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 of the Student: Brnch: Unit I (Mtrices) Cle Hmilton Theorem ) Verif tht the mtri A stisfies its chrcteristic eqution nd hence find A ) Verif Cle Hmilton theorem nd find the inverse of the mtri A 0 0 0 ) Using Cle Hmilton theorem, find ) If 5) Find A find A nd A of the mtri 0 0 A using Cle Hmilton theorem n A using Cle Hmilton theorem, tking A 6) Use Cle Hmilton theorem to find the vlue of A 5A 8A A I where Hence find A A 5A 7A A 8 7 6 5 8 5 5 A 0 0 0 0 5 5 8 Eigen Vlues nd Eigen Vectors of given mtri ) Find the Eigenvlues nd Eigenvectors of the mtri Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 ) Find the eigenvlues nd eigenvectors of A 0 0 ) Find the eigenvlues nd eigenvectors of A 6 0 ) Find ll the eigenvlues nd eigenvectors of the mtri 7 5) Find the eigenvlues nd eigenvectors of the mtri A 7 5 0 6 Digonlistion of Mtri ) Digonlise the mtri orthogonl trnsformtion ) Digonlise the mtri Qudrtic form to Cnonicl form 0 0 0 0 0 0 8 6 A 6 7 orthogonl trnsformtion 0 0 0 0 0 0 0 5 ) Reduce the qudrtic form 6 z z z to cnonicl form orthogonl reduction ) Reduce the qudrtic form to the cnonicl form through n orthogonl trnsformtion Also find the rnk, inde, signture nd nture of the qudrtic form ) Reduce the qudrtic form z z into cnonicl form mens of orthogonl trnsformtion Find its nture Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 ) Reduce the qudrtic form 5 z to cnonicl form orthogonl reduction nd stte its nture 5) The eigenvectors of X rel smmetric mtri A corresponding to the eigen T T T vlues,, 6 re, 0,,,,,,, mtri A respectivel, find the Unit II (Three Dimensionl Anlticl Geometr) Sphere 5 ) Find the eqution of the sphere which psses through the points0,0,0, 0,,,,,0 nd,, ) Find the eqution of the sphere pssing through the points,, 0,, nd hving its centre on the plne z 6, 5,,, ) Find the centre nd rdius of the circle given z z 9 0 nd z 7 0 ) Find the eqution of the sphere through the circle z 6 0, z 9 0 nd the centre of the sphere z z 6 5 0 5) Find the equtions of the spheres which psses through the circle nd z nd touch the plne 5 z 5 6) Find the eqution of the sphere hving the circle z 0 z 8 0, z s gret circle 7) Find the eqution of the sphere tht psses through the circle z z 0, 5 z 7 0 nd cuts orthogonll the sphere z 5 7z 6 0 Eqution of tngent plne to sphere: 8) Show tht the plne z 0 touches the sphere z z lso find the point of contct 6 7 z 9) Show tht the line touches the sphere 5 z 0 nd find the coordintes of the point of contct 0) Find the equtions of the tngent plnes to the sphere z 6z 5 0 which re prllel to the plne 8z 0 Find lso their points of contct Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 ) Find the equtions of the two tngent plnes to the sphere z 6z 0 which re prllel to the coordinte plne 0 Cone ) Find the eqution of the cone whose verte is,, nd se curve, z ) Find the eqution of the cone with verte t,, nd the guiding curve is the circle z, z ) Find the eqution of the right circulr cone whose verte is the origin, whose is is z the line nd which hs semi verticl ngle of 0 ) Find the eqution of the right circulr cone whose verte is,,, semi verticl z ngle 0 nd the is the line 7 7 z 6 z 8z 8 88 6z 0 5) Find the semi verticl ngle nd the eqution of the right circulr cone hving its verte t the origin nd pssing through the circle z 5, 5 6 6z 0 6) Find the eqution of the right circulr cone generted the stright lines drwn,,,,, nd from the origin to cut the circle through the three points,, 8 z 5 z 5z 0 Clinder ) Find the eqution of the right circulr clinder of rdius nd hving s is of the z line 5 8 5z z 8z 6 z 0 0 0 ) Find the eqution of the right circulr clinder whose is is z nd which psses through the point0,0, 0 5z 6z z 6 8 0z 5 0 ) Find the right circulr clinder which hs the circle z z 0, z 0 s the guiding curve 5 8 5z z 8z 8 0z 56 0 Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 Unit III (Differentil Clculus) Rdius of Curvture nd Circle of curvture ) Find the circle of the curvture of the curve t/, / ) Find the rdius of curvture t the point /, / on the curve 6 ) Find the rdius of curvture of the curve ) Find the rdius of curvture t the origin for the curve t(,0) 0 5) For the curve if is the rdius of curvture n point (, ), show tht / Hint:, 6) Show tht the rdius of curvture t the point ( cos, sin ) on the curve / / / is sincos 7) Prove tht the rdius of curvture t n point of the ccloid ( sin ), ( cos ) is cos 8) Find the rdius of curvture of the curve (cost t sin t), (sint tcos t) t n point t 9) Find the circle of curvture t (,) on 57 5 6 8 Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 5
Engineering Mthemtics 0 0) Fin the eqution of circle of curvture t (,6) on 5 6 88 ) Find the eqution of the circle of curvture t ( cc, ) on c c c c ) Find the rdius of curvture nd centre of curvture t n point (, ) on the curve logsec c csec c c ) Find the rdius of curvture of the curve ccosh t the point(0, ) c c c Evolute ) Find the eqution of evolute of the prol 7 ) Find the eqution of evolute of the prol 7 ) Find the eqution of the evolute of the ellipse ) Find the eqution of the evolute of the hperol / / / / / / 5) Show tht the evolute of the ccloid ( sin ), ( cos ) is nother ccloid 6) Find the evolute of the curve / / / 7) Find the evolute of the rectngulr hperol / / / c c / / / Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 6
Engineering Mthemtics 0 t 8) Show tht the evolute of the trctri cos t log tn, sin t is the ctenr cosh Envelope ) Find the envelope of the fmil lines m m, where m is the prmeter ) Find the envelope of the fmil of stright lines cos sin, eing / / the prmeter / ) Find the envelope of cos sin, where is the prmeter ) Find the envelope of the fmil of stright lines, where the prmeters nd re relted the eqution c, c eing constnt 5) Find the envelope of the fmil of stright lines c, where the prmeters nd re relted the eqution c, c eing constnt 6) Find the envelope of the fmil of stright lines, where the / / / c prmeters nd re relted the eqution n n c n, c eing n n n n n n constnt c Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 7
Engineering Mthemtics 0, where 7) Find the envelope of the fmil of stright lines re, prmeters 8) Find the envelope of prmeters nd c is constnt 9) Find the envelope of c, where c, nd re the c, / / / c, where c, nd re the prmeters nd c is constnt c 0) Find the envelope of, where n n c n, nd re the prmeters nd c is constnt n n n n n n c ) Find the envelope of, where c nd c is constnt Evolute s the envelope of normls, nd re the prmeters ) Find the evolute of the prol, treting it s the envelope of 7 normls ) Find the evolute of the prol, treting it s the envelope of 7 normls ) Find the evolute of the ellipse s envelope of it s normls / / / Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 8
Engineering Mthemtics 0 ) Find the evolute of the hperol s envelope of it s normls / / / 5) Find the evolute of the curve c s envelope of normls c / / / Unit IV (Functions of severl vriles) Euler s Theorem ) If u is homogeneous function of degree n in nd Show tht u u u nn u ) If u cos, Prove tht u u cot u 0 ) If u sin, prove tht () u u () tn u nd u u u tn u ) If u sin, Prove tht sin cos u u u u u cos u Totl derivtives prove tht u u 0 ) If u log tn / ) If z f (, ) where r cos, r sin, Show tht z z z z r r Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 9
Engineering Mthemtics 0 ) u is function of nd, r cos, r sin Show tht u u u u u r r r r u u u ) If u f (, z, z ), Show tht 0 z z u u u 5) If u,, Show tht z 0 z z 6) If z e function of u & v nd u & v re other two vriles &, such tht u m, v m Show tht z z z z m u v 7) Given tht the trnsformtions u e cos, v e sin nd tht is the function of u nd v nd lso of nd, Prove tht 8) If z f (, ), where u v nd uv Prove tht z z u v z z u v Tlor s epnsion u v u v ) Epnd Tlor s series the function f (, ) in powers of nd upto the third powers ) Epnd e cos out 0, upto the third term using Tlor s series ) Otin terms upto the third degree in the Tlor series epnsion of e sin round the point, ) Find the Tlor series epnsion of e sin t the point, upto rd degree terms 5) Epnd e log in powers of nd upto the terms of third degree 6) Epnd f (, ) e in Tlor series in power of nd upto second dgree 7) Epnd the function sin in powers of nd upto second degree terms Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge 0
Engineering Mthemtics 0 Mim nd Minim ) Find the etreme vlues of the function f (, ) 0 ) Emine the function f, ) Find the mim nd minim of for etreme vlues ) Discuss the mim nd minim of the function 5) In plne tringle ABC, find the mimum vlue ofcos Acos BcosC Prolems of Lgrngin Multipliers: 6) A rectngulr o open t the top, is to hve volume of cc Find the dimension of the o, tht requires the lest mteril for its construction 7) Find the dimension of the rectngulr o without top of mimum cpcit with surfce re squre meter 8) Find the Mimum vlue of the lrgest rectngulr prllelepiped tht cn e z inscried in n ellipsoid c 9) The temperture u(,, z) t n point in spce is u 00z Find the highest temperture on surfce of the sphere z 0) Find the mimum nd minimum vlues of z suject to the condition z ) Find the mimum vlue of Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge m n p z, when z ) Find the shortest nd the longest distnce from the point,, to the sphere z, using Lgrnge s method of constrined mim nd minim Jcoins ) If u, v while r cos, r sin Prove tht ( uv, ) r ( r, ) ) If z u, z uv, z uvw prove tht ) Find the Jcoin (,, z) ( r,, ) r sin sin, z r cos (,, z) ( u, v, w) uv of the trnsformtion r sin cos, ) Find the Jcoin of,, with respect to,, if,,
Engineering Mthemtics 0 Unit V (Multiple Integrls) Simple prolems on doule integrl ) Evlute ) Evlute 0 0 0 0 dd dd Chnge of order of integrtion ) Chnge the order of integrtion of ) Chnge the order of integrtion in ) Chnge the order of integrtion in 0 0 0 0 log dd nd evlute it dd nd evlute it e dd nd evlute it ) Chnge the order of integrtion in 0 dd nd evlute it 6 5) B chnge of order of integrtion evlute Note: Do the sme prolem putting 6) Chnge the order of integrtion nd evlute Note: Do the sme prolem putting 7) Chnge the order of integrtion nd evlute 0 0 0 dd dd 6 dd 8 8) Evlute chnging the order of integrtion Chnge into polr coordintes 0 dd Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 ) B chnging into polr coordintes, evlute ) Evlute chnging to polr coordintes ) Evlute chnging to polr coordintes 0 0 0 0 dd dd ) B chnging into polr coordintes, evlute 5) B chnging into polr coordintes, evlute e t dt 0 Are s doule integrl 0 0 e 0 0 / log dd dd dd Hence prove tht ) Using doule integrtion find the re enclosed the curves nd ) Using doule integrl, find the re ounded nd 6 ) Find the smller of the res ounded nd ) Evlute R dd where R is the region enclosed 0, 0 nd 6 5) Evlute R dd where R is the domin ounded X is, ordinte nd the curve over the re etween 6) Evlute ( ) dd R nd Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge
Engineering Mthemtics 0 Triple integrl ) Evlute 0 0 0 dddz z ) Find the volume of the portion of the ellipsoid first octnt using triple integrl ) Find the volume of the sphere z which lies in the c z using triple integrls z ) Find the volume of the tetrhedron ounded the plne nd the c coordinte plnes log 5) Evlute z e dddz 0 0 0 ---- All the Best---- Prepred CGnesn, MSc, MPhil, (Ph:986897) Pge