ENGINEERING APTITUDE

ENGINEERING APTITUDE (QUANTATIVE APTITUDE AND ANALYTICAL ABILITY) (For ESE & GATE Exam) (CE, ME, PI, CH, EC, EE, IN, CS, IT) Salient Features : 171 topics under 25 chapters in 5 units 531 theoretical examples for comprehensive understanding 388 chapterwise solved examples for practice 633 Questions segregated chapterwise from last 38 years of UPSC/GATE/ESE Exams with detailed solutions Office : F-126, (Lower Basement), Katwaria Sarai, New Delhi-110016 Phone : 011-26522064 Mobile : 8130909220, 9711853908 E-mail: info@iesmasterpublications.com, info@iesmaster.org Web : iesmasterpublications.com, iesmaster.org

IES MASTER PUBLICATION F-126, (Lower Basement), Katwaria Sarai, New Delhi-110016 Phone : 011-26522064, Mobile : 8130909220, 9711853908 E-mail : info@iesmasterpublications.com, info@iesmaster.org Web : iesmasterpublications.com, iesmaster.org All rights reserved. Copyright 2018, 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 : 2018 Typeset at : IES Master Publication, New Delhi-110016

PREFACE Union Public Services Commission (UPSC) in its quest for best engineering minds looks for the very basic pride of an engineer, which it tests through your quantitative and analytical abilities. The profession itself calls for putting you in situations, both human and technical, where things are tied into hundred knots. As an engineer you are expected to think critically, detect systematic themes while analysing data, and achieve thoroughness with accuracy in deriving solutions under challenging circumstances. To test these qualities in an ESE aspirant, in the year 2016, UPSC introduced Engineering Aptitude as a part of the syllabus for common paper of ESE in 2016. With an objective to develop these abilities, IES Master has come up with this Engineering Aptitude book that brings you face-to-face with thousands of problems under various subheads such as probability, polynomials, speed-time, work-time, clock and calendar, as well as geometry and measurements that you might encounter as a professional. Covering 171 topics under 25 chapters in 5 units, this book is an effort by IES Master to expose you to the complete theory of ESE syllabus along with previous years questions from UPSC (last 38 years), GATE (last 9 years), and ESE (last 2 years). As you flip through the pages of this book, it captures your imagination with subtleness, and exposes you to more than 1,200 problems, enough to give your pen the required strength to take on any competitive exams including ESE, GATE and PSUs. Under the expert guidance of Mr Kanchan Kumar Thakur (Ex-IES), Mr Puneet Sharma, who has more than 14 years of teaching, has tamed the tornado that will blow you away, with its concept clarity and conciseness. Having gone through it, we hope that as your fingers follow your command, the brain will engineer solutions no matter how difficult the challenge is. IES Master Publication New Delhi

CONTENTS UNIT 1 : QUANTITATIVE REASONING 1.1 Number System... 01 40 (i) Number Line... 01 (ii) Definition of Various Types of Numbers... 01 (iii) Various forms of Number Tree... 02 (iv) Concept of Prime and Composite Numbers... 04 (v) Factorization... 06 (vi) Factorial... 06 (vii) Factorization of Factorial... 07 (viii) Last Digit Problems... 09 (ix) Number of Zeros at the End of any Product... 10 (x) Divisibility Rules... 11 (xi) Relations between Dividend, Divisor and Quotient... 13 (xii) Successive Division... 14 (xiii) Remainder Theorem... 15 (xiv) LCM and HCF... 18 (xv) Indices or Powers... 24 (xvi) Surds... 25 (xvii) Simplification... 28 Solved Examples... 31 Previous Years Questions UPSC/GATE/ESE... 34 1.2 Ratio and Proportion... 41 54 (i) Ratio... 41 (ii) Proportion... 43 (iii) Partnership... 46 Solved Examples... 47 Previous Years Questions UPSC/GATE/ESE... 48

VI ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE 1.3 Percentage... 55 68 (i) Percentage... 55 (ii) Concept of Multiplying Factors... 56 (iii) Successive Percentage Change... 57 Solved Examples... 60 Previous Years Questions UPSC/GATE/ESE... 61 1.4 Profit and Loss... 69 82 (i) Profit or Gain... 69 (ii) Loss... 69 (iii) Concept of Multiplying Factor Related to Profit and Loss... 69 (iv) Discount... 73 (v) Three Special Cases of Profit and Loss... 74 Solved Examples... 76 Previous Years Questions UPSC/GATE/ESE... 79 1.5 Simple Interest and Compound Interest... 83 90 (i) Simple Interest... 83 (ii) Compound Interest... 83 (iii) Difference between CI and SI for First Two Years... 85 Solved Examples... 87 Previous Years Questions UPSC/GATE/ESE... 89 1.6 Average and Alligation... 91 116 (i) Average... 91 (ii) Problems based on Ages... 93 (iii) Weighted Average... 95 (iv) Alligation... 96 (v) Problems based on Mixture of Two Liquids... 101 (vi) Addition of Pure Solution in a Mixture... 102 (vii) Removal and Replacement... 103 Solved Examples... 107 Previous Years Questions UPSC/GATE/ESE...111

GS AND ENGINEERING APTITUDE CONTENTS VII 1.7 Time and Work... 117 134 (i) Concept of Variation... 117 (ii) Work-Time... 119 (iii) Men Days... 119 (iv) Types of Questions... 120 Solved Examples... 125 Previous Years Questions UPSC/GATE/ESE... 129 1.8 Speed, Distance and Time... 135 162 (i) Speed... 135 (ii) Analysis of Speed, Distance and Time Relationship... 136 (iii) Average Speed... 138 (iv) Relative Speed... 140 (v) Boats and Streams... 141 (vi) Linear and Circular Races... 142 Solved Examples... 147 Previous Years Questions UPSC/GATE/ESE... 152 1.9 Geometry and Mensuration... 163 218 (i) Basic Concepts... 163 (ii) Polygon... 167 (iii) Triangle... 170 (iv) Triangle Classification... 171 (v) Similar Triangles... 177 (vi) Quadrilaterals... 182 (vii) Perimeter and Areas of Quadrilaterals... 184 (viii) Hexagon... 186 (ix) Circle... 186 (x) Properties of Circle... 187 (xi) Mensuration... 193 Solved Examples... 198 Previous Years Questions UPSC/GATE/ESE... 206

VIII ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE UNIT 2 : ANALYTICAL REASONING 2.1 Ranking Test... 219 224 (i) Introduction... 219 Solved Examples... 221 Previous Years Questions UPSC/GATE/ESE... 223 2.2 Dices and Cubes... 225 242 (i) Cubes... 225 (ii) Types of Problems... 225 (iii) Type 1 : Counting of Cubes... 225 (iv) Type 2 : Concept of Dice/Cube with Numbers... 226 (v) Type 3 : (Unfolded Cube/Dice)... 228 (vi) Type 4 : (Maximum Number of Pieces with x Cuts)... 229 (vii) Type 5 : (Minimum Number of Cuts with n Pieces)... 230 (viii) Type 6 : (Colouring of Cubes)... 231 Solved Examples... 235 Previous Years Questions UPSC/GATE/ESE... 239 2.3 Direction Sense... 243 262 (i) Introduction... 243 (ii) Reference Compass... 243 (iii) Problem Solving Technique... 243 (iv) Types of Problems... 244 (v) Type I : Shortest Distance Based Questions... 244 (vi) Type II : Direction Based Questions... 247 (vii) Type III : Shadow Based Questions... 248 (viii) Type IV : Clocks Based Questions... 249 (ix) Type V : Faulty Compass Based Questions... 250 (x) Type VI : Rotation Based Questions... 250 Solved Examples... 254 Previous Years Questions UPSC/GATE/ESE... 258

GS AND ENGINEERING APTITUDE CONTENTS IX 2.4 Blood Relationship... 263 276 (i) Introduction... 263 (ii) Symbols Used in Family Diagram... 263 (iii) Types of Questions... 264 (iv) Blood Relation Based on Conversation... 264 (v) Blood Relation Based on Puzzle... 267 (vi) Coded Blood Relationship... 268 Solved Examples... 271 Previous Years Questions UPSC/GATE/ESE... 274 2.5 Seating Arrangement... 277 294 (i) Introduction... 277 (ii) Linear Seating Arrangements... 277 (iii) Circular Arrangement... 281 (iv) Two Row Sitting Arrangements... 283 (v) Complex Arrangements... 285 Solved Examples... 287 Previous Years Questions UPSC/GATE/ESE... 292 2.6 Coding Decoding... 295 304 (i) Introduction... 295 (ii) Type I : Change in Relative Position of Letters... 295 (iii) Type II : Shifting of Letters to Form New Word... 295 (iv) Type III : Coding Letter to Numbers... 297 (v) Type IV : Substitution Based Coding... 298 (vi) Type V : Mixed Number Coding... 298 (vii) Type VI : Mixed Letter Coding... 299 (ix) Type VII : Direct Letter Coding... 300 (x) Type VIII : Mixed Type Coding... 300 Solved Examples... 302 Previous Years Questions UPSC/GATE/ESE... 304 2.7 Puzzles... 305 328 (i) Introduction... 305 Solved Examples... 311 Previous Years Questions UPSC/GATE/ESE... 314

X ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE 2.8 Clocks... 329 340 (i) Clocks... 329 (ii) Types of Problems... 330 (iii) Type 1: Angle Between Two Hands (when time is given)... 330 (iv) Type 2 : Time (when angle between two hands is given)... 330 (v) Type 3 : Slow and Fast Clocks... 332 (vi) Type 4 : Overall Gain or Loss of Time... 332 (v) Type 5 : Mirror Based Problems... 333 Solved Examples... 324 Previous Years Questions UPSC/GATE/ESE... 336 2.9 Calendars... 341 347 (i) Concepts of Extra Days (or) Odd Days... 341 (ii) Important Points... 341 (iii) Types of Problems... 342 Solved Examples... 345 Previous Years Questions UPSC/GATE/ESE... 347 UNIT 3 : CRITICAL REASONING 3.1 Basic Concepts of Syllogism... 348 372 (i) Introduction... 348 (ii) Basic Terminology... 348 (iii) Classification of Propositions... 350 (iv) Elimination Techniques of Venn Diagrams... 353 (v) Adjectives in Syllogism... 354 (v) Conclusions from Multiple Statements... 354 (vi) Types of Conclusions... 355 (vii) Special Case (The Case of EITHER... OR... )... 356 Solved Examples... 358 Previous Years Questions UPSC/GATE/ESE... 371 3.2 Critical Reasoning... 373 392 (i) Introduction... 373 (ii) Basic Terminology... 373 (iii) Steps Followed in Critical Reasoning... 373 (iv) Types of Questions... 374 Solved Examples... 380 Previous Years Questions UPSC/GATE/ESE... 389

GS AND ENGINEERING APTITUDE CONTENTS XI UNIT 4 : MODERN MATHS 4.1 Sequences and Series... 393 422 (i) Sequence... 393 (ii) Series... 393 (iii) Arithmetic Progression (AP)... 393 (iv) Geometric Progression (GP)... 396 (v) Arithmetico Geometric Progression (AGP)... 400 (vi) Harmonic Progression (HP)... 401 (vii) Sum of General Series... 402 (viii) Hidden Sequences... 405 Solved Examples... 406 Previous Years Questions UPSC/GATE/ESE... 413 4.2 Polynomials... 423 458 (i) Introduction... 423 (ii) Algebraic Expression and Various Terms... 423 (iii) Graph of Elementary Functions... 425 (iv) Linear Equation... 426 (v) Quadratic Equations... 428 (vi) Roots of a Polynomial of Higher Degree... 432 (vii) Maximum and Minimum Values of a Polynomial... 434 (ix) Inequalities... 435 (x) Quadratic Inequalities... 437 (xi) Modulus... 439 (xii) Logarithms... 440 (xiii) Logarithmic Inequalities... 443 Solved Examples... 445 Previous Years Questions UPSC/GATE/ESE... 450 4.3 Set Theory... 459 478 (i) Set... 459 (ii) Types of Set... 459 (iii) Venn Diagrams of Different Sets... 461 (iv) Standard Results Based on Venn Diagrams... 462 (v) Algebraic Laws of Sets... 463 (vi) Concept of maximizing or Minimizing the Intersection and Union... 465 Solved Examples... 467 Previous Years Questions UPSC/GATE/ESE... 470

XII ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE 4.4 Permutation and Combination... 479 518 (i) Fundamental Principle of Counting... 479 (ii) Permutation... 480 (iii) Combination... 482 (iv) Total Number of Combinations... 484 (v) Difference between Permutation and Combination... 485 (vi) Permutation of Alike Items... 487 (vii) Division of Items into Groups of Different Sizes... 489 (viii) Division of Different Items into Groups of Equal Size... 489 (ix) Distribution of Different Items... 491 (x) Distribution of Identical Items into Groups... 492 (xi) Circular Permutation... 494 (xii) Sum of all Numbers formed from given Digits... 495 (xiii) Rank of a Word... 496 (xiv) Number of Dearrangements... 498 (xv) If Only Selections or Rejections is to be Considered... 499 (xvi) Formation of Words / Formation of Numbers... 500 Solved Examples... 502 Previous Years Questions UPSC/GATE/ESE... 508 UNIT 5 : DATA INTERPRETATION 5.1 Data Interpretation... 519 562 (i) Introduction... 519 (ii) Some Theoretical Concpets... 519 (iii) Tables... 520 (iv) Line Graph... 524 (v) Bar Graph... 527 (v) Pie Charts... 536 (vi) Double pie charts... 543 (vii) Combination of Graphs... 546 Solved Examples... 551 Previous Years Questions UPSC/GATE/ESE... 555

1.1 NUMBER SYSTEM NUMBER LINE Number line is a line on which all the positive and negative numbers can be represent in a sequence. It stretches from negative infinity to positive infinity. 3 2 1 0 1 2 3 + DEFINITION OF VARIOUS TYPES OF NUMBERS Natural Numbers Counting Numbers 1, 2, 3, 4,... are called Natural Numbers. The symbolic representation is N, i.e., N = {1, 2, 3, 4, 5,...}. Whole Numbers All the natural numbers together with 0 are called Whole Numbers and the symbolic representation is W, i.e., W = {0, 1, 2, 3, 4,...} Integers An integer is a number that can be written without a fractional component, it is represented by Z. Integers are further classified into positive integers (2, 4, 5 etc.), zero (0) and negative integers ( 2, 5 etc.). Rational Numbers A number which can be expressed in the form of p where, p and q are integers and q 0, is called a rational number. q For example : Any integer number is a rational number since, it can be written as the ratio of two integer numbers, one the integer number itself and another number is 1. Other examples of rational number are 2, 3, etc. 3 7 Note : A decimal represents a rational number if and only if it has a finite number of digits. But recurring decimals are exceptions as they are also assumed as Rational Numbers, i.e., all recurring decimals are rational numbers. Irrational Numbers A real number, which is not rational, is called irrational number. An irrational number has non-terminating and nonrecurring decimal part. Between any two numbers, there are infinite numbers of irrational numbers. Examples of irrational numbers are : 3 4 3, 5, 7, 11 Numbers and e are also irrational number because both have non-terminating and non-recurring decimal part. = 3.14159265358979 and e = 2.71828182859045 where, e is called Euler s number.

2 ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE Note : Any terminating or recurring decimal is a rational number. Any non-terminating and non-recurring decimal is an irrational number. Example 1 Which one of the following is not a rational number? (a) 3 8 (b) 111 (c) 23 2 (d) None of these Sol. (c) The number is option (a) and (b) are rational numbers, as they are the ratio of two integers. The number non-recurring, so it is not a rational number. 2 is Real Numbers The real numbers include all the measuring numbers. The symbol for the real number is R. All the numbers which can be represented on the number line are called real numbers. Complex Numbers All the numbers that can be represented in a + ib form where a & b are real numbers and i = 1 are called Complex Numbers. VARIOUS FORMS OF NUMBER TREE 2 C = {a ib; a, b R & i 1} Form - 1 : Form - 2 : Numbers Real Numbers Complex Numbers Numbers Rational Numbers Integers Factions Whole numbers Natural numbers Irrational Numbers Integers Terminating Rational Numbers Decimals Non-Terminating Recurring Non-Recurring (Irrational Numbers) Form - 3 : Complex Numbers Real Number Irrational Rational Fractions Integers ( ve Integers) Whole Number Natural Numbers

GS AND ENGINEERING APTITUDE NUMBER SYSTEM 3 Example 2 Sol. Consider the following statements : I. Every natural number is a real number. II. Every real number is a rational number. III. Every integer is a real number. IV. Every rational number is a real number. Which of the above statements are correct? (a) I, II and III (b) I, III and IV (c) II and III (d) III and IV (b) From the number tree, given above, all natural numbers are real numbers but its converse is not true. So, statement (I) is true. Every real number is not a rational number, some may be irrational numbers. Hence, statement (II) is wrong. Similarly, from number tree, we can say about statement (III) and (IV) that both the statements are true. Recurring Decimals A decimal in which a digit or a set of digits is repeated continuously is called a recurring decimal. For representing recurring decimal, we place bar on the repeated numbers. For example : (i) The number 0.247632476324763 can be represented as 0.24763 Example 3 Sol. (ii) and similarly, the number 0.1555 can be represented as 0.15 Express the recurring decimal 0.230 in the form of a fraction. The given decimal can be written as 0.230 = 0.230230230 (i) As the bar is placed on three digits, so, we will multiply the above equation by 10 3 From equation (i) and (ii), we can write 3 0.230 10 = 230.230230 (ii) 3 230 (10 1) 0.230 = 230 or 0.230 = 3 10 1 = 230 999 Example 4 The value of 1.34 4.12 is (a) 133 90 (b) 371 90 (c) 219 5 990 (d) 461 5 990 Sol. (d) The decimal 1.34 can be written as 1.34 = 1.343434 (i) As the bar is placed on two digits after decimal point, so, we will multiply the above equation by 10 2 From equation (i) and (ii) 2 1.34 10 = 134.3434 (ii) 2 (10 1) 1.34 = 133 1.34 = 133 99 In 4.12, the bar is place on one digit so it can be written as 4.12 = 4.1222 by multiplying 10 2 2 in above equation 4.12 10 = 412.222 (ii) (i) IES MASTER Publications

34 ENGINEERING APTITUDE GS AND ENGINEERING APTITUDE 1. Product of 82540027 and 43253 is (a) 3570403787831 (b) 3570403787832 (c) 3570403787833 (d) 3570403787834 2. In binary system, 1010 is equivalent to (a) 8 (b) 9 [UPSC 1985] (c) 10 (d) 11 [UPSC 1986] 3. If n is an integer between 20 and 80, then any of the following could be (n + 7) except (a) 47 (b) 58 (c) 84 (d) 88 [UPSC 1988] 4. Zero was invented by (a) Aryabhata (c) Bhaskara I (b) Varahamihira (d) An unknown Indian [UPSC 1995] 5. A person has to completely put each of three liquids: 403 litres of petrol, 465 litres of diesel and 496 litres of mobil oil in bottles of equal size without mixing any of the above three types of liquids such that each bottle is completely filled. What is the least possible number of bottles required? (a) 34 (b) 44 (c) 46 (d) None of the above [UPSC 2007] 6. Four metal rods of lengths 78 cm, 104 cm, 117cm and 169 cm are to be cut into parts of equal length. Each part must be as long as possible. What is the maximum number of pieces that can be cut? (a) 27 (b) 36 (c) 43 (d) 480 [UPSC 2009] 7. Three bells toll at intervals of 9, 12 and 15 minutes respectively. All the three begin to toll at 8am. At what time will they first toll together again? (a) 08:45 am (c) 11:00 am (b) 10:30 am (d) 01:30 pm [UPSC 2013] 8. There are five hobby clubs in a college viz, photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days? (a) 3 (b) 5 (c) 10 (d) 18 [UPSC 2013] 9. A gardener has 1000 plants. He wants to plan them in such a way that the number of rows and the number of columns remains the same. What is the minimum number of plants that he needs more for this purpose? (a) 14 (b) 24 (c) 32 (d) 34 [UPSC 2013] 10. Three persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps? (a) 25.20 m (c) 75.60 m (b) 50.40 m (d) 100.80 m [UPSC 2013] 11. Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 sec. respectively. The number of times they would fire the bullets together at the target in an hour is. (a) 6 (b) 7 (c) 8 (d) 9 [UPSC 2014] 12. The question is followed by two statements I and II. Mark the answer as (a) if the question can be answered with the help of statement I alone. (b) if the question can be answered with the help of statement II, alone. (c) if both statement I and statement II are needed to answer the question. (d) if the question cannot be answered even with the help of both the statements. If x, y and z are real numbers, is z x even or odd? (I) xyz is odd. (II) xy + yz + zx is even. [UPSC 2014]

GS AND ENGINEERING APTITUDE NUMBER SYSTEM 37 ANSWER KEY 1. (a) 7. (c) 13. (b) 19. (d) 25. (d) 31. (d) 2. (c) 8. (a) 14. (c) 20. (c) 26. (a) 32. (b) 3. (d) 9. (b) 15. (d) 21. (c) 27. (d) 33. (d) 4. (d) 10. (a) 16. (b) 22. (7) 28. (b) 34. (b) 5. (b) 11. (b) 17. (c) 23. (a) 29. (b) 35. (c) 6. (b) 12. (a) 18. (d) 24. (c) 30. (b) EXPLANATIONS 1. (a) 2. (c) 3. (d) 4. (d) 5. (b) 6. (b) In this problem, we will see only last digits. These are 7 and 3 respectively. Hence, last digit of the product of two numbers will be 1. Hence, option (a) is correct. (1001) 2 =1 2 3 + 0 2 2 + 0 2 1 + 1 2 0 = 9 Given, 20 < n < 80 So, 20 + 7 < (n + 7) < 80 + 7 or, 27 < (n + 7) < 87. Hence, option (d) is not possible. It is not clear whether Brahm Gupta or Aryabhata was the inventor of zero. So, option (d). The size of bottle required to be filled = HCF of (403, 465 and 1496 litres) = 31 litres The minimum possible number of bottles required 403 465 496 = 31 31 31 = 13 + 15 + 16 = 44 The maximum equal size of each piece = HCF of (78 cm, 104 cm, 117 cm and 169 cm) By factorisation method, HCF can be found as below 78 = 2 3 13 104 = 2 2 2 13 117 = 3 3 13 169 = 13 13 HCF = 13 cm 7. (c) The required number of pieces = 78 104 117 169 13 13 13 13 = (6 + 8 + 9 + 13) = 36 The time direction after which each bell toll 8. (a) = LCM of (9, 12 and 15 min.) = 180 min. = 3 hrs If all the three begin to toll at 8:00 AM then the next toll together again = 8 + 3 = 11:00 AM The number of days after which all the five groups meet same day = LCM of (2, 3, 4, 5 and 6 days) = 60 days 9. (b) 10. (a) The number of times all the groups meet on the same day within 180 days = 180 60 = 3 times As the number of columns and rows are same so, the number of plants will be a perfect square (i.e. n 2 ). The perfect square just greater than 1000 is 1024 so, the number of plants needed = 1024 1000 = 24 plants The required distance = LCM of (40 cm, 42 cm and 45 cm). By factorisation method, the LCM can be found as below. 40 = 2 2 2 5 42 = 2 3 7 IES MASTER Publications