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Title

: GATE 2022 - Mathematics - 22 Years Chapter-wise Solved Papers 2000-2021

Language

: English

Editor’s Name

: Vinit Garg

Copyright ©

: 2021 CLIP

No part of this book may be reproduced in a retrieval system or transmitted, in any form or by any means, electronics, mechanical, photocopying, recording, scanning and or without the written permission of the Author/Publisher. Typeset & Published by : Career Launcher Infrastructure (P) Ltd. A-45, Mohan Cooperative Industrial Area, Near Mohan Estate Metro Station, New Delhi - 110044 Marketed by : G.K. Publications (P) Ltd. Plot No. 9A, Sector-27A, Mathura Road, Faridabad, Haryana-121003 ISBN : 978-93-91061-22-7 Printer’s Details : Made in India, New Delhi. For product information : Visit www.gkpublications.com or email to [email protected]

 Preface

(vii)

 About GATE

(ix)

 GATE Syllabus

(xix)

Solved Papers (Chapter-Wise) Verbal Ability

Numerical Ability

1. English Grammar

1.1 - 1.4

MCQ Type Questions

1.1



Answers

1.4



Explanations

1.4

2. Sentence Completion MCQ Type Questions

2.1 - 2.4 2.1



Answers

2.4



Explanations

2.4

3. Synonyms MCQ Type Questions

3.1

Answers

3.2



Explanations

3.2

MCQ Type Questions

1.1 - 1.12

MCQ Type Questions

1.1

Numerical Type Questions

1.4



Answers

1.6



Explanations

1.7

2. Percentage and Its Applications

3.1 - 3.2



4. Antonyms

1. Numbers and Algebra

2.1 - 2.2

MCQ Type Questions

2.1

Numerical Type Questions

2.1



Answers

2.2



Explanations

2.2

3. Time and Work

3.1 - 3.4

MCQ Type Questions

3.1

4.1

Numerical Type Questions

3.2

4.1 - 4.2



Answers

4.1



Answers

3.2



Explanations

4.2



Explanations

3.3

5. Reasoning Ability

5.1 - 5.14

4. Ratio, Proportion and Mixtures

4.1 - 4.2

MCQ Type Questions

5.1

MCQ Type Questions

Numerical Type Questions

5.9



Answers

4.1



Explanations

4.2



Answers

5.10



Explanations

5.10

(iii)

4.1

5. Algebra

5. Permutations and Combinations & Probability

5.1 - 5.4

5.1 - 5.14

Answers

5.7

Explanations

5.8

MCQ Type Questions

5.1

Numerical Type Questions

5.2

6. Functional Analysis

6.1 - 6.14



Answers

5.2

Answers

6.9



Explanations

5.3

Explanations

6.9

6. Miscellaneous

6.1 - 6.6

7. Numerical Analysis

MCQ Type Questions

6.1

Answers

Numerical Type Questions

6.3

Explanations



Answers

6.4



Explanations

6.4

1.12

Explanations

1.13

8.9

Explanations

8.9

Answers Explanations

3. Real Analysis

2.1 - 2.22 2.9 2.10

9.6

Explanations

9.7

10.1 - 10.18

Answers

10.13

Explanations

10.13

11. Linear Programming

3.1 - 3.28

9.1 - 9.9

Answers

10. Probability & Statistics 2. Complex Analysis

7.10

Answers

9. Topology

1.1 - 1.28

Answers

7.9

8. Partial Differential Equations 8.1 - 8.18

M athematics 1. Linear Algebra

7.1 - 7.22

11.1 - 11.17

Answers

11.10

Explanations

11.11

Answers

3.15

Explanations

3.16



Solved Paper 2017

1 – 15

4. Ordinary Differential Equations 4.1-4.22



Solved Paper 2018

1 – 15



Solved Paper 2019

1 – 16



Solved Paper 2020

1 – 18



Solved Paper 2021

1 – 21

Answers Explanations

4.9 4.10

(iv)

IIT Institutes

Indian Institute of Technology, Roorkee

(v)

Preface The Graduate Aptitude Test in Engineering (GATE) is an online exam conducted by the IITs for admissions to PG courses in IITs, IISc Bangalore, NITs and many state run universities as well as private universities. Also there are more than 37 PSUs that use GATE score for recruitments. A large number of corporates are also using GATE score as a tool to screen students for placements. GK Publications is well known as the ‘‘publisher of choice’’ to students preparing for GATE and other technical examinations in the country. We published the first set of books in 1994 when GATE exam, both objective and conventional, was conducted in the paper and pencil environment, and used as a check point for entry to post graduate courses in IITs and IISCs. At that time, students had little access to technology and relied mainly on instructor led learning followed by practice with books available for these examinations. A lot has changed since then! Today, GATE is conducted in an online only mode with multiple choice and numerical based questions. The score is valid for three years and is used not only for post graduate courses but is also used by major PSUs for recruitment. Today’s students have easy access to technology and the concept of a monologue within the classroom has changed to dialogue where students come prepared with concepts and then discuss topics. They learn a lot of things on the go with their mobile devices and practice for mock tests online. We, as a leading publisher of GATE books, have also embraced change. Today, our books are no more guides and papers only but come with a fully supported mobile app and a web portal. The mobile app provides access to video lectures, short tests and regular updates about the exam. The web portal in additional to what is available on the app provides full length mock tests to mimic the actual exam and help you gauge your level of preparedness. The combination of practice content in print, video lectures, and short and full length tests on mobile and web makes this product a complete courseware for GATE preparation. This book includes previous years GATE questions along with detailed solution of each question for better understanding. It will help the GATE aspirants to know an idea about the pattern of questions asked in GATE examination. We also know that improvement is a never ending process and hence we welcome your suggestions and feedback or spelling and technical errors if any. Please write to us at [email protected] We hope that our small effort will help you prepare well for the examination. We wish you all the best! GK Publications Pvt. Ltd.

(vii)

About GATE The Graduate Aptitude Test in Engineering (GATE) conducted by IISc and IITs has emerged as one of the bench mark tests for engineering and science aptitude in facilitating admissions for higher education (M.Tech./Ph.D.) in IITs, IISc and various other Institutes/Universities/Laboratories in India. With the standard and high quality of the GATE examination in 27 disciplines of engineering and science, Humanities and Social Sciences subjects, it identifies the candidate's understanding of a subject and aptitude and eligibility for higher studies. During the last few years, GATE score is also being used as one of the criteria for recruitment in Government Organizations such as Cabinet Secretariat, and National/State Public Sector Undertakings in India. Because of the importance of the GATE examination, the number of candidates taking up GATE exams has increased tremendously. GATE exams are conducted by the IITs and IISc as a computer based test having multiple choice questions and numerical answer type questions. The questions are mostly fundamental, concept based and thought provoking. From 2017 onwards GATE Exam is being held in Bangladesh, Ethiopia, Nepal, Singapore, Sri Lanka and United Arab Emirates. An Institute with various nationalities in its campus widens the horizons of an academic environment. A foreign student brings with him/her a great diversity, culture and wisdom to share. Many GATE qualified candidates are paid scholarships/assistantship, especially funded by Ministry of Human Resources Development, Government of India and by other Ministries. IIT, Kharagpur is the Organizing Institute for GATE 2022.

Why GATE? Admission to Post Graduate and Doctoral Programmes Admission to postgraduate programmes with MHRD and some other government scholarships/ assistantships in engineering colleges/institutes is open to those who qualify through GATE. GATE qualified candidates with Bachelor’s degree in Engineering/Technology/Architecture or Master’s degree in any branch of Science/Mathematics/Statistics/Computer Applications are eligible for admission to Master/Doctoral programmes in Engineering/Technology/Architecture as well as for Doctoral programmes in relevant branches of Science with MHRD or other government scholarships/ assistantships. Candidates with Master’s degree in Engineering/Technology/Architecture may seek admission to relevant Ph.D programmes with scholarship/assistantship without appearing in the GATE examination.

Financial Assistance A valid GATE score is essential for obtaining financial assistance during Master’s programs and direct Doctoral programs in Engineering/Technology/Architecture, and Doctoral programs in relevant branches of Science in Institutes supported by the MHRD or other Government agencies. As per the directives of the MHRD, the following procedure is to be adopted for admission to the post-graduate programs (Master’s and Doctoral) with MHRD scholarship/assistantship. Depending upon the norms adopted by a specific institute or department of the Institute, a candidate may be admitted directly into a course based on his/her performance in GATE only or based on his/her performance in GATE and an admission test/interview conducted by the department to which he/she has applied and/or the

(ix)

candidate’s academic record. If the candidate is to be selected through test/interview for post-graduate programs, a minimum of 70% weightage will be given to the performance in GATE and the remaining 30% weightage will be given to the candidate’s performance in test/interview and/or academic record, as per MHRD guidelines. The admitting institutes could however prescribe a minimum passing percentage of marks in the test/interview. Some colleges/institutes specify GATE qualification as the mandatory requirement even for admission without MHRD scholarship/assistantship. To avail of the financial assistance (scholarship), the candidate must first secure admission to a program in these Institutes, by a procedure that could vary from institute to institute. Qualification in GATE is also a minimum requirement to apply for various fellowships awarded by many Government organizations. Candidates are advised to seek complete details of admission procedures and availability of MHRD scholarship/assistantship from the concerned admitting institution. The criteria for postgraduate admission with scholarship/assistantship could be different for different institutions. The management of the post-graduate scholarship/assistantship is also the responsibility of the admitting institution. Similarly, reservation of seats under different categories is as per the policies and norms prevailing at the admitting institution and Government of India rules. GATE offices will usually not entertain any enquiry about admission, reservation of seats and/or award of scholarship/ assistantship.

PSU Recruitments As many as 37 PSUs are using GATE score for recruitment. It is likely that more number of PSUs may start doing so by next year. Below is the list of PSUs: MDL, BPCL, GAIL, NLC LTD, CEL, Indian Oil, HPCL, NBPC, NECC, BHEL, WBSEDCL, NTPC, ONGC, Oil India, Power Grid, Cabinet Secretariat, Govt. of India, BAARC, NFL, IPR, PSPCL, PSTCL, DRDO, OPGC Ltd., THDC India Ltd., BBNL, RITES, IRCON, GHECL, NHAI, KRIBHCO, Mumbai Railway Vikas Corporation Ltd. (MRVC Ltd.), National Textile Corporation, Coal India Ltd., BNPM, AAI, NALCO, EdCIL India. Important :

1. Admissions in IITs/IISc or other Institutes for M.Tech./Ph.D. through GATE scores shall be advertised separately by the Institutes and GATE does not take the responsibility of admissions. 2. Cabinet Secretariat has decided to recruit officers for the post of Senior Field Officer (Tele) (From GATE papers of EC, CS, PH), Senior Research Officer (Crypto) (From GATE papers of EC, CS, MA), Senior Research Officer (S&T) (From GATE papers EC, CS, CY, PH, AE, BT) in the Telecommunication Cadre, Cryptographic Cadre and Science & Technology Unit respectively of Cabinet Secretariat. The details of the scheme of recruitment shall be published in National Newspaper/Employment News by the concerned authority. 3. Some PSUs in India have expressed their interest to utilize GATE scores for their recruitment purpose. The Organizations who intend to utilize GATE scores shall make separate advertisement for this purpose in Newspapers etc.

Who Can Appear for GATE? Eligibility for GATE Before starting the application process, the candidate must ensure that he/she meets the eligibility criteria of GATE given in Table. (x)

Eligibility Criteria for GATE 2022 Degree/Program

B.E. / B.Tech. / B.Pharm.

B. Arch.

B.Sc. (Research) / B.S.

Pharm. D. (after 10+2)

M.B.B.S.

M. Sc. / M.A. / MCA or equivalent

Int. M.E./ M.Tech. (Post-B.Sc.) Int. M.E./ M.Tech. or Dual Degree (after Diploma or 10+2)

B.Sc. / B.A. / B.Com.

Int. M.Sc. / Int. B.S. / M.S.

Professional Society Examinations (equivalent to B.E. / B.Tech. / B.Arch.)

Description of Eligible Candidates

Expected Year of Completion

Currently in the 3rd year or higher grade or already completed

2022

Bachelor’s degree of Architecture (5- year course) / Naval Architecture (4- year course) / Planning (4- year course)

Currently in the 3rd year or higher grade or already completed

2023 (for 5-year program), 2022 (for 4-year program)

Bachelor’s degree in Science (Post-Diploma/4 years after 10+2)

Currently in the 3rd year or higher grade or already completed

2022

Qualifying Degree/Examination

Bachelor’s degree in Engineering / Technology (4 years after 10+2 or 3 years after B.Sc. / Diploma in Engineering / Technology)

6 years degree program, consisting of internship or residency training, during third year onwards Degree holders of M.B.B.S. and those who are in the 5th/6th/7th semester or higher semester of such programme.

Currently in the 3rd/4th/5th/6th year or already completed

2024

5th, 6th, 7th or higher semester or already completed

2022

Currently in the first year or higher or already Completed

2022

Post-B.Sc Integrated Master’s degree programs in Engineering/ Technology (4-year program)

Currently in the 1st / 2nd/3rd/4th year or already completed

2024

Integrated Master’s degree program or Dual Degree program in Engineering/Technology (5-year program)

Currently in the 3rd /4th/5th year or alreadycompleted

2023

Master’s degree in any branch of Arts/Science/Mathematics/Statistics/ Computer Applications or equivalent

Bachelor degree in any branch of Science / Arts / Commerce (3 years program)

Integrated M.Sc. or 5-year integrated B.S.-M.S. program

B.E./B.Tech./B.Arch. equivalent examinations of Professional Societies, recognized by MoE/UPSC/AICTE (e.g. AMIE by Institution of EngineersIndia, AMICE by the Institute of Civil Engineers-India and so on)

Currently in the 3rd year or already completed Currently in the 3rd year or higher or already completed

Completed Section A or equivalent of such professional courses

2021

2022

NA

In case a candidate has passed one of the qualifying examinations as mentioned above in 2020 or earlier, the candidate has to submit the degree certificate / provisional certificate / course completion certificate / professional certificate / membership certificate issued by the society or institute. In case, the candidate is expected to complete one of the qualifying criteria in 2022 or later as mentioned above, he/she has to submit a certificate from Principal or a copy of marks card for section A of AMIE. (xi)

Certificate From Principal Candidates who have to submit a certificate from their college Principal have to obtain one from his/ her institution beforehand and upload the same during the online submission of the application form.

Candidates With Backlogs Candidates, who have appeared in the final semester/year exam in 2022, but with a backlog (arrears/ failed subjects) in any of the papers in their qualifying degree should upload a copy of any one of the mark sheets of the final year, OR obtain a declaration from their Principal along with the signature and seal beforehand and upload the same during the online submission of the application form.

GATE Structure Structure of GATE GATE 2022 will be conducted on 27 subjects (papers). Table below shows the list of papers and paper codes for GATE 2022. A candidate is allowed to appear in ANY ONE or UP TO TWO papers of the GATE examination. However, note that the combination of TWO papers in which a candidate can appear MUST be from the pre-defined list as given in Table. Also note that for a paper running in multiple sessions, a candidate will be mapped to appear for the examination in one of the sessions ONLY. List of GATE Papers and Corresponding Codes GATE Paper

Code

GATE Paper

Code

Aerospace Engineering

AE

Instrumentation Engineering

IN

Agricultural Engineering

AG

Mathematics

MA

Architecture and Planning

AR

Mechanical Engineering

ME

Bio-medical Engineering

BM

Mining Engineering

MN

Biotechnology

BT

Metallurgical Engineering

MT

Civil Engineering

CE

Petroleum Engineering

PE

Chemical Engineering

CH

Physics

PH

Computer Science and Information Technology

CS

Production and Industrial Engineering

PI

Chemistry

CY

Statistics

ST

Electronics and Communication Engineering

EC

Textile Engineering and Fibre Science

TF

Electrical Engineering

EE

Engineering Sciences

XE*

Environmental Science & Engineering

ES

Humanities & Social Sciences

XH**

Ecology and Evolution

EY

Life Sciences

XL***

Geology and Geophysics

GG

Note: Environmental Science and Engineering (ES) and Humanities and Social Sciences (XH) are two new papers introduced in GATE 2022. (xii)

*XE Paper Sections Engineering Mathematics (Compulsory) (15 marks)

Cod e A

**XH Paper Sections Reasoning and Comprehension (Compulsory)

Code

B1

***XL Paper Sections

Code

Chemistry (Compulsory)

P

(25 marks)

(25 marks)

Any TWO optional Sections (2x35 = 70 marks)

Any ONE optional Section (60 marks)

Any TWO optional Sections (2x30 = 60 marks)

Fluid Mechanics

B

Economics

C1

Biochemistry

Q

Materials Science

C

English

C2

Botany

R

Solid Mechanics

D

Linguistics

C3

Microbiology

S

E

Philosophy

C4

Zoology

T

Polymer Science and Engineering

F

Psychology

C5

Food Technology

U

Food Technology

G

Sociology

C6

Atmospheric and

H

Thermodynamics

Oceanic Sciences

*XE (Engineering Sciences), **XH (Humanities & Social Sciences), ***XL (Life Sciences), papers are of general nature and will be comprised of Sections listed in the above table Note: Each subject/paper is of total 100 marks. General Aptitude (GA) section of 15 marks is common for all papers. Hence remaining 85 marks are for the respective subject/paper code. Combination of Two Papers Allowed to Appear in GATE 2022 (subject to availability of infrastructure and schedule) Code of The First (Primary) Paper AE AG AR BM BT CE CH CS CY EC EE ES EY GG IN MA ME MN MT PE PH PI ST TF XE XH XL

Codes of Papers Allowed as The Second Paper XE ES CE BT / XL BM / XL AR / ES CY / PE / XE MA CH / XL IN / PH IN AG / CE XL MN / PE / PH EC / EE / PH CS / PH / ST XE GG / XE PH / XE CH / GG / XE EC / GG / IN / MA / MT / ST XE MA / PH XE AE / CH / ME / MN / MT / PE / PI / TF ---BM / BT / CY / EY

(xiii)

General Aptitude Questions All the papers will have a few questions that test the General Aptitude (Language and Analytical Skills), apart from the core subject of the paper.

Duration and Examination Type All the papers of the GATE 2022 examination will be for 3 hours duration and they consist of 65 questions for a total of 100 marks. Since the examination is an ONLINE computer based test (CBT), at the end of the stipulated time (3-hours), the computer screen will automatically close the examination inhibiting any further action. Candidates will be permitted to occupy their allotted seats 40 minutes before the scheduled start of the examination. Candidates can login and start reading the instructions 20 minutes before the start of examination. The late login time (if any) recorded by the computer system MUST NOT be beyond 30 minutes from the actual starting time of the examination. Under NO circumstances, will a candidate be permitted to login after 30 minutes from the actual examination starting time. Candidates will NOT be permitted to leave the examination hall before the end of the examination.

Pattern of Question Papers GATE 2022 may contain questions of THREE different types in all the papers: (i) Multiple Choice Questions (MCQ) carrying 1 or 2 marks each, in all the papers and sections. These questions are objective in nature, and each will have choice of four answers, out of which ONLY ONE choice is correct. Negative Marking for Wrong Answers: For a wrong answer chosen in a MCQ, there will be negative marking. For 1-mark MCQ, 1/3 mark will be deducted for a wrong answer. Likewise, for 2-mark MCQ, 2/3 mark will be deducted for a wrong answer. (ii) Multiple Select Questions (MSQ) carrying 1 or 2 marks each in all the papers and sections. These questions are objective in nature, and each will have choice of four answers, out of which ONE or MORE than ONE choice(s) is / are correct. Note: There is NO negative marking for a wrong answer in MSQ questions. However, there is NO partial credit for choosing partially correct combinations of choices or any single wrong choice. (iii) Numerical Answer Type (NAT) Questions carrying 1 or 2 marks each in most of the papers and sections. For these questions, the answer is a signed real number, which needs to be entered by the candidate using the virtual numeric keypad on the monitor (keyboard of the computer will be disabled). No choices will be shown for these types of questions. The answer can be a number such as 10 or –10 (an integer only). The answer may be in decimals as well, for example, 10.1 (one decimal) or 10.01 (two decimals) or –10.001 (three decimals). These questions will be mentioned with, up to which decimal places, the candidates need to present the answer. Also, for some NAT type problems an appropriate range will be considered while evaluating these questions so that the candidate is not unduly penalized due to the usual round-off errors. Candidates are advised to do the rounding off at the end of the calculation (not in between steps). Wherever required and possible, it is better to give NAT answer up to a maximum of three decimal places. Note: There is NO negative marking for a wrong answer in NAT questions. Also, there is NO partial credit in NAT questions.

Marking Scheme – Distribution of Marks and Questions General Aptitude (GA) Questions In all papers, GA questions carry a total of 15 marks. The GA section includes 5 questions carrying 1mark each (sub-total 5 marks) and 5 questions carrying 2-marks each (sub-total 10 marks). (xiv)

Question Papers other than GG, XE, XH and XL These papers would contain 25 questions carrying 1-mark each (sub-total 25 marks) and 30 questions carrying 2-marks each (sub-total 60 marks) consisting of some MCQ type questions, while the remaining may be MSQ and / or NAT questions.

GG (Geology and Geophysics) Paper Apart from the General Aptitude (GA) section, the GG question paper consists of two parts: Part A and Part B. Part A is compulsory for all the candidates. Part B contains two sections: Section 1 (Geology) and Section 2 (Geophysics). Candidates will have to attempt questions in Part A and questions in either Section 1 or Section 2 of Part B. The choice of Section 1 OR Section 2 of Part B has to be made at the time of online registration in GOAPS. At the examination hall, candidate cannot request for change of section. Part A consists of 25 questions carrying 1-mark each (sub-total 25 marks and some of these will be MSQ and/or numerical answer type questions while remaining questions will be MCQ type). Either section of Part B (Section 1 and Section 2) consists of 30 questions carrying 2-marks each (sub-total 60 marks and some of these will be MSQ and/or numerical answer type questions while remaining questions will be MCQ type).

XE Paper (Engineering Sciences) A candidate appearing in the XE paper has to answer the following: • GA – General Aptitude carrying a total of 15 marks. • Section A – Engineering Mathematics (Compulsory): This section contains 11 questions carrying a total of 15 marks: 7 questions carrying 1-mark each (sub-total 7 marks), and 4 questions carrying 2-marks each (sub-total 8 marks). Some questions will be of numerical answer type while remaining questions will be MCQ type. • Any two of XE Sections B to H: The choice of two sections from B to H can be made during the examination after viewing the questions. Only TWO optional sections can be answered at a time. A candidate wishing to change midway of the examination to another optional section must first choose to deselect one of the previously chosen optional sections (B to H). Each of the optional sections of the XE paper (Sections B through H) contains 22 questions carrying a total of 35 marks: 9 questions carrying 1-mark each (sub-total 9 marks) and 13 questions carrying 2-marks each (subtotal 26 marks). Some questions will be of MSQ and/or numerical answer type while remaining questions will be MCQ type.

XH Paper (Humanities and Social Sciences) A candidate appearing in the XH paper has to answer the following: • GA – General Aptitude carrying a total of 15 marks. • Section B1 – Reasoning and Comprehension (Compulsory): This section contains 15 questions carrying a total of 25 marks: 5 questions carrying 1-mark each (sub-total 5 marks) and 10 questions carrying 2-marks each (sub-total 20 marks). Some questions will be of MSQ and/or numerical answer type while remaining questions will be MCQ type. • Any ONE of XH Sections C1 to C6: The ONE choice of section from C1 to C6 has to be made at the time of online registration in GOAPS. At the examination hall, candidate cannot request for change of section. Each of the optional sections of the XH paper (Sections C1 through C6) contains 40 questions carrying a total of 60 marks: 20 questions carrying 1-mark each (sub-total 20 marks) and 20 questions carrying 2-marks each (sub-total 40 marks). Some questions will be of MSQ and/or numerical answer type while remaining questions will be MCQ type. (xv)

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