book

Business Mathematics

Name: Business Mathematics
Author: P. Mariappan
ISBN: 9789332558380

by P. Mariappan

April 2015

Intermediate to advanced

800 pages

18h 38m

English

Pearson Education India

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Cover
Title page
Brief Contents
Contents
About the Author
Dedication
Foreword
Preface
CHAPTER 1 Introduction to Business Mathematics
1.1 Introduction to Business Mathematics 1.2 Scope and Importance 1.2.1 Steps in Quantitative Analysis Approach 1.3 Number System 1.3.1 Natural Numbers 1.3.2 Whole Numbers 1.4 Positive and Negative Integers 1.4.1 Positive Integers 1.4.2 Negative Integers 1.5 Fractions 1.5.1 Proper Fraction 1.5.2 Improper Fraction 1.6 Rational and Irrational Numbers 1.6.1 Rational Numbers 1.6.2 Irrational Numbers 1.7 Real Numbers 1.8 Sum of First n Natural Numbers 1.9 Sum of Square and Sum of Cube of First n Natural Numbers
CHAPTER 2 Algebra
2.1 Algebra 2.2 Algebraic Identities 2.3 Simple Factorizations 2.4 Equations 2.5 Linear and Quadratic Equations 2.5.1 Linear Equations 2.5.2 Simultaneous Linear Equations 2.5.3 Quadratic Equation 2.6 Indices and Surds 2.6.1 Indices 2.6.2 Surds 2.7 Logarithms and Their Properties 2.7.1 Definition 2.7.2 Properties 2.7.3 Common Logarithms 2.8 Problems Based on Logarithms 2.9 Inequality in Graphs 2.10 Remainder Theorem and Factor Theorem 2.10.1 Remainder Theorem 2.10.2 Factor Theorem Exercise

CHAPTER 3 Theory of Sets and Its Business Applications
3.1 Set Theory 3.2 Representation of Sets 3.2.1 Tabulation Method 3.2.2 Set Builder Form 3.2.3 Cardinal Number 3.2.4 Element of a Set 3.2.5 Order of a Set 3.2.6 Finite Set 3.2.7 Infinite Set 3.2.8 Singleton Set 3.3 Union of Sets 3.4 Intersection of Two Sets 3.5 Complement of a Set 3.6 Subset 3.6.1 Identical/Equal Set 3.6.2 Equivalent Set 3.6.3 Disjoint Set 3.6.4 Universal Set 3.7 Null Set/Empty Set/Void Set 3.8 Difference of a Set 3.9 De Morgan’s Law 3.10 Universal Set 3.11 Basic Operations on Set 3.11.1 Union of Sets 3.11.2 Intersection of Two Sets 3.11.3 Difference of a Set 3.11.4 Complement of a Set 3.12 Cartesian Products 3.13 Laws of Set Algebra 3.14 Cardinal Numbers 3.15 Venn Diagram 3.16 Business Applications Exercise
CHAPTER 4 Ratio
4.1 Ratio 4.2 Proportion 4.3 Types of Proportion 4.3.1 Direct Proportion 4.3.2 Inverse Proportion 4.3.3 Indirect Proportion 4.4 Ratio, Proportion and Its Type of Comparison 4.4.1 Addition and Subtraction of Ratios 4.5 Variations 4.5.1 Direct Variation 4.5.2 Inverse Variation 4.5.3 Joint Variation 4.6 Percentage 4.6.1 Uses of Percentages 4.6.2 Converting Fractions into Percentage 4.6.3 Problem Based on Quantity Application 4.6.4 Problem Based on Election Application 4.6.5 Problem Based on Examination Application 4.6.6 Price Mixture 4.6.7 Decimal Fraction 4.7 Annexing Zeros and Removing Decimal Signs 4.7.1 Annexing Zeros 4.8 Average (Arithmetic Mean) 4.8.1 Discrete Data 4.8.2 Discrete Data with Frequency 4.9 Computation of Mean for Grouped Series 4.10 Mean of Composite Group Exercise
CHAPTER 5 Functions and Graphs
5.1 Introduction 5.2 Function 5.2.1 Constant 5.2.2 Variable 5.3 Domain and Range of a Function 5.4 Continuous and Discrete Functions 5.4.1 Continuous Function 5.4.2 Discrete Function 5.5 Composition and Inverse Functions 5.5.1 Composition Functions 5.5.2 Order of Composition 5.5.3 Decomposition of a Function 5.5.4 Inverse Function 5.6 Classification of Functions 5.7 Exponential and Logarithmic Functions 5.7.1 Exponential Functions 5.7.2 Logarithmic Functions 5.8 Graphs 5.8.1 The General Idea of Curve Sketching and Graphs of Straight Lines 5.9 Modulus Function Exercise
CHAPTER 6 Quadratic Functions and Theory of Quadratic Equations
6.1 Quadratic Functions 6.1.1 Quadratic Modeling 6.2 Evaluating the Solutions of the Quadratic Equation ax2 + bx + c = 0 6.3 Nature of the Roots of Quadratic Equation 6.4 Sum and Product of the Roots 6.5 Relations Between the Roots 6.6 Condition for the Existence of a Common Root 6.7 To Form the Equation Whose Roots Are r1 and r2 6.8 Graph of the Function f (x) = ax2 + bx + c; a ≠ 0 6.9 Sign of the Expression f (x) = ax2 + bx + c 6.10 Maximum/Minimum Values of a Quadratic Function 6.11 Solution of Quadratic Inequalities 6.12 Additional Problems Exercise
CHAPTER 7 Permutation and Combination
7.1 Fundamental Counting Principl 7.2 Factorial 7.3 Permutation 7.3.1 Permutation of n Different Things 7.3.2 Permutation of n Things Not All Different (Taken All Together) 7.4 Restricted Permutation 7.5 Circular Permutations 7.6 Combination 7.6.1 To Find the Number of Ways of Selecting r Things Out of n Things 7.6.2 Complementary Combination 7.6.3 Different Formulas on Combination 7.7 Restricted Combinations 7.8 Division into Groups 7.9 Mixed Problems on Permutation and Combination Exercise
CHAPTER 8 Mathematical Induction and Binomial Theorem
8.1 Principle of Mathematical Induction 8.2 Binomial Theorem for a Positive Integral Index 8.3 Properties of the Binomial Coefficients 8.4 General Term in Binomial Expansion 8.5 Middle Term in the Binomial Expansion 8.6 Greatest Term in the Binomial Expansion 8.7 Binomial Theorem for Any Index 8.7.1 Binomial Theorem for a Rational Index Exercise
CHAPTER 9 Sequence and Series–Its Application to Business
9.1 General Idea and Different Types of Sequences 9.2 Different Types of Sequence 9.2.1 Finite Sequence 9.2.2 Infinite Sequence 9.2.3 Fibonacci Sequence 9.3 Kinds of Sequence 9.3.1 Arithmetic Sequence 9.3.2 Geometric Sequence 9.4 Arithmetic and Geometric Means 9.5 Arithmetic Progression (AP) 9.6 Geometric Progression (GP) 9.6.1 Sum to Infinity of a Geometric Series 9.6.2 Recurring Decimals as Geometric Progression 9.7 Harmonic Progression (HP) 9.7.1 Harmonic Mean 9.7.2 Special Case 9.7.3 Method of Differences Exercise
CHAPTER 10 Compound Interest and Annuities
10.1 Interest 10.1.1 Simple Interest 10.1.2 Compound Interest/Continuous Compounding 10.2 Present Value and Amount of a Sum 10.3 Annuity and Its Types 10.4 Present Value of an Annuity (PVA) 10.5 Debenture Exercise
CHAPTER 11 Coordinate Geometry
11.1 Fundamentals 11.2 Rectangular Cartesian Coordinates/Cartesian Coordinates System 11.3 Polar Coordinates 11.4 Distance Formula 11.5 Section Formula 11.5.1 To Find the Mid-point of the Line Joining (x1, y1) and (x2, y2) 11.6 Centroid of a Triangle 11.7 Area of a Triangle Exercise
CHAPTER 12 Locus and Equation to a Locus
12.1 Locus 12.2 Equation to a Locus Exercise
CHAPTER 13 Straight Line
13.1 Introduction 13.2 Definitions of the Terms 13.2.1 To Find the Equation of a Line Parallel to the x-axis 13.3 Different Forms of the Equation of a Straight Line 13.3.1 Gradient/Slope-intercept Form 13.3.2 Point Slope Form 13.3.3 Intercept Form of the Equation of a Straight Line 13.3.4 Normal Form of the Equation of a Straight Line 13.3.5 Distance Form of the Equation of a Straight Line 13.3.6 General Form ax + by + c = 0 13.4 Distance of a Point from a Straight Line 13.4.1 To Find the Length of the Perpendicular from the Origin on the Line ax + by + c = 0 13.5 Angle Between Two Lines and Condition of Parallelism and Perpendicularityof Lines 13.5.1 Angle Between Two Lines 13.5.2 Condition for Parallelism of Two Lines 13.5.3 Condition for Perpendicularity of Two Lines 13.5.4 Equations of Straight Lines Parallel or Perpendicular to the Given Line 13.6 Point of Intersection of Two Lines 13.6.1 General Form of the Equation of a Line 13.7 Condition for Concurrency of Three Given Straight Lines 13.8 Position of a Point with Respect to a Given Line Exercise
CHAPTER 14 Circle
14.1 Introduction 14.2 Standard Form/General Form/Center Radius Form of the Equation of a Circle 14.3 Finding the Centre and Radius of a Circle 14.4 Position of a Point with Respect to a Given Circle 14.5 Condition for a Line to be a Tangent 14.5.1 Condition for a Line to Touch the Circle with Centre O(0, 0) 14.5.2 Condition for a Line to Be Tangent to the Circle with Centre C(g, f ) 14.6 Diameter Form of the Equation of a Circle 14.7 Tangents 14.7.1 Equations of Tangents to the Circle of the Form x2 + y2 = a2 14.7.2 Equations of Tangents to the Circle of the Form x2 + 2gx + y2 + 2fy + c = 0 14.8 Points of Intersection of a Circle and a Straight Line 14.9 To Find the Length of the Tangent from P(x1, y1) to the Circle 14.10 Condition for Two Circles to Touch Each Other 14.11 Normal 14.12 Equations of the Circles in Some Special Cases 14.12.1 The Equation of a Circle of Radius r, Passing Through the Origin and Having Its Center on the (i) x-axis and (ii) y-axis 14.12.2 Equation of a Circle of Radius r and Touching Both the Axes 14.12.3 Equation of a Circle on the Join of A (x1, y1) and B (x2, y2) as a Diameter 14.12.4 Equation of a Circle Passing Through the Origin and Making Intercepts a and b on the Axis 14.13 Angle of Intersection of Two Circles 14.13.1 Orthogonal Circles 14.14 Equation of the Common Chord (Radical Axis) of the Two Circles 14.15 Family of Circles Exercise
CHAPTER 15 Matrix and Its Applications to Business
15.1 Introduction 15.2 Definitions and Notations 15.2.1 Matrix 15.2.2 General Form of the Matrix 15.2.3 Square Matrix 15.2.4 Rectangular Matrix 15.2.5 Row Matrix 15.2.6 Column Matrix 15.2.7 Transpose of a Matrix 15.2.8 Diagonal Matrix 15.2.9 Scalar Matrix 15.2.10 Identity Matrix 15.2.11 Null Matrix 15.2.12 Upper Triangular Matrix 15.2.13 Lower Triangular Matrix 15.2.14 Triangular Matrix 15.2.15 Matrix Representation of Multivariable Data 15.3 Operations on Matrices 15.3.1 Rules for Matrices Operation 15.3.2 Properties of Matrix Addition 15.3.3 Properties of Matrix Multiplication 15.3.4 Scalar Multiplication of Matrix 15.4 Conversion of a Business Problem into a Linear System of Equations (LSE) (Figure 15.1) 15.4.1 Equation 15.4.2 Linear Equation 15.5 Determinant of a Square Matrix 15.5.1 Non-singular Matrix and Singular Matrix 15.5.2 Sarrus Diagram for Expansion of Determinant of Order 3 15.5.3 Properties of Determinants 15.5.4 Rank of a Matrix 15.5.5 Application of Rank Concept 15.5.6 Minor and Cofactor 15.5.7 Adjoint of a Square Matrix 15.5.8 Inverse of a Square Matrix (Figure 15.2) 15.5.9 Finding Inverse Using Row Transformation Technique (Figure 15.3) 15.5.10 The Echelon Matrix: Row Echelon Forms (REF) 15.5.11 Depreciation Lapse Schedule 15.6 Matrix Equation 15.6.1 Conversion 15.7 Methods to Solve Linear System of Equations 15.7.1 Solution to the Linear System of Equations 15.7.2 Types of Solutions 15.7.3 Cramer’s Rule (Figure 15.4) 15.7.4 Matrix Inversion Method (Figure 15.5) 15.7.5 Gaussian Elimination Method (Figure 15.6) 15.7.6 Gauss–Jordan Method (Figure 15.7) 15.7.7 Variance Analysis Exercise
CHAPTER 16 Application of Matrix Concept
16.1 Input-Output Analysis 16.1.1 Assumptions of a Model 16.2 Markov Analysis 16.2.1 Stochastic Process 16.2.2 Markov Process 16.2.3 One Stage Transition Probability 16.2.4 Markov Chain 16.2.5 State Transition Matrix 16.2.6 Steady State 16.2.7 Existence of the Steady State Exercise
CHAPTER 17 Differential Calculus and Its Business Applications
17.1 Introduction 17.2 Limit of a Function and Rules for Evaluating the Limit of a Function 17.2.1 Theorems on Limit 17.2.2 List of Important Results 17.3 Continuous Function 17.4 Slope and Rate of Change 17.4.1 Derivative 17.5 Method for Evaluating the Differential Coefficient Using the First Principle and Standard Results 17.5.1 Derivative of a Sum/Difference of Two/More Functions 17.5.2 Derivative of a Product of Two Functions 17.5.3 Derivative of a Quotient of Two Functions 17.5.4 Derivative of the nth Power of a Function 17.5.5 Derivative of a Logarithmic Function 17.5.6 Derivative of an Exponential Function 17.6 Differentiation of an Implicit function 17.7 Parametric Differentiation 17.8 Logarithmic Differentiation 17.8.1 Properties of Logarithms 17.8.2 Differentiation of ax w.r.t. x 17.9 Applications of Differential Calculus 17.9.1 Average Concept 17.9.2 Marginal Concept 17.9.3 Average Cost 17.9.4 Marginal Cost 17.9.5 Elasticity 17.9.6 Price Elasticity of Supply 17.9.7 Price Elasticity of Demand 17.9.8 Marginal Revenue and Elasticity of Demand 17.10 Successive Differentiation 17.10.1 Definition and Notations 17.11 Concept of Maxima and Minima 17.12 Lagrangian Method for Equality Constraints 17.13 Constrained Optimization with Inequality Constraints: Kuhn−Tucker Conditions 17.14 Differential Equation 17.14.1 Formation of a Differential Equation 17.14.2 Solution to a Given Linear Differential Equation 17.14.3 Method for Solving the Equation of the Type dy/dx = f(x)/f(y) 17.14.4 Method for Solving the Differential Equation of the Type dy/dx = f1(x,y)/f2(x,y) Exercise
CHAPTER 18 Applications of Integral Calculus to Business
18.1 Introduction 18.2 Constant of Integration 18.3 List of Standard Integrals 18.4 Standard Results 18.5 Definite Integral 18.5.1 Properties of Definite Integral 18.6 Integration by Substitution 18.6.1 Some Special Types 18.7 Method of Partial Fractions 18.8 Method of Integration by Parts 18.9 Application of Integral Calculus to Business 18.9.1 Cost Function 18.9.2 Revenue Function 18.9.3 Consumer’s Surplus and Producer’s Surplus 18.10 Learning Curves 18.10.1 Arithmetic Approach 18.10.2 Logarithmic Approach Exercise
CHAPTER 19 Functions and Their Applications to Business
19.1 Introduction 19.2 Function 19.2.1 Constant 19.2.2 Variable 19.2.3 Single Valued Function 19.2.4 Many Valued Function 19.2.5 Explicit and Implicit Functions 19.2.6 Even and Odd Functions 19.2.7 Algebraic Function 19.2.8 Transcendental Function 19.2.9 Inverse Function 19.3 Functions Related to Commerce and Economic Theory 19.3.1 Demand Function 19.3.2 Cost Function 19.3.3 Total Cost Function 19.3.4 Average Cost Function 19.3.5 Marginal Cost Function 19.3.6 Revenue Function 19.3.7 Profit Function Exercise
CHAPTER 20 Linear Programming and Its Application to Business
20.1 Introduction 20.2 General Model of the Linear Programming Problem 20.3 Characteristics of a Linear Programming Problem 20.4 Assumptions of Linear Programming 20.4.1 Limitations of Linear Programming Program 20.5 Formulation of Linear Programming Program 20.6 Standard Form of an LPP 20.6.1Conversion of an LPP into Standard Form with Maximization Objective (Figure: 20.1) 20.7 Solution to an LPP 20.7.1 Types of Possible Solutions to an LPP 20.8 Convex Set and Extreme Point 20.9 Graphical Solution to an LPP 20.10 Simplex Methods 20.10.1 Simplex Method-I/Ordinary Simplex Method 20.10.2 Penalty Method/Big-M Method/Charnes Method (Figure 20.5) 20.10.3 Two-Phase Method (Figure 20.6) 20.10.4 Duality Concept in a Linear Programming 20.10.5 Dual Simplex Method (DSM) (Figure 20.7) Exercise
CHAPTER 21 Transportation Problem
21.1 Introduction 21.2 Conversion of a TP into an Equivalent LPP Form 21.3 Formulation of a Transportation Problem 21.4 Concept of Feasibility, Basicness and Degeneracy 21.5 Methods Used to Find the Solution to a Transportation Problem 21.6 Description of Various Methods to Find the Initial Basic Feasible Solution 21.6.1 North–West Corner Rule 21.6.2 Row Minima Method 21.6.3 Column Minima Method 21.6.4 Least Cost Method/Matrix Minima Method 21.6.5 Vogel’s Approximation Method 21.6.6 Effectiveness of Various Methods 21.7 Stepping Stone Method/Modified Distributive Method 21.8 Transshipment Problems 21.9 Sensitivity Analysis for Transportation Problem 21.9.1 Change in the Objective Function Coefficient by a Non-basic Variable 21.9.2 Change in the Objective Function Coefficient of a Basic Variable 21.9.3 Increasing Both Supply Si and Demand dj by ∆ 21.10 Introduction to Assignment Programming Problem 21.10.1 General Model of the Assignment Problem 21.10.2 Conversion into an Equivalent LPP 21.10.3 Solution to the Assignment Problem 21.10.4 Travelling Salesman Problem 21.11 Introduction to Game Theory 21.11.1 Game 21.11.2 Algebraic Method to Solve a 2 × 2 Game 21.11.3 Algebraic Method to Solve a 3 × 3 Game 21.11.4 Graphical Method to Solve a 2 × n or n × 2 Game 21.11.5 Dominance Property 21.11.6 Linear Programming Application to Solve a Game Exercise
CHAPTER 22 Probability Theory
22.1 Introduction 22.2 Definition for Certain Key Terms 22.3 Meaning of Probability 22.3.1 Addition Rules for Probability 22.3.2 Multiplication Rule on Probability When Events Are Independent 22.3.3 Compound Probability or Conditional Probability 22.4 Baye’s Theorem 22.5 Random Variables and Expectation 22.5.1 Introduction 22.5.2 Random Variable 22.5.3 Discrete Random Variable 22.5.4 Continuous Random Variable 22.6 Probability Distribution 22.6.1 Discrete Probability Distribution 22.6.2 Characteristics of a Discrete Distribution 22.6.3 Probability Function 22.7 Mathematical Expectation 22.8 Discrete Probability Distribution (Binomial and Poisson Distributions) 22.8.1 Introduction 22.8.2 Binomial Distribution 22.8.3 Characteristics of a Bernoulli Process 22.8.4 Definition of Binomial Distribution 22.8.5 Conditions of Binomial Distribution 22.8.6 Properties of Binomial Distribution 22.8.7 Mean of Binomial Distribution 22.8.8 Variance of Binomial Distribution 22.9 Poisson Distribution 22.9.1 Definition of Poisson Distribution 22.9.2 Properties of Poisson Distribution 22.9.3 Mean of Poisson Distribution 22.9.4 Variance of Poisson Distribution 22.10 Continuous Probability Distribution (Normal Distribution) 22.10.1 Introduction 22.10.2 Definition of Normal Distribution 22.10.3 Standard Normal Distribution 22.10. 4 Properties of Normal Distribution Exercise
CHAPTER 23 Correlation and Regression Analysis
23.1 Introduction 23.2 Correlation 23.2.1 Simple Correlation/Correlation 23.2.2 Rank Correlation 23.2.3 Group Correlation 23.2.4 Assumptions for Karl Pearson’s Coefficient of Correlation 23.2.5 Limitations 23.2.6 Properties 23.2.7 Scatter Diagram 23.3 Karl Pearson Coefficient of Correlation 23.4 Coefficient of Correlation for a Grouped Data 23.5 Probable Error of the Coefficient of Correlation 23.6 Rank Correlation 23.7 Regression Equations Exercise
CHAPTER 24 Concept of Shares
24.1 Shares 24.2 Stock Exchange 24.2.1 Role of Stock Exchanges in the Economy 24.3 Nature of Shares 24.3.1 Face Value 24.3.2 Book Value 24.3.3 Market Value 24.3.4 Difference Between Face Value and Market Value 24.3.5 Brokerage 24.3.6 Dividend 24.4 Different Types of Shares 24.4.1 Equity Shares 24.4.2 Preferential Shares 24.5 Bonus Share
Acknowledgements
Copyright
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Content preview from Business Mathematics

CHAPTER 15 Matrix and Its Applications to Business

15.1 INTRODUCTION

This matrix theory was developed and introduced by British mathematician Arthur Caley in 1857.

15.2 DEFINITIONS AND NOTATIONS

In this section we are going to see variables definitions related to different form of matrix.

15.2.1 Matrix

An arrangement of numbers in the form of rows and columns is called MATRIX denoted by the capital alphabets like A, B, C etc.

Note: The numbers used in the matrix can be a real value or complex value, i.e., A = [a_ij]_m_×_n where a_ij refers to the element which lies at the intersection of ith row and jth column.

ith rows: 1, 2, ..., m; jth columns: 1, 2, ..., n

It indicates that A refers to a matrix which contains m rows and n columns where ...

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CHAPTER 15