College Algebra For First Year And Pre-Degree Students

410 Pages·15.875 MB·English

by  T. G. Kulkarni, M. K. Kelkar

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C O L L E GE
A L G E B RA
• For First Year
and
• Pre-Degree Student
BY
T. G. KULKARNI, M. A.,
Professor of Mathematics
Jai Hind College and
Basantsing Institute of Science, Bombay
and
M. K. KELKAR, M.SC., B.T.,
Professor of Mathematics and
Head of the Department of Mathematics and
Statistics,
Ismail Yusuf College, Bombay
1973
Published by :
Smt. SlNDHU KULKARNI, B. a , T D.,
Umasadan, 4th Lane,
102, Hindu Colony.
Dadar, Bombay 14
First Edition ; 1958
Second Edition 1959
Third Edition 1961
Fourth Edition 1962
Fifth Edition 1964
Revised Sixth Edition .1965
Revised Seventh Edition 1968
Revised Eighth Edition 1970
Revised Nineth Edition 1973
All rights including those of translation and reproduction as well as
those of preparing and publishing a key giving the solutions
of the examples in this book are reserved by the Authors.
Printed by :
R. S. Gupte,
Aryabhushan Press,
915/1, Shivajinagar, Poona 4.
REVISED SYLLABUS IN ALGEBRA
for
F. Y. Arts and Science, Bombay University
1. Elements of set theory : sets, subsets, empty sets, union
and intersection of sets, complementation, Venn
diagrams.
2. Number systems : Natural numbers, integers, rational
numbers, real numbers, complex numbers.
The real number system : correspondence of real numbers
with points on a straight line ; concept of order and
existence of real number between two given numbers to be
introduced informally. Approximation of irrational numbers
by rational numbers.
Surds: Rational operations with binomial quadratic
surds ; conjugate surds and rationalising factors;
Theorem : If a + yjb = c + \[d, then a '= c and b = d
under prescribed conditions. Properties of real numbers
with reference to closure for elementary operations,
commutativity, associativity and distributivity. The complex
number system : Correspondence between complex numbers
and points in a co-ordinate plane referred to rectangular axes;
conjugate complex numbers ; rational operations with com-
plex numbers ; reduction of given complex expressions to the
form a + ib (a, b real ).
3. Theory of Quadratic Equations with real coefficients:
Solution of quadratic equations ; nature of roots; relations
between roots and coefficients; generalisation to cubic and
quartic equations (statement only). Simple symmetric
functions of roots of a quadratic. Equations with roots
related in a simple way to roots of a given quadratic equation.
iii
Exponents and Logarithms:
Definition of a" for a > 0 and m rational, Theorems:
m n m + n , t \m rn im /• "l <
a Xa =a ,{ab) —a b , (a ) =a , m, n being
rational numbers (proofs for positive integral exponents
only). Informal discussion of a" when x is irrational.
Definition of log,, x, a > 0, a 1 and x > 0. Theorems on
logarithms of product, quotient, power and change of base.
Permutations and Combinations : Linear permutations
with distinct objects. Combinations (case of repetitions
excluded ).
Theorems:
M+V _V 4- V • "r — "r
Relation between nP and "C,.
r
Simple Illustrations for the use of Mathematical Induction :
Formulae for
2 a + ( r - l ) d, 2 ^/ ;
r=t r—0
n2 r, 2n r2, f2t r3.
r= 1 r=l r=l
The Binomial Theorem for a positive integral exponent
( Exclude determination of greatest coefficient, greatest term,
properties of coefficients).
iv
Preface to the Revised Eighth Edition, 1973
This book, first published in 1958, is today running nto
its tkineth edition. All along we have been very keen to incrase
the utility of this book to the students. In view of the Itest
decisions taken by all the Indian Universities to introduce :)me
essential mathematical concepts to the First Year student: we
had introduced an elementary treatment of the theory of ses in
the very first chapter. The chapter on the number system had
been thoroughly revised and made more logical to suit the £eds
of the present tendencies in developing this topic. The ch pter
on Method of Induction now has been taken earlier so as to :take
the use of this method wherever possible. Determinants, th>ugh
not included in the syllabus of the University of Bombay,'nave
been considered in the last chapter to increase the utility cf the
book to a genera] reader of the First Year standard. Tt is sujgest -
ed that the first two chapters on Set Theory and the Nunber
System need not be studied in detail at the first reading. Thf new
ideas introduced in these chapters can be better understood f the
book is read " backwards and forwards" as suggested by
G. Chrystal a famous author of a book on Algebra.
Some Test Papers have been added at the end ii the
Appendix to give the students some practice for the examiiation
under the new course. We are thankful to the Authorities of
Bombay University for allowing us to include their papers it our
book. The copy right of these papers vests with the University of
Bombay.
Our thanks are due to Shri. S. A. Bhide and the staff of
Aryabhushan Press for helping us in all respects in the printing
of this book.
Any suggestions for the further improvement of the book
will be gratefully received.
T. G. KULKARNI,
M. K. KELKAR.
v
PREFACE TO THE FIRST EDITION
The Bombay University, from this academic year, has revised
he syllabus in mathematics for the First Year students. The
Poona University and some other Universities have started the
Pre-degree classes. The present book has been written so as to
meet the needs of the new syllabus in Algebra prescribed for the
First Year and Pre-degree Courses.
The Authors have kept in view the standard of the present
students passing the S. S. C. Examination with Elementary
Mathematics in their regional languages. Special effort has been
made to present the subject matter in simple language without
sacrificing the mathematical rigour. Necessary important results
from School Algebra are given in the Appendix for immediate
reference. Ample illustrative problems have been worked out to
illustrate every new principle and formula. Few examples have
been given immediately at the end of every important article to
enable the students to use and remember the result of the article.
Every effort has been made to prepare carefully the gradation and
selection of examples for every chapter. Attempt has been made
to complete the subject matter in every chapter instead of
restricting ourselves completely to the exact syllabus of any
particular University. Teachers may omit some articles to suit
their needs. The arrangements of the various chapters is flexible
and permits of any sequence desired by the teacher. We hope
that this book will enable the students to understand the subject
matter and also will create interest in the subject.
Our sincere thanks are due to the Manager Shri V. A.
Patwardhan, and the staff of the Aryabhushan Press for the care
they have taken throughout the printing of the book.
T. G. KULKARNI,
M. K. KELKAR.
vi
CONTENTS
Chapter Page
1 Sets .... t
2 Real Numbers ..... 43
3 Complex Numbers 79
4 Indices ( Exponents ) ... 92
5 Logarithms ••• 115
6 Surds — 141
7 Quadratic Equations -- 177
8 Method of Induction ...213
9 Progressions -•• 222
10 Summation of Series '- ... 265
11 Permutations and Combinations ... 280
12 Binomial Theorem --• 307
13 Determinants ••• 325
Appendix
1 Logarithmic and A.nti-logarithmic Tables ... 355
2 Important Formulae and Results ... 359
3 Test Papers ••• 369
Bombay University Papers
vii
Symbols and Abbreviations used in this book
A — \ a, b, c, ••• A is a set of elements a, b, c, ...
A = \ x j x satisfies a given property P
A is a set of elements x such that x satisfies a given
property P.
s. t. such that
iff if and only if
0 the empty set
A' complement a set A
e belongs to
£ does not belong to
g there exists
V for all
c subset
C proper subset
=> implies
o implies and is implied by
U union
f) intersection
N set of positive integers
JV set of negative integers
J set of integers
Q set of rational numbers
Q set of irrational numbers
i
R set of real numbers
C set of co7nplex numbers
A discriminant of a quadratic
equation
2 summation
n ! or ]_» 1 • 2 • • 3 • • • n
"p number of permutations of n
distinct objects taken r at a
time
*C number of combinations of n
r
distinct objects taken r at a
time.