Table Of ContentBUSINESS
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.
Reviewers
Thiru. N. RAMESH, Dr. M.R. SRINIVASAN,
Selection Grade Lecturer Reader in Statistics
Department of Mathematics Department of Statistics
Govt. Arts College (Men) University of Madras,
Nandanam, Chennai - 600 035. Chennai - 600 005.
Thiru. S. GUNASEKARAN,
Headmaster,
Govt. Girls Hr. Sec. School,
Tiruchengode, Namakkal Dist.
Authors
Thiru. S. RAMACHANDRAN, Thiru. S. RAMAN,
Post Graduate Teacher Post Graduate Teacher
The Chintadripet Hr. Sec. School, Jaigopal Garodia National Hr. Sec. School
Chintadripet, Chennai - 600 002. East Tambaram, Chennai - 600 059.
Thiru. S.T. PADMANABHAN, Tmt. K. MEENAKSHI,
Post Graduate Teacher Post Graduate Teacher
The Hindu Hr. Sec. School, Ramakrishna Mission Hr. Sec. School (Main)
Triplicane, Chennai - 600 005. 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 S n, S n2 and S n3.
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 sinq = sina , cosq = cosa and tanq = tan a -
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 - 2x and ex - 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
Lt xn- an , Lt (1+ 1 )x, Lt ex- 1 , Lt log(1+x) ,
xfi a x-a xfi 0 x xfi 0 x xfi 0 x
Lt sinq (statement only)
xfi 0 q
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
MATRICES 1
AND DETERMINANTS
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
(cid:230) a a ....a (cid:246)
(cid:231) 11 12 1n (cid:247)
(cid:231) a a ....a (cid:247)
21 22 2n
(cid:231) (cid:247)
. . .
(cid:231) (cid:247)
is called a MATRIX
(cid:231) . . . (cid:247)
(cid:231) (cid:247)
(cid:231) . . . (cid:247)
(cid:231) (cid:247)
Ł a a ....a ł
m1 m2 mn
The numbers or functions a of this array are called elements, may be
ij
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 = (cid:231)(cid:231)Ł(cid:230) 12 42 (cid:247)(cid:247)ł(cid:246) and B = (cid:231)(cid:231)Ł(cid:230) xx2 sin1x(cid:247)(cid:247)ł(cid:246) are the matrices
x
1
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 ) is a matrix of order m x n. The first subscript i in (a )
ij mxn 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
(cid:230) 1 2 3(cid:246)
A = (cid:231)(cid:231) (cid:247)(cid:247) is a Matrix of order 2 x 3 and
Ł 4 5 6ł
(cid:230) 1 2 (cid:246)
B = (cid:231)(cid:231) (cid:247)(cid:247) is a Matrix of order 2 x 2
Ł 2 4 ł
(cid:230) sinq cosq (cid:246)
(cid:231) (cid:247)
C = (cid:231) (cid:247) is a Matrix of order 2 x 2
Ł cosq sinq ł
(cid:230) 0 22 30 (cid:246)
(cid:231) (cid:247)
D = (cid:231) - 4 5 - 67(cid:247) is a Matrix of order 3 x 3
(cid:231)Ł 78 - 8 93 (cid:247)ł
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
(cid:230) 5 7(cid:246)
(cid:231) (cid:247)
A = (cid:231) (cid:247) is a Square Matrix of order 2
Ł 6 3ł
(cid:230) 3 1 5(cid:246)
(cid:231) (cid:247)
B = (cid:231) 4 1 6(cid:247) is a Square Matrix of order 3
(cid:231) (cid:247)
Ł 2 4 9ł
(cid:230) sina sinb sind (cid:246)
(cid:231) (cid:247)
C = (cid:231) cosa cosb cosd (cid:247) is a Square Matrix of order 3
(cid:231) (cid:247)
Ł coseca cosecb cosecd ł
2
(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
(cid:230) 2 (cid:246)
(cid:231) (cid:247)
A = (cid:231) 0 (cid:247) is a column matrix of order 3 x 1 and
(cid:231) (cid:247)
Ł 1 ł
(cid:230) 1(cid:246)
(cid:231) (cid:247)
B = (cid:231) (cid:247) is a column matrix of order 2 x 1
Ł 0ł
(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
(cid:230) 0 0(cid:246)
(cid:231) (cid:247)
O = (cid:231) (cid:247) is a Null Matrix of order 2 x 2 and
Ł 0 0ł
(cid:230) 0 0 0(cid:246)
(cid:231) (cid:247)
O = (cid:231) (cid:247) is a Null Matrix of order 2 x 3
Ł 0 0 0ł
(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
(cid:230) 5 0(cid:246)
(cid:231) (cid:247)
A = (cid:231) (cid:247) is a Diagonal Matrix of order 2 and
Ł 0 9ł
(cid:230) 1 0 0(cid:246)
(cid:231) (cid:247)
B = (cid:231) 0 2 0(cid:247) is a Diagonal Matrix of order 3
(cid:231) (cid:247)
Ł 0 0 3ł
3
Consider the square matrix
(cid:230) 1 3 7 (cid:246)
(cid:231) (cid:247)
A = (cid:231) 5 - 2 - 4(cid:247)
(cid:231) (cid:247)
Ł 3 6 5 ł
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
(cid:230) 2 0 0(cid:246)
(cid:231) (cid:247)
A = (cid:231) 0 2 0(cid:247) is a Scalar Matrix of order 3 and the value of scalar K = 2
(cid:231) (cid:247)
Ł 0 0 2ł
(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
(cid:230) 1 0(cid:246)
(cid:231) (cid:247)
I = (cid:231) (cid:247) is a Unit Matrix of order 2
2 Ł 0 1ł
(cid:230) 1 0 0(cid:246)
(cid:231) (cid:247)
I = (cid:231) 0 1 0(cid:247) is a Unit Matrix of order 3
3 (cid:231) (cid:247)
Ł 0 0 1ł
1.1.4 Multiplication of a marix by a scalar
If A = (a ) is a matrix of any order and if K is a scalar, then the Scalar
ij
Multiplication of A by the scalar k is defined as
KA= (Ka ) for all i, j.
ij
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 ) is defined by - A = (-a ) for all
ij mxn ij mxn
i, j and is obtained by changing the sign of every element.
4
