ENGINEERING MATHEMATICS (For ESE & GATE Exam) (CE, ME, PI, CH, EC, EE, IN, CS, IT)

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2 ENGINEERING MATHEMATICS (For ESE & GATE Exam) (CE, ME, PI, CH, EC, EE, IN, CS, IT) Salient Features : 89 topics under 31 chapters in 8 units 67 Solved Examples for comprehensive understanding 1386 questions from last 5 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 Office : F-16, (Lower Basement), Katwaria Sarai, New Delhi Phone : Mobile : , info@iesmasterpublications.com, info@iesmaster.org Web : iesmasterpublications.com, iesmaster.org

3 IES MASTER PUBLICATION F-16, (Lower Basement), Katwaria Sarai, New Delhi Phone : , Mobile : , info@iesmasterpublications.com, info@iesmaster.org Web : iesmasterpublications.com, iesmaster.org All rights reserved. Copyright 017, by IES MASTER Publications. No part of this booklet may be reproduced, or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior permission of IES MASTER, New Delhi. Violates are liable to be legally prosecuted. First Edition : 017 Typeset at : IES Master Publication, New Delhi

4 PREFACE We, the IES MASTER, have immense pleasure in placing the first edition of Engineering Mathematics before the aspirants of GATE & ESE exams. Dear Students, as we all know that in 016 UPSC included Engineering Mathematics as a part of syllabus of common paper 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 previously asked questions in GATE & ESE exams. Since most of the books focus on only some selected main topics, students have not been able to answer more than 60-65% of 1386 questions that have been asked in GATE & ESE exams so far. Hence to overcome this problem, we have tried our best by covering more than 89 topics under 31 chapters in 8 units. (One should not be in dilemma that 89 topics are more than sufficient. These are the minimum topics from where GATE & ESE have already asked questions). Since we have covered every previous year questions from last 5 years of each topic, students can easily decide, how much time to allocate on each chapter based on the number of questions asked in that particular exam. Again, we have included only those proofs that are necessary for concept building 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 there are sufficient number of solved examples for concept building & easy learning. The book includes such types of 67 examples. It also covers all the previously asked questions in which conceptual questions are marked with * sign so that 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 excel in 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 strongly believe 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 Publications New Delhi

5 CONTENTS UNIT 1 : CALCULUS 1.1 Limits, Continuity and Differentiability (i) Function (ii) Limit of a Function (iii) Theorem on Limits... 0 (iv) Indeterminate Forms (v) L-Hospital Rule (vi) Fundamentals of Continuity (vii) Kinds of Discontinuities (viii) Properties of Continuous Functions (ix) Saltus of a Function (x) Function of Two Variables (xi) Limit of a Function of Two Variables (xii) Continuity of Function of TwoVariables... (xiii) Differentiability... 3 Previous Years GATE & ESE Questions Partial Differentiation (i) Parial Derivatives (ii) Homogeneous Functions (iii) Euler s Theorem on Homogenous Function (iv) Total Differential Coefficients... 5 (v) Change of Variables... 5 (vi) Jacobian (vii) Chain Rule of Jacobian... 59

7 (vi) (viii) Functional Dependence Previous Years GATE & ESE Questions Infinite Series (i) Expansion of Functions (ii) Maclaurin s theorem (iii) Taylor s Theorem (iv) Convergence and Divergence of Infinite Series (v) Methods to Find Convergence of Infinite Series (vi) Power Series Previous Years GATE & ESE Questions Mean Value Theorems (i) Rolle s Theorem (ii) Lagrange s First Mean Value Theorem (iii) Cauchy s Mean Value Theorem (iv) Bolzanos Theorem (v) Intermediate Value Theorem (vi) Darboux Theorem Previous Years GATE & ESE Questions Maxima and Minima (i) Increasing and Decreasing Functions (ii) Maxima and Minima of Function of Single Variable (iii) Sufficient Condition (Second Derivate Test) (iv) Maxima-Minima of Functions of Two Variables (v) Sufficient Condition (Lagrange s Conditions) (vi) Lagrange s Method of Undetermined Multipliers Previous Years GATE & ESE Questions Multiple Integral (i) Concept of Integration (ii) Some Standard Formulae

8 (vii) (iii) Some Important Integration and their Hints (iv) Definite integrals (v) Fundamental Properties of Definite Integrals (vi) Fundamental Theorem of Integral Calculus (vii) Definite integral as the Limit of a Sum (viii) Double Integrals (ix) Double Integrals in Polar Coordinates... 1 (x) Triple Integrals (xi) Change of Order of Integration (xii) Use of Jacobian in Multiple Integrals (xiii) Multiple Integral using Change of Variables Concept (xiv) Area and Volume in Different Coordinates Systems (xv) Arc Length of Curves (Rectification) (xvi) Intrinsic Equation of a Curve (xvii) Volumes of Solids of Revolution (xviii) Surfaces of Solids of Revolution (xix) Beta and Gamma Functions (xx) Properties of Beta & Gamma Functions (xxi) Leibnitz Rule of Differentiation under the sign of Integration (xxii) Improper Integral Previous Years GATE & ESE Questions Functions and their Graphs (i) Cartesian Product (ii) Relations (iii) Types of Relations (iv) Functions (v) Types of Functions (vi) Some Basic Graphs (vii) Graph Transformation Previous Years GATE & ESE Questions

9 (viii) UNIT : VECTOR CALCULUS.1 Vector & their Basic Properties (i) Vector (ii) Addition or Subtraction of Two Vectors (iii) Dot Product (iv) Angle between Two Vectros (v) Triangle Inequality (vi) Cross Product (vii) Area of a Triangle (viii) Area of a Paraellelogram (ix) Direction Cosines of a Vector (x) Direction Ratio of a Line joining Two Points Previous Years GATE & ESE Questions Gradient, Divergence and Curl (i) Partial Derivatives of Vectors (ii) Point Functions (iii) Del Operator (iv) The Laplacian Operator (v) Gradient (vi) Directional Derivative (vii) Level Surface (viii) Divergence (ix) Curl (Rotation) (x) Vector Identities Previous Years GATE & ESE Questions Vector Integration (i) Line Integral (ii) Surface Integral... 3 (iii) Volume Integral... 33

10 (ix) (iv) Gauss Divergence Theorem (v) Stoke s Theorem (vi) Green s Theorem Previous Years GATE & ESE Questions UNIT 3 : COMPLEX ANALYSIS 3.1 Complex Number (i) Complex Numbers as Ordered Pairs (ii) Euler Notation (iii) Algebraic Properties (iv) Geometrical Representation (v) Modulus and Argument (vi) Complex Conjugate Numbers... 5 (vii) Cube Roots of Unity... 5 Previous Years GATE & ESE Questions Analytic Functions (i) Neighbourhood of Complex Number z (ii) Function of a Complex Variable (iii) Types of Complex Function (iv) Continuity of Complex Function (v) Differentiability of Complex Function (vi) Analytic Functions (vii) Cauchy-Riemann Equations (viii) Polar Form of Cauchy-Riemann Equations (ix) Harmonic Functions Previous Years GATE & ESE Questions Complex Integration (i) Contour (ii) Elementary Properties of Complex Integrals... 8 (iii) Cauchy Integral Theorem... 85

11 (x) (iv) Cauchy s Integral Formula (v) Taylor Series of Complex Function (vi) Laurent s Series (vii) Zeros of an Analytic Function... 9 (viii) Singularities of an Analytic Function (ix) Types of Isolated Singular Points (x) Residue (xi) Cauchy-Residue Theorem (xii) Methods of Evaluating Residues Previous Years GATE & ESE Questions UNIT 4 : DIFFERENTIAL EQUATIONS 4.1 Ordinary Differential Equations (i) Definition of Differential Equation (ii) Order and Degree of Ordinary Differential Equation (iii) Non-Linear Differential Equation (iv) Solution of Differential Equation (v) Formation of Differential Equation (vi) Wronksian (vii) Linearly dependent & Linearly independent solutions (viii) Methods of Solving Differential Equations (ix) Differential Equations of First Order and First Degree (x) Exact Differential Equation (xi) Equations Reducible to Exact Form Previous Years GATE & ESE Questions Linear Differential Equations of Higher Order with Constant Coefficients (i) Complementary Function (ii) Rules for Finding the Complementary Function (iii) Particular Integral

12 (xi) (iv) Methods of Evaluating Particular Integral (v) Cauchy s Homogeneous Linear Differential Equation (vi) Legender s Homogeneous Linear Differential Equation (vii) Simultaneous Linear Differential Equation (viii) Variation of Parameters (ix) Differential Equation of First Order and Higher Degree (x) Clairaut s Equation Previous Years GATE & ESE Questions Partial Differential Equations (i) Order and Degree of Partial Differential Equation (ii) Linear Partial Differential Equation (iii) Classification of nd Order Linear P.D.E. in Two Variables (iv) One Dimensional Wave Equations (v) One Dimensional Heat Equation (vi) Two Dimensional Heat Equation (vii) Laplace Equation Previous Years GATE & ESE Questions UNIT 5 : NUMERICAL METHODS 5.1 Numerical Solutions of Linear Equations (i) Gauss-Jacobi Method (ii) Gauss-Seidel Method Previous Years GATE & ESE Questions Numerical Solution of Algebraic and Transcendental Equations (i) Graphical Method (ii) Bisection Method (iii) Regula-Falsi Method (iv) The Secant Method (v) Newton-Raphson Method Previous Years GATE & ESE Questions

14 (xii) 5.3 Interpolation (i) Interpolation and Extrapolation (ii) Methods of Interpolation (iii) Notation of Finite Difference Calculus (iv) Newton-Gregory Forward Interpolation Formula (for equal intervals) (v) Newton-Gregory Backward Interpolation Formula (for equal intervals) (vi) Newton s Divided Difference Interpolation Formula (for Unequal Intervals) (vii) Lagrante s Interpolation Formula (for Unequal Intervals) Previous Years GATE & ESE Questions Numerical Integration (Quadrature) (i) General Quadrature Formula (ii) The Trapezoidal Rule (iii) Simpson s One-Third Rule (iv) Simpsons 3/8 th Rule Previous Years GATE & ESE Questions Numerical Solution of a Differential Equation (i) Picard s Method (ii) Euler s Method (iii) Improved Euler s Method (iv) Modified Euler s Method (v) Runge-Kutta Methods Previous Years GATE & ESE Questions UNIT 6 : TRANSFORM THEORY 6.1 Laplace Transform (i) Integral Transform (ii) Definition of Laplace Transform (iii) Properties of Laplace Transform (iv) Laplace Transform of Periodic Function (v) Types of Laplace Transform

15 (xiii) (vi) Evaluation of Integrals with the help of Laplace Transform (vii) Inverse Laplace Transform (viii) Properties of Inverse Laplace Transform (ix) Types of Inverse Laplace Transform (x) Convolution of Two Functions (xi) Unit Step Function (xii) Unit Impulse Function (Dirac Delta Function) (xiii) Existence Theorem for Laplace Transform (xiv) Initial Value Theorem (xv) Final Value Theorem (xvi) Application of Laplace Transform in Differencial Equation Previous Years GATE & ESE Questions Fourier Series (i) Periodic Function (ii) Fourier Series (Main Definition) (iii) Euler s Formula of Fourier Series (iv) Dirichlet s Conditions (v) Fourier Expansion of Even and Odd Function (vi) Half Range Fourier Series Previous Years GATE & ESE Questions UNIT 7 : LINEAR ALGEBRA 7.1 Algebra of Matrices (i) Definition of Matrix (ii) Types of Matrices (iii) Product of Matrix by a Scalar (iv) Addition and Subtraction of Matrices (v) Multiplication of Matrices (vi) Minors of Matrix (vii) Cofactors of a Matrix

16 (xiv) (viii) Properties of Determinants (ix) Adjoint of Matrix (x) Inverse of a Matrix (xi) Conjugate of a Matrix (xii) Conjugate Transpose of a Matrix (xiii) Special Types of Matrices Previous Years GATE & ESE Questions Rank of Matrices (i) Definition of Rank (ii) Elementary Transformations (iii) Equivalent Matrices (iv) Properties of Rank (v) Echelon Form (vi) Normal Form (vii) Elementary Matrices (viii) Computation of the Inverse of Matrix by Elementary Transformation (ix) Row Rank and Column Rank of a Matrix Previous Years GATE & ESE Questions System of Equations (i) Linear Dependence and Independence of Vectors (ii) System of Linear Equations (iii) Methods of Solving Non-Homogenous System (iv) Methods of Solving Homogenous System (v) Diagonally Dominant System (vi) Gauss Elimination Method with Pivoting (vii) Gauss-Jordan Method (viii) LU Decomposition (Factorization) Method (ix) Dolittle Method (x) Crout s Method Previous Years GATE & ESE Questions

17 (xv) 7.4 Eigen Values and Eigen Vectors (i) Definition (ii) Properties of Eigen Values & Eigen Vectors (iii) Similar Matrices... 6 (iv) Diagonalisation... 6 (v) Cayley-Hamilton Theorem Previous Years GATE & ESE Questions UNIT 8 : PROBABILITY & STATISTICS 8.1 Permutation and Combination (i) Set (ii) Types of Set (iii) Venn Diagrams of Different Sets (iv) Fundamental Principle of Counting (v) Permutation (vi) Combination (vii) Difference between Permutation and Combination (viii) Total Number of Combinations (ix) Permutation of Alike Items (x) Division of Different Items into Groups of Equal Size (xi) Circular Permutation (xii) Sum of All Numbers formed from Given Digits (xiii) Division of Identical Items into Groups (xiv) Rank of a Word (xv) Number of Dearrangements (xvi) Number of Factors Previous Years GATE & ESE Questions Probability (i) Some Basic Concepts (ii) Definition of Probability

18 (xvi) (iii) AdditionTheorem of Probability (iv) Odds in Favour and Odds Against (v) Compound Events (vi) Conditional Probability (vii) Important Results on Independence of Events (viii) Binomial Theorem on Probability (ix) Total Probability Theorem (x) Baye s Theorem Previous Years GATE & ESE Questions Statistics - I (Probability Distribution) (i) Random Experiment (ii) Discrete Random Variable (iii) Cumulative Distribution Function for Discrete Random Variable (iv) Bernoulli Random Variable (v) Binomial Random Variable (vi) Geometric Random Variable (vii) Poisson Random Variable (viii) Continuous Random Variable (ix) Cumulative Distribution Function for Continuous Random Variable (x) Uniform Random Variable (xi) Exponential Random Variable (xii) Gamma Random Variable (xiii) Normal Random Variable (xiv) Standard Normal Distribution (xv) Skewness in Normal Curve (xvi) Expectation of Random Variable (xvii) Variance (xviii) Standard Deviation (xix) Covariance (xx) Mean

19 (xvii) (xxi) Median (xxii) Mode Previous Years GATE & ESE Questions Statistics - II (Correlation & Regression) (i) Curve Fitting (ii) Methods of Least Suqares (iii) Correlation (iv) Methods of Estimating Correlation (v) Regression Analysis (vi) Line of Regression of y on x (vii) Line of Regression of x on y (viii) Properties of Regression Coefficients (ix) Angle between Two Lines of Regression Previous Years GATE & ESE Questions

20 Engineering Mathematics 9 Questions marked with aesterisk (*) are Conceptual Questions 1.*. x x Lim e 1 cos x 1 x0 x1 cos x 1 1 e Lim x e j5x jx (a) 0 (b) 1.1 (c) 0.5 (d) 1 sinx 3. The value of lim x is: x (a) (b) (c) 1 (d) 0 4. The value of 1 1 lim is : x sin x tan x (a) 0 (b) 1 (c) (d) 5. The limit of 6.* (a) 1 (b) 0 (c) (d) x Lt 0 sin None of the above 3 x cos x lim equal x sin x is (a) (b) 0 [GATE-1993 (ME)] [GATE-1993] [GATE-1994; 1 Mark] [GATE-1994; 1 Mark] [GATE-1994; 1 Mark] (c) (d) does not exist [GATE-1995-CS; Marks] 7.* The f unction f(x) x + 1 on the interv al [, 0] is (a) (b) (c) (d) continuous and differentiable continuous on the integers but not differentiable at all points neither continuous nor differentiable differentiable but not continuous [GATE-1995; 1 Mark] 8. 1 lim x sin x 0 x is (a) (b) 0 (c) 1 (d) Non-existant [GATE-1995; 1 Mark] 9. If a function is continuous at a point its first derivative (a) (b) (c) (d) may or may not exist exists always will not exist has unique value [GATE-1996; 1 Mark] 10. If y x for x < 0 and y x for x 0, then 11. (a) dy dx is discontinuous at x 0 (b) y is discontinuous at x 0 (c) y is not defined at x 0 (d) 0 None of these [GATE-1997; 1 Mark] sinm lim, where m is an integer, is one of the following: (a) m (b) m (c) m (d) 1 [GATE-1997; 1 Mark] 1.* Find the limiting value of the ratio of the square of the sum of a natural numbers to n times the sum of squares of the n natural number as, n approaches infinity. [GATE-1997] 13.* The profile of a cam in a particular zone is given by x profile at axis): (a) (c) 3 cos and y sin. The normal to the cam 4 3 is at an angle (with respect to x 4 (b) (d) 0 [GATE-1998]

21 30 Limits, Continuity and Differentiability 14. Limit of the function lim n (a) 1 (b) 0 (c) (d) Value of the function lim x a 16. (a) 1 (b) 0 (c) (d) a Lt x 1 is x1 x 1 (a) (b) 0 (c) (d) 1 n n n is [GATE-1999; 1 Mark] (x a) (x a) is [GATE-1999; 1 Mark] [GATE-000; 1 Mark] 4 a 17. The limit of the function f(x) 1 x 4 as x is given by (a) 1 (b) exp[ a 4 ] (c) (d) Zero [GATE-000; 1 Mark] 18. W hat is the deriv ativ e of f(x) x at x 0? (a) 1 (b) 1 (c) 0 (d) Does not exist sin x lim x 4 x 4 (a) 0 (b) (c) 1 (d) 1 [GATE-001; 1 Mark] [GATE-001 (IN)] 0.* Which of the following functions is not differentiable in the domain [ 1,1]? (a) f(x) x (b) f(x) x 1 (c) f(x) (d) f(x) maximum(x, x) [GATE-00; 1 Mark] 1.* The limit of the following sequence as n is: xn n 1/n. (a) 0 (b) 1 (c) (d) x0 sin x lim is equal to x (a) 0 (b) (c) 1 (d) 1 3. The value of the function f(x) (a) zero (b) (c) 1 7 [GATE-00-CE; 1 Mark] 1 7 (d) infinite [GATE-003; 1 Mark] x0 3 x x lim x 7x 3 is [GATE-004; 1 Mark] 4. If x a sin and y a1 cos, then dy/dx will be equal to (a) (c) sin tan (b) (d) cos cot [GATE-004] 5.* With a 1 unit change in b, what is the change in x in the solution of the system of equations x + y, 1.01x y b? (a) zero (b) units (c) 50 units (d) 100 units [GATE-005] 6.* Equation of the line normal to function f(x) (x 8) /3 + 1 at P(0, 5) is (a) y 3x 5 (b) y 3x + 5 (c) 3y x + 15 (d) 3y x 5 7. If f(x) [GATE-006; Marks] x 7x 3 5x 1x 9, then lim f(x) is x3 (a) 1/3 (b) 5/18 (c) 0 (d) /5 [GATE-006; Marks] 8.* If, y x x x x..., then y()

22 Engineering Mathematics (See Sol.) 1. (b) 41. (b) 61. (c). (c). (a) 4. (d) 6. (c) 3. (d) 3. (b) 43. (c) 63. (c) 4. (d) 4. (c) 44. (c) 64. (c) 5. (a) 5. (c) 45. (b) 65. (c) 6. (a) 6. (b) 46. (b) 66. (0.5) 7. (b) 7. (b) 47. (a) 67. (1) 8. (b) 8. (b) 48. (b) 68. (d) 9. (a) 9. (b) 49. (d) 69. (5) 10. (a) 30. (a) 50. (a) 70. (b) 11. (a) 31. (c) 51. (c) 71. (d) 1. (0.75) 3. (c) 5. (c) 7. (b) 13. (c) 33. (b) 53. (a) 73. (a) 14. (d) 34. (a) 54. (c) 74. (d) 15. (a) 35. (b) 55. ( ) 75. (c) 16. (c) 36. (c) 56. (c) 76. (1) 17. (a) 37. (c) 57. ( 0.33) 18. (d) 38. (c) 58. (d) 19. (d) 39. (a) 59. (a) 0. (d) 40. (c) 60. (c) SOLUTIONS Sol 1: Using L' Hospital Rule Req. Limit x0 x Again using L' Hospital rule (e 1) xe sin x 0 lim (1 cos x) x(sin x) 0 x0 x x (e e xe cos x 0 lim sinx sinx x cos x 0 Again using L' Hospital Rule x x x x lim e e e xe sin x x 0 cos x cos x cos x x sinx Sol : (c) Using L' Hospital rule Sol 3: (d) Sol 4: (d) j5x lim 1 e x0 10(1 e jx ) x sin x lim x sin x lim 0 x x j5x 1 5je 5 lim 0.5 je x010 jx 10 sin1/y lim 1/y y0 1 lim y sin 0 y 0 y ( sinx < 1)

24 36 Limits, Continuity and Differentiability 1 1 lim sin x tan x x Sol 5: (a) Sol 6: (a) x sin Lim x cos x lim x (sin x) lim x Sol 7: (b) x x 3 cos x sinx 1 cos x lim sinx x x sin lim x x sin cos x x lim tan x x cos x 1 3 lim x x (sinx) 3 3 x x x cos x 1 3 lim x 1 (sin x) x 3 x f(x) x + 1 Only concern is x 1 Left limit Right limit (x + 1) for x < 1 (x + 1) for x > 1 lim (x + 1) 0 x1 x1 lim (x + 1) 0 f(x) is continuous in theinterval[,0] LD 1 Right Derivative + 1 Hence f(x) is not differentiableat x 1 Alternative Solution : 1 It is clear from graph f(x) is continuous every where but not differentiate at x 1 because there is sharp change in slope at x 1. Sol 8: (b) Sol 9: (a) 1 lim x sin x x 0 0 sin 0 [ 1, 1] 0 If f (x) is continuous at a point x a then it may or may not be differentiable at x a. Sol 10: (a) Sol 11: (a) y x dy and x, for x 0 1, x 0 x, for x 0 dx 1, x 0 left limit Right limit lim ( x) 0 x 0 lim (x) 0 x 0 Left limit Right limit y iscontinuous at x 0 Left derivative 1 Right derivative +1 Left derivative Right derivative Apply LH Rule dy is discontinuous at x 0 dx sinm 0 Lim 0 0 mcosm Lim m 0 1 Alternative Solution: Sol 1: (0.75) sinm Lim 0 sinm Lim m m(1) m 0 m Sum of the squares of natural numbers S S n S n(n 1)(n 1) 6 Sum of natural numbers S 1 S n Hence, n S 1 n(n 1) n(n 1) lim n(n 1)(n 1) n 6

25 Engineering Mathematics 37 n n (n 1) lim 4 n n(n 1)(n 1) 6 3 (n 1) lim (n 1) n lim n n 1 n Sol 13: (c) x 3 cos y sin Slope of cam profile at an angle is dy dx dy d 1 dx d cos 3 sin cot 3 Slope of Normal to cam profile at angle is m m 1 dy dx 3 tan 4 m 3 Angle of Normal with x-axis is Sol 14: (d) Lim n n n n 3 tan tan 1 (m) tan n Lim n 1 n 1 n 1 Lim 1 n 1 1 n Apply L H Rule logx a Lim xa 1 x a 1 Lim x a xa 1 x a log y Lim x a xa log y 0 y e 1 Sol 16: (c) Sol 17: (a) Sol 18: (d) x 1 Lt x 1 x1 x1 x 1 Lt x 1 Lim f(x) x Left derivative 1 x1 x 1 x 1 Lt x 1 Lt x Right derivative + 1 x1 Lim a 1 x x 4 a f(x) x x, x 0 x, x 0 LD RD The derivative does not exist at x 0. Alternative Solution : It is clear from graph f(x) is not differentiable at x 0. Sol 19: (d) Sol 15: (a) Let y Limx a xa xa Taking logarithm on both sides. log y Lim log x x a a xa Lim x a log(x a) xa lim x 4 sin x 4 0 form 0 x 4 cos x 4 lim 1 x 4

Engineering. Mathematics. GATE 2019 and ESE 2019 Prelims. For. Comprehensive Theory with Solved Examples

Engineering. Mathematics. GATE 2019 and ESE 2019 Prelims. For. Comprehensive Theory with Solved Examples Thoroughly Revised and Updated Engineering Mathematics For GATE 2019 and ESE 2019 Prelims Comprehensive Theory with Solved Examples Including Previous Solved Questions of GATE (2003-2018) and ESE-Prelims