PROBLEMS AND SOLUTIONS IN ENGINEERING MATHEMATICS

PROBLEMS AND SOLUTIONSINENGINEERIN G MATHEMATICSMATHEMATICS

PROBLEMS AND SOLUTIONSINENGINEERING MATHEMATICSENGINEERINGMATHEMATICSForB.E. / B.Tech. 1st Year (I & II Semesters)(Volume-I)ByDr. T.C. GUPTAM.A., Ph. DDeputy Director-General (Retired),Indian Council of Medical ResearchAnsari Nagar, New DelhiFormerly Scientist,Defence Research and Development Organisation,New DelhiUNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)BANGALOREl CHENNAIl COCHINl GUWAHATIl HYDERABADJALANDHARl KOLKATAl LUCKNOWl MUMBAIl PATNARANCHI l NEW DELHI

Copyright 2012 by Laxmi Publications Pvt. Ltd. All rights reserved. Nopart of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical, photocopying,recording or otherwise without the prior written permission of the publisher.Published by :UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)113, Golden House, Daryaganj,New Delhi-110002Phone : 011-43 53 25 00Fax : 011-43 53 25 comPrice : 550.00 Only.First Edition : 2007 ; Second Edition : erabadJalandhar&&0484-237 70 04, 405 13 03 &0361-251 36 69, 251 38 81&040-24 65 23 33&080-26 75 69 30044-24 34 47 26KolkataLucknow033-22 27 43 840522-220 99 16Mumbai022-24 91 54 15, 24 92 78 69PatnaRanchi0612-230 00 970651-221 47 640181-222 12 72UPS-9659-550-PROB SOL ENGG MATH I&II-GUPTypeset at : Goswami Associates, Delhi.Printed at :

ite SeriesMatrices and Its ApplicationsApplications of DifferentiationPartial DifferentiationApplications of Single IntegrationMultiple IntegrationVector CalculusOrdinary Differential Equations and Its ApplicationsLaplace Transforms and Its ApplicationsPartial Differential EquationsApplications of Partial Differential Equations(v).170139274383407468505642707763

PREFACEI have no words to express my gratitude towards my worthy students on account ofwhose keen interest and continuous suggestions this book is appearing in its present form asthe new Revised Second Edition (Part-I) Keeping in view the changes done by Some Universitiesin the Syllabus of First and Second Semesters, I have revised it thoroughly to make aComprehensive Book by rearranging some topics/chapters, adding new and important problemsand all the questions asked/set in the previous university examinations.The response to the First Edition of this book (All the three Volumes for respectiveSemesters), has been overwhelming and very encouraging which amply indicates that thisbook has proved extremely useful and helpful to all the B.E./B.Tech. students of Engineeringcolleges and Institutes throughout the country. Obviously it has helped them to be betterequipped and more confident in solving the problems asked in several university examinations.All the problems have been solved systematically and logically so that even an averagestudent can become familiar with the techniques to solve the mathematical problemsindependently. Mathematics has always been a problematic subject for the students, hencethey have been depending and relying upon private tuitions and coaching academies. It ishoped that this book in its new form will provide utmost utility to its readers.—Author(vii)

SYMBOLSGreek AlphabetsABΓDEZHΘαβγδεζηθ AlphaBetaGammaDeltaEpsilonZetaEtaThetaThere uNuXiOmicronPiFor PhiChiPsiOmegaMetric Weights and MeasuresLENGTH10 millimetres 1 centimetreCAPACITY10 millilitres 1 centilitre10 centimetres10 decimetres 1 decimetre 1 metre10 centilitres10 decilitres 1 decilitre 1 litre10 metres10 decametres 1 decametre 1 hectometre10 litres10 decalitres 1 dekalitre 1 hectolitre10 hectometres 1 kilometre10 hectolitres 1 kilolitreVOLUME1000 cubic centimetres 1 centigramAREA100 square metres 1 are1000 cubic decimetres 1 cubic metre100 ares100 hectares 1 hectare 1 square KilometreWEIGHT10 milligrams 1 centigramABBREVIATIONSkilometrekmtonne10 centigrams10 decigrams 1 decigram 1 grammetrecentimetremcmquintalkilogram10 grams10 dekagrams 1 decagram 1 hectogrammillimetrekilolitremmkl10 hectograms100 kilograms 1 kilogram 1 quintallitremillilitre10 quintals 1 metric ton (tonne)lmlgramarehectarecentiaretqkggahaca

1Infinite SeriesIMPORTANT DEFINITIONS AND FORMULAE1. Convergent, Divergent and Oscillating Sequences:A Sequence {an} is said to be convergent or divergent if Lt an is finite or not finiten respectively.For example, consider the sequenceHerean 1 1 1,,, .2 22 2311, Lt a Lt n 0 which is finite.n n n n22 The sequence {an} is convergent.Consider the sequences {n2} or {– 2n}.Herean n2 or – 2nLt an n or – . Both these sequences are divergent.If a sequence {an} neither converges to a finite number nor diverges to or – , it is called anOscillatory sequence.Oscillatory sequences are of 2 types:(i) A bounded sequence which does not converge, is said to oscillate finitely.nFor example, consider the sequence {(– 1) }.nHere an (– 1) . It is a bounded sequence because there exist two real numbers k and K (k K)such that k an K n N.{an} {– 1, 1, – 1, 1, – 1, .} – 1 an 1Lt a2n Lt ( 1)2n 1Nown n Lt a2n 1 Lt ( 1)2n 1 – 1n n Thus Lt an does not exist The sequence does not converge. Hence this sequence osciln lates finitely.(ii) An unbounded sequence which does not diverge, is said to oscillate infinitely.Note: When we say Lt an l, it meansn Lt a2n Lt a2n 1 l.n n 1

2PROBLEMS AND SOLUTIONS IN ENGINEERING MATHEMATICS2. Infinite Series: If {un} is a sequence of real numbers, then the expression u1 u2 . un .[i.e., the sum of the terms of the sequence, which are infinite in number] is called an infiniteseries, usually denoted by n 1un or more briefly, by Σun.3. Partial Sums: If Σun is an infinite series where the terms may be ve or –ve, then Sn u1 u2 . un is called the nth partial sum of Σun. Thus, the nth partial sum of an infinite series is thesum of its first n terms. S1, S2, S3, . are the first, second, third, . partial sums of the series. Sincen N (set of natural numbers), {Sn} is a sequence called the sequence of partial sums of theinfinite series Σun. Therefore, to every infinite series Σun, there corresponds a sequence {Sn} of itspartial sums.4. Behaviour of an Infinite Series: An infinite series Σun converges, diverges or oscillates (finitelyor infinitely) according as the sequence {Sn} of its partial sums converges, diverges or oscillates(finitely or infinitely).5. Geometric Series: The geometric series 1 x x2 x3 . to (i) converges if – 1 x 1 i.e., x 1(ii) diverges if x 1(iii) oscillates finitely if x – 1(iv) oscillates infinitely if x – 16. Theorem: If a series Σun is convergent, thenLt un 0.n However, converse of the above theorem is not always true i.e., the nth term may tend to zero asn even if the series is not convergent.Thus Lt un 0 Σun may or may not be convergent.n Also Lt un 0n Σun is not convergent.7. A positive term series either converges or diverges to .8. There are six different comparison tests which can be used to examine the nature of infiniteseries. These are described in detail in question number 18 of this chapter.9. General procedure for testing a series for convergence is given under question 127, dependingupon the type of series whether it is alternating, positive term series or a power series.1. Give an example of a monotonic increasing sequence which is (i) convergent (ii) divergent.SOLVED PROBLEMSSol. (i) Consider the sequenceSince1 2 3n, , , .,, .2 3 4n 11 2 3 . , the sequence is monotonic increasing.2 3 4nn1, lim an lim lim 1, which is finite.1n n 1n n 1 n 1 n The sequence is convergent.(ii) Consider the sequence 1, 2, 3, ., n, .Since 1 2 3 . n ., the sequence is monotonic increasing,an

3INFINITE SERIESan n, lim an lim n n n The sequence diverges to .2. Give an example of a monotonic decreasing sequence which is (i) convergent (ii) divergent.1 11, , . , .2 3nSol. (i) Consider the sequence 1,Since 1 1 1 ., the sequence is monotonic decreasing.2 311, lim an lim 0n nn n The sequence converges to 0.(ii) Consider the sequence –1, –2, –3, ., –n, .an Since –1 –2 –3 ., the sequence is monotonic decreasing.an –n, lim an lim ( n ) n n The sequence diverges to – .3. Discuss the convergence of the sequence {an}, where(i) an n(ii) an 1 n2 1Sol. (i) Here, an an 1 – an 1 11 . n3 323(iii) an n 1.nnn2 1n 1 2(n 1) 1n 2n 1 n 2 n 1(n2 2n 2)(n2 1)(n 1)(n2 1) n(n2 2n 2)(n2 2n 2)(n2 1) 0 n an 1 an {an} is a decreasing sequencen 0 n {an} is bounded below by 0.n2 1ä {an} is decreasing and bounded below, it is convergent.an Also,lim ann limn nn2 1 limn 1n1 1n2 0 The sequence {an} converges to zero.(ii) Here,1 11 2 . n3 33 sum of (n 1) terms of a G.P. whose first term is 1 andan 1 common ratio is1 1 1 n 1 3 1 1 3 13 Sn a(1 r n )1 r

4PROBLEMS AND SOLUTIONS IN ENGINEERING MATHEMATICS an 1 1 Now, 3 1 1 n 1 2 3 1 111 . n n 13 32331an 1 – an 0 n3n 1{an} is an increasing sequence. an 1 an n3 1 331 n 1 n {an} is bounded above by .2 223 ä {an} is increasing and bounded above, it is convergent.Also,an lim ann limn The sequence {an} converges to(iii)an an 1 – an 3 1 331 n 1 (1 0) 2 23 23.2n 1nn 2 n 1 1 0 nn 1nn( n 1)an 1 an n{an} is a decreasing sequence.an Also,n 11 1 1 nnn {an} is bounded below by 1.ä {an} is decreasing and bounded below, it is convergent.lim ann 1 lim 1 1n n The sequence {an} converges to 1.4. What is an infinite series ? When does it converge, diverge or oscillates (finitely or infinitely) ?Sol. If {un} is a sequence of real numbers, then the expression u1 u2 u3 . un . (i.e., thesum of the terms of the sequence, which are infinite in number) is called an infinite series. Theinfinite series u1 u2 . un . is denoted by unn 1or more briefly, by Σun.To every infinite series Σun , there corresponds a sequence {Sn}, where Sn u1 u2 u3 . un iscalled the partial sum of its first n terms.The infinite series Σun converges, diverges or oscillates (finitely or infinitely) according as thesequence {Sn} of its partial sums converges, diverges or oscillates (finitely or infinitely)(i) Series unis convergent if lim Sn finite.(ii) Series unis divergent if lim Sn or – (iii) Series unoscillates finitely if {Sn} is bounded and neither converges nor diverges.(iv) Series unoscillates infinitely if {Sn} is unbounded and neither converges nor diverges.n n

5INFINITE SERIES5. Discuss whether the following series converges or otherwise,Sol. Here,1111 . . 1.2 2.3 3.4n(n 1)111 n(n 1) n n 1un Putting n 1, 2, 3, .n, we haveu1 1 u4 111 1 , . un n n 14 5Sn 1 Adding,11 11 1, u2 , u3 22 33 41n 1lim Sn 1 – 0 1n {Sn} converges to 1 Σ un converges to 1.6. Examine the convergence of the series n 1Sol. Let, un 1.n(n 2)1 11 11 1111 11 2 n n 1 n 1 n 2 n(n 2) 2n 2(n 2) 2 n n 2 Putting n 1, 2, 3, . n, we obtainAdding,u1 1 1 1 1 1 1 1 1 1 1 , u2 2 2 2 3 2 2 3 3 4 u3 12un 1 11 11 2 n n 1 n 1 n 2 Sn 1 1 11 1 2 n 1 2 n 2 1 1 1 1 , 3 4 4 5 .lim Sn 1 1 0 1 0 2 2 n to3, a finite quantity.4the given sequence Sn converges to3.43. Hence the given infinite series4 unn 1converges