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GATE 2020 and ESE 2020 Prelims Comprehensive Theory with Solved Examples Including Previous Solved Questions of GATE (2003-2019) and ESE-Prelims (2017-2019) Note: Syllabus of ESE Mains Electrical Engineering also covered Thoroughly Revised and Updated Publications For Engineering Mathematics

Thoroughly Revised and Updated Engineering

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Page 1: Thoroughly Revised and Updated Engineering

GATE 2020 and ESE 2020 Prelims

Comprehensive Theory with Solved ExamplesIncluding Previous Solved Questions of

GATE (2003-2019) and ESE-Prelims (2017-2019)

Note: Syllabus of ESE Mains Electrical Engineering also covered

Thoroughly Revised and Updated

Publications

For

EngineeringMathematics

Page 2: Thoroughly Revised and Updated Engineering

Engineering Mathematics for GATE 2020 and ESE 2020 Prelims © Copyright, by MADE EASY Publications.All rights are reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photo-copying, recording or otherwise), without the prior written permission of the above mentioned publisher of this book.

1st Edition : 2009

2nd Edition : 2010

3rd Edition : 2011

4th Edition : 2012

5th Edition : 2013

6th Edition : 2014

7th Edition : 2015

8th Edition : 2016

9th Edition : 2017

10th Edition : 2018

11th Edition : 2019

MADE EASY PUBLICATIONS has taken due care in collecting the data and providing the solutions, before publishing this book. Inspite of this, if any inaccuracy or printing error occurs then MADE EASY PUBLICATIONS owes no responsibility. We will be grateful if you could point out any such error. Your suggestions will be appreciated.

MADE EASY Publications

Corporate Office: 44-A/4, Kalu Sarai (Near Hauz Khas Metro Station), New Delhi-110016E-mail: [email protected]: 011-45124660, 8860378007

Visit us at: www.madeeasypublications.org

Publications

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Over the period of time the GATE and ESE examination have

become more challenging due to increasing number of

candidates. Though every candidate has ability to succeed

but competitive environment, in-depth knowledge, quality

guidance and good source of study is required to achieve

high level goals.

The new edition of Engineering Mathematics for GATE 2020 and ESE 2020 Prelims has been fully

revised, updated and edited. The whole book has been divided into topicwise sections.

I have true desire to serve student community by way of providing good source of study and

quality guidance. I hope this book will be proved an important tool to succeed in GATE and ESE

examination. Any suggestions from the readers for the improvement of this book are most welcome.

B. Singh (Ex. IES)

Chairman and Managing Director

MADE EASY Group

Preface

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SYLLABUSGATE and ESE Prelims: Civil EngineeringLinear Algebra: Matrix algebra; Systems of linear equations; Eigen values and Eigen vectors.Calculus: Functions of single variable; Limit, continuity and differentiability; Mean value theorems, local maxima and minima, Taylor and Maclaurin series; Evaluation of definite and indefinite integrals, application of definite integral to obtain area and volume; Partial derivatives; Total derivative; Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.Ordinary Differential Equation (ODE): First order (linear and non-linear) equations; higher order linear equations with constant coefficients; Euler-Cauchy equations; Laplace transform and its application in solving linear ODEs; initial and boundary value problems.Partial Differential Equation (PDE): Fourier series; separation of variables; solutions of one-dimensional diffusion equation; first and second order one-dimensional wave equation and two-dimensional Laplace equation.Probability and Statistics: Definitions of probability and sampling theorems; Conditional probability; Discrete Random variables: Poisson and Binomial distributions; Continuous random variables: normal and exponential distributions; Descriptive statistics - Mean, median, mode and standard deviation; Hypothesis testing.Numerical Methods: Accuracy and precision; error analysis. Numerical solutions of linear and non-linear algebraic equations; Least square approximation, Newton’s and Lagrange polynomials, numerical differentiation, Integration by trapezoidal and Simpson’s rule, single and multi-step methods for first order differential equations.

GATE and ESE Prelims: Mechanical EngineeringLinear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors.Calculus: Functions of single variable, limit, continuity and differentiability, mean value theorems, indeterminate forms; evaluation of definite and improper integrals; double and triple integrals; partial derivatives, total derivative, Taylor series (in one and two variables), maxima and minima, Fourier series; gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, applications of Gauss, Stokes and Green’s theorems.Differential equations: First order equations (linear and nonlinear); higher order linear differential equations with constant coefficients; Euler-Cauchy equation; initial and boundary value problems; Laplace transforms; solutions of heat, wave and Laplace’s equations.Complex Variables: Analytic functions; Cauchy-Riemann equations; Cauchy’s integral theorem and integral formula; Taylor and Laurent series.Probability and Statistics: Definitions of probability, sampling theorems, conditional probability; mean, median, mode and standard deviation; random variables, binomial, Poisson and normal distributions.Numerical Methods: Numerical solutions of linear and non-linear algebraic equations; integration by trapezoidal and Simpson’s rules; single and multi-stepmethods for differential equations..

GATE and ESE Prelims: Electrical EngineeringLinear Algebra: Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors.Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem, Green’s theorem.Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of variables.Complex Variables: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals.Probability and Statistics: Sampling theorems, Conditional probability, Mean, Median, Mode, Standard Deviation, Random variables, Discrete and Continuous distributions, Poisson distribution, Normal distribution, Binomial distribution, Correlation analysis, Regression analysis.

Numerical Methods: Solutions of nonlinear algebraic equations, Single and Multi-step methods for differential equations.Transform Theory: Fourier Transform, Laplace Transform, z-Transform.Electrical Engineering ESE MainsMatrix theory, Eigen values & Eigen vectors, system of linear equations, Numerical methods for solution of non-linear algebraic equations and differential equations, integral calculus, partial derivatives, maxima and minima, Line, Surface and Volume Integrals. Fourier series, linear, nonlinear and partial differential equations, initial and boundary value problems, complex variables, Taylor’s and Laurent’s series, residue theorem, probability and statistics fundamentals, Sampling theorem, random variables, Normal and Poisson distributions, correlation and regression analysis.

GATE and ESE Prelims: Electronics EngineeringLinear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations – existence and uniqueness.Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Taylor series.Differential equations: First order equations (linear and nonlinear), higher order linear differential equations, Cauchy’s and Euler’s equations, methods of solution using variation of parameters, complementary function and particular integral, partial differential equations, variable separable method, initial and boundary value problems.Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss’s, Green’s and Stoke’s theorems.Complex Analysis: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula; Taylor’s and Laurent’s series, residue theorem.Numerical Methods: Solution of nonlinear equations, single and multi-step methods for differential equations, convergence criteria.Probability and Statistics: Mean, median, mode and standard deviation; combinatorial probability, probability distribution functions - binomial, Poisson, exponential and normal; Joint and conditional probability; Correlation and regression analysis.

GATE: Instrumentation EngineeringLinear Algebra : Matrix algebra, systems of linear equations, Eigen values and Eigen vectors.Calculus : Mean value theorems, theorems of integral calculus, partial derivatives, maxima and minima, multiple integrals, Fourier series, vector identities, line, surface and volume integrals, Stokes, Gauss and Green’s theorems.Differential Equations : First order equation (linear and nonlinear), higher order linear differential equations with constant coefficients, method of variation of parameters, Cauchy’s and Euler’s equations, initial and boundary value problems, solution of partial differential equations: variable separable method.Analysis of complex variables: : Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and Laurent’s series, residue theorem, solution of integrals.Complex Variables : Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and Laurent’ series, Residue theorem, solution integrals.Probability and Statistics : Sampling theorems, conditional probability, mean, median, mode and standard deviation, random variables, discrete and continuous distributions: normal, Poisson and binomial distributions.Numerical Methods : Matrix inversion, solutions of non-linear algebraic equations, iterative methods for solving differential equations, numerical integration, regression and correlation analysis.

GATE: Computer Science & IT EngineeringLinear Algebra: Matrices, determinants, system of linear equations, eigenvalues and eigenvectors, LU decomposition.Calculus: Limits, continuity and differentiability. Maxima and minima. Mean value theorem. Integration.Probability: Random variables. Uniform, normal, exponential, poisson and binomial distributions. Mean, median, mode and standard deviation. Conditional probability and Bayes theorem.

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1. Linear Algebra ................................................................................................... 1-99 1.1 Introduction .................................................................................................................................. 1 1.2 Algebra of Matrices ..................................................................................................................... 1 1.3 Determinants ..............................................................................................................................10 1.4 Inverse of Matrix ........................................................................................................................13 1.5 Rank of A Matrix .........................................................................................................................14 1.6 Sub-Spaces : Basis and Dimension ......................................................................................19 1.7 System of Equations .................................................................................................................24 1.8 Eigenvalues and Eigenvectors ..............................................................................................29 Previous GATE and ESE Questions ...........................................................................................38

2. Calculus .......................................................................................................100-264 2.1 Graphs of Basic Functions.................................................................................................... 100 2.2 Basic Trigonometric Relations ............................................................................................ 104 2.3 Limit ............................................................................................................................................ 105 2.4 Continuity ................................................................................................................................. 108 2.5 Differentiability ....................................................................................................................... 109 2.6 Mean Value Theorems ........................................................................................................... 111 2.7 Computing the Derivative ................................................................................................... 118 2.8 Applications of Derivatives ................................................................................................. 125 2.9 Partial Derivatives .................................................................................................................. 136 2.10 Total Derivatives ..................................................................................................................... 138 2.11 Maxima and Minima (of Function of Two Independent Variables)........................ 139 2.12 Theorems of Integral Calculus ............................................................................................ 141 2.13 Definite Integrals .................................................................................................................... 145 2.14 Applications of Integration ................................................................................................. 150 2.15 Multiple Integrals and Their Applications ...................................................................... 157 2.16 Vectors ....................................................................................................................................... 163 Previous GATE and ESE Questions ........................................................................................ 189

3. Differential Equations .................................................................................265-321 3.1 Introduction ......................................................................................................................265 3.2 Differential Equations of First Order ..........................................................................265 3.3 Linear Differential Equations (Of nth Order) .............................................................277 3.4 Two Other Methods of Finding P.I. .............................................................................286 3.5 Equations Reducible to Linear Equation with Constant Coefficient ...............287 Previous GATE and ESE Questions ........................................................................................ 289

4. Complex Functions .....................................................................................322-369 4.1 Properties of IOTA (i) .......................................................................................................322 4.2 Complex Functions .........................................................................................................324 4.3 Limit of a Complex Function ........................................................................................327 4.4 Singularity ..........................................................................................................................328

ContentsSl. Units Pages

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4.5 Derivative of f(z) ...............................................................................................................328 4.6 Analytic Functions ...........................................................................................................329 4.7 Complex Integration ......................................................................................................333 4.8 Cauchy’s Theorem ...........................................................................................................336 4.9 Cauchy’s Integral Formula ............................................................................................337 4.10 Series of Complex Terms ...............................................................................................338 4.11 Zeros and Singularities or Poles of an Analytic Function ...................................339 4.12 Residues .............................................................................................................................340 Previous GATE and ESE Questions ........................................................................................ 343

5. Probability and Statistics ...........................................................................370-440 5.1 Probability Fundamentals .............................................................................................370 5.2 Statistics .............................................................................................................................376 5.3 Probability Distributions ...............................................................................................382 Previous GATE and ESE Questions ........................................................................................ 396

6. Numerical Method ......................................................................................441-490 6.1 Introduction ......................................................................................................................441 6.2 Numerical Solution of System of Linear Equations ..............................................443 6.3 Numerical Solutions of Nonlinear Algebraic and Transcendental Equations by Bisection, Regula-Falsi, Secant and Newton-Raphson Methods ................448 6.4 Numerical Integration (Quadrature) by Trapezoidal and Simpson’s Rules ....456 6.5 Numerical Solution of Ordinary Differential Equations.......................................461 Previous GATE and ESE Questions ........................................................................................ 466

7. Transform Theory ........................................................................................491-511 7.1 Laplace Transform ................................................................................................................. 491 7.2 Definition .................................................................................................................................. 491 7.3 Transforms of Elementary Functions ............................................................................... 492 7.4 Properties of Laplace Transforms ...................................................................................... 492 7.5 Evaluation of Integrals by Laplace Transforms .............................................................. 495 7.6 Inverse Transforms – Method of Partial Fractions ........................................................ 497 7.7 Unit Step Function ................................................................................................................. 498 7.8 Second Shifting Property ..................................................................................................... 498 7.9 Unit Impulse Function .......................................................................................................... 498 7.10 Periodic Functions.................................................................................................................. 499 7.11 Fourier Series ........................................................................................................................... 499 7.12 Dirichlet’s Conditions ............................................................................................................ 500 Previous GATE and ESE Questions ........................................................................................ 503

8. Second Order Linear Partial Differential Equations ..................................512-526 8.1 Classification of Second Order Linear PDEs ................................................................... 512 8.2 Undamped One-Dimensional Wave Equation: Vibrations of an Elastic String ... 512 8.3 The One-Dimensional heat Conduction Equation ...................................................... 518 8.4 Laplace Equation for a Rectangular Region ................................................................... 520 Previous GATE and ESE Questions ........................................................................................ 523

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