabilities
	Review summary
	Probabilities of combined events
		Or
		Non-mutually exclusive events
		And – multiplication rule
		Dependent events
		Independent events
		Probability trees
	Conditional probability
	Review summary
	Probability distributions
		Random variables
		Expectation
		Variance and standard deviation
		Bernoulli trials
		Binomial probability distribution
		Expectation and standard deviation
		The Poisson probability distribution
		Binomial and Poisson compared
	Continuous probability distributions
		Normal distribution curve (or normal curve)
	Standard normal curve
	Review summary
	Can you?
		Checklist 29
	Test exercise 29
	Further problems 29
Answers
	Part I
		Quiz F.1 (page 4)
		Test exercise F.1 (page 59)
		Further problems F.1 (page 60)
		Quiz F.2 (page 64)
		Test exercise F.2 (page 94)
		Further problems F.2 (page 95)
		Quiz F.3 (page 98)
		Test exercise F.3 (page 121)
		Further problems F.3 (page 121)
		Quiz F.4 (page 124)
		Test exercise F.4 (page 154)
		Further problems F.4 (page 154)
		Quiz F.5 (page 158)
		Test exercise F.5 (page 1704)
		Further problems F.5 (page 170)
		Quiz F.6 (page 174)
		Test exercise F.6 (page 185)
		Further problems F.6 (page 185)
		Quiz F.7 (page 188)
		Test exercise F.7 (page 212)
		Further problems F.7 (page 212)
		Quiz F.8 (page 216)
		Test exercise F.8 (page 232)
		Further problems F.8 (page 232)
		Quiz F.9 (page 236)
		Test exercise F.9 (page 256)
		Further problems F.9 (page 257)
		Quiz F.10 (page 260)
		Test exercise F.10 (page 277)
		Further problems F.10 (page 277)
		Quiz F.11 (page 280)
		Test exercise F.11 (page 305)
		Further problems F.11 (page 306)
		Quiz F.12 (page 310)
		Test exercise F.12 (page 345)
		Further problems F.12 (page 346)
		Quiz F.13 (page 348)
		Test exercise F.13 (page 374)
		Further problems F.13 (page 375)
	Part II
		Test exercise 1 (page 403)
		Further problems 1 (page 404)
		Test exercise 2 (page 428)
		Further problems 2 (page 429)
		Test exercise 3 (page 451)
		Further problems 3 (page 452)
		Test exercise 4 (page 480)
		Further problems 4 (page 481)
		Test exercise 5 (page 515)
		Further problems 5 (page 516)
		Test exercise 6 (page 541)
		Further problems 6 (page 542)
		Test exercise 7 (page 560)
		Further problems 7 (page 561)
		Test exercise 8 (page 582)
		Further problems 8 (page 582)
		Test exercise 9 (page 604)
		Further problems 9 (page 605)
		Test exercise 10 (page 638)
		Further problems 10 (page 639)
		Test exercise 11 (page 663)
		Further problems 11 (page 664)
		Test exercise 12 (page 689)
		Further problems 12 (page 689)
		Test exercise 13 (page 732)
		Further problems 13 (page 733)
		Test exercise 14 (page 754)
		Further problems 14 (page 755)
		Test exercise 15 (page 771)
		Further problems 15 (page 771)
		Test exercise 16 (page 797)
		Further problems 16 (page 797)
		Test exercise 17 (page 826)
		Further problems 17 (page 826)
		Test exercise 18 (page 838)
		Further problems 18 (page 839)
		Test exercise 19 (page 856)
		Further problems 19 (page 857)
		Test exercise 20 (page 878)
		Further problems 20 (page 878)
		Test exercise 21 (page 907)
		Further problems 21 (page 908)
		Test exercise 22 (page 926)
		Further problems 22 (page 927)
		Test exercise 23 (page 949)
		Further problems 23 (page 950)
		Test exercise 24 (page 974)
		Further problems 24 (page 975)
		Test exercise 25 (page 1009)
		Further problems 25 (page 1009)
		Test exercise 26 (page 1034)
		Further problems 26 (page 1034)
		Test exercise 27 (page 1052)
		Further problems 27 (page 1052)
		Test exercise 28 (page 1081)
		Further problems 28 (page 1082)
		Test exercise 29 (page 1122)
		Further problems 29 (page 1123)
Index