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https://iesmasterpublications.com/GATE-Books/engineering-mathematics-ese-gate-psus-2018/?utm_source=engineeringmathematics&utm_medium=EBOOK&utm_campaign=engineeringmathematics
ENGINEERING MATHEMATICS
Office : Phone : F-126, (Lower Basement), Katwaria Sarai, New Delhi-110016 011-26522064
Mobile : E-mail:
Web :
8130909220, 9711853908 [email protected], [email protected]
iesmasterpublications.com, iesmaster.org
Salient Features :
289 topics under 31 chapters in 8 units 672 Solved Examples for comprehensive understanding 1386 questions from last 25 years of GATE & ESE exams with detailed solutions Only book having complete theory on ESE & GATE Pattern Comprising conceptual questions marked with * to save the time while revising
(For ESE & GATE Exam)
(CE, ME, PI, CH, EC, EE, IN, CS, IT)
First Edition : 2017
Typeset at : IES Master Publication, New Delhi-110016
IES MASTER PUBLICATIONF-126, (Lower Basement), Katwaria Sarai, New Delhi-110016Phone : 011-26522064, Mobile : 8130909220, 9711853908E-mail : [email protected], [email protected] : iesmasterpublications.com, iesmaster.org
All rights reserved.Copyright 2017, by IES MASTER Publications. No part of this booklet may be reproduced, ordistributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwiseor stored in a database or retrieval system without the prior permission of IES MASTER, New Delhi.Violates are liable to be legally prosecuted.
PREFACE
We, the IES MASTER, have immense pleasure in placing the first edition of Engineering Mathematics before theaspirants of GATE & ESE exams.
Dear Students, as we all know that in 2016 UPSC included Engineering Mathematics as a part of syllabus of commonpaper for ESE exam as well as of a technical paper for EC/EE branch, while Engineering Mathematics already has 15%weightage in GATE exam. We have observed that currently available books cover neither all the topics nor all previouslyasked questions in GATE & ESE exams. Since most of the books focus on only some selected main topics, studentshave not been able to answer more than 60-65% of 1386 questions that have been asked in GATE & ESE exams sofar. Hence to overcome this problem, we have tried our best by covering more than 289 topics under 31 chapters in8 units. (One should not be in dilemma that 289 topics are more than sufficient. These are the minimum topics fromwhere GATE & ESE have already asked questions). Since we have covered every previous year questions from last 25years of each topic, students can easily decide, how much time to allocate on each chapter based on the number ofquestions asked in that particular exam. Again, we have included only those proofs that are necessary for conceptbuilding of topics and we have stressed on providing elaborate solution to all the questions.
It is the only book in the market which has complete theory exactly on ESE & GATE Pattern. After each topic thereare sufficient number of solved examples for concept building & easy learning. The book includes such types of 672examples. It also covers all the previously asked questions in which conceptual questions are marked with * sign sothat students can save their time, while revising.
Having incorporated my teaching experience of more than 13 years, I believe this book will enable the students to excelin Engineering Mathematics.
My source of inspiration is Mr. Kanchan Thakur Sir (Ex-IES). He has continuously motivated me while writing this book.
My special thanks to the entire IES MASTER Team for their continuous support in bringing out the book. I stronglybelieve that this book will help students in their journey of success. I invite suggestions from students, teachers &educators for further improvement in the book.
Dr. Puneet Sharma(M.Sc., Ph.D.)
IES Master PublicationsNew Delhi
https://iesmaster.org/?utm_source=ebook&utm_medium=Admissionopen&utm_campaign=Publication
UNIT 1 : CALCULUS
1.1 Limits, Continuity and Differentiability ....................................................... 01 44
(i) Function ........................................................................................................... 01
(ii) Limit of a Function ............................................................................................ 01
(iii) Theorem on Limits ........................................................................................... 02
(iv) Indeterminate Forms ........................................................................................ 04
(v) L-Hospital Rule ................................................................................................ 05
(vi) Fundamentals of Continuity .............................................................................. 08
(vii) Kinds of Discontinuities .................................................................................... 10
(viii) Properties of Continuous Functions .................................................................. 15
(ix) Saltus of a Function .......................................................................................... 16
(x) Function of Two Variables ................................................................................ 17
(xi) Limit of a Function of Two Variables ................................................................. 17
(xii) Continuity of Function of TwoVariables ............................................................. 22
(xiii) Differentiability ................................................................................................. 23
Previous Years GATE & ESE Questions ...................................................... 29
1.2 Partial Differentiation ................................................................................... 45 65
(i) Parial Derivatives............................................................................................. 45
(ii) Homogeneous Functions ................................................................................. 49
(iii) Eulers Theorem on Homogenous Function ...................................................... 50
(iv) Total Differential Coefficients ............................................................................ 52
(v) Change of Variables ........................................................................................ 52
(vi) Jacobian .......................................................................................................... 56
(vii) Chain Rule of Jacobian .................................................................................... 59
CONTENTS
(vi)
(viii) Functional Dependence ................................................................................... 60
Previous Years GATE & ESE Questions ...................................................... 63
1.3 Infinite Series ................................................................................................ 66 78
(i) Expansion of Functions .................................................................................... 66
(ii) Maclaurins theorem......................................................................................... 66
(iii) Taylors Theorem ............................................................................................. 66
(iv) Convergence and Divergence of Infinite Series ................................................ 69
(v) Methods to Find Convergence of Infinite Series ................................................ 70
(vi) Power Series ................................................................................................... 71
Previous Years GATE & ESE Questions ...................................................... 72
1.4 Mean Value Theorems ................................................................................. 79 86
(i) Rolles Theorem............................................................................................... 79
(ii) Lagranges First Mean Value Theorem............................................................. 80
(iii) Cauchys Mean Value Theorem........................................................................ 80
(iv) Bolzanos Theorem ........................................................................................... 81
(v) Intermediate Value Theorem............................................................................. 81
(vi) Darboux Theorem ............................................................................................ 81
Previous Years GATE & ESE Questions ...................................................... 84
1.5 Maxima and Minima .................................................................................... 87 114
(i) Increasing and Decreasing Functions ............................................................... 87
(ii) Maxima and Minima of Function of Single Variable........................................... 88
(iii) Sufficient Condition (Second Derivate Test)...................................................... 89
(iv) Maxima-Minima of Functions of Two Variables ................................................. 89
(v) Sufficient Condition (Lagranges Conditions).................................................... 91
(vi) Lagranges Method of Undetermined Multipliers ............................................... 94
Previous Years GATE & ESE Questions ............