Wiley Acing the Gate: Engineering Mathematics and General Aptitude

860 Pages·2018·32.714 MB·English

by  Anil Kumar Maini, Varsha Agrawal, Nakul Maini

WILEY
GATE
ENGINEERING MATHEMATICS
AND GENERAL APTITUDE
SECOND EDITION

WILEY
GATE
ENGINEERING MATHEMATICS
AND GENERAL APTITUDE
SECOND EDITION
Dr Anil Kumar Maini
Senior Scientist and former Director of
Laser Science and Technology Centre
Defence Research and Development Organization, New Delhi
Varsha Agrawal
Senior Scientist, Laser Science and Technology Centre
Defence Research and Development Organization, New Delhi
Nakul Maini
WILEY ACING THE GATE
ENGINEERING MATHEMATICS AND GENERAL APTITUDE
SECOND EDITION
Copyright © 2017 by Wiley India Pvt. Ltd., 4435-36/7, Ansari Road, Daryaganj, New Delhi-110002.
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John Wiley & Sons (Asia) Pte Ltd, 1 Fusionpolis Walk #07-01 Solaris, South Tower Singapore 138628
John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada, M9W 1L1
First Edition: 2015
Second Edition: 2017
ISBN: 978-81-265-6655-6
ISBN: 978-81-265-8347-8 (ebk)
www.wileyindia.com
Printed at:
PREFACE
Wiley Acing the GATE: Engineering Mathematics and General Aptitude is intended to be the complete
book for those aspiring to compete in the Graduate Aptitude Test in Engineering (GATE) in various engineering
disciplines, including Electronics and Communication, Electrical, Mechanical, Computer Science, Civil, Chemical and
Instrumentation, comprehensively covering all topics as prescribed in the syllabus in terms of study material and an
elaborate question bank. There are host of salient features offered by the book as compared to the content of the other
books already published for the same purpose. Some of the important ones include the following.
One of the notable features of the book includes presentation of study material in simple and lucid language
and in small sections while retaining focus on alignment of the material in accordance with the
requirements of GATE examination. While it is important for a book that has to cover a wide range of topics in
Engineering Mathematics, General Aptitude including Verbal Ability and Numerical Ability to be precise in the treat-
ment of different topics, the present book achieves that goal without compromising completeness. The study material
and also the question bank have all the three important `C’ qualities, Conciseness, Completeness and Correctness, for
communicating effectively with the examinees.
Another important feature of the book is its comprehensive question bank. The question bank is organized in
three different categories, namely the Solved Examples, Practice Exercise and Solved GATE Previous Years’ Questions.
Solved Examples contain a large number of questions of varying complexity. Each question in this category is followed
by its solution. Under Practice Exercise, again there are a large number of multiple choice questions. The answers to
these questions are given at the end of the section. Each of the answers is supported by an explanation unlike other
books where solutions to only selected questions are given. The third category contains questions from previous GATE
examinations from 2003 onwards. Each question is followed by a complete solution.
Briefly outlining the length and breadth of the material presented in the book, it is divided into two broad sections,
namely Engineering Mathematics and General Aptitude. Section on General Aptitude is further divided into two
sections covering Verbal Ability and Numerical Ability.
Engineering Mathematics is covered in eleven different chapters. These include Linear Algebra covering impor-
tant topics such as matrix algebra, systems of linear equations, and eigenvalues and eigenvectors; Calculus covering
important topics such as functions of single variable, limit, continuity and differentiability, mean value theorem, defi-
nite and improper integrals, partial derivatives, maxima and minima, gradient, divergence and curl, directional deriva-
tives, line, surface and volume integrals, Stokes’ theorem, Gauss theorem and Green’s theorem, and Fourier series;
Differential Equations covering linear and non-linear differential equations, Cauchy’s and Euler’s equations, Laplace
transforms, partial differentiation, solutions of one-dimensional heat and wave equations, Laplace equation, and method
of variation of parameters; Complex Variables covering analytic functions, Cauchy’s integral theorem, and Taylor and
Laurent series; Probability and Statistics covering conditional probability, random variables, discrete and continuous
distributions, Poisson, normal, uniform, exponential and binomial distributions, correlation and regression analyses,
residue theorem, and solution integrals; Numerical Methods covering numerical solutions of linear and non-linear alge-
braic equations, integration by trapezoidal and Simpson’s rules, single and multi-step methods for differential equa-
tions, numerical solutions of non-linear algebraic equations by secant, bisection, Runge—Kutta and Newton—Raphson
methods; Mathematical Logic including first-order logic and proportional logic; Set Theory and Algebra including
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vi     PREFACE 
sets, relations, functions and groups, partial orders, lattice, and Boolean algebra; Combinatory covering permutations
and combinations, counting, summation, generating functions, recurrence relations, and asymptotics; Graph Theory
covering connectivity and spanning trees, cut vertices and edges, covering and matching, independent sets, colouring,
planarity, and isomorphism; and Transform Theory covering Fourier transform, Laplace transform and z-transform.
General Aptitude comprises two sub-sections namely Verbal Ability and Numerical Ability. Important topics
covered under Verbal Ability include English grammar, synonyms, antonyms, sentence completion, verbal analogies,
word groups, and critical reasoning and verbal deduction. Under Numerical Ability, important topics covered include
basic arithmetic, algebra, and reasoning and data interpretation. Under Basic Arithmetic, we have discussed number
system; percentage; profit and loss; simple interest and compound interest; time and work; average, mixture and
allegation; ratio, proportion and variation; and speed, distance and time. Under Algebra, we have discussed permu-
tation and combination; progression; probability; set theory; and surds, indices and logarithm. The topics covered
under Reasoning and Interpretation are cubes and dices, line graph, tables, blood relationship, bar diagram, pie chart,
puzzles, and analytical reasoning.
The Graduate Admission Test in Engineering (GATE) is an All-India level competitive examination for
engineering graduates aspiring to pursue Master’s or Ph.D. programs in India. The examination evaluates the exam-
inees in General Aptitude, Engineering Mathematics and the subject discipline. Though majority of questions is asked
from the subject discipline; there are sizable number of questions set from Engineering Mathematics and General
Aptitude. It is an examination where close to ten lakh students appear every year. The level of competition is therefore
very fierce. While admission to a top institute for the Master’s programme continues to be the most important reason
for working hard to secure a good score in the GATE examination; another great reason to appear and handsomely
qualify GATE examination is that many Public Sector Undertakings (PSUs) are and probably in future almost all
will be recruiting through GATE examination. And it is quite likely that even big private sector companies may start
considering GATE seriously for their recruitment as GATE score can give a bigger clue about who they are recruiting.
The examination today is highly competitive and the GATE score plays an important role. This only reiterates the
need to have a book that prepares examinees not only to qualify the GATE examination by getting a score just above
the threshold but also enabling them to achieve a competitive score. In a competition that is as fierce as the GATE is,
a high score in Engineering Mathematics and General Aptitude section can be a great asset.
The present book is written with the objective of fulfilling this requirement. The effort is intended to offer to the
large section of GATE aspirants a self-study and do-it-yourself book providing comprehensive and step-by-step treat-
ment of each and every aspect of the examination in terms of concise but complete study material and an exhaustive
set of questions with solutions. The authors would eagerly look forward to the feedback from the readers through pub-
lishers to help them make the book better.
Dr ANIL KUMAR MAINI
VARSHA AGRAWAL
NAKUL MAINI
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CONTENTS
Preface v
ENGINEERINGMATHEMATICS 1
1 LinearAlgebra 3
Matrix 3
Types of Matrices 3
Types of a Square Matrix 4
Equality of a Matrix 4
Addition of Two Matrices 5
Multiplication of Two Matrices 5
Multiplication of a Matrix by a Scalar 5
Transpose of a Matrix 5
Adjoint of a Square Matrix 6
Inverse of a Matrix 6
Rank of a Matrix 6
Determinants 7
Minors 7
Cofactors 7
Solutions of Simultaneous Linear Equations 8
Solution of Homogeneous System of Linear Equations 9
Solution of Non-Homogeneous System of Simultaneous Linear Equations 9
Cramer’s Rule 10
Augmented Matrix 10
Gauss Elimination Method 10
Cayley—Hamilton Theorem 11
Eigenvalues and Eigenvectors 11
Properties of Eigenvalues and Eigenvectors 12
Solved Examples 12
Practice Exercise 15
Answers 18
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viii     CONTENTS 
Explanations and Hints 19
Solved GATE Previous Years’ Questions 26
2 Calculus 73
Limits 73
Left-Hand and Right-Hand Limits 73
Properties of Limits 74
L’Hospital’s Rule 74
Continuity and Discontinuity 74
Differentiability 74
Mean Value Theorems 74
Rolle’s Theorem 74
Lagrange’s Mean Value Theorem 75
Cauchy’s Mean Value Theorem 75
Taylor’s Theorem 75
Maclaurin’s Theorem 75
Fundamental Theorem of Calculus 75
Differentiation 75
Applications of Derivatives 76
Increasing and Decreasing Functions 76
Maxima and Minima 77
Partial Derivatives 78
Integration 78
Methods of Integration 78
Integration Using Table 78
Integration Using Substitution 79
Integration by Parts 79
Integration by Partial Fraction 80
Integration Using Trigonometric Substitution 80
Definite Integrals 80
Improper Integrals 80
Double Integration 81
Change of Order of Integration 82
Triple Integrals 82
Applications of Integrals 82
Area of Curve 82
Length of Curve 82
Volumes of Revolution 83
Fourier Series 83
Conditions for Fourier Expansion 83
Fourier Expansion of Discontinuous Function 84
Change of Interval 84
Fourier Series Expansion of Even and Odd Functions 84
Half Range Series 84
Vectors 85
Addition of Vectors 85
Multiplication of Vectors 85
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CONTENTS     ix
Multiplication of Vectors Using Cross Product 86
Derivatives of Vector Functions 86
Gradient of a Scalar Field 86
Divergence of a Vector 86
Curl of a Vector 86
Directional Derivative 87
Scalar Triple Product 87
Vector Triple Product 87
Line Integrals 87
Surface Integrals 88
Stokes’ Theorem 88
Green’s Theorem 88
Gauss Divergence Theorem 88
Solved Examples 89
Practice Exercise 99
Answers 103
Explanations and Hints 103
Solved GATE Previous Years’ Questions 112
3 DifferentialEquations 171
Introduction 171
Solution of a Differential Equation 171
Variable Separable 171
Homogeneous Equation 172
Linear Equation of First Order 172
Exact Differential Equation 172
Integrating Factor 172
Clairaut’s Equation 172
Linear Differential Equation 173
Particular Integrals 173
Homogeneous Linear Equation 174
Bernoulli’s Equation 175
Euler—Cauchy Equations 175
Homogeneous Euler—Cauchy Equation 175
Non-homogeneous Euler—Cauchy Equation 175
Solving Differential Equations Using Laplace Transforms 176
Variation of Parameters Method 176
Separation of Variables Method 176
One-Dimensional Diffusion (Heat Flow) Equation 177
Second Order One-Dimensional Wave Equation 178
Two-Dimensional Laplace Equation 178
Solved Examples 179
Practice Exercises 187
Answers 192
Explanations and Hints 192
Solved GATE Previous Years’ Questions 198
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