BUSINESS MATHEMATICS Higher Secondary - First Year Untouchability is a sin Untouchability is a crime Untouchability is inhuman Tamilnadu Textbook Corporation College Road, Chennai - 600 006. © Government of Tamilnadu First Edition - 2004 Chairperson Thiru. V. THIRUGNANASAMBANDAM, Retired Lecturer in Mathematics Govt. Arts College (Men) Nandanam, Chennai - 600 035. Thiru. S. GUNASEKARAN, Headmaster, Govt. Girls Hr. Sec. School, Tiruchengode, Namakkal Dist. Reviewers Dr. M.R. SRINIVASAN, Reader in Statistics Department of Statistics University of Madras, Chennai - 600 005. Thiru. N. RAMESH, Selection Grade Lecturer Department of Mathematics Govt. Arts College (Men) Nandanam, Chennai - 600 035. Authors Thiru. S. RAMACHANDRAN, Post Graduate Teacher The Chintadripet Hr. Sec. School, Chintadripet, Chennai - 600 002. Thiru. S. RAMAN, Post Graduate Teacher Jaigopal Garodia National Hr. Sec. School East Tambaram, Chennai - 600 059. Thiru. S.T. PADMANABHAN, Post Graduate Teacher The Hindu Hr. Sec. School, Triplicane, Chennai - 600 005. Tmt. K. MEENAKSHI, Post Graduate Teacher Ramakrishna Mission Hr. Sec. School (Main) T. Nagar, Chennai - 600 017. Thiru. V. PRAKASH, Lecturer (S.S.), Department of Statistics, Presidency College, Chennai - 600 005. Price : Rs. This book has been prepared by the Directorate of School Education on behalf of the Government of Tamilnadu This book has been printed on 60 GSM paper Laser typeset by : JOY GRAPHICS, Chennai - 600 002. Printed by : Preface This book on Business Mathematics has been written in conformity with the revised syllabus for the first year of the Higher Secondary classes. The aim of this text book is to provide the students with the basic knowledge in the subject. We have given in the book the Definitions, Theorems and Observations, followed by typical problems and the step by step solution. The society’s increasing business orientation and the students’ preparedness to meet the future needs have been taken care of in this book on Business Mathematics. This book aims at an exhaustive coverage of the curriculum and there is definitely an attempt to kindle the students creative ability. While preparing for the examination students should not restrict themselves only to the questions / problems given in the self evaluation. They must be prepared to answer the questions and problems from the entire text. We welcome suggestions from students, teachers and academicians so that this book may further be improved upon. We thank everyone who has lent a helping hand in the preparation of this book. Chairperson The Text Book Committee iii SYLLABUS 1) Matrices and Determinants (15 periods) Order - Types of matrices - Addition and subtraction of matrices and Multiplication of a matrix by a scalar - Product of matrices. Evaluation of determinants of order two and three - Properties of determinants (Statements only) - Singular and non singular matrices - Product of two determinants. 2) Algebra (20 periods) Partial fractions - Linear non repeated and repeated factors - Quadratic non repeated types. Permutations - Applications - Permutation of repeated objects - Circular permutaion. Combinations - Applications - Mathematical induction - Summation of series using Σn, Σn 2 and Σn 3 . Binomial theorem for a positive integral index - Binomial coefficients. 3) Sequences and series (20 periods) Harnomic progression - Means of two positive real numbers - Relation between A.M., G.M., and H.M. - Sequences in general - Specifying a sequence by a rule and by a recursive relation - Compound interest - Nominal rate and effective rate - Annuities - immediate and due. 4) Analytical Geometry (30 periods) Locus - Straight lines - Normal form, symmetric form - Length of perpendicular from a point to a line - Equation of the bisectors of the angle between two lines - Perpendicular and parallel lines - Concurrent lines - Circle - Centre radius form - Diameter form - General form - Length of tangent from a point to a circle - Equation of tangent - Chord of contact of tangents. 5) Trigonometry (25 periods) Standard trigonometric identities - Signs of trigonometric ratios - compound angles - Addition formulae - Multiple and submultiple angles - Product formulae - Principal solutions - Trigonometric equations of the form sinθ = sinα, cosθ = cosα and tanθ = tan α - Inverse trigonometric functions. 6) Functions and their Graphs (15 Periods) Functions of a real value - Constants and variables - Neighbourhood - Representation of functions - Tabular and graphical form - Vertical iv line test for functions - Linear functions - Determination of slopes - Power function - 2 x and e x - Circular functions - Graphs of sinx, ,cosx and tanx - Arithmetics of functions (sum, difference, product and quotient) Absolute value function, signum function - Step function - Inverse of a function - Even and odd functions - Composition of functions 7) Differential calculus (30 periods) Limit of a function - Standard forms ax Lt → a-x ax nn − , 0x Lt → (1+ x 1 ) x , 0x Lt → x 1e x − , 0x Lt → x x)log(1+ , 0x Lt → θ θsin (statement only) Continuity of functions - Graphical interpretation - Differentiation - Geometrical interpretation - Differtentiation using first principles - Rules of differentiation - Chain rule - Logarithmic Differentitation - Differentiation of implicit functions - parametric functions - Second order derivatives. 8) Integral calculus (25 periods) Integration - Methods of integration - Substitution - Standard forms - integration by parts - Definite integral - Integral as the limit of an infinite sum (statement only). 9) Stocks, Shares and Debentures (15 periods) Basic concepts - Distinction between shares and debentures - Mathematical aspects of purchase and sale of shares - Debentures with nominal rate. 10) Statistics (15 Periods) Measures of central tendency for a continuous frequency distribution Mean, Median, Mode Geometric Mean and Harmonic Mean - Measures of dispersion for a continuous frequency distribution - Range - Standard deviation - Coefficient of variation - Probability - Basic concepts - Axiomatic approach - Classical definition - Basic theorems - Addition theorem (statement only) - Conditional probability - Multiplication theorem (statement only) - Baye’s theorem (statement only) - Simple problems. v Contents Page 1. MATRICES AND DETERMINANTS 1 2. ALGEBRA 25 3. SEQUENCES AND SERIES 54 4. ANALYTICAL GEOMETRY 89 5. TRIGONOMETRY 111 6. FUNCTIONS AND THEIR GRAPHS 154 7. DIFFERENTIAL CALCULUS 187 8. INTEGRAL CALCULUS 229 9. STOCKS, SHARES AND DEBENTURES 257 10. STATISTICS 280 vi 1 1.1 MATRIX ALGEBRA Sir ARTHUR CAYLEY (1821-1895) of England was the first Mathematician to introduce the term MATRIX in the year 1858. But in the present day applied Mathematics in overwhelmingly large majority of cases it is used, as a notation to represent a large number of simultaneous equations in a compact and convenient manner. Matrix Theory has its applications in Operations Research, Economics and Psychology. Apart from the above, matrices are now indispensible in all branches of Engineering, Physical and Social Sciences, Business Management, Statistics and Modern Control systems. 1.1.1 Definition of a Matrix A rectangular array of numbers or functions represented by the symbol                     mnm2m1 2n2221 1n1211 aaa aaa aaa is called a MATRIX The numbers or functions a ij of this array are called elements, may be real or complex numbers, where as m and n are positive integers, which denotes the number of Rows and number of Columns. For example A =         42 21 and B =         x 1 2 x sinxx are the matrices MATRICES AND DETERMINANTS 1 2 1.1.2 Order of a Matrix A matrix A with m rows and n columns is said to be of the order m by n (m x n). Symbolically A = (a ij ) mxn is a matrix of order m x n. The first subscript i in (a ij ) ranging from 1 to m identifies the rows and the second subscript j in (a ij ) ranging from 1 to n identifies the columns. For example A =         654 321 is a Matrix of order 2 x 3 and B =         42 21 is a Matrix of order 2 x 2 C =         θθ θθ sincos cossin is a Matrix of order 2 x 2 D =           − −− 93878 6754 30220 is a Matrix of order 3 x 3 1.1.3 Types of Matrices (i) SQUARE MATRIX When the number of rows is equal to the number of columns, the matrix is called a Square Matrix. For example A =         36 75 is a Square Matrix of order 2 B =           942 614 513 is a Square Matrix of order 3 C =           δβα δβα δβα coseccoseccosec coscoscos sinsinsin is a Square Matrix of order 3 3 (ii) ROW MATRIX A matrix having only one row is called Row Matrix For example A = (2 0 1) is a row matrix of order 1 x 3 B = (1 0) is a row matrix or order 1 x 2 (iii) COLUMN MATRIX A matrix having only one column is called Column Matrix. For example A =           1 0 2 is a column matrix of order 3 x 1 and B =         0 1 is a column matrix of order 2 x 1 (iv) ZERO OR NULL MATRIX A matrix in which all elements are equal to zero is called Zero or Null Matrix and is denoted by O. For example O =         00 00 is a Null Matrix of order 2 x 2 and O =         000 000 is a Null Matrix of order 2 x 3 (v) DIAGONAL MATRIX A square Matrix in which all the elements other than main diagonal elements are zero is called a diagonal matrix For example A =         90 05 is a Diagonal Matrix of order 2 and B =           300 020 001 is a Diagonal Matrix of order 3 4 Consider the square matrix A =           −− 563 425 731 Here 1, -2, 5 are called main diagonal elements and 3, -2, 7 are called secondary diagonal elements. (vi) SCALAR MATRIX A Diagonal Matrix with all diagonal elements equal to K (a scalar) is called a Scalar Matrix. For example A =           200 020 002 is a Scalar Matrix of order 3 and the value of scalar K = 2 (vii) UNIT MATRIX OR IDENTITY MATRIX A scalar Matrix having each diagonal element equal to 1 (unity) is called a Unit Matrix and is denoted by I. For example I 2 =         10 01 is a Unit Matrix of order 2 I 3 =           100 010 001 is a Unit Matrix of order 3 1.1.4 Multiplication of a marix by a scalar If A = (a ij ) is a matrix of any order and if K is a scalar, then the Scalar Multiplication of A by the scalar k is defined as KA= (Ka ij ) for all i, j. In other words, to multiply a matrix A by a scalar K, multiply every element of A by K. 1.1.5 Negative of a matrix The negative of a matrix A = (a ij ) mxn is defined by - A = (-a ij ) mxn for all i, j and is obtained by changing the sign of every element. [...]... Evaluate 1 3   then determinant of A is   -1 - 2 |A| and the determinant value is = ad - bc Example 9 Solution: 1 3 -1 - 2 = 1 x (-2 ) - 3 x (-1 ) = -2 + 3 = 1 Example 10 2 0 4 − 1 1 7 Evaluate 5 9 8 Solution: 2 0 4 5 −1 1 9 7 8 =2 −1 7 1 8 -0 5 9 1 8 +4 5 9 −1 7 = 2 (-1 x 8 - 1 x 7) - 0 (5 x 8 -9 x 1) + 4 (5x7 - (-1 ) x 9) = 2 (-8 -7 ) - 0 (40 - 9) + 4 (35 + 9) = -3 0 - 0 + 176 = 146 1.2.2 Properties Of Determinants... −2 8 −2 −4 -2 -4 8 = 1(24) - x (40) -4 (-2 0 +6) = 24 -4 0x + 56 = -4 0x + 80 ⇒ -4 0 x + 80 = 0 ∴ x=2 Example : 16 1 b+c b2 + c2 Show 1 c+a a+b c 2 + a 2 = (a-b) (b-c) (c-a) a 2 + b2 1 Solution : 1 b+c b 2 + c2 1 c+a c2 + a2 1 a + b a 2 + b2 R2 → R2 - R1 , R3 → R3 - R1 1 b + c b 2 + c2 = 0 a-b a 2 + b2 0 a -c a2 - c2 1 b+c = 0 a -b 0 a -c b 2 + c2 (a + b)(a - b) taking out (a-b) from R and (a-c) from R... Solution: AB  1 (-1 ) + 2 (-1 ) + 3(1)  =  2 (-1 ) + 4 (-1 ) + 6(1)  3 (-1 ) + 6 (-1 ) + 9(1)  0  = 0 0  0 0 0 1 (-2 ) + 2 (-2 ) + 3x2 2 (-2 ) + 4 (-2 ) + 6(2) 3 (-2 ) + 6 (-2 ) + 9(2) 1 (-4 ) + 2 (-4 ) + 3x4   2 (-4 ) + 4 (-4 ) + 6x4  3 (-4 ) + 6 (-4 ) + 9x4   0  0 0 3x 3   − 17 − 34 − 51   Similarly BA =  − 17 − 34 − 51  17 34 51    ∴ AB ≠ BA Example 5 1 If A =  3  Solution: − 2  , then compute A 2-5 A + 3I... x 2 Solution :  2 AB −3   1 − 4 2  0 1  =   4 0 1       − 4 −2  1x2 + (-4 )x0 + 2 (-4 ) 1x (-3 ) + (-4 )x1 + 2x (-2 )   =   4x2 + 0x0 + 1x (-4 ) 4x (-3 ) + 0x1 + 1x (-2 )     2+ 0- 8 - 3- 4 - 4  = 8 + 0 - 4 -1 2 + 0 - 2  =     −6 ∴ L.H.S = (AB)T =   4  R.H.S = BT A T  - 6 - 11    4 - 14      −6 4  − 11   =  −11 −14   − 14    T  2 0 − 4 = −3 1 − 2  ... = 0 a-b a 2 + b2 0 a -c a2 - c2 1 b+c = 0 a -b 0 a -c b 2 + c2 (a + b)(a - b) taking out (a-b) from R and (a-c) from R 2 3 (a + c)(a - c) 19 1 b + c b2 + c 2 0 1 a+b 0 = (a-b) (a-c) 1 a+c = (a-b) (a-c) [a+c-a-b] (Expanding along c1) = (a-b) (a-c) (c-b) = (a-b) (b-c) (c-a) EXERCISE 1.2 1) Evaluate (i) 4 6 −2 3 1 2) Evaluate 3 1 (ii) 3 4 2 5 (iii) −2 −1 −4 −6 2 0 1 0 0 −1 2 4 4 Evaluate 0 0 1 0 0 1 3)... 2 3 3  1  is non-singular 2  1 2 If the value of 4 6 7) 3 1 2 1 Evaluate 2 3 5 2 0 = -6 0, then evaluate 4 7 6 4 −2 2 4 −1 6 6 2 5 0 4 7 1 2 3 1 8 3 If the value of 1 1 3 = 5, then what is the value of 1 7 3 2 0 1 2 12 1 20 10) Show that 2 + 4 6 +3 1 5 a - b b -c Prove that b - c 1 a-b a 3 1 5 1 Prove that c + a b a + b c 1 =0 1 1 13) 5 4 + a -b = 0 b- c b+ c 12) 6 c -a c-a c -a 11) 2 = 1 1 Show... Solution: Step 1: Step 2: 4x +1 A B Let ( x -2 )( x +1) = + x -2 x +1 Taking L.C.M on R.H.S 4x + 1 ( x - 2 )( x +1) = (1) A (x + 1 ) + B ( x - 2 ) ( x - 2 )( x + 1 ) Step 3: Equating the numerator on both sides 4x+1 = A(x+1) + B(x-2) = Ax+A + Bx-2B = (A+B)x + (A-2B) Step 4: Equating the coefficient of like terms, A+B =4 -( 2) A-2B = 1 -( 3) Step 5: Solving the equations (2) and... a non-singular matrix Example 13  1 2 Show that   2 4  is a singular matrix    Solution: 1 2 = 4-4 =0 2 4 ∴ The matrix is singular Example 14 2 5  Show that   9 10 is a non-singular matrix    Solution : 2 5 9 10 = 29 - 45 = -2 5 ≠ 0 ∴ The given matrix is non singular 18 Example : 15 1 x −4 Find x if 5 3 0 = 0 - 2 -4 8 Solution : Expanding by 1st Row, 1 x −4 3 5 3 0 = 1 3 0 -x 5 0 + (-4 )... if A =  2 − 1  B= 6 7   3 x 2 3  then AB =  2 6  5   − 1 7    5 − 7  − 2 4    2 x 2  5 −7  − 2 4     3x (-7 ) + 5x(5)   5  3 x 5 + 5x (-2 ) −1      2 x 5 + (-1 ) x (-2 ) 2 x (-7 ) + (-1 ) x (4)  =  12 − 18  =   6 x 5 + 7x (-2 ) 6x (-7 ) + 7x (4)   16 − 14      1.1.11 Properties of matrix multiplication (i) Matrix Multiplication is not commutative i.e for the... 1 ( x - 2) (x+1) = x − 2 + x + 1 Example 2 1 (x - 1) (x + 2) 2 Resolve into partial fractions Solution: 1 A B Step 1: Let Step 2: C Taking L.C.M on R.H.S we get (x -1 )(x + 2 )2 = x −1 + x + 2 + (x + 2 )2 1 A ( x + 2 ) + B ( x − 1 )( x + 2 )+ C ( x − 1 ) = (x -1 )(x + 2 )2 ( x -1 ) ( x + 2 )2 2 Step 3: Equating Numerator on either sides we get 1 = A(x+2)2+B(x-1)(x+2)+C(x-1) Step 4: Puting x = -2 we get .           ++++++ ++++++ ++++++ 9x46 (-4 )3 (-4 )9(2)6 (-2 )3 (-2 )9(1)6 (-1 )3 (-1 ) 6x44 (-4 )2 (-4 )6(2)4 (-2 )2 (-2 )6(1)4 (-1 )2 (-1 ) 3x42 (-4 )1 (-4 )3x22 (-2 )1 (-2 )3(1)2 (-1 )1 (-1 ) = 3 x 3 000 000 000           Similarly.         ++++ ++++ 1x (-2 )0x14x (-3 )1x (-4 )0x04x2 2x (-2 ) (-4 )x11x (-3 )2 (-4 ) (-4 )x01x2 =         ++ + 2-0 1 2-4 -0 8 4-4 - 3-8 -0 2 =         1 4-4 1 1-6 - ∴ L.H.S.