GATE 2020 - Mechanical Engineering (33 Years Solution) by IES Master Publication

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

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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 ÂŠ 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

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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

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ENGINEERING APTITUDE

CONTENTS

| VII

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

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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

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ENGINEERING APTITUDE

CONTENTS

| IX

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

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2.8

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ENGINEERING APTITUDE

Clocks .............................................................................................................. 329–340 (i) (ii) (iii) (iv) (v) (vi) (v)

Clocks .............................................................................................................. 329 Types of Problems .......................................................................................... 330 Type 1: Angle Between Two Hands (when time is given) ........................... 330 Type 2 : Time (when angle between two hands is given) ........................... 330 Type 3 : Slow and Fast Clocks ..................................................................... 332 Type 4 : Overall Gain or Loss of Time ......................................................... 332 Type 5 : Mirror Based Problems ................................................................... 333

Solved Examples ...................................................................................................... 324 Previous Years Questions UPSC/GATE/ESE .......................................................... 336 2.9

Calendars ........................................................................................................ 341–347 (i) (ii) (iii)

Concepts of Extra Days (or) Odd Days ........................................................ 341 Important Points .............................................................................................. 341 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) (ii) (iii) (iv) (v) (v) (vi) (vii)

Introduction ...................................................................................................... 348 Basic Terminology ........................................................................................... 348 Classification of Propositions ......................................................................... 350 Elimination Techniques of Venn Diagrams .................................................... 353 Adjectives in Syllogism ................................................................................... 354 Conclusions from Multiple Statements ........................................................... 354 Types of Conclusions ..................................................................................... 355 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) (ii) (iii) (iv)

Introduction ...................................................................................................... 373 Basic Terminology ........................................................................................... 373 Steps Followed in Critical Reasoning ............................................................ 373 Types of Questions ......................................................................................... 374

Solved Examples ...................................................................................................... 380 Previous Years Questions UPSC/GATE/ESE .......................................................... 389

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CONTENTS

| XI

UNIT 4 : MODERN MATHS 4.1

Sequences and Series ................................................................................... 393–422 (i) (ii)

Sequence ........................................................................................................ 393 Series .............................................................................................................. 393

(iii) (iv)

Arithmetic Progression (AP) ........................................................................... 393 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) (ii)

Introduction ...................................................................................................... 423 Algebraic Expression and Various Terms ...................................................... 423

(iii) (iv) (v)

Graph of Elementary Functions ..................................................................... 425 Linear Equation ............................................................................................... 426 Quadratic Equations ....................................................................................... 428

(vi) Roots of a Polynomial of Higher Degree ...................................................... 432 (vii) Maximum and Minimum Values of a Polynomial ........................................... 434 (ix) (x)

Inequalities ...................................................................................................... 435 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) (iii)

Types of Set ................................................................................................... 459 Venn Diagrams of Different Sets ................................................................... 461

(iv) (v) (vi)

Standard Results Based on Venn Diagrams ................................................. 462 Algebraic Laws of Sets .................................................................................. 463 Concept of maximizing or Minimizing the Intersection and Union ............... 465

Solved Examples ...................................................................................................... 467 Previous Years Questions UPSC/GATE/ESE .......................................................... 470

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ENGINEERING APTITUDE

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) (ii) (iii) (iv) (v) (v) (vi) (vii)

Introduction ...................................................................................................... 519 Some Theoretical Concpets ........................................................................... 519 Tables .............................................................................................................. 520 Line Graph ...................................................................................................... 524 Bar Graph ....................................................................................................... 527 Pie Charts ....................................................................................................... 536 Double pie charts ........................................................................................... 543 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

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 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. 2 3 Other examples of rational number are ,  ,… etc. 3 7 A number which can be expressed in the form of

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, 5, 3 7, 4 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.

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ENGINEERING APTITUDE

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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) Sol.

3 8

(b)



111 23

(c)

(d)

None of these

(c)

The number is option (a) and (b) are rational numbers, as they are the ratio of two integers. The number

2 is

non-recurring, so it is not a rational number. 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 = Numbers.

1 are called Complex

C = {a  ib; a, b  R & i2  1}

VARIOUS FORMS OF NUMBER TREE Form - 1 :

Form - 2 :

Numbers

Numbers Complex Numbers

Real Numbers

Decimals

Integers Rational Numbers

Irrational Numbers

Terminating Integers

Non-Terminating

Factions

Recurring Whole numbers

Rational Numbers

Natural numbers

Form - 3 : Complex Numbers Real Number

Irrational

Rational Fractions Integers (–ve Integers) Whole Number

Natural Numbers

Non-Recurring (Irrational Numbers)

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ENGINEERING APTITUDE

| 3

NUMBER SYSTEM

Example 2

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) Sol.

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

(ii) and similarly, the number 0.1555  can be represented as 0.15 Example 3

Express the recurring decimal 0.230 in the form of a fraction. Sol.

The given decimal can be written as 0.230 = 0.230230230… As the bar is placed on three digits, so, we will multiply the above equation by 103

…(i)

0.230  103 = 230.230230…

…(ii)

From equation (i) and (ii), we can write

(103  1) 0.230 = 230

0.230 =

230 3

10  1

230 999

Example 4

The value of 1.34  4.12 is (a) Sol.

133 90

(b)

371 90

(c)

219 990

(d)

461 990

(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 102 1.34  10 2 = 134.3434…

…(ii)

From equation (i) and (ii) 133 1.34 = 99 In 4.12 , the bar is place on one digit so it can be written as 4.12 = 4.1222 …

…(i)

by multiplying 102 in above equation 4.12  10 2 = 412.222…

…(ii)

(10 2  1) 1.34 = 133



IES MASTER Publications

34 |

ENGINEERING APTITUDE

Product of 82540027 and 43253 is (a) 3570403787831

(b) 3570403787832

(d) 3570403787834

In binary system, 1010 is equivalent to

(a) 3

(b) 5

(a) 8

(b) 9

(d) 18

(d) 11

[UPSC–1986]

If n is an integer between 20 and 80, then any of the following could be (n + 7) except (a) 47

(b) 58

(d) 88

[UPSC–1988]

Zero was invented by (a) Aryabhata

(b) Varahamihira

(d) An unknown Indian

10.

[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? 11.

(a) 34

(b) 44

(d) None of the above

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

(d) 480

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

(d) 34

[UPSC–2013]

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

(b) 50.40 m

(d) 100.80 m

12.

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

(d) 9

[UPSC–2014]

The question is followed by two statements I and II. Mark the answer as

[UPSC–2009]

(a) if the question can be answered with the help of statement I alone.

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?

(b) if the question can be answered with the help of statement II, alone.

(a) 08:45 am

(b) 10:30 am

(d) 01:30 pm [UPSC–2013]

[UPSC–2013]

[UPSC–2007] 6.

ENGINEERING APTITUDE

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?

[UPSC–1985] 2.

AND

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,

(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]

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ENGINEERING APTITUDE

| 37

NUMBER SYSTEM

ANSWER KEY 1.

(a)

(c)

13.

(b)

19.

(d)

25.

(d)

31.

(d)

(c)

(a)

14.

(c)

20.

(c)

26.

(a)

32.

(b)

(d)

(b)

15.

(d)

21.

(c)

27.

(d)

33.

(d)

10.

(a)

16.

(b)

22.

(7)

28.

(b)

34.

(b)

11.

(b)

17.

(c)

23.

(a)

29.

(b)

35.

(c)

(b)

12.

(a)

18.

(d)

24.

(c)

30.

(b)

EXPLANATIONS 1.

(a)



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. 2.

(c)

 78 104 117 169     =   13 13   13 13 = (6 + 8 + 9 + 13) = 36

= LCM of (9, 12 and 15 min.)

(d)

= 180 min. = 3 hrs

Given, 20 < n < 80 So, 20 + 7 < (n + 7) < 80 + 7

 If all the three begin to toll at 8:00 AM then the next toll together again = 8 + 3 = 11:00 AM

or, 27 < (n + 7) < 87. Hence, option (d) is not possible. 4.

(d)

= LCM of (2, 3, 4, 5 and 6 days) = 60 days

So, option (d). (b) The size of bottle required to be filled = HCF of (403, 465 and 1496 litres)

The minimum possible number of bottles required

 The number of times all the groups meet on the 180 same day within 180 days = = 3 times 60 (b) As the number of columns and rows are same so, the number of plants will be a perfect square (i.e. n2).

= 31 litres

(a) The number of days after which all the five groups meet same day

It is not clear whether Brahm Gupta or Aryabhata was the inventor of zero. 5.

(1001)2 =1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 = 9 3.

403 465 496   = 13 + 15 + 16 = 44 31 31 31

The perfect square just greater than 1000 is 1024 so, the number of plants needed

(b)

= 1024 – 1000

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

The required number of pieces

= 24 plants 10.

(a) 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