PANIMALAR INSTITUTE OF TECHNOLOGY

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1 PIT/QB/MATHEMATICS/I/MA5/M PANIMALAR INSTITUTE OF TECHNOLOGY (A Christi Miorit Istittio JAISAKTHI EDUCATIONAL TRUST (A ISO 9: 8 Certified Istittio No.: 9, Bglore Trk Rod, Vrdhrjprm, Nrthpetti, CHENNAI. DEPARTMENT OF MATHEMATICS MA5-MATHEMATICS - I QUESTION BANK I YEAR B.E./ B.TECH. (SEMESTER I COMMON TO ALL BRANCHES

2 Qestio Bk Semester-I MA5-Mthemtics - I PART A UNIT I - MATRICES. Fid the chrcteristic eqtio of the mtri. Let The chrcteristic eqtio is. If d re the eige vles of mtri, wht re the eige vles of d. Give The eige vles of e d The eige vles of e d. Fid the sm d prodct of the eige vles of the mtri Let =++ = 5 4. Fid the eige vles of the mtri. Hece form the mtri whose eige vles re. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

3 Qestio Bk Semester-I MA5-Mthemtics - I Let The chrcteristic eqtio of is Hece the eige vles of re d. The mtri whose eige vles re d will be 5. The prodct of two eige vles of the mtri is. Fid the third eige vle. Let the three eige vles re Give. If d re the eige vles of, fid the eige vles of d. Let the eige vles be The eige vles of re The eige vles of re Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

4 Qestio Bk Semester-I MA5-Mthemtics - I 7. Two eige vles of re eql d the re doble the third. Fid the eige vle of. Let the eige vles re Give The eige vles of re 8. Fid the sm of the sqres of the eige vles of. The digol elemets re the eige vles of the give mtri. The eige vles re 9. Oe of the eige vles of is. Fid the other two eige vles. Let the eige vles be Also prodct of the eige vles Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

5 Qestio Bk Semester-I MA5-Mthemtics - I The eige vles re. Two eige vles of the mtri re eql to ech. Fid the eige vle of. Let the eige vles be Give The eige vles of re. The eige vles of re. If, wht will be the eige vles of. Sice is triglr, the eige vles of re. If the mtri is give b where, the sig Cle Hmilto theorem prove tht. The bove mtri is lower triglr mtri d hece the eige vles re Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

6 Qestio Bk Semester-I MA5-Mthemtics - I The chrcteristic eqtio is B Cle Hmilto theorem. If d re the eige vles of sqre mtri of order, epress i terms of d. Sice d re the eige vles of, the chrcteristic eqtio of is B Cle Hmilto Theorem, 4. Show tht is orthogol. Therefore is orthogol. 5. Stte Cle Hmilto Theorem. Ever sqre mtri stisfies its ow chrcteristic eqtio.. Verif Cle Hmilto theorem for. The chrcteristic eqtio of To Prove: is Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

7 Qestio Bk Semester-I MA5-Mthemtics - I 7. Write the qdrtic form correspodig to the mtri. The qdrtic form is 8. Determie the tre of the qdrtic form. The mtri of the qdrtic form is The eige vles of re The qdrtic form is semi- defiite 9. Fid the mtri of the qdrtic form. The mtri of the qdrtic form is. For give mtri of order, d two of its eige vles re 8 d. Fid the sm of the eige vles. Usig the propert the prodct of the eige vles of mtri is eql to Sm of the eige vles of the mtri is. If the sm of the two eige vles d trce of mtri re eql. Fid the vle of. B propert, sm of the eige vles = Trce of the mtri, Give tht Sm of two eige vles = Trce of the mtri. Third eige vle is B propert, Prodct of the eige vles is Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

8 Qestio Bk Semester-I MA5-Mthemtics - I. Use Cle Hmilto theorem, to fid whe. The Chrcteristic eqtio of is B Cle Hmilto theorem, To fid the vle of. Fid the costts d sch tht the mtri hs d s its eige vles. Solvig ( & ( 4. C be digolied? Wh? c be digolied. We c fid distict eige vectors for 5. If,, re the Eigevles of,the Fid the Vle of b. Prodct of eige vles = Determit of A ** = (4-+(-b Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 8

9 Qestio Bk Semester-I MA5-Mthemtics - I = 8-b Sm of the eige vles = Trce of A ++ = = = We get, ( = 8-b b = b = UNIT II SEQUENCES AND SERIES. Wht do o me b coverget series? Coverget series: If the seqece S is Coverget d lim S seists s mber, the the series is clled Coverget.Otherwise the series is clled diverget.. If the series Let S... & Sice the is coverget the prove tht lim S... The S S. lim S s Hece lim S s. is coverget, the seqece lim lim( S S o. S is Coverget. o. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 9

10 Qestio Bk Semester-I MA5-Mthemtics - I. Defie series of positive terms A series... for which >, N is clled series of Positive terms. 4. Epli the compriso test. Let d v be two give series of Positive mbers sch tht kv where k is fied positive mber. The (i If (ii If v is coverget the is diverget the v is coverget is diverget., N 5. Wht is the limit compriso test for series of positive mbers? Let d v be two give series of Positive mbers sch tht lim v, ( is o-ero fiite o the both d v coverge (or diverge.. Test the covergeces of the series The th term of the series is ( Let v. lim lim. lim (which is fiite d ot eql to v B limit compriso test both the series d v coverges (or diverges together Bt v is coverget (sice p>. is coverget Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

11 Qestio Bk Semester-I MA5-Mthemtics - I 7. Usig compriso test prove tht the series is diverget To fid I the me to :,,, e i AP, = I the deomi to :,,5, e i AP, = I the deomitor:,5,7, e i AP, = = = = Let v =, p=, q= V = = d fiite is of the form with p=. diverges the lso diverges 8. Test the covergeces of the series ( ( ( let v lim lim v ( ( ( 4 ( lim (which is fiite& ot eql to B compriso test both the 4 ( ( ( series d v coverges (or diverges together v (sice p=is diverget. is diverget. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

12 Qestio Bk Semester-I MA5-Mthemtics - I 9. Test the covergeces of the series sice p>. is coverget. Test the covergeces of the series Sice Let v. lim lim (which is fiite& ot eql to. v B limit compriso test both the series d v (sice p= is diverget. v is diverget. coverges (or diverges together. Write D Alembert s R tio test. If is series of Positive mbers sch tht lim the the series (i is coverget if <. (ii is diverget if >. (iii the test fils whe =. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

13 Qestio Bk Semester-I MA5-Mthemtics - I. Wht is Altertig series? A series i which the terms re ltertivel positive d egtive is clled Altertig series. A ltertig series is of the form Stte Leibit s rle. A ltertig series 4... coverges if (i Ech term is mericll less th its precedig term. (ii lim 4. Usig itegrl test determie the covergece of Hece is decresig. Now, d = [ ] = Hece we hve the give series be diverget 5. Emie the chrcter of the series...!! 4! & (! (! (! (!. ( ( (!,. lim (! Hece the give series is coverges. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

14 Qestio Bk Semester-I MA5-Mthemtics - I. Wht do o me b bsoltel coverget series?is coverget series bsoltel coverget? If the series of rbitrr terms be sch tht the series... is coverget, the the series is sid to bsoltel coverget. A bsoltel coverget is ecessril coverget bt coverse is ot tre. 7. Show tht... is coverget..... =... =, which is coverges (p>. is coverges bsoltel.hece is coverges. cos 8. Determie whether the series is coverget (or diverget.. cos The series hs both positive d egtive terms bt ot ltertig cos,, bt, which is coverges( for p>. cos B compriso test is coverget. 9. Wht is the Itegrl test? A positive te ms of se ies f( +f( +f(+..f(+.. Whe e f( dec e ses s ic e ses, coverges (or diverges ccordig s the itegrl f ( d is fiite (or ifiite Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

15 Qestio Bk Semester-I MA5-Mthemtics - I. Show tht the hrmoic series of order p, diverges for p. If p>, coverges. d = p m d lim = m p m lim m p = p p... coverges for p> d p p p for p> (which is fiite Hece series If p=, the Hece series diverges. d log UNIT III APPLICATIONS OF DIFFERENTIAL CALCULUS. Defie ivoltes d evoltes. The locs of the ceter of crvtre of the give crve is clled the evolte of the Crve. The give crve is clled the ivolte of its evolte. Stte two properties of evolte. i.the orml t poit of crve is tget to its evolte tochig t the correspodig ceter of crvtre. ii.the differece betwee the rdii of crvtres t two poits of crve is eql to the legth of the rc of the evolte betwee two correspodig poits. Defie the circle of crvtre t poit o the crve. Circle of crvtre is circle with ceter t Cetre of crvtre d rdis s rdis of crvtre 4. Fid the crvtre of. Give This is circle with rdis (Rdis of crvtre Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

16 Qestio Bk Semester-I MA5-Mthemtics - I [The rdis of the circle is ] Rdis of crvtre Hece crvtre of the crve is 5. Fid the crvtre of t the verte. The verte of is. Differetite prtill with respect to, Therefore we cosider * + [ ] Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

17 Qestio Bk Semester-I MA5-Mthemtics - I. Fid the rdis of crvtre of t the poit where it cts the is. * + 7. Write dow the forml for rdis of crvtre i terms of prmetric coordite sstem. 8. Defie the crvtre of ple crve d wht is the crvtre of stright Lie? The rte t which the ple crve chges its directio with respect to rcl distce (Rte of bedig of crve is clled the crvtre of crve. The crvtre of stright lie is ero 9. Wht is the crvtre of the crve o it? For the circle, with ceter t the origi d rdis 5 its, the rdis of crvtre is its rdis 5 its. Therefore the crvtre of the crve. For the crve fid the rdis of crvtre t. Give crve is Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

18 Qestio Bk Semester-I MA5-Mthemtics - I Differetite with respect to Differetite with respect to * +. Fid the rdis of the crvtre of the crve t. Give, Differetitig with respect to [ ]. Fid the ceter of crvtre of t the origi. Ceter of crvtre is ( where Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 8

19 Give: Qestio Bk Semester-I MA5-Mthemtics - I At Ceter of crvtre is. Defie evelope of fmil of crves. The locs of the ltimte poits of itersectio of cosective members of fmil of crves is clled the evelope of the fmil of crves 4. Fid the evelope of the fmil of stright lies beig the prmeter. Differetite prtill with respect to, Dividig b Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 9

20 Qestio Bk Semester-I MA5-Mthemtics - I 5. Fid the evelope of the crve where is the prmeter. The evelope of crve of the form Here is.. Fid the evelope of the fmil of stright lies, where is the prmeter. This is qdrtic eqtio i. The evelope is give b. Here 7. Fid the evelope of where the prmeters re relted b where is kow. Hece the stright lie becomes This is qdrtic eqtio i. Hece the evelope is. Here. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

21 Qestio Bk Semester-I MA5-Mthemtics - I 8. Fid the evelope of the fmil of lies, beig the prmeter. Give crve is This is qdrtic eqtio i. The eqtio of the evelope is 9. Fid the evelope of the fmil of stright lies where beig the prmeter. Divide b, This is qdrtic eqtio i The eqtio of the evelope is. Fid the Evelope of the lies cosec cot =, beig Prmter cosec cot = Differette with respect to -coses cot = -cot cosec cot coses - + cot = X = cosec Y= cot Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

22 Qestio Bk Semester-I MA5-Mthemtics - I UNIT IV DIFFERENTIAL CALCULUS OF SEVERAL VARIABLES. If, show tht. From. If prove tht.. Fid whe. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

23 Qestio Bk Semester-I MA5-Mthemtics - I 4. Fid if. Prtil differetite with respect to & 5. If, show tht. Prtil differetite with respect to, & Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

24 Qestio Bk Semester-I MA5-Mthemtics - I. If fid. Let 7. If, fid. Let Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

25 Qestio Bk Semester-I MA5-Mthemtics - I 8. If d the fid. ( 9. Defie Jcobi i two dimesios. If d re fctios of two idepedet vribles d, the the determit is clled the Jcobim of with respect to d d is deoted b or. Fid the Jcobi of. (Nov/Dec 4(M/Je 4 = r. If evlte. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

26 Qestio Bk Semester-I MA5-Mthemtics - I. If d, fid the Jcobi of d with respect to d. (J.. Fid the vle of if Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

27 Qestio Bk Semester-I MA5-Mthemtics - I 4. If the compte? 5. If fid the Jcobi of i terms of. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

28 Qestio Bk Semester-I MA5-Mthemtics - I. If fid. 7. If fctios of idepedet vribles of d if, the fid the vle of. 8. Fid the Tlor Series epsio of er the poit p to first degree terms. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 8

29 Qestio Bk Semester-I MA5-Mthemtics - I 9. Epd i powers of d sig T lor s theorem pto first degree terms. [ ] [ ]. Epd s Tlor series epsio i powers of d. [ ] [ ] Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 9

30 Qestio Bk Semester-I MA5-Mthemtics - I. St te Eler s Theorem for homogeeos fctio. If is homogeeos fctio of degree i d the. Stte sfficiet coditios for fctio of two vribles to hve etremm t poit. Let be the reqired fctio d be the sttior poit. If d the (i is mimm vle if. (ii is miimm vle if (iii is ot etrem if. (iv if, the test is icoclsive. Obti the sttior poits of. Give The sttior poits re d Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

31 Qestio Bk Semester-I MA5-Mthemtics - I 4. Fid the sttior poits of. Whe, whe The sttior poits re d 5. Fid the mimm d miimm vle of. Let The sttior poit is is miimm t Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

32 Qestio Bk Semester-I MA5-Mthemtics - I UNIT V MULTIPLE INTEGRALS. Evlte [ ]. Evlte. [ ] [ ] ( -. Evlte. [ ] Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

33 Qestio Bk Semester-I MA5-Mthemtics - I * + * + * + 4. Evlte over the positive qdrt of the circle. Let Whe Whe Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

34 Qestio Bk Semester-I MA5-Mthemtics - I * + 5. Evlte. * + [ ] [ ]. Evlte. * + *( + * + * + Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

35 Qestio Bk Semester-I MA5-Mthemtics - I 7. Evlte. 8. Evlte. * + [ ] [ ] Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

36 Qestio Bk Semester-I MA5-Mthemtics - I 9. Evlte the itegrl Pt [ ] [ ][ ] [ ][ ]. Evlte * + [ ] * + [ ] Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

37 Qestio Bk Semester-I MA5-Mthemtics - I * +. Evlte. [ ] [ ] [ ] [ ] [ ][ ] [ ][ ][ ]. Evlte over the regio boded b. [ ] [ ]. Write dow the doble itegrl, to fid the re betwee the circles. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

38 Qestio Bk Semester-I MA5-Mthemtics - I 4. Epress i polr co-ordites. 5. Trsform the itegrl to polr co-ordites.. Fid the re eclosed b the stdrd ellipse. [ ] * + * + Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 8

39 Qestio Bk Semester-I MA5-Mthemtics - I 7. Fid the re of circle of rdis b doble itegrtio i polr co-ordites. The eqtio of circle of rdis d cetre is. * + * + [ ] 8. Epress volme of solid s triple itegrl. 9.. Chge the order of itegrtio i. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 9

40 Qestio Bk Semester-I MA5-Mthemtics - I. Chge the order of itegrtio i.. Chge the order of itegrtio i.. Evlte where is the pth from to. * + 4. Chge the itegrtio ito sphericl polr coordites. To chge the Crtesi coordites ito sphericl polr coordites, we se the trsformtio Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

41 Qestio Bk Semester-I MA5-Mthemtics - I 5. Sketch the cotor of the itegrl. r=4si r=4si Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

42 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4 PART B UNIT I MATRICES. Fid the Eige vles d Eige vectors of the mtri. A Let To fid the chrcteristic eqtio D D D For the give mtri 45 ; ; D D D 45 The Eige vles re,, 5 The Eige vectors re give b which is For 5, Solvig b cross mltiplictio method X For, ( X A I

43 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge Solvig b cross mltiplictio method Which does ot eist? If & X The give mtri is ot smmetric; therefore the Eige vectors eed ot be orthogol. Therefore we choose the Eige vector which m depedet or idepedet of other Eige vectors. Let & X. Fid the Eige vles d Eige vectors of the mtri. Let A = which is smmetric mtri For the give mtri ; ; D D D The chrcteristic eqtio is give b The Eige vles re give b 8,, The Eige vectors re give b For 8, 5 Solvig b cross mltiplictio method ( X A I

44 Qestio Bk Semester-I MA5-Mthemtics - I X For, 4 4 Solvig b cross mltiplictio method which is impossible eige vector. 4 If &, X To choose the eige vector, remember we re delig with smmetric mtri, i which the eige vectors re orthogol i pirs. Let X b be the third eige vector which is orthogol to X& X c X X b c Similrl X X c Solvig the bove two eqtios b cross mltiplictio method, we get X 5. 4 Fid A if A sig Cle Hmilto Theorem. 4 A For the give mtri D, D 5, D 5 A A 5A I The chrcteristic eqtio is B Cle Hmilto theorem, Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 44

45 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 45 Mltiplig b 5 5, A I A A A I A A A A I A A 4. Verif tht A stisfies its ow chrcteristic eqtio d hece fid 4 A. For the give mtri 8,, D D D The chrcteristic eqtio is give b 8 B Cle Hmilto theorem, 8 I A A A From the give mtri & A A I A A A Mltiplig b A A A A A A A A A 8 8, A I A A A A I A A A

46 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4 5. The Eige vectors of rel smmetric mtri A correspodig to the Eige vles,, d re T T [,,], ],, [ d T ],, [ respectivel, fid A. The eige vles re,, The eige vectors re, & The give set of eige vectors re orthogol i pir. Therefore the give mtri is smmetric mtri. Therefore we se orthogol trsformtio to Digolise tht smmetric mtri. B orthogol trsformtio, T N AN D, where N is mtri whose colms re ormlied eige vector d D is digol mtri whose digol elemets re the eige vles. Therefore the ormlied modl mtri, N To fid tht smmetric mtri A, T D N N A A N N D N D T 5

47 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 47. Redce the qdrtic form 4 5 to coicl form throgh orthogol trsformtio d hece fid rk, ide, sigtre, tre d lso give o-ero set of vles,, ( {if the eist}, tht will mke the qdrtic form ero The mtri correspodig to the give qdrtic form is A = which is smmetric mtri For the give mtri 4,, 7 D D D The chrcteristic eqtio is 4 7 The eige vles re 4,, The eige vectors re give b i.e For, Solvig b cross mltiplictio method 4 Therefore 4 5 X is the first Eige vector. For, Solvig b cross mltiplictio method ( X A I

48 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 48 Therefore X is the secod Eige vector For 4, Solvig b cross mltiplictio method 44 Therefore X is the third Eige vector All the eige vectors occrs orthogol i pirs. Therefore sig orthogol trsformtio D AN N T we c digolise the give mtri. Hece the ormlied modl mtri is N= AN N T To fid the coicl form of the Qdrtic form, we se o- lier trsformtio NY X, X T AX = (NY T A(NY = Y T (N T ANY = Y T (DY

49 Qestio Bk Semester-I MA5-Mthemtics - I 4 Ide p =. Rk r = Sigtre s = p-r = Here ll the Eige vles re positive. Therefore it is positive semi-defiite. 4 Y T DY= is the reqired coicl form. The orthogol trsformtio X = NY will redce the give Q.F to the coicl form The coicl form of the Q.F. is ero, whe, d is rbitrr. Let 4,,, we get, 5, 4. These vles of,, mke Q.F. ero. 7. Redce the qdrtic form to coicl form throgh orthogol trsformtio. The mtri correspodig to the give qdrtic form is A = which is smmetric mtri For the give mtri D, D, D 4 4 The chrcteristic eqtio is The Eige vles re,, The Eige vectors re give b ( A I X i.e. For, Solvig b cross mltiplictio method Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 49

50 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5 Therefore X is the first Eige vector. For, Sbstittig this vle i, we get, solvig this eqtio,,, if X To choose the third Eige vector, remember we re delig with smmetric mtri, i which the Eige vectors re orthogol i pirs. Let c b X be the third Eige vector which is orthogol to X & X c b X X Similrl c b X X Solvig the bove two eqtios, the third Eige vector is X Sice the eige vectors re orthogol i pirs, sig orthogol trsformtio D AN N T, we c digolise the give mtri. Hece the ormlied modl mtri is ( X A I

51 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5 D AN N T To fid the coicl form of the Qdrtic form, we se o- lier trsformtio NY X, X T AX = (NY T A(NY = Y T (N T ANY = Y T (DY Y T DY= is the reqired coicl form. UNIT II SEQUENCE AND SERIES. Test the covergece of the series Soltio : Step : To fid the th term ( ( Step : q p v, p = (Degree of i deomitor, q =(Degree of i mertor

52 Qestio Bk Semester-I MA5-Mthemtics - I v Step : To fid = lim = lim = lim ( v - ( ( ( ( - lim ( ( ( - = (fiite Step 4 : B Comprisio test, & re coverget d diverget together = is of the form Where p = (ie P is coverget Hece is lso coverget Fid the tre of the series... e e e Soltio : 4 9 Give =... e e e e Step : To fid f e - e ' Step : verifig f f ( = e - ' f e - e - = - e - ( Step :To fid f( d Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

53 Qestio Bk Semester-I MA5-Mthemtics - I f( d = - e d, dv e d ', v =- e '', v = e, v = -e ' ''' sig berollis rle dv v v v... e - e - - e - = ( e - e - e - = 5 e - (fiite Step 4 : f( d = fiite, is coverget. =...5. Test the covergece of the series Soltio :..5...( Step : (..9...( Step : lim lim ( lim ( lim Step : B D Alembe t s R tio test is coverget. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

54 Qestio Bk Semester-I MA5-Mthemtics - I 4. Test for covergece of the series... ( 5 Omittig the first term, Cosider the series Step : To fid, ( Step : lim 5 lim (. (i Step : lim. ( ( lim B D Alembe t s test If <, the is coverget (ii If >, the is diverget (iii If =, the thetest fils Bt =, is of the form where p => p is coverget. ( ( ( 4 5. Test for covergece of the series To fid, & Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 54

55 Qestio Bk Semester-I MA5-Mthemtics - I Step : Step : ( 4... ( (( ( (, - - ( ( - ( ( ( ( ( ( - ( Pt = Give > - 4 = 7 Step : To fid lim lim ( lim ( lim F om &, Leibit s le s tisfied. Hece the give series is coverget.. Discss the covergece of the series... log log log 4 Step : STEP : log(, log( log( log( Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 55

56 Qestio Bk Semester-I MA5-Mthemtics - I Step : To fid = log log( log( log = log log( lim lim log( = F om &, Leibit s le s tisfied. Hece the give series is oscilltor. UNIT-III DIFFERENTIAL CALCULUS. Fid the rdis of crvtre of the crve t ( / 4, / 4. Give: ( Diff. ( w.r.t, Diff. ( w.r.t, d d d d d d /4, / ( d d d d = Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

57 Qestio Bk Semester-I MA5-Mthemtics - I d d /4 /4, /4 The rdis of crvtre t ( / 4, / 4 is ( /4, /4 = d d d d =. 4/. Fid for the crve ccosh t the poit (,c. c Soltio Give: ccosh c ( Diff.( w.r.t we get, csih. sih c c c ( (, c Diff.( w.r.t we get, cosh. c c (, c c The rdis of crvtre t (, c is 4 = c. c Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 57

58 Qestio Bk Semester-I MA5-Mthemtics - I. Prove tht the rdis of crvtre t poit of the ccloid ( si, ( cos is 4 cos (or Fid the rdis of crvtre t poit of the ccloid ( si, ( cos Give : ( si ( cos ' ( cos ' si '' si '' cos The rdis of crvtre i prmetric form is ' ' ' '' ' '' ( cos si = ( cos cos si ( si cos cos si = [cos cos si ] ( cos = [ cos ] cos = [cos ] cos = cos = 4 cos. 4. Fid the ceter d circle of crvtre of the crve t ( / 4, / 4. Give: ( Diff. ( w.r.t, d d Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 58

59 Qestio Bk Semester-I MA5-Mthemtics - I d d d d /4, / ( d d d Diff. ( w.r.t, d = d 4 = d /4 /4, /4 The rdis of crvtre t ( / 4, / 4 is ( /4, /4 d d d d = 4/. The cetre of crvtre is X, Y where X = ( ; Y = ( = = = = = = 4 4 Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 59

60 Qestio Bk Semester-I MA5-Mthemtics - I The cetre of crvtre is, 4 4 X X X X d the eqtio of circle of crvtre i.e X Y Fid the eqtio of circle of crvtre of the prbol t the poit (,. Give : ( Diff. ( w.r.t, Diff. At (, ; The rdis of crvtre t (, is Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge = = The cetre of crvtre is X, Y where X = ( ; Y = ( = = =5 = The eqtio of the circle of crvtre is At, X X X X ( X 5 Y. Fid the eqtio of the evolte of the prbol 4.(or Show tht the evolte of the prbol The prmetric eqtios of the prbol 4 is the crve 4 re 7 4( t, t. is

61 Qestio Bk Semester-I MA5-Mthemtics - I First we hve to fid the cetre of crvtre t t d d t dt dt d ; d t t d d d d d dt d d d dt d d t t = t The cetre of crvtre is X, Y where X = ( Y = ( = t t = t t t t t t = t t = t t t t = t t = t ( = t ( Elimite 't' betwee ( d ( we get X = t Y = t t X X t = X X Y = 4 The eqtio of the give crve is ( The prmetric form of the give crve is cos ; si Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7 7 Y 4 X. Sice, the evolte is the locs of ceter of crvtre 7 Y = = 4 X is the evolte of the prbol Fid the evolte of the crve.

62 Qestio Bk Semester-I MA5-Mthemtics - I d d ' cos si ; ' cos si d d d d Now d t d d d d d d d d d d d d d d d 4 sec cos ec = sec cos si The cetre of crvtre is X, Y where X = ( ( t = cos t 4 sec cos ec t = cos sec cos ec = cos si cos ( Y = ( = si t 4 sec cos ec = si sec cos ec = si cos si ( Elimite ' ' betwee ( d ( we get X +Y = cos si cos + si cos si = (si cos si + cos si cos ( 4 (5 = (cos si (4 X -Y = cos si cos + si cos si = (cos si cos - si cos si = (cos si (5 X Y X Y cos si cos si Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

63 Qestio Bk Semester-I MA5-Mthemtics - I cos si. Sice the evolte is the locs of the cetre of crvtre, The locs of X, Y is obtied b replcig X b X d Y b Y. X Y X Y is the evolte of the give crve. 8. Fid the evolte of the crve (cos si ; (si cos Give: (cos si ; (si cos d d cos ; si d d d d si d t d d cos d d d d ( (t d d d sec = sec cos. The cetre of crvtre is X, Y where X = ( t = (cos si ( t cos = cos si si cos ( sec = cos X + Y = ( = (si cos ( t sec = si cos si sec Y = cos si Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

64 Qestio Bk Semester-I MA5-Mthemtics - I Sice the evolte is the locs of the cetre of crvtre, The locs of X, Y is obtied b replcig X b X d Y b Y X + Y = is the evolte of the give crve. 9. Fid the evolte of the rectglr hperbol c. The prmetric eqtios of the prbol c re ct, c. t d d c c dt dt t c d t ; d c t d d d d d dt d d d dt d d = t c ct The cetre of crvtre is X, Y where X = ( Y = ( = 4 t ct t = c ct 4 t t ct 4 ct t = c ct ct = ( t t ct c = ( t Elimite 't' betwee ( d ( we get c c X +Y = t + t t t c = t t c c X -Y = t + t t t = c t t c = t t t t Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 4

65 Qestio Bk Semester-I MA5-Mthemtics - I X Y X Y c t t t t c = (4 c = (4 = c Sice the evolte is the locs of the cetre of crvtre, The locs of X, Y is obtied b replcig X b X d Y b Y. X Y X Y c is the evolte of the rectglr hperbol c. UNIT IV DIFFERENTIAL CALCULUS OF SEVERAL VARIABLES. A rectglr bo ope t the top is to hve volme cc. Fid the dimesios of the bo reqirig lest mteril for its costrctio. Give volme Let (,, Srfce Are S f (,, The f ( ( Differetite w.r.to : ( ( Differetite w.r.to : ( ( Differetite w.r.to : ( ( Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 5

66 Qestio Bk Semester-I MA5-Mthemtics - I ( ( ( (,, ( (, Pt, i d Dimesios of the bo is legth 4, bredth 4, height. The tempertre U (,, t poit i spce isu 4 tempertre o the srfce of the sphere. (M 5 Let (,, ( f (,, 4 (. Fid the highest f f f ( ( ( ( (4 (5 ( (4 (5 ( 4 4 ( 8 Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge

67 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge (4 We hve T The highest tempertre is c 5. Fid the volme of the lrgest rectglr solid which c be iscribed i the ellipsoid c b. Give the ellipsoid is,, ( c b ( The volme of the prllelepiped is f 8,, ( ( At the mimm poit or miimm poit (5 8 (4 8 ( 8 c f b f f B solvig these eqtios 4 (5 (4 ( c b

68 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 8 the 8 8 ( Similrl, c b Sbstitte i (, we get Mimm Volme = 8 bc 4. Fid the mimm vle of p m whe.,, ( ( ( ( p m p m p m p f f m f From ( (, ( d we get p m p m p m e i p m p m p m p m.,.. Hece the mimm vle of f occrs whe p m p p m p m m,,

69 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 9 The mimm vle of * + * + * + 5. If,, show tht the Jcobi of,, with respect to,, is 4. (M 7 4,, (,, ( J. If log( show tht ( 9. Give log( ( ( ( Addig

70 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7 ( ( ( ( ( Differetitig both the sides prtill w.r.to,, ( ( ( Addig ( 9 ( 9 Hece proved. 7. Emie f, ( for mim d miim. Give f, ( f t f s f r f q f p f f Solvig these eqtios

71 Qestio Bk Semester-I MA5-Mthemtics - I if the ( 4 (,,, if the The poits (, d (, m be mimm poit or miimm poit. At (, rt s ( 9, r (, is either mimm or miimm. This poit is sddle poit. At (, rt s 9 7, r The the fctio is miimm t (,. positive 8. If f (, d v re two vribles sch tht l m, v l m the prove tht ( l m. v Let be fctio of d v d v re fctios of d l m v l m v l m v m l Now v v. l.( m v Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

72 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7 ( ( v m v m l l m v m l v m l v l m m v l v l m v l v m v l v m v l v v Similrl ( v l v m l m Addig ( d ( ( v m l UNIT-V MULTIPLE INTEGRALS. Chge the order of itegrtio i dd d the evlte it. Let I = dd The limit defied i the itegrtio I is,, &

73 Qestio Bk Semester-I MA5-Mthemtics - I tht is ( which is circle with ceter t the (, ( I dd d d ( ( d d 4 4 d ( d. Chge the order of itegrtio d hece evlte dd. Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

74 Qestio Bk Semester-I MA5-Mthemtics - I Give order of itegrtio is first with respect to d the with respect to Sice is with vrible limit. Chgig the order of itegrtio, we do itegrtio first with respect to, i.e. b selectig the vrible limit for Sice the give regio of itegrtio c be split ito two, R d R. I R, vries betwee to, vries betwee to I R, vries betwee to, vries betwee to - - dd d d+ d d 4 d d ( d d 4 4 d B chgig to polr co-ordites show tht e t dt. To prove tht e ( Usig the trsformtio, r cos, r si dd 4 e ( dd. Hece evlte 4 d d r dr d we trsform the Crtesi coordites ito polr coordites, the the give itegrl is Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 74

75 Qestio Bk Semester-I MA5-Mthemtics - I r e r dr d To evlte e t dt Let t t dt d d dt B e t dt defiitio e e of d d r e e t r dr dt 4 Whe Gmm it egrl, e if t t d e t t r t Limits : t : to dt dt r dr 4. Fid the re commo to 4, 4 sig doble itegrtio. The poit of itersectio of 4, 4 is ( 4, 4 Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 75

76 Qestio Bk Semester-I MA5-Mthemtics - I Are 4 4 d d 4 4 d Fid the re iside the circle r si bt lig otside the crdioid r ( cos. r ( cos r si To fid the limits for r d, si ( cos si cos cos si si cos si si si or 4 4 or or si 4 si 4 Are si r (cos dr d * + Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 7

77 Qestio Bk Semester-I MA5-Mthemtics - I * [ ] [ ] + * + * +. Fid the volme of the ellipsoid. b c Volme of the whole ellipsoid = 8 Volme of theellipsoid i the First octt Volme of the ellipsoid i the First octt d Limits: : to : : to to c Volme b b b b c c b b d d d d d d d Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 77

78 Qestio Bk Semester-I MA5-Mthemtics - I Pimlr Istitte of Techolog Deprtmet of Mthemtics Pge 78 si bc d b b c d b b c d b b b b c d d b b c b b Volme of the whole ellipsoid = octt theellipsoid i the First of Volme 8 8 bc