SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)

Transcription

1 QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering Mhemic-I (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem: I-B.Tech & I-Sem UNIT I Regulion: R6. ) Solve b) Solve. ) Solve. b) Solve. ) Solve. b) Solve. ) Solve. b) Show h he fmily of Confocl conic, (where λ i Prmeer), i Self orhogonl. 5. ) Solve ( b) Solve. 6. ) A circui h in erie on elecromoive force given by vol reior of ohm nd n inducor of.5 H. If he iniil curren i.find he curren ime >. b) Solve 7. )Solve b) Find he orhogonl rjecorie of he fmily of he prbol. 8. ) Find he orhogonl rjecorie of he fmily of curve. b) A body i originlly nd cool down o in min. If he emperure of he ir i, find he emperure of he body fer min.,? 9) ) Solve b) A rdiocive ubnce diinegre re proporionl o i m. When he m i mg, he re of diinegrion i.5 mg per dy. How long will i ke for he m of mg o reduce o i hlf? ). ) Solve by he mehod of vriion of prmeer. b) A body kep in ir wih emperure 5 C cool from C o 8 C in min. Find when he body cool down o 5 C. Mhemic I Pge

2 QUESTION BANK 6 UNIT-II. Uing Mclurin erie epnd upo he fifh power of nd hence find erie for. )If yz z y ( u, v, w) u, v, w hen how h. y z (, y, z) b)if u y, v y where r co, y r in hen how h [M] ( u, v) r ( r, ) )S.T. in...! 5! 7! b)s.t. log( e ) log )Epnd log e in power of (-) nd hence evlue log(.) correc o deciml plce. b) Clcule he pproime vlue of correc o deciml plce uing Tylor erie. 5. ) For he crdioid r ( co ),P.T r i conn where i he rdiu of curvure. b) Find he ionry poin ofu(, y) in.in y.in( y),, y nd find he mimum of u. m n p 6. () Prove h he mimum vlue of y z under he condiion y z i m n p m n p ( m n p) mn p mn p b) Find he minimum vlue of y z given y z. 7. )Find hore nd longe dince from he poin (,,-) o he phere y z b)find he volume of he lrge recngulr prllelopiped h cn be incribed in he ellipoid y 9z 6 8. )Find he rdiu of curvure ny poin on he curve y c coh( ) c b)find he rdiu of curvure of he curve y ( y ) (-,). ( ) 9. )Find he rdiu of curvure he origin of he curve y. b)find he rdiu of curvure he origin for he curve y ( y ) y.. )Verify wheher he following funcion re funcionlly dependen, if o find he relion y beween hem, u, v n n y. y b) Emine he funcion for ereme vlue y y y (, y ). Mhemic I Pge

3 QUESTION BANK 6 UNIT III. )Evlue c b e y dyd b) Evlue ( y z ) d dy dz cb. ) Evlue ( y ) ddy b) Evlue dz d dy y. ) Evlue ddy ( )( y ) in b) Evlue r dr d ( y ). ) Evlue e ddy b) Evlue e y e log z dz d dy log 5. ) Evlue b) Evlue y ( y) dyd in r r dz dr d 6. ) Evlue ( y ) d dy b) Evlue ( y ) d dy y over he re bounded by he ellipe b over he poiive qudrn for which y 7. ) Evlue he inegrl by chnging he order of inegrion dyd y Mhemic I Pge

4 QUESTION BANK 6 ( y ) b) Evlue he following inegrl by chnging o polr coordine e ddy 8. ) Evlue he inegrl by chnging he order of inegrion ( y ) ddy b) Evlue yddy where R i he domin bounded by -i ordine R he curve nd y 9. ) Evlue he inegrl by chnging he order of inegrion dyd in over he crdioid ( co ) b) Evlue r dr d r bove he iniil line.). Evlue he inegrl by chnging he order of inegrion y dyd b) Show h he double inegrion, he re beween he prbol y 6 nd y i UNIT IV. ) Find he Lplce rnform of in & co b). Find he Lplce rnform of co.co. ) Find he Lplce rnform of. b) Se nd prove fir hifing heorem.. ) Find he Lplce rnform of e co5 in 5 b) Find he Lplce rnform of f coh. in b. ) find Lplce rnform of f e inh n n n n b) To prove f f f 5. ) Find uing chnge of cle propery n f f L Co L, if LSin e / b) Find he Lplce rnform of n 6. ) To prove L f f n n d f e co d. where,, n n d f in f in.co b) Find he Lplce rnform of 7. ) Find he Lplce rnform of Mhemic I Pge

5 QUESTION BANK 6 b) Find he Lplce rnform of f 8. ) Show h e.in d 5 co, Uing Lplce rnform b) Find he Lplce rnform of f in co cob 9. ) Uing Lplce rnform, evlue d. b) Find Lplce Trnform of Squre-wve funcion of periodic,defined k f k, f wih period T, where. Find Lplce Trnform of periodic funcion f E E T E E, T T T T [M] UNIT V 5. ) Find he Invere Lplce rnform of ( )( ) 5 8 b). Find L by uing liner propery ) Find L by uing fir hifing heorem b) Se nd prove chnge of cle propery.. Ue rnform mehod o olve y y y e where y, y [M]. ) find he invere Lplce rnform of b b) Find L log 5. ) Evlue L log u du u b) Find he invere Lplce rnform of log. 6. ) Se nd Prove Convoluion heorem b) Uing Convoluion heorem, find L 7. Ue rnform mehod o olve 5 in, where y, y [M] y y y e 8. Find L, uing Convoluion heorem. [ M] 9 Mhemic I Pge 5

6 QUESTION BANK 6 9. ) Find L e uing econd hifing heorem. b) Find L, uing Convoluion heorem. 5. Uing Lplce Trnform mehod olve when [M] Mhemic I Pge 6

7 QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (OBJECTIVE) Subjec wih Code : Engineering Mhemic-I (6HS6)Coure & Brnch : B.Tec Common o ll Yer &Sem : I-B.Tech& I-Sem Regulion : R6 UNIT I ) Which of he following i condiion for ec differenil equion --- ) Which of he uible form of ) Inegring fcor of --- ) Which of he uible form of ) The equion i of he ype yf(y)d+g(y)dy= hen he I.F i None 6) The equion i no ec bu i homogeneou in nd y hen he I.F i None 7) Given h i no n ec, hen I.F i None 8) The generl oluion of = i None Mhemic I Pge 7

8 QUESTION BANK 6 9) The order of i None ) The differenil equion i liner non-liner homogeneou None ) An inegring fcor of ) The I.F of i None ) The Inegring Fcor of i None ) An inegring fcor of 5) An equion of he form dy d y ` Liner Ec Bernoulli none 6) The differenil equion homogeneou D.E Leibniz liner equion Bernoulli D.E Non liner D.E 7) The differenil equion of orhogonl rjecorie of he fmily of curve, where i he prmeer i y y None 8) The fmily of righ line ping hrough he origin i repreened by he differenil equion y 9) The equion repreen he orhogonl rjecorie of he fmily Mhemic I Pge 8

9 QUESTION BANK 6 y= ) A curve which cu every member of given fmily of curve righ ngle i clled rjecory orhogonl rjecory Oblique rjecory crdiod ) The orhogonl rjecorie of he circle re righ line elllipe hyperbol None ) If he differenil equion of he fmily of given curve i me he D.E of heir orhogonl rjecorie hen he fmily i clled iogonl ioherml elf-orhogonl ioropic ) The orhogonl rjecorie of he curve i r=c None ) An orhogonl rjecorie in polr co-ordine replce =---- 5) If he C.F. i C cob C in b hen he roo re--- Comple Ec Rel &equl Rel &diinc 6) If n Auiliry equion h he vlue m hen he roo re Comple Ec Rel &equl Rel &diinc 7) Auiliry equion of differenil equion m 6 m m m 6 m 6 co 8) The Priculr Inegrl of he differenil equion ( D ) y in i ---- (in co ) (co in ) (co in ) co 9) The Auiliry Equion of he differenil equion y 6y 9y i---- m m 6m m Mhemic I Pge 9 6m 9 6m 9 m 6m 9m ) The Priculr Inegrl of he differenil equion ( D 5D 6) e i e e 5

10 QUESTION BANK 6 6 e ) Find he generl oluion of y y C Ce ( C C ) e ( C C ) e None ) The uni for Cpcince ( C ) i Frd henry Ohm coulomb ) If f ( b ) hen P.I of in b i D b e ) none Le be he curren nd be he chrge in he condener ple ime. Then Volge drop cro he reince V = qi None 5) Which of he following i oluion o he differenil equion ) The vlue of i None 7) The differenil equion of L-C circui wih elecro moive force (e.m.f) i ) The C. F of he equion i None 9) The complemenry funcion of ( y= + ) The P.I of he equion i None Mhemic I Pge

11 QUESTION BANK 6 UNIT II n ' '' n. If f ( ) f () f (). f ().... f (). hen he erie i clled! n! Mclurin erie Tylor erie Cuchy erie Lgrnge erie ' '' '''. If f (), f (), f (), f () hen he Mclurin erie epnion of i given by y. If,,hen, re r co, r co co, in co, ec co,r coec. In Tylor erie epnion,he hird erm i. ' ( ) '' ( ) ''' f() ( ) f ( ) f ( ) f ( )!! 5. Mclurin erie epnion for log The fir erm of Tylor erie of Sin bou i. 7. The econd erm of Mclurin erie of Co bou i If = If The mimum or minimum vlue of funcion i clled i ereme vlue ddle poin ec vlue criicl poin. If ln m & l hen he funcion f (, y) i No concluion Neiher M nor Min Mimum Minimum. If ln m poin (,b) hen (,b) i clled Mhemic I Pge

12 QUESTION BANK 6 poin of mimum poin of minimum ddle poin ereme vlue. If m, hen h mimum vlue hen ln m, l ln m, l. If r & r hen he funcion f (, y) i Minimum Mimum Neiher M nor Min Undecided 5. Jcobin i. rnk conn funcion deerminn vlue 6. If ;,hen = -r /r r -/r 7. The rdiu of curvure in Crein co-ordine i y y y y y y y y 8. The polr form formul for he rdiu of curvure i r r r r r r r r r r r r r rr r r rr r r rr 9. Curvure ny poin on he righ line i conn. If y-i i he ngen he origin o he curve hen he rdiu of curvure i None. Rdiu of curvure (,) of he curve = i A ). The re of chnge of bending of curve ny poin i clled lengh volume curvure re. The ionry vlue of he funcion f()= re,,5,,-. Find he poin on he plne which i nere o he origin for hi wrie he Lgrngin funcion. ( y z ) ( y z ) none 5. If l =, m=,n=, hen he funcion h. eiher m (or) min m min undecided 6. If u =, v = re funcionl dependence, hen find he relion. v = u v = u = v u 7. If u= y z, w y yz z re funcionl dependence, hen find relion beween hem. u =v+w v =u +w w =v+u uv=w 8. Ifl =, m=,n=, hen he funcion h. m min no ereme vlue no concluion 9. If he ionry poin re (,) (,) (,-) (,) u. If u y hen i equl o y Mhemic I Pge

13 QUESTION BANK 6 +y +y ( u, v) (, y).. equl o (, y) ( u, v) - none of hee. The curvure ny poin of circle ny poin on i i conn doe no ei. The rdiu of curvure of he curve r he poin (, ) i r r r r r r r r. The rdiu of curvure of he curve y e (,) i 5..If ;,hen r r r r D r r r y u 6. Ifu hen = y y log 7. The fourh derivive of e i e e e 8. D ( )! If he curvure of he curve i K, he rdiu of curvure i.. k /k k. Reciprocl of curvure poin i clled Rdiu of curvure curvure ngen curve e. UNIT III ydyd 8 e. d e e e e y. ydyd 6 5 Mhemic I Pge

14 QUESTION BANK 6. The vlue of double inegrl 5. The vlue of double inegrl ydyd i drd π π/ π None 6. The vlue of he riple inegrl dz dy d i The vlue of double inegrl The vlue of he riple inegrl dyd e yz ( e ) ( e ) Mhemic I Pge d dy dz i ( ) e 9. y dy d. ( 5 5 y) ddy 5. The vlue of he riple inegrl The vlue of double inegrl. The vlue of double inegrl \ d dy dz i dyd ( ) dyd The vlue of double inegrl y dyd 5. The vlue of d

15 QUESTION BANK 6 6. If f() = f ( ) hen f ( ) d f ( ) d f ( ) d f ( ) d f ( ) d 7. e d 8. ( y) dyd The re of region R bounded by he given curve i ddy ddy ddy R R R. If he region i repreened in polr coordine hen he re i given by R rdrd r drd R R. If he region R i bounded by, y y drd nd if he vericl rip i conidered fir hen he limi of Y re.,,, y, y. If he region R i bounded by, y y nd if he vericl rip i conidered fir hen he limi of X re.,,,,. If he region R i bounded by, y y nd if he vericl rip i conidered fir hen he limi of Y re.,,,,. r drd over he region included beween he circle r in, r in i in r drd r drd in / in in in in r drd None 5. If he region R i bounded by, y, y nd if vericl rip i conider fir hen he limi of re,,,-y,- 6. The re encloed by he prbol = y nd y = i 7. dz dy d yzd dy dz Mhemic I Pge 5

16 QUESTION BANK co 9. r in drd. Uing he Double inegrl we cn find. Lengh Are Volume None. Uing he Triple Inegrl we cn find. Lengh Are Volume None. Suppoe he region of inegrion i hen he region lie in nd qudrn nd qudrn nd qudrn nd qudrn. By chnge of vrible mehod, d dy = dr d r dr d dr d None. Uing he ingle Inegrl we cn find Lengh Are Volume None 5. Suppoe he region of inegrion i hen by chnge of order of inegrion mehod y vrie from o o o o 6. To ge y limi in, drw he rip prllel o X Y Any none 7. Evlue y ddy y 5 8. The limi of inegrion of over he domin bounded by re o ; y o y o y ; y o None 9. To ge limi in y drw he rip prllel o X Y Any none. Find he vlue of None UNIT IV. L { e } Mhemic I Pge 6

17 QUESTION BANK 6. L {Co}. L {}. L {Coh} 5. L{ e inb} b b b b 6. If L f f hen L e f f f f 7. The Lplce rnform of f i defined e f d e f d e f d None 8. L {in} b b 9. L {inh}. e L e b b log log b log b log b. L {k} k. If H i uni ep funcion hen k L H Mhemic I Pge 7

18 QUESTION BANK 6 e e e e. L { e }. L{ e } 5. L{ e co} 6. If L f f d b b b b b f f, hen L f d f d b f d 7. If hen. None 8. When 9. Find he vlue of =. None. If hen. None. L {coh }. Find =.. Find he vlue of =. None. If hen. 5. L in = L co 6. Find Mhemic I Pge 8

19 QUESTION BANK 6 7. Find L e L hen 8. If f f f f f f f f f f f 9. c f c f L L cl f cl f Thi propery in repec of Lplce rnform I clled Shifing propery Diribuive propery Symmeric propery Lineriy propery. L {}. L. L { }. L inh log log log log. L co 5. L{ e } e 6. L log log log log Mhemic I Pge 9

20 QUESTION BANK 6 7. L5 e 8. L { } L {in.co} L {in.} UNIT V. The vlue of L ( None. If L e e e e. If L f f, hen L f ( f f f None. If L e e e e 5. If ) hen =. None 6. i poible only when n i Poiive ineger zero Negive ineger No Mhemic I Pge

21 QUESTION BANK 6 7. Find 8. If ) nd f() =, hen 9. Find he vlue of.. Find he vlue of. None. If L f f nd f, hen f f f f f L. If L n i poible only when n i Poiive ineger Zero Negive ineger All of hee. If n i poiive ineger,hen None. If ) nd hen L f d f() None 5. If L ( in 6. If L f ( f co n d nd n,,, hen L f n d in co n f n n f n n f n f 7. If L ( inh co in coh 8. If L f f hen L f ( n f f f f 9. If L Mhemic I Pge

22 QUESTION BANK 6 ( in co in co. If ) hen L f. None. If L ( e e 5 e e. If L ( co in co. The vlue of 5 ( S ) e e e e. Find he vlue of If L ( in co in co 6. If L ( in co in co 7. If L e f = f H f H f H None 8. If f n hen f L in co in in ( 9. If f log hen L f ( in coh inh co. If L Mhemic I Pge

23 QUESTION BANK 6 ( in. If ( in. If L f f co in co L co in co n hen L f n ( f n n f n f. If L f f hen f ( f. If f co hen in ( 5. If f log hen f ( e 6. If L f ( 7. If L f f None L f f None L f co in in f f f L e e None L f d f f None L f, hen nd f, hen f f f ( 8. If L f f nd L g g, hen L f. g ( f * g f * g f g None 9. If L e e e e. Find he vlue of. None Mhemic I Pge