A Textbook of Engineering Mathematics For B.TECH. 2nd YEAR, SEMESTER III

708 Pages·2020·22.193 MB·English

by  Manish Goyal, Satinder Bal Gupta

ABOUT THE BOOK A

The underlying object of this book is to provide the reader with the thorough understanding of topics included in the T
E Tenth Edition
syllabus of Mathema�cs for engineering students. X
The salient features of the book are as follows: T
ABOUT THE BOOK B
The underlying object of this book is to provide the reader with the thorough understanding of topics included in the It exactly covers the prescribed syllabus. Nothing undesirable has been included and nothing essen�al has been O
syllabus of Mathema�cs for engineering students. le�. O
K
The salient features of the book are as follows: Its approach is explanatory and language is lucid and communicable.
O
It exactly covers the prescribed syllabus. Nothing undesirable has been included and nothing essen�al has been The exposi�on of the subject ma�er is systema�c and the students are be�er prepared to solve the problems. F

le�. All fundamentals of the included topics are explained with a micro-analysis.
E
Its approach is explanatory and language is lucid and communicable. Sufficient number of solved examples have been given to let the students understand the various skills
N
The exposi�on of the subject ma�er is systema�c and the students are be�er prepared to solve the problems. necessary to solve the problems. These examples are well-graded.
G
All fundamentals of the included topics are explained with a micro-analysis. Unsolved exercises of mul�-varie�es have been given in a well-graded style. A�emp�ng these on their own, will
I
Sufficient number of solved examples have been given to let the students understand the various skills enable a student to create confidence and independence in him/her regarding the understanding of the subject. N
necessary to solve the problems. These examples are well-graded. Daily life problems and prac�cal applica�ons have been incorporated in the body of the text. E
Unsolved exercises of mul�-varie�es have been given in a well-graded style. A�emp�ng these on their own, will A large number of a�rac�ve and accurate figures have been drawn which enable a student to grasp the subject in E
R
enable a student to create confidence and independence in him/her regarding the understanding of the subject. an easier way.
I
Daily life problems and prac�cal applica�ons have been incorporated in the body of the text. All the answers have been checked and verified. N
A large number of a�rac�ve and accurate figures have been drawn which enable a student to grasp the subject in ABOUT THE AUTHORS G
an easier way.

M
All the answers have been checked and verified.
ABOUT THE AUTHORS A
Dr. Manish Goyal has been associated with the G.L.A. University, Mathura for the last two decades as a competent, T
dynamic, successful and responsible faculty member in Mathema�cs. He is working as Associate Head of the Department H
Dr. Manish Goyal has been associated with the G.L.A. University, Mathura for the last two decades as a competent, A TEXTBOOK OF
of Mathema�cs in Ins�tute of Applied Sciences and Humani�es there. dynamic, successful and responsible faculty member in Mathema�cs. He is working as Associate Head of the Department E
He is equipped with an extraordinary calibre and appreciable academic potency. He has always been synonymous with of Mathema�cs in Ins�tute of Applied Sciences and Humani�es there. M
the word merit and excellence. He is awarded with the degree of Ph.D by Dr. B.R. Ambedkar University (Formerly Agra He is equipped with an extraordinary calibre and appreciable academic potency. He has always been synonymous with ENGINEERING
A
University), Agra in 2009. He qualified CSIR-NET Examina�on in Mathema�cal Sciences with an All India Rank (A.I.R) – 09 the word merit and excellence. He is awarded with the degree of Ph.D by Dr. B.R. Ambedkar University (Formerly Agra
T
in June 2011. University), Agra in 2009. He qualified CSIR-NET Examina�on in Mathema�cal Sciences with an All India Rank (A.I.R) – 09
I
He has so far authored 73 (SEVENTY THREE) Books. Out of them, he has wri�en about FOUR DOZEN books on in June 2011. C
Engineering Mathema�cs for various Technical Universi�es, the prominent among them are A.K.T.U., Lucknow; R.G.P.V., He has so far authored 73 (SEVENTY THREE) Books. Out of them, he has wri�en about FOUR DOZEN books on
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Bhopal; Anna University; Punjab Technical University; U�arakhand Technical University; M.D.U., K.U. etc. His book �tled Engineering Mathema�cs for various Technical Universi�es, the prominent among them are A.K.T.U., Lucknow; R.G.P.V., MATHEMATICS
Computer Based Numerical and Sta�s�cal Techniques has widely been acclaimed. Bhopal; Anna University; Punjab Technical University; U�arakhand Technical University; M.D.U., K.U. etc. His book �tled
Computer Based Numerical and Sta�s�cal Techniques has widely been acclaimed.
He is an ardent researcher. He has, to his credit many research papers, published in reputed journals. He regularly a�ends
He is an ardent researcher. He has, to his credit many research papers, published in reputed journals. He regularly a�ends D D N
workshops/STCs as IITs and NITs. His area of research includes frac�onal differen�al equa�ons and numerical methods.
hSSHoaies�c Biinse. Tdtaeye .crrhe Bv (aiCelo wGmeuprp utoatfe hTra HSscR dieEonEnc eien h&teis Er PnnhagD�go. ()Cn fraoolm mrep Spuautentert dLSoc SineCgnIo ciwen)da fler Ixonemsd� Ktjuoutureur noksaf hlEsen. tg rHga. eU & ni sTi vetehchresn iltoiyfle,o Kgmuyre, uLmkosbnhegero tworafa .Il Fn audnridath npe oMrs, tha gethr haedamsu aad��oconanel SSHwoaeo�c riinkse sdtahye .orrep Bvsai/elS wGTeuCrps toaafs hTIIaHTssR daEonEnd ei Nn htIeiTsrs Pn. ha HD�io s( Cnaaorelm arep opufut reteers dSec aSireCcnhI c iienn)dc felruxodemeds Kj foururaurcnk�saohlnse.a tl r Had eiUff niesir vetehnre�s ialtilyf ee, Kqmuureaum�kosbnheser atornafd .I Fnnudurimathnee rMri,c haaetl hm heaemst hado�odcnase.l r. Satind r. Manis .P. Bali Strictly As Per the New AICTE Model Syllabus (KAS-402)
from Maharshi Dayanand University, Rohtak. He qualified UGC-NET, conducted by University Grants Commission and fhrios mB. TMecahh (aCroshmi pDuatyearn Sacniedn Ucen &iv eErnsgitgy.,) Rfroohmta Ska. nHte L oqnugaoliwfieadl I nUsG�tCu-tNeE oTf, Ecnogngd. u&c Tteedch bnyo Ulongiyv,e Lrosnitgyo Gwraaln atns dC poomsmt girsasdiouna �aondn er h G FOR Dr. A.P.J. ABDUL KALAM TECHNICAL UNIVERSITY, LUCKNOW
published many research papers in Interna�onal & Na�onal Journals and Conferences. B o
published many research papers in Interna�onal & Na�onal Journals and Conferences. a y SEMESTER-III
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He is presently working as Associate Professor in Computer Science and Engineering Department at Indira Gandhi He is presently working as Associate Professor in Computer Science and Engineering Department at Indira Gandhi
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University, Meerpur, Rewari. His Area of interest are Discrete Mathema�cs, Theory of Automata, Compiler Design, So� University, Meerpur, Rewari. His Area of interest are Discrete Mathema�cs, Theory of Automata, Compiler Design, So� u
Compu�ng and Search Engines etc. With a more than 20 years of teaching experience, he has wri�en about 20 books and p
Compu�ng and Search Engines etc. With a more than 20 years of teaching experience, he has wri�en about 20 books and t
more than 30 research papers in Journals of Na�onal and Interna�onal repute. a
more than 30 research papers in Journals of Na�onal and Interna�onal repute. N.P. Bali
ISBN:978-93-5274-554-8
Dr. Manish Goyal
Dr. Satinder Bal Gupta
(An ImpArinn tI SofO L 9a0x0m1i: 2P0u1b5li cCaotimonpsa nPyvt. Ltd.) 9 789352 745548 9789352745548-0499
M.R.P. ` 499.00
UEM-9845-499-ENGG MATH III (UP)-GOY

A TEXTBOOK OF
ENGINEERING MATHEMATICS

A TEXTBOOK OF
ENGINEERING
MATHEMATICS
For
B.TECH. 2nd YEAR, SEMESTER III
Strictly as per the New AICTE Model Syllabus (KAS-402)
FOR Dr. A.P.J. ABDUL KALAM TECHNICAL UNIVERSITY, LUCKNOW
By
Dr. MANISH GOYAL Dr. SATINDER BAL GUPTA
M.Sc. (Mathematics), Ph.D., CSIR-NET Ph. D (Computer Science)
Associate Professor Associate Professor
Department of Mathematics Deptt. of Computer Science & Engineering
Institute of Applied Sciences & Humanities Indira Gandhi University, Meerpur, Rewari
G.L.A. University, Mathura Haryana
U.P.
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CONTENTS
Preface ... (vii)
Syllabus ... (viii)
Chapter Pages
Module – 1
1. Laplace Transforms ... M-1.1 – M-1.96
Module – 2
1. Integral Transforms ... M-2.1 – M-2.84
Module – 3
1. Formal Logic ... M-3.1 – M-3.57
2. Algebraic Structures ... M-3.58 – M-3.117
Module – 4
1. Sets ... M-4.1 – M-4.44
2. Principle of Inclusion and Exclusion ... M-4.45 – M.4.57
3. Relations ... M-4.58 – M-4.123
4. Functions ... M-4.124 – M-4.166
5. Methods of Proof ... M-4.167 – M-4.191
6. Mathematical Induction ... M-4.192 – M-4.215
7. Recurrence Relations and Generating Functions ... M-4.216 – M-4.254
8. Counting Techniques ... M-4.255 – M-4.291
Module – 5
1. Posets and Lattices ... M-5.1 – M-5.49
2. Boolean Algebra ... M-5.50 – M-5.106
(v)

PREFACE TO THE TENTH EDITION
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The Tenth edition of this textbook is an outcome of new syllabus of Mathematics–III
(KAS 402) for the students of B. Tech. II year, proposed and implemented by Dr. A.P.J. Abdul
Kalam Technical University (A.K.T.U), Lucknow recently. The book has been renovated in
the light of the latest syllabus. It will work as the latest ready reckoner for the readers.
The subject matter has been made more lucid and easier to understand. A large number
of new solved examples and questions have been added. All the answers have been checked
and verified. All the questions of latest university papers have been added in the body of the
text. The suggestions from our colleagues and readers have been incorporated at the proper
places. An appreciably heavy demand of the book ensures its utility to the users.
Separate exercise (CHAPTER END PROBLEMS) have been given at the end of each
chapter.
This book is written with a unique style only meant for the welfare of dear students. We
hope that this book will be a strength for them and it will serve their very purpose of attaining
excellent results.
We are indebted to GOD for shower of blessing. The suggestions with a view to enhance
the utility of the book are always welcome.
—AUTHORS
(vii)
SYLLABUS
DR. A.P.J.A.K.T.U., LUCKNOW
MATHEMATICS-III
(Integral Transform and Discrete Maths)
(To be offered to CE and Allied Branches)
Subject Code KAS402
Category Basic Science Course
Subject Name Mathematics-III (Integral Transform and Discrete Maths)
L-T-P Theory Sessional Total Credit
Scheme and Credits Marks Test Assig/Att.
3—1—0 100 30 20 150 4
Pre-requisites (if any) Knowledge of Mathematics I and II of B. Tech or equivalent
Module I (8)
Laplace Transforms: Laplace transform, Existence theorem, Laplace transforms of
derivatives and integrals, Initial and final value theorems, Unit step function, Dirac-delta
function, Laplace transform of periodic function, Inverse Laplace transform, Convolution
theorem, Application to solve simple linear and simultaneous differential equations.
Module II (9)
Integral Transforms: Fourier integral, Fourier Transform, Complex Fourier transform,
Inverse Transforms, Convolution Theorems, Fourier sine and cosine transform, Applications
of Fourier transform to simple one dimensional heat transfer equations, wave equations and
Laplace equations, Z-Transform and its application to solve difference equations.
Module III (8)
Formal Logic, Group, Ring and Field: Introduction to First order logic, Proposition,
Algebra of Proposition, Logical connectives, Tautologies, contradictions and contingency,
Logical implication, Argument, Normal form, Rules of inferences, semigroup, Monoid Group,
Group, Cosets, Lagrange’s theorem, Congruence relation, Cyclic and permutation groups,
Properties of groups, Rings and Fields (definition, examples and standard results only).
Module IV (10)
Set, Relation, Function and Counting Techniques: Introduction of Sets, Relation
and Function, Methods of Proof, Mathematical Induction, Strong Mathematical Induction,
Discrete numeric function and Generating functions, recurrence relations and their solution,
Pigeonhole principle.
Module V (10)
Lattices and Boolean Algebra: Introduction, Partially ordered sets, Hasse Diagram,
Maximal and Minimal element, Upper and Lower bounds, Isomorphic ordered sets, Lattices,
Bounded Lattices and, Distributive Lattices.
Duality, Boolean Algebras as Lattices, Minimization of Boolean Expressions, prime
Implicants, Logic Gates and Circuits, Truth Table, Boolean Functions, Karnaugh Maps.
(viii)