Wiley Acing the GATE: Engineering Mathematics and General Aptitude PDF

 WILEY GATE ENGINEERING MATHEMATICS AND GENERAL APTITUDE SECOND EDITION  WILEY ENGINEERING MATHEMATICS GATE AND GENERAL APTITUDE...

 WILEY GATE ENGINEERING MATHEMATICS AND GENERAL APTITUDE SECOND EDITION  WILEY ENGINEERING MATHEMATICS GATE 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. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or scanning without the written permission of the publisher. Limits of Liability: While the publisher and the author have used their best efforts in preparing this book, Wiley and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this book, and specifically disclaim any implied warranties of merchantability or fitness for any particular purpose. There are no warranties which extend beyond the descriptions contained in this paragraph. No warranty may be created or extended by sales representatives or written sales materials. Disclaimer: The contents of this book have been checked for accuracy. Since deviations cannot be precluded entirely, Wiley or its author cannot guarantee full agreement. As the book is intended for educational purpose, Wiley or its author shall not be responsible for any errors, omissions or damages arising out of the use of the information contained in the book. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. Other Wiley Editorial Offices: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030, USA Wiley-VCH Verlag GmbH, Pappellaee 3, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia 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 Gate_FM.indd 5 8/25/2017 5:38:27 PM  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 Gate_FM.indd 6 8/25/2017 5:38:27 PM  CONTENTS Preface v ENGINEERING MATHEMATICS 1 1 Linear Algebra 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 Gate_FM.indd 7 8/25/2017 5:38:27 PM  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 Gate_FM.indd 8 8/25/2017 5:38:27 PM  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 Differential Equations 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 Gate_FM.indd 9 8/25/2017 5:38:27 PM  x   CONTENTS 4 Complex Variables 225 Introduction 225 Complex Functions 225 Exponential Function of Complex Variables 225 Circular Function of Complex Variables 226 Hyperbolic Functions of Complex Variables 226 Logarithmic Function of Complex Variables 226 Limit and Continuity of Complex Functions 227 Derivative of Complex Variables 227 Cauchy—Riemann Equations 227 Integration of Complex Variables 228 Cauchy’s Theorem 228 Cauchy’s Integral Formula 228 Taylor’s Series of Complex Variables 229 Laurent’s Series of Complex Variables 229 Zeros and Poles of an Analytic Function 229 Residues 230 Residue Theorem 230 Calculation of Residues 230 Solved Examples 230 Practice Exercise 233 Answers 234 Explanations and Hints  234 Solved GATE Previous Years’ Questions 236 5 Probability and Statistics 253 Fundamentals of Probability 253 Types of Events 254 Approaches to Probability 254 Axioms of Probability 254 Conditional Probability 254 Geometric Probability 254 Rules of Probability 255 Statistics 255 Arithmetic Mean 255 Median 256 Mode 256 Relation Between Mean, Median and Mode 256 Geometric Mean 257 Harmonic Mean 257 Range 257 Mean Deviation 257 Standard Deviation 258 Coefficient of Variation 258 Probability Distributions 258 Random Variable 258 Properties of Discrete Distribution 259 Gate_FM.indd 10 8/25/2017 5:38:27 PM  CONTENTS xi Properties of Continuous Distribution 259 Types of Discrete Distribution 259 Types of Continuous Distribution 260 Correlation and Regression Analyses 261 Correlation 261 Lines of Regression 261 Hypothesis Testing 262 Hypothesis Testing Procedures for Some Common Statistical Problems 262 Hypothesis Testing of a Proportion 262 Hypothesis Testing of a Mean 263 Hypothesis Testing of Difference Between Proportions 263 Hypothesis Testing of Difference Between Means 264 Bayes’ Theorem 265 Solved Examples 265 Practice Exercise 272 Answers 275 Explanations and Hints 275 Solved GATE Previous Years’ Questions 281 6 Numerical Methods 313 Introduction 313 Numerical Solution of System of Linear Equations 313 Gauss Elimination Method 313 Matrix Decomposition Methods (LU Decomposition Method) 314 Gauss—Jordan Method 315 Iterative Methods of Solution 315 Numerical Solution of Algebraic and Transcendental Equations 316 Bisection Method 316 Regula—Falsi Method (Method of False Position Method) 317 Newton—Raphson Method 317 Secant Method 317 Jacobian 318 Numerical Integration 318 Newton—Cotes Formulas (General Quadrature) 318 Rectangular Rule 319 Trapezoidal Rule 319 Simpson’s Rule  320 Numerical Solution of Ordinary Differential Equation (O.D.E.)  320 Picard’s Method 321 Euler’s Method 321 Runge—Kutta Method 322 Euler’s Predictor—Corrector Method 322 Accuracy and Precision 322 Classification of Errors 323 Significant Figures 323 Measuring Error 324 Propagation of Errors 324 Gate_FM.indd 11 8/25/2017 5:38:27 PM  xii   CONTENTS Method of Least Square Approximation 324 Lagrange Polynomials 325 Numerical Differentiation 326 Newton’s Forward Formula 326 Newton’s Backward Formula 326 Stirling’s or Bessel’s Formula 327 Solved Examples  327 Practice Exercise 340 Answers 342 Explanations and Hints 342 Solved GATE Previous Years’ Questions 351 7 Mathematical Logic 371 Introduction 371 Statements 371 Atomic Statements 371 Molecular Statements 371 Truth Table 371 Truth Values 371 Connectives 372 Types of Connectives 372 Well-Formed Formulas 373 Duality Law 373 Equivalent Well-Formed Formula 373 Logical Identities 374 Normal Form 375 Disjunction Normal Form 375 Conjunctive Normal Form 376 Propositional Calculus 376 Rules of Inference 376 Predicate Calculus 378 Predicates 378 Quantifier 378 Solved Examples 379 Practice Exercise  381 Answers  382 Explanations and Hints  382 Solved GATE Previous Years’ Questions 385 8 Set Theory and Algebra 395 Introduction to Set Theory 395 Subsets and Supersets 395 Equal Sets 395 Comparable Sets 395 Universal Set 396 Power Set 396 Types of Sets 396 Gate_FM.indd 12 8/25/2017 5:38:27 PM  CONTENTS xiii Operations on Sets 396 Important Laws and Theorems 396 Venn Diagrams 397 Operations on Sets Using Venn Diagrams 397 Application of Sets 397 Cartesian Product of Sets  398 Relations  398 Types of Relations  398 Properties of Relation 398 Functions 399 Types of Functions 399 Compositions of Functions  400 Introduction to Algebra 400 Semigroups 401 Some Important Theorems  401 Group  402 Some Important Theorems  402 Residue Classes  402 Residue Class Addition 402 Residual Class Multiplication 402 Partial Ordering 402 Hasse Diagram 402 Lattice 403 Sublattice 403 Bounds 403 Boolean Algebra of Lattices 404 Solved Examples 405 Practice Exercise  411 Answers 416 Explanations and Hints 416 Solved GATE Previous Years’ Questions 422 9 Combinatory 427 Introduction  427 Counting 427 Fundamental Principle of Addition 427 Fundamental Principle of Multiplication 427 Permutations 428 Conditional Permutations  428 Permutations When all Objects are not Distinct  428 Circular Permutations  428 Combinations 428 Properties of Combinations  428 Conditional Combinations  429 Generating Functions  429 Ordinary Generating Function 429 Gate_FM.indd 13 8/25/2017 5:38:27 PM  xiv   CONTENTS Exponential Generating Function  429 Poisson Generating Function 429 Recurrence Relation  429 Logistic Map 429 Fibonacci Numbers  430 Binomial Coefficients 430 Summation  430 Asymptotic Analysis  430 Solved Examples  431 Practice Exercise  432 Answers  433 Explanations and Hints  434 Solved GATE Previous Years’ Questions 437 10 Graph Theory 439 Introduction 439 Fundamental Concepts of Graph 439 Common Terminologies 439 Degree of a Vertex 440 Multigraph 440 Walks, Paths and Connectivity 440 Subgraph 441 Types of Graphs 441 Complete Graph 441 Regular Graph 441 Bipartite Graph 441 Tree Graph 442 Trivial Graph 442 Cycle 443 Operations on Graphs 443 Matrix Representation of Graphs 443 Adjacent Matrix 443 Incidence Matrix 444 Cuts 444 Spanning Trees and Algorithms 444 Kruskal’s Algorithm 444 Prim’s Algorithm 445 Binary Trees 445 Euler Tours 445.. Konigsberg Bridge Problem 445 Hamiltonian Graphs 445 Closure of a Graph 446 Graph Isomorphism 446 Homeomorphic Graphs 446 Planar Graphs 447 Matching 447 Covering 447 Gate_FM.indd 14 8/25/2017 5:38:27 PM  CONTENTS xv Independent Set 447 Graph Coloring 447 Solved Examples 448 Practice Exercise 452 Answers 455 Explanations and Hints 455 Solved GATE Previous Years’ Questions 457 Chapter 11 Transform Theory  465 Introduction 465 Laplace Transform 465 Laplace Transform of Common Signals 466 Properties of Laplace Transform 466 Inverse Laplace Transform 467 z -Transform 467 z -Transform of Common Sequences 468 Properties of z -Transform 468 Inverse z-Transform 469 Relationship between z-Transform and Laplace Transform 470 Fourier Transform 470 Convergence of Fourier Transforms 470 Properties of Fourier Transform 471 Solved Examples 472 Practice Exercise 474 Answers  475 Explanations and Hints 475 Solved GATE Previous Years’ Questions 476 GENERAL APTITUDE: VERBAL ABILITY 487 1 English Grammar 489 Articles 489 Noun 491 Use of Nouns in Singular Form 491 Use of Nouns in Plural Form 491 Conversion of Nouns from Singular to Plural 491 Collective Nouns 491 Pronoun 492 Personal Pronoun 492 Reflexive and Emphatic Pronoun 492 Demonstrative Pronoun 493 Indefinite Pronoun 493 Distributive Pronoun 493 Relative Pronoun 493 Interrogative Pronoun 493 Use of Pronouns 493 Gate_FM.indd 15 8/25/2017 5:38:28 PM  xvi   CONTENTS Adjective 494 Use of Adjectives 494 Preposition 494 Preposition of Time 494 Preposition of Position 495 Preposition of Direction 495 Other Uses of Preposition 495 Verbs 496 Use of Verb 496 Use of Infinitives 496 Use of Gerunds 497 Tenses 497 Use of Tenses 498 Adverbs 498 PracticeExercise 499 Answers 502 2 Synonyms 503 Tips to Solve Synonym Based Questions 503 Practice Exercise 508 Answers 513 3 Antonyms 515 Graded Antonyms 515 Complementary Antonyms 516 Relational Antonyms 516 Practice Exercise 516 Answers 522 4 Sentence Completion 523 Tips to Solve Sentence Completion Based Questions 523 Practice Exercise 525 Answers 528 5 Verbal Analogies 529 Types of Analogies 529 Tips to Solve Verbal Analogies Questions 530 Practice Exercise 530 Answers 531 Explanations and Hints 532 6 Word Groups 533 Practice Exercise 535 Answers 536 Explanations and Hints 536 Gate_FM.indd 16 8/25/2017 5:38:28 PM  CONTENTS xvii 7 Verbal Deduction 537 Tips and Tricks to Solve Verbal Deduction Questions 537 Practice Exercise 538 Answers 540 Explanations and Hints 541 GENERAL APTITUDE: NUMERICAL ABILITY 543 Unit 1: Basic Arithmetic 545 1 Number System 547 Numbers 547 Classification of Numbers 547 Progressions 548 Arithmetic Progressions 548 Geometric Progressions 548 Infinite Geometric Progressions 548 Averages, Mean, Median and Mode 548 Relation between AM, GM and HM 549 Some Algebraic Formulas 549 The Remainder Theorem 549 The Polynomial Factor Theorem 549 Base System 550 Counting Trailing Zeros 550 Inequations 550 Quadratic Equations 551 Even and Odd Numbers 551 Prime Numbers and Composite Numbers 551 HCF and LCM of Numbers 551 Cyclicity 552 Test for Divisibility 552 Solved Examples 553 Practice Exercise 556 Answers  557 Explanations and Hints  558 2 Percentage 561 Introduction 561 Some Important Formulae 561 Solved Examples 562 Practice Exercise 566 Answers 568 Explanations and Hints 568 3 Profit and Loss 573 Introduction 573 Gate_FM.indd 17 8/25/2017 5:38:28 PM  xviii   CONTENTS Some Important Formulae 573 Margin and Markup 574 Solved Examples 574 Practice Exercise 577 Answers 579 Explanations and Hints 579 4 Simple Interest and Compound Interest 583 Introduction 583 Some Important Formulae 583 Solved Examples 584 Practice Exercise 588 Answers 591 Explanations and Hints 591 5 Time and Work 597 Introduction 597 Important Formulas and Concepts 597 Solved Examples 598 Practice Exercise 601 Answers 604 Explanations and Hints 605 6 Average, Mixture and Alligation 611 Average 611 Weighted Average 611 Mixture and Alligation 611 Solved Examples 612 Practice Exercise 615 Answers 617 Explanations and Hints 617 7 Ratio, Proportion and Variation 621 Ratio 621 Proportion 622 Variation 622 Solved Examples 623 Practice Exercise 625 Answers 627 Explanations and Hints 627 8 Speed, Distance and Time 631 Introduction 631 Some Important Formulas 631 Solved Examples 632 Practice Exercise 636 Answers 640 Explanations and Hints 640 Gate_FM.indd 18 8/25/2017 5:38:28 PM  CONTENTS xix Unit 2: Algebra 647 1 Permutation and Combination 649 Permutation 649 Combination 649 Partitions 650 Counting 650 Fundamental Principle of Addition 650 Fundamental Principle of Multiplication 650 Solved Examples 650 Practice Exercise 652 Answers 653 Explanations and Hints 653 2 Progression 657 Arithmetic Progression 657 Geometric Progression 657 Infinite Geometric Progression 658 Harmonic Series 658 Relation between AM, GM and HM 658 Solved Examples 658 Practice Exercise 661 Answers 662 Explanations and Hints 662 3 Probability 665 Introduction 665 Some Basic Concepts of Probability 665 Some Important Theorems 666 Solved Examples 666 Practice Exercise 671 Answers 673 Explanations and Hints 674 4 Set Theory 679 Introduction 679 Venn Diagrams 679 Operation on Sets 679 Venn Diagram with Two Attributes 680 Venn Diagram with Three Attributes 680 Solved Examples 681 Practice Exercise 682 Answers 684 Explanations and Hints 684 5 Surds, Indices and Logarithms 687 Surds 687 Gate_FM.indd 19 8/25/2017 5:38:28 PM  xx   CONTENTS Indices 687 Logarithm 687 Solved Examples 688 Practice Exercise 691 Answers 692 Explanations and Hints 693 Unit 3: Reasoning and Data Interpretation 697 1 Cubes and Dices 699 Cubes 699 Dices 700 Solved Examples 700 Practice Exercise 702 Answers  703 Explanations and Hints 703 2 Line Graph 705 Introduction 705 Solved Examples 706 Practice Exercise 707 Answers 709 Explanations and Hints 709 3 Tables 711 Introduction 711 Solved Examples 711 Practice Exercise 713 Answers 715 Explanations and Hints 715 4 Blood Relationship 717 Introduction 717 Standard Coding Technique 718 Solved Examples 718 Practice Exercise 719 Answers  721 Explanations and Hints 721 5 Bar Diagram 723 Introduction 723 Solved Examples 724 Practice Exercise 728 Answers 729 Explanations and Hints 729 Gate_FM.indd 20 8/25/2017 5:38:28 PM  CONTENTS xxi 6 Pie Chart 731 Introduction 731 Types of Pie Charts 731 3-D Pie Chart 731 Doughnut Chart  732 Exploded Pie Chart  732 Polar Area Chart  732 Ring Chart/Multilevel Pie Chart 732 Solved Examples 733 Practice Exercise 736 Answers 737 Explanations and Hints 738 7 Puzzles 739 Introduction 739 Types of Puzzles 739 Solved Examples 739 Practice Exercise 742 Answers  744 Explanations and Hints 744 8 Analytical Reasoning 747 Introduction 747 Solved Examples 747 Practice Exercise 749 Answers  751 Explanations and Hints 751 GENERAL APTITUDE: SOLVED GATE PREVIOUS YEARS’ QUESTIONS 753 Gate_FM.indd 21 8/25/2017 5:38:28 PM Gate_FM.indd 22 8/25/2017 5:38:28 PM  ENGINEERING MATHEMATICS Chapter 1.indd 1 8/22/2017 2:14:56 PM Chapter 1.indd 2 8/22/2017 2:14:56 PM  CHAPTER 1 LINEAR ALGEBRA MATRIX Types of Matrices A set of mn number (real or imaginary) arranged in the 1. Row matrix: A matrix having only one row is form of a rectangular array of m rows and n columns called a row matrix or a row vector. Therefore, for is called an m × n matrix. An m × n matrix is usually a row matrix, m = 1. written as For example, A = [1 2 −1] is a row matrix with m = 1 and n = 3.  a11 a12 a13 … a1n   2. Column matrix: A matrix having only one  a21 a22 a23 … a2n   column is called a column matrix or a column A=         vector. Therefore, for a column matrix, n = 1.         2 a     m1 am2 am3 … amn  For example, A = 3 is a column matrix with  1   In compact form, the above matrix is represented by m = 3 and n = 1. A = [aij]m×n or A = [aij]. 3. Square matrix: A matrix in which the number The numbers a11, a12, …, amn are known as the ele- of rows is equal to the number of columns, say n, is ments of the matrix A. The element aij belongs to ith called a square matrix of order n. row and jth column and is called the (ij)th element of 1 2 the matrix A = aij . For example, A =   is a square matrix of   order 2. 3 4 Chapter 1.indd 3 8/22/2017 2:14:58 PM  4     LINEAR ALGEBRA 4. Diagonal matrix: A square matrix is called a that An = 0, then n is called index of the nilpotent diagonal matrix if all the elements except those in matrix A. the leading diagonal are zero, i.e. aij = 0 for all i ≠ j. 2. Symmetrical matrix: It is a square matrix in 1 0 0  which aij = aji for all i and j. A symmetrical matrix   For example, A = 0 5 0  is a diagonal matrix is necessarily a square one. If A is symmetric, then  0 0 10 AT = A.   1 2 3 and is denoted by A = diag[1, 5, 10].   For example, A = 2 5 4. 3 4 6  5. Scalar matrix: A matrix A = [aij]n×n is called a   scalar matrix if (a) aij = 0, for all i ≠ j. 3. Skew-symmetrical matrix: It is a square matrix in which aij = −aji for all i and j. In a skew- (b) aii = c, for all i, where c ≠ 0. symmetrical matrix, all elements along the diago-  5 0 0   nal are zero. For example, A = 0 5 0 is a scalar matrix of  0 2 3  0 0 5     For example, 2 0 4. order 5.  3 4 0   6. Identity or unit matrix: A square matrix A = 4. Hermitian matrix: It is a square matrix A in [aij]n×n is called an identity or unit matrix if which (i, j)th element is equal to complex conju- (a) aij = 0, for all i ≠ j. gate of the (j, i)th element, i.e. aij = aji for all (b) aij = 1, for all i. i and j. A necessary condition for a matrix A to be  1 0 Hermitian is that A = Aq, where Aq is transposed For example, A =   is an identity matrix of  0 1 conjugate of A. order 2.  1 1 + 4i 2 + 3i    For example, 1 − 4i 2 5 + i . 7. Null matrix: A matrix whose all the elements are  2 − 3i 5 − i 4  zero is called a null matrix or a zero matrix.   0 0 5. Skew-Hermitian matrix: It is a square matrix For example, A =   is a null matrix of order A = [aij] in which aij = −aij for all i and j.  0 0 2 × 2. The diagonal elements of a skew-Hermitian 8. Upper triangular matrix: A square matrix matrix must be pure imaginary numbers or A = [aij] is called an upper triangular matrix if zeroes. A necessary and sufficient condition for a aij = 0 for i > j. matrix A to be skew-Hermitian is that 1 2 6 3 Aq = −A   0 5 7 4 6. Orthogonal matrix: A square matrix A is called For example, A =  3 is an upper tri- 0 0 9 orthogonal matrix if AAT = ATA = I. 0 2  0 0 angular matrix. For example, if  cos q − sin q   cos q sin q  9. Lower triangular matrix: A square matrix A = A=  and AT =   [aij] is called a lower triangular matrix if aij = 0 for  sin q cos q  − sin q cos q  i < j.  1 0  then AAT =   = I. 1 0 0 0  0 1   2 5 0 0 For example, A =   is a lower trian- 6 2 9 0  3 7 4 3   Equality of a Matrix gular matrix. Two matrices A = [aij]m×n and B = [bij]x×y are equal if Types of a Square Matrix 1. m = x, i.e. the number of rows in A equals the number of rows in B. 1. Nilpotent matrix: A square matrix A is called a 2. n = y, i.e. the number of columns in A equals the nilpotent matrix if there exists a positive integer n number of columns in B. such that An = 0. If n is least positive integer such 3. aij = bij for i = 1, 2, 3, …, m and j = 1, 2, 3, …, n. Chapter 1.indd 4 8/22/2017 10:52:29 AM  MATRIX     5 Addition of Two Matrices For example, if A =  1 2  and B =   4 3  3 4 2 1 Let A and B be two matrices, each of order m × n.  1 × 4 + 2 × 2 1 × 3 + 2 × 1 Then their sum (A + B) is a matrix of order m × n and Then A × B =   is obtained by adding the corresponding elements of A 3 × 4 + 4 × 2 3 × 3 + 4 × 1 and B. 8 5 C = A×B =   Thus, if A = [aij]m×n and B = [bij]m×n are two matri- 20 13 ces of the same order, their sum (A + B) is defined to be Some important properties of matrix multiplication are: the matrix of order m × n such that 1. Matrix multiplication is not commutative. (A + B)ij = aij + bij for i = 1, 2, …, m and 2. Matrix multiplication is associative, i.e. (AB)C = j = 1, 2, …, n A(BC). The sum of two matrices is defined only when they 3. Matrix multiplication is distributive over matrix are of the same order. addition, i.e. A(B + C) = AB + AC.  2 1 7 8  For example, if A =   and B =   4. If A is an m×n matrix, then ImA = A = AIn. 3 5 1 2 5. The product of two matrices can be the null matrix  9 9 while neither of them is the null matrix. Hence, A + B =    4 7   1 3 1 3 1 Multiplication of a Matrix by a Scalar However, addition of   and   is not possible. 2 1 2 2 1 If A = [aij] be an m × n matrix and k be any scalar Some of the important properties of matrix addition constant, then the matrix obtained by multiplying every are: element of A by k is called the scalar multiple of A by k 1. Commutativity: If A and B are two m × n and is denoted by kA.  1 3 1   matrices, then A + B = B + A, i.e. matrix addi- tion is commutative. For example, if A = 2 7 3 and k = 2.  4 9 8 2. Associativity: If A, B and C are three matrices of   the same order, then 2 6 2    (A + B) + C = A + (B + C) ⇒ kA = 2 14 6  3 18 16 i.e.matrix addition is associative.   3. Existence of identity: The null matrix is the Some of the important properties of scalar multiplica- identity element for matrix addition. Thus, A + O = tion are: A=O+A 4. Existence of inverse: For every matrix A = 1. k(A + B) = kA + kB [aij]m×n, there exists a matrix [aij]m×n, denoted 2. (k + l) ⋅ A = kA + lA by −A, such that A + (−A) = O = ( −A) + A 3. (kl) ⋅ A = k(lA) = l(kA) 5. Cancellation laws: If A, B and C are matrices of 4. (−k) ⋅ A = −(kA) = k(−A) the same order, then 5. 1⋅A=A 6. −1 ⋅ A = −A A+B=A+C⇒B=C B+A=C+A⇒B=C Here A and B are two matrices of same order and k and l are constants. If A is a matrix and A2 = A, then A is called idempo- Multiplication of Two Matrices tent matrix. If A is a matrix and satisfies A2 = I, then A is called involuntary matrix. If we have two matrices A and B, such that A = [aij]m×n and B = [bij]x×y, then Transpose of a Matrix A × B is possible only if n = x, i.e. the columns of the Consider a matrix A, then the matrix obtained by inter- pre-multiplier is equal to the rows of the post multiplier. changing the rows and columns of A is called its trans- Also, the order of the matrix formed after multiplying pose and is represented by AT. will be m × y.  1 2 5 1 3 7  n     (AB)ij = Σ air brj = ai1b1j + ai2b2 j +  + ain bnj For example, if A = 3 9 8 , AT = 2 9 6. 7 6 4   5 8 4     r =1 Chapter 1.indd 5 8/22/2017 10:52:34 AM  6     LINEAR ALGEBRA Some of the important properties of transpose of a 7. If A and B are non-singular square matrices of the matrix are: same order, then 1. For any matrix A, (AT)T = A adj (AB) = (adj B)(adj A) 2. For any two matrices A and B of the same order 8. If A is an invertible square matrix, then (A + B)T = AT + BT adj AT = (adj A)T 9. If A is a non-singular square matrix, then 3. If A is a matrix and k is a scalar, then adj (adj A) = |A|n−2 A (kA)T = k(AT) 10. If A is a non-singular matrix, then 4. If A and B are two matrices such that AB is |A−1| = |A|−1, i.e. | A−1 | = 1 defined, then |A| (AB)T = BTAT 11. Let A, B and C be three square matrices of same type and A be a non-singular matrix. Then Adjoint of a Square Matrix AB = AC ⇒ B = C and      BA = CA ⇒ B = C Let A = [aij] be a square matrix of order n and let Cij be the cofactor of aij in A. Then the transpose of the matrix Rank of a Matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A. The column rank of matrix A is the maximum number of Thus, adj A = [Cij]T ⇒ (adj A)ij = Cji = cofactor of linearly independent column vectors of A. The row rank aji in A. of A is the maximum number of linearly independent  a11 a13  row vectors of A.   a12 If A = a21 a22 a23  In linear algebra, column rank and row rank are always   a31 a32 a33  equal. This number is simply called rank of a matrix.  c11 c12 c13   c11 c31  The rank of a matrix A is commonly denoted by     c21 then adj(A) = c21 c22 c23  = c12 c22 c32  rank (A). Some of the important properties of rank of     c31 c32 c33  c13 c33  c23 a matrix are: 1. The rank of a matrix is unique. Inverse of a Matrix 2. The rank of a null matrix is zero. 3. Every matrix has a rank. A square matrix of order n is invertible if there exists a 4. If A is a matrix of order m × n, then rank (A) ≤ square matrix B of the same order such that m × n (smaller of the two) 5. If rank (A) = n, then every minor of order n + 1, AB = In = BA n + 2, etc., is zero. In the above case, B is called the inverse of A and is 6. If A is a matrix of order n × n, then A is non- denoted by A−1. singular and rank (A) = n. A−1 = (adj A) 7. Rank of IA = n. |A| 8. A is a matrix of order m × n. If every kth order Some of the important properties of inverse of a minor (k < m, k < n) is zero, then matrix are: rank (A) < k 1. A−1 exists only when A is non-singular, i.e. |A| ≠ 0. 9. A is a matrix of order m × n. If there is a minor of 2. The inverse of a matrix is unique. order (k < m, k < n) which is not zero, then 3. Reversal laws: If A and B are invertible matrices of rank (A) ≥ k the same order, then 10. If A is a non-zero column matrix and B is a non- (AB)−1 = B−1A−1 zero row matrix, then rank (AB) = 1. 4. If A is an invertible square matrix, then (AT)−1 = 11. The rank of a matrix is greater than or equal to the (A−1)T rank of every sub-matrix. 5. The inverse of an invertible symmetric matrix is a 12. If A is any n-rowed square matrix of rank, n − 1, then symmetric matrix. adj A ≠ 0 6. Let A be a non-singular square matrix of order n. 13. The rank of transpose of a matrix is equal to rank Then of the original matrix. |adj A| = |A|n−1 rank (A) = rank (AT) Chapter 1.indd 6 8/22/2017 10:52:35 AM  DETERMINANTS     7 14. The rank of a matrix does not change by 1 3 For example, say A =  , then minors of A will be ­pre-multiplication or post-multiplication with a 2 4 non-singular matrix. M11 = 4 15. If A − B, then rank (A) = rank (B). M12 = 2 16. The rank of a product of two matrices cannot exceed rank of either matrix. M21 = 3 rank (A × B) ≤ rank A M22 = 1 or     rank ( A × B) ≤ rank B  1 2 3   Say       A =  3 5 1 −4 4 7  17. The rank of sum of two matrices cannot exceed sum of their ranks.   18. Elementary transformations do not change the  5 1 M11 =   = 35 − 4 = 31  4 7  rank of a matrix.  3 1   = 21 − (−4) = 25 DETERMINANTS M12 = −4 7   3 5 M13 =   = 12 − (−20) = 32 Every square matrix can be associated to an expression −4 4 or a number which is known as its determinant. If A =  2 3 [aij] is a square mat