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Egieerig Mthemtics 5 NAME OF THE SUBJECT : Mthemtics I SUBJECT CODE : MA65 MATERIAL NAME : Additiol Prolems MATERIAL CODE : HGAUM REGULATION : R UPDATED ON : M-Jue 5 (Sc the ove QR code for the

1 Egieerig Mthemtics 5 NAME OF THE SUBJECT : Mthemtics I SUBJECT CODE : MA65 MATERIAL NAME : Additiol Prolems MATERIAL CODE : HGAUM REGULATION : R UPDATED ON : M-Jue 5 (Sc the ove QR code for the direct dowlod of this mteril) Nme of the Studet: Brch: Uit I (Mtrices) Cle Hmilto Theorem ) Verif Cle Hmilto theorem d fid the iverse of the mtri ) Usig Cle Hmilto theorem, fid A of the mtri ) Usig Cle Hmilto theorem, fid the vlue of ) If 6 5 A 5A 8A A 9A A 6I if A fid A d A A usig Cle Hmilto theorem As: A Prepred CGes, MSc, MPhil, (Ph:986897) Pge

2 Egieerig Mthemtics 5 Eige Vlues d Eige Vectors of give mtri ) Fid the eigevlues d eigevectors of the mtri ) Fid the eigevlues d eigevectors of the mtri ) Fid the eigevlues d eigevectors of the mtri A 7 A 5 As:,,,, 6, A 5 As:,, 6,,, ) Fid the eigevlues d eigevectors of dja, give tht the mtri A As: 6,,,,, Digolistio of Mtri ) Digolise the mtri similrit trsformtio Also fid the fourth power of this mtri As: D, A ) Digolise the mtri usig orthogol trsformtio As: Prepred CGes, MSc, MPhil, (Ph:986897) Pge

3 Egieerig Mthemtics 5 ) Digolise the mtri orthogol trsformtio As: 8 6 ) Digolise the mtri A 6 7 orthogol trsformtio As: 5 Qudrtic form to Coicl form ) Reduce the qudrtic form 6 z z z to coicl form orthogol reductio ) Reduce the qudrtic form to the coicl form through orthogol trsformtio Also fid the rk, ide, sigture d ture of the qudrtic form ) Reduce the qudrtic form z z ito coicl form mes of orthogol trsformtio Fid its ture Uit II (Sequeces d Series) Compriso Test ) Test the covergece of the series 5 ) Emie the covergece of the series 56 ) Test the covergece of the series 56 ) Test the covergece of the series ) Emie the covergece of the series 5 Prepred CGes, MSc, MPhil, (Ph:986897) Pge

4 Egieerig Mthemtics 5 6) Test the covergece of the series 7) Emie the covergece of the series 8) Test the covergece of the series 9) Test the covergece of the series 5 ) Emie the covergece of the series ) Test the covergece of the series ) Emie the covergece of the series ) Test the covergece of the series ) Emie the covergece of the series si Itegrl Test ) Test the covergece of the Hrmoic series ) Discuss the covergece of the series ) Test the covergece of the series ) Test the covergece of the Hrmoic series 5) Emie the covergece of the series 6) Discuss the covergece of the series 7) Emie the covergece of the series log 8) Discuss the covergece of the series e Prepred CGes, MSc, MPhil, (Ph:986897) Pge

5 Egieerig Mthemtics 5 D Alemert s Rtio Test d Altertig Series ) Discuss the covergece of the series ) Discuss the covergece of the series ) Discuss the covergece of the series! ) Test the covergece of the series! 5) Discuss the covergece of the series! 6) Discuss the covergece of the series 7) Test the covergece of the series, 8) Emie the covergece of the series, 5 9) Discuss the covergece of the series, 6 ) Emie the covergece of the series, 5 ) Discuss the covergece of the series ( ) ) Discuss the covergece of the series! 5 ) Emie the covergece of the series ) Discuss the covergece of the series ) Discuss the covergece of the series 6) Test the covergece of the series log log log ( ) 7) Discuss the covergece of the series Prepred CGes, MSc, MPhil, (Ph:986897) Pge 5

6 Egieerig Mthemtics 5 Asolute d Coditiol Covergece ) Discuss the covergece of the series!! 6! ) Discuss the covergece of the series! 5! 7! ( ) ) Discuss the covergece of the series (log ) ( ) ) Emie the covergece of the series 5) Discuss the covergece of the series 6) Discuss the covergece of the series 5 7) Discuss the covergece of the series Uit III (Applictios of Differetil Clculus) Rdius of Curvture d Circle of curvture ) Fid the rdius of curvture t the origi for the curve ) Fid the equtio of the circle of curvture t cc, o c c c c As: ) Fid the equtio of circle of curvture of the curve t /, / ) Fid the rdius of curvture d cetre of curvture t poit (, ) o the curve As: c sec c clogsec c 5) Fid the cetre of curvture /o the curve cos t cos t, si t si t As: /, / Prepred CGes, MSc, MPhil, (Ph:986897) Pge 6

7 Egieerig Mthemtics 5 t t 6) Show tht the rdius of curvture of the curve e cos t, e si t t poit t is e t 7) If the cetre of curvture of ellipse t oe ed of the mior is lies t the other ed, prove tht the eccetricit of the ellipse is Evolute ) Fid the equtio of the evolute of the ellipse ) Fid the evolute of the curve / / / ) Fid the evolute of the rectgulr hperol Prepred CGes, MSc, MPhil, (Ph:986897) Pge 7 / / As: / As: c / / / As: c / / / ) Fid the equtio of the evolute of the curve t t t si t t cos t cos si, t 5) Show tht the evolute of the trctri cos t log t, si t is the cter cosh Evelope ) Fid the evelope of the fmil lies m m, where m is the prmeter ) Fid the evelope of the fmil of stright lies cos si As:, eig the / / prmeter As: /

8 Egieerig Mthemtics 5 ) Fid the evelope of cos si, where is the prmeter As: ) Fid the evelope of the sstem of lies, where l d m re coected l m l m the reltio ( l d m re the prmeters) As: 5) Fid the evelope of d c is costt 6) Fid the evelope of, where c, d re the prmeters As: / / / c, where c, d re the prmeters d c is costt As: c 7) Fid the evelope of, where c, d re the prmeters d c is costt 8) Fid the evelope of c is costt, where c As: Evolute s the evelope of ormls ) Fid the evolute of the prol As: c, d re the prmeters d, tretig it s the evelope of 7 ormls As: ) Fid the evolute of the prol, tretig it s the evelope of 7 ormls As: Prepred CGes, MSc, MPhil, (Ph:986897) Pge 8

9 Egieerig Mthemtics 5 ) Fid the evolute of the ellipse ) Fid the evolute of the curve s evelope of it s ormls c / / As: / s evelope of ormls As: c / / / Uit IV (Differetil Clculus of Severl Vriles) Totl derivtives ) u is fuctio of d, r cos, r si Show tht u u u u u r r r r u u u ) If u f (, z, z ), Show tht z z u u u ) If u,, Show tht z z z ) If z e fuctio of u & v d u & v re other two vriles &, such tht u m, v m Show tht z z z z m u v 5) Give tht the trsformtios u e cos, v e si d tht is the fuctio of u d v, d lso of d, Prove tht u v Tlor s epsio u v ) Epd e cos out, upto the third term usig Tlor s series ) Oti terms upto the third degree i the Tlor series epsio of e si the poit, roud ) Epd f (, ) e i Tlor series i power of d upto secod dgree Prepred CGes, MSc, MPhil, (Ph:986897) Pge 9

10 Egieerig Mthemtics 5 ) Epd the fuctio si i powers of d upto secod degree terms Mim d Miim ) Discuss the mim d miim of the fuctio ) Fid the mimum d miimum vlues of si si si( );, ) I ple trigle ABC, fid the mimum vlue of cos Acos Bcos C Prolems of Lgrgi Multipliers: ) The temperture u(,, z) t poit i spce is u z Fid the highest temperture o surfce of the sphere z 5) Fid the shortest d the logest distce from the poit,, to the sphere z, usig Lgrge s method of costried mim d miim Jcois ) If u, v while r cos, r si Prove tht ( uv, ) r ( r, ) ) If r cos, r si, verif tht z z ) If u, v, w, prove tht z (, ) ( r, ) ( r, ) (, ) ( u, v, w) (,, z) Uit V (Multiple Itegrls) Simple prolems o doule itegrl ) Evlute ) Evlute dd dd As: As: log Chge of order of itegrtio Prepred CGes, MSc, MPhil, (Ph:986897) Pge

11 Egieerig Mthemtics 5 ) Chge the order of itegrtio of ) B chge of order of itegrtio evlute dd d evlute it Note: Do the sme prolem puttig ) Chge the order of itegrtio d evlute ) Evlute chgig the order of itegrtio Chge ito polr coordites ) B chgig ito polr coordites, evlute ) Evlute chgig to polr coordites dd As: As: dd As: dd As: dd As: dd ) B chgig ito polr coordites, evlute ) B chgig ito polr coordites, evlute e t dt Are s doule itegrl e 6 As: log dd As: dd Hece prove tht As: ) Usig doule itegrtio fid the re eclosed the curves d As: ) Usig doule itegrl, fid the re ouded d As: 6 ) Fid the smller of the res ouded d As: Prepred CGes, MSc, MPhil, (Ph:986897) Pge

12 Egieerig Mthemtics 5 ) Evlute R dd where R is the regio eclosed, d As: 6 5) Evlute R dd where R is the domi ouded X is, ordite d the curve As: over the re etwee 6) Evlute ( ) dd Triple itegrl ) Evlute R dddz z d ---- All the Best---- Prepred CGes, MSc, MPhil, (Ph:986897) Pge

Eigen Values and Eigen Vectors of a given matrix

Eigen Values and Eigen Vectors of a given matrix Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme