BUSINESS MATHEMATICS
BUSINESSMATHEMATICSHIGHER SECONDARY - SECOND YEARRevised based on the recommendation of theTextbook Development CommitteeA Publication underGovernment of TamilnaduDistribution of Free Textbook Programme(NOT FOR SALE)Untouchability is a sinUntouchability is a crimeUntouchability is inhumanTAMILNADU TEXTBOOK ANDEDUCATIONAL SERVICES CORPORATIONCollege Road, Chennai - 600 006.
Government of TamilnaduRevised Edition - 2008, 2009Reprint- 2017Text Book CommitteeChairpersonReviewerDr.S.ANTONY RAJDr. M.R. SRINIVASANPrincipalGovt. Thirumagal Mills CollegeGudiyattam, Vellore Dist - 632 602.Reader in StatisticsUniversity of Madras,Chennai - 600 005.Reviewers - cum - AuthorsThiru. N. RAMESHDr. R.MURTHYSelection Grade LecturerDepartment of MathematicsGovt. Arts College (Men)Nandanam, Chennai - 600 035.Reader in MathematicsDepartment of MathematicsPresidency CollegeChennai 600005.AuthorsDr. V. PRAKASHThiru. S. RAMACHANDRANLecturer (S.G.L),Department of StatisticsPresidency CollegeChennai - 600 005.HeadmasterThe Chintadripet Hr. Sec. SchoolChintadripet, Chennai - 600 002.Thiru. S.T. PADMANABHANThiru. S. RAMANAssistant HeadmasterThe Hindu Hr. Sec. SchoolTriplicane, Chennai - 600 005.Post Graduate TeacherJaigopal Garodia National Hr. Sec. SchoolEast Tambaram, Chennai - 600 059.Tmt. AMALI RAJAPost Graduate TeacherGood Shepherd MatriculationHr. Sec. School,Chennai 600006.Tmt. M.MALINIPost Graduate TeacherP.S. Hr. Sec. School (Main)Mylapore, Chennai 600004.Price : Rs.This book has been prepared by the Directorate of School Educationon behalf of the Government of TamilnaduThis book has been printed on 60 GSM paperPrinted by web offset at :ii
Preface‘The most distinct and beautiful statement of any truth must atlast take theMathematical form’ -Thoreau.Among the Nobel Laureates in Economics more than 60% were Economistswho have done pioneering work in Mathematical Economics.These Economists notonly learnt Higher Mathematics with perfection but also applied it successfully in theirhigher pursuits of both Macroeconomics and Econometrics.A Mathematical formula (involving stochastic differential equations) wasdiscovered in 1970 by Stanford University Professor of Finance Dr.Scholes andEconomist Dr.Merton.This achievement led to their winning Nobel Prize for Economicsin 1997.This formula takes four input variables-duration of the option,prices,interestrates and market volatility-and produces a price that should be charged for the option.Not only did the formula work ,it transformed American Stock Market.Economics was considered as a deductive science using verbal logic groundedon a few basic axioms.But today the transformation of Economics is complete.Extensive use of graphs,equations and Statistics replaced the verbal deductivemethod.Mathematics is used in Economics by beginning with a few variables,graduallyintroducing other variables and then deriving the inter relations and the internal logicof an economic model.Thus Economic knowledge can be discovered and extended bymeans of mathematical formulations.Modern Risk Management including Insurance,Stock Trading and Investmentdepend on Mathematics and it is a fact that one can use Mathematics advantageouslyto predict the future with more precision!Not with 100% accuracy, of course.But wellenough so that one can make a wise decision as to where to invest money.The ideaof using Mathematics to predict the future goes back to two 17th Century FrenchMathematicians Pascal and Fermat.They worked out probabilities of the variousoutcomes in a game where two dice are thrown a fixed number of times.In view of the increasing complexity of modern economic problems,the need tolearn and explore the possibilities of the new methods is becoming ever more pressing.If methods based on Mathematics and Statistics are used suitably according to theneeds of Social Sciences they can prove to be compact, consistent and powerful toolsespecially in the fields of Economics, Commerce and Industry. Further these methodsnot only guarantee a deeper insight into the subject but also lead us towards exact andanalytical solutions to problems treated.This text book has been designed in conformity with the revised syllabus ofBusiness Mathematics(XII) (to come into force from 2005 - 2006)-http:/www.tn.gov.in/schoolsyllabus/. Each topic is developed systematically rigorously treated fromiii
first principles and many worked out examples are provided at every stage to enablethe students grasp the concepts and terminology and equip themselves to encounterproblems. Questions compiled in the Exercises will provide students sufficient practiceand self confidence.Students are advised to read and simultaneously adopt pen and paper forcarrying out actual mathematical calculations step by step. As the Statistics componentof this Text Book involves problems based on numerical calculations,BusinessMathematics students are advised to use calculators.Those students who succeedin solving the problems on their own efforts will surely find a phenomenal increase intheir knowledge, understanding capacity and problem solving ability. They will find iteffortless to reproduce the solutions in the Public Examination.We thank the Almighty God for blessing our endeavour and we do hope that theacademic community will find this textbook triggering their interests on the subject!“The direct application of Mathematical reasoning to the discovery of economictruth has recently rendered great services in the hands of master Mathematicians”– Alfred Marshall.MaliniAmali hSrinivasanAntony Rajiv
CONTENTSPage1.APPLICATIONS OF MATRICES AND DETERMINANTS1.1Inverse of a Matrix1Minors and Cofactors of the elements of a determinant Adjoint of a square matrix - Inverse of a non singular matrix.1.2Systems of linear equationsSubmatrices and minors of a matrix - Rank of a matrix Elementary operations and equivalent matrices - Systemsof linear equations - Consistency of equations - Testing theconsistency of equations by rank method.1.3Solution of linear equationsSolution by Matrix method - Solution by Cramer’s rule1.4Input - Output Analysis1.5Transition Probability Matrices2.ANALYTICAL GEOMETRY2.1ConicsThe general equation of a conic2.2ParabolaStandard equation of parabola - Tracing of the parabola2.3EllipseStandard equation of ellipse - Tracing of the ellipse2.4HyperbolaStandard equation of hyperbola - Tracing of the hyperbola Asymptotes - Rectangular hyperbola - Standard equation ofrectangular hyperbolav42
3.APPLICATIONS OF DIFFERENTIATION - I3.1Functions in economics and commerce72Demand function - Supply function - Cost function - Revenuefunction - Profit function - Elasticity - Elasticity of demand Elasticity of supply - Equilibrium price - Equilibrium quantity Relation between marginal revenue and elasticity of demand.3.2Derivative as a rate of changeRate of change of a quantity - Related rates of change3.3Derivative as a measure of slopeSlope of the tangent line - Equation of the tangent - Equationof the normal4.APPLICATIONS OF DIFFERENTIATION - II4.1Maxima and MinimaIncreasing and decreasing functions - Sign of the derivative- Stationary value of a function - Maximum and minimumvalues - Local and global maxima and minima - Criteria formaxima and minima - Concavity and convexity - Conditionsfor concavity and convexity - Point of inflection - Conditionsfor point of inflection.4.2Application of Maxima and MinimaInventory control - Costs involved in inventory problems Economic order quantity - Wilson’s economic order quantityformula.4.3Partial DerivativesDefinition - Successive partial derivatives - Homogeneousfunctions - Euler’s theorem on Homogeneous functions.4.4Applications of Partial DerivativesProduction function - Marginal productivities - PartialElasticities of demand.vi102
5.APPLICATIONS OF INTEGRATION5.1Fundamental Theorem of Integral Calculus138Properties of definite integrals5.2Geometrical Interpretation of Definite Integral as AreaUnder a Curve5.3Application of Integration in Economics and CommerceThe cost function and average cost function from marginalcost function - The revenue function and demand functionfrom marginal revenue function - The demand function fromelasticity of demand.5.4Consumers’ Surplus5.5Producers’ Surplus6.DIFFERENTIAL EQUATIONS6.1Formation of Differential EquationsOrder and Degree of a Differential Equation-Family of curvesFormation of Ordinary Differential Equation6.2First order Differential EquationsSolution of a differential equation-Variables SeparableHomogeneous differential equations-Solving first orderhomogeneous differential equations-First order lineardifferential equation-Integrating factor6.3Second order Linear Differential Equations with constantcoefficientsAuxiliary equations and Complementary functions-ParticularIntegral -The General solutionvii165
7.INTERPOLATION AND FITTING A STRAIGHT LINE7.1Interpolation192Graphic method of interpolation-Algebraic methods ofinterpolation-Finite differences-Derivation of Gregory Newton's forward formula-Gregory-Newton's backwardformula-Lagrange's formula7.2Fitting a straight lineScatter diagram-Principle of least squares-Derivation ofnormal equations by the principle of least squares8.PROBABILITY DISTRIBUTIONS8.1Random variable and Probability function217Discrete Random Variable-Probability function and Probabilitydistribution of a Discrete random variable-CumulativeDistribution function-Continuous Random Variable-Probabilityfunction-Continuous Distribution function8.2Mathematical expectation8.3Discrete DistributionsBinomial distribution-Poisson distribution8.4Continuous DistributionsNormal Distribution-PropertiesStandard Normal DistributionofNormalDistribution-9.SAMPLING TECHNIQUES AND STATISTICAL INFERENCE9.1Sampling and Types of ErrorsSampling and sample-Parameter and Statistic-Need forSampling-Elements of Sampling Plan-Types of SamplingSampling and non-sampling errors9.2EstimationEstimator-Point Estimate and Interval Estimate-ConfidenceInterval for population mean and proportion9.3Hypothesis TestingNull Hypothesis and Alternative Hypothesis-Types of ErrorCritical region and level of significance-Test of significanceviii252
10. APPLIED STATISTICS27210.1 Linear ProgrammingStructure of Linear programming problem-Formulation ofthe Linear Programming Problem-Applications of Linearprogramming-Some useful Definitions-Graphical method10.2 Correlation and RegressionMeaning of Correlation-Scatter Diagram-Co-efficient dentVariable-Two Regression Lines10.3 Time Series AnalysisUses of analysis of Time Series-Components of Time SeriesModels-Measurement of secular trend10.4 Index NumbersClassification of Index Numbers-Uses of Index NumbersMethod of construction of Index Numbers-Weighted IndexNumbers-Test of adequacy for Index Number-Cost of livingindex-Methods of constructing cost of living index-Uses ofcost of living index number10.5 Statistical Quality ControlCauses for variation-Role and advantages of SQC - Processand Product control-Control Charts326ANSWERSSTANDARD NORMAL DISTRIBUTION-TABLECalculator should be used for solving problems in chapters 7- 10ix341
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1APPLICATIONS OF MATRICESAND DETERMINANTSThe concept of matrices and determinants has extensive applications inmany fields such as Economics, Commerce and Industry. In this chapter we shalldevelop some new techniques based on matrices and determinants and discuss theirapplications.1.1 INVERSE OF A MATRIX1.1.1 Minors and Cofactors of the elements of a determinant.The minor of an element aij of a determinant A is denoted by Mij and is thedeterminant obtained from A by deleting the row and the column where aij occurs.The cofactor of an element aij with minor Mij is denoted by Cij and isdefined asC ij {M ij , if i j is even - M ij ,if i j is oddThus, cofactors are signed minors.In the case ofAlsoa11a21a12we havea22M11 a22,M12 a21,M21 a12,M22 a11C11 a22,C12 – a21,C21 – a12,C22 a11In the case ofa11 a12a13a21 a22a23 , we havea33a31 a32M11 a22 a23,a32 a33C11 a 22 a 23;a 32 a 33M12 a 21 a 23,a 31 a 33C12 -a 21 a 23;a 31 a 33M13 a 21 a 22,a 31 a 32C13 a 21 a 22;a 31 a 32M 21 a12 a13,a 32 a 33C21 -a12 a13and so on.a 32 a 331
1.1.2 Adjoint of a square matrixThe transpose of the matrix got by replacing all the elements of a square matrix Aby their corresponding cofactors in A is called the Adjoint of A or Adjugate of A and isdenoted by Adj A.Thus, Adj A AtcNote(i) d c a b then Ac Let A b a c d d b Adj A Atc c a a b Thus the Adjoint of a 2 x 2 matrix c d d b can be written instantly as c a (ii)Adj I I, where I is the unit matrix.(iii)A (Adj A) (Adj A) A A I(iv)Adj (AB) (Adj B) (Adj A)(v)If A is a square matrix of order 2, then Adj A A If A is a square matrix of order 3 , then Adj A A 2Example 1Ê 1 -2ˆWrite the Adjoint of the matrix A ÁË 4 3 Solution : 3 2 Adj A 4 1 Example 2Ê 0 1 2ˆÁ Find the Adjoint of the matrix A Á 1 2 3 Ë 3 1 1 2
Solution :0 1 2tA 1 2 3 , Adj A A c3 1 1Now, C11 2 31 31 2 1, C12 8, C13 5, 1 13 13 1 C21 C31 1 20 20 1 1, C22 6, C23 3, 1 13 13 1 1 20 20 1 1, C32 2, C33 1, 2 31 31 2 1 8 5 A c 1 63 1 2 1 Hencet1 1 1 8 5 1 Adj A 1 63 8 6 2 3 1 1 2 1 51.1.3 Inverse of a non singular matrixThe inverse of a non singular matrix A is the matrix B such that AB BA I. B isthen called the inverse of A and denoted by A-1.Note(i)A non square matrix has no inverse.(ii)The inverse of a square matrix A exists only when A 0 that is, if A is a singularmatrix then A–1 does not exist.(iii)If B is the inverse of A then A is the inverse of B. That is B A–1 A B–1.(iv)A A–1 I A–1 A(v)The inverse of a matrix, if it exists, is unique. That is, no matrix can have more thanone inverse.(vi)The order of the matrix A–1 will be the same as that of A.(vii)I–1 I(viii)(AB)–1 B–1 A–1, provided the inverses exist.(ix)A2 I implies A–1 A3
(x)If AB C then(a) A CB–1 (b) B A–1C, provided the inverses exist.(xi)We have seen thatA(Adj A) (Adj A) A A I A11(Adj A) (Adj A)A I A A ( A 0)This suggests thatA 1 11 tA(Adj A). That is, A 1 A A c(xii) (A–1)–1 A, provided the inverse exists. a b Let A with A ad – bc 0 c d d -c,-b ad b c a d bad bc c a a b can be written instantly asThus the inverse of a 2 2 matrix c d 1 d b provided ad – bc 0ad bc c a Example 3Ê 5 3ˆ, if it exists.Find the inverse of A ÁË 4 2 Solution : A 5 3–14 2 – 2 0 A exists.A 1 1 2 3 1 2 3 2 4 5 2 4 5 4
Example 4Show that the inverses of the following do not exist:Ê 3 1 - 2ˆ(ii) A Á 2 7 3 Á Ë 6 2 - 4 Ê-2 6 ˆ(i) A ÁË 3 - 9 Solution :(i) A (ii) 263 9 0 A–1 does not exist. A–1 does not exist.Example 5Ê2 3 4 ˆFind the inverse of A Á 3 2 1 , if it exists.Á Ë 1 1 - 2 Solution :2 34 A 3 2 1 15 0 A–1 exists.1 1 2We have, A–1 1 tA A cNow, the cofactors areC11 2 1 5,1 2C21 C31 3 4 10,1 23 4 5,2 1C12 C22 3 1 7,1 22 4 8,1 2C32 2 4 10,3 1Hence5C13 3 2 1,1 1C23 C33 2 3 1,1 12 3 5,2 2
Example 6Ê 3 -2 3 ˆShow that A ÁÁ 2 1 -1 and B Ë 4 -3 2 1 17 8 17 10 175176171 171 17 9 17 7 17 are inverse of each other.Solution : 3 2 3 AB 2 1 1 4 3 2 1 17 8 17 10 175176171 171 17 9 17 7 17 3 2 31 5 117 0 011 2 1 18 6 9 0 17 0 17 174 3 210 1 70 0 171 0 0 0 1 0 I 0 0 1Since A and B are square matrices and AB I, A and B are inverse of each other.EXERCISE 1.1 1 3 1. Find the Adjoint of the matrix . 2 1 2 0 1 2. Find the Adjoint of the matrix 5 1 0 . 0 1 3 4 3 3 3. Show that the Adjoint of the matrix A 10 1 is A itself. 4 3 4 1 1 1 4. If A 1 2 3 , verify that A (Adj A) (Adj A) A A I. 2 1 3 6
1 0 3 1 5. Given A 4 2 , B 2 1 , verify that Adj (AB) (Adj B) (Adj A).6. In the second order matrix A (aij), given that aij i j, write out the matrix A and verifythat Adj A A . 1 1 1 7. Given A 2 1 1 , verify that Adj A A 2. 3 1 1 2 4 8. Write the inverse of A 3 2 . 1 0 2 9. Find the inverse of A 3 1 1 . 2 1 2 1 0 a 10. Find the inverse of A 0 1 b and verify that AA–1 I. 0 0 1 a111. If A 0 00a200 0 and none of the a’s are zero, find A–1. a3 1 2 2 12. If A 4 3 4 , show that the inverse of A is itself. 4 4 5 13.If A–1 1 3 4 3 2 2 , find A. 1 1 1 2 3 1 14. Show that A 1 2 3 and B 3 1 2 1 18 7 18 5 18518118718 77 18 5 18 1 18 are inverse of each other.
2 3 15. If A , compute A–1 and show that 4A–1 10 I – A. 4 8 4 3 verify that (A–1)–1 A.16. If A 2 1 3 1 17. Verify (AB)–1 B–1 A–1, when A and B 2 1 6 0 0 9 . 6 7 1 18. Find λ if the matrix 3 l 5 has no inverse. 9 11 l 1 2 3 19. If X 2 4 5 and Y 3 5 6 4 3 X 20. If 5 2 1 3 2 3 3 1 find p, q such that Y X–1. p q 2 14 29 , find the matrix X.1.2 SYSTEMS OF LINEAR EQUATIONS1.2.1 Submatrices and minors of a matrixMatrices obtained from a given matrix A by omitting some of its rows and columnsare called sub matrices of A. 3 2e.g. If A 2 32011411 10 4415 4 , some of the submatrices of A are:2 2 3 2 3 5 2 4 1 4 2 0 , 2 4 , 3 2 , 0 2 ,1 0 1 4 4 3 2 4 5 1 2 4 0 1 1 , 1 0 2 , 1 1 and 2 0 1 4 1 1 4 2 4 3 1 4 2 The determinants of the square submatrices are called minors of the matrix.8
Some of the minors of A are :2 40 1,1 41 1,3 22 0,3 53 2,3 4 11 1 42 1 1 and 0 4 22 044121.2.2 Rank of a matrixA positive integer ‘r’ is said to be the rank of a non zero matrix A, denoted by ρ(A), if(i)there is at least one minor of A of order ‘r’ which is not zero and(ii)every minor of A of order greater than ‘r’ is zero.Note(i)The rank of a matrix A is the order of the largest non zero minor of A.(ii)If A is a matrix of order m n then ρ(A) minimum (m, n)(iii)The rank of a zero matrix is taken to be 0.(iv)For non zero matrices, the least value of the rank is 1.(v)The rank of a non singular matrix of order n n is n.(vi)ρ(A) ρ(At)(vii)ρ(I2) 2, ρ(I3) 3Example 7Ê 2 1 3ˆFind the rank of the matrix A Á -1 0 2 Á Ë 0 1 5 Solution :Order of A is 3 3. ρ(A) 3Consider the only third order minor2 1 3 1 0 2 – 2 0.0 1 5There is a minor of order 3 which is not zero. ρ(A) 3.9
Example 8Ê 4 5 6ˆFind the rank of the matrix A Á 1 2 3 Á Ë 3 4 5 Solution :Order of A is 3 3. ρ(A) 3Consider the only third order minor4 5 61 2 3 03 4 5The only minor of order 3 is zero. ρ(A) 2Consider the second order minors.We find,4 5 3 01 2There is a minor of order 2 which is non zero. ρ(A) 2Example 95 ˆÊ2 4Find the rank of the matrix A Á 48 10 .Á Ë -6 -12 -15 Solution :Order of A is 3 3. ρ(A) 3Consider the only third order minor2454810 0 6 12 15(R1 R2)The only minor of order 3 is zero. ρ(A) 2Consider the second order minors. Obviously they are all zero. ρ(A) 1 Since A is a non zero matrix. ρ (A) 110
Example 10Ê 1 -3 4 7ˆFind the rank of the matrix A ÁË 9 1 2 0 .Solution :Order of A is 2 4. ρ(A) 2Consider the second order minors.We find,1 3 28 09 1There is a minor of order 2 which is not zero. ρ(A) 2Example 11Ê 1 2 -4 5ˆFind the rank of the matrix A Á 2 -1 3 6 Á Ë 8 1 9 7 Solution :Order of A is 3 4. ρ(A) 3Consider the third order minors.We find,1 2 42 1 3 40 08 19There is a minor of order 3 which is not zero. ρ(A) 3.1.2.3 Elementary operations and equivalent matrices.The process of finding the values of a number of minors in our endeavour to find thera
BUSINESS MATHEMATICS HIGHER SECONDARY - SECOND YEAR Revised based on the recommendation of the Textbook Development Committee A Publication under Government of Tamilnadu Distribution of Free Textbook Programme (NOT FOR SALE) TAMILNADU TEXTBOOK AND EDUCATIONAL SERVICES CORPORATION
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
as HSC Year courses: (in increasing order of difficulty) Mathematics General 1 (CEC), Mathematics General 2, Mathematics (‘2 Unit’), Mathematics Extension 1, and Mathematics Extension 2. Students of the two Mathematics General pathways study the preliminary course, Preliminary Mathematics General, followed by either the HSC Mathematics .
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
2. Further mathematics is designed for students with an enthusiasm for mathematics, many of whom will go on to degrees in mathematics, engineering, the sciences and economics. 3. The qualification is both deeper and broader than A level mathematics. AS and A level further mathematics build from GCSE level and AS and A level mathematics.
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The Nature of Mathematics Mathematics in Our World 2/35 Mathematics in Our World Mathematics is a useful way to think about nature and our world Learning outcomes I Identify patterns in nature and regularities in the world. I Articulate the importance of mathematics in one’s life. I Argue about the natu
1.1 The Single National Curriculum Mathematics (I -V) 2020: 1.2. Aims of Mathematics Curriculum 1.3. Mathematics Curriculum Content Strands and Standards 1.4 The Mathematics Curriculum Standards and Benchmarks Chapter 02: Progression Grid Chapter 03: Curriculum for Mathematics Grade I Chapter 04: Curriculum for Mathematics Grade II
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