INSTITUTE OF AERONAUTICAL ENGINEERING Dundgl, Hyderbd - 5 3 FRESHMAN ENGINEERING TUTORIAL QUESTION BANK Nme : MATHEMATICS II Code : A6 Clss : II B. Te II Semester Brn : FRESHMAN ENGINEERING Yer : 5 Fulty : Mr J. Sures Goud, Assote Professor, Ms C. Rn Reddy, Assstnt Professor COURSE OBJECTIVES: Te gol of ts ourse s to provde students wt better understndng of nd preprton for mtemts w re pplble n most of engneerng brnes.. To mke wre students bout te mportne nd symboss between Mtemts nd Engneerng.. Ts ourse elps n trnsltng pysl or oter problem n to mtemtl model. 3. To provde n overvew of dsoverng te epermentl spet of modern ppled mtemts. Ts ourse retes te blty to model, solve nd nterpret ny pysl or engneerng problem GROUP-I (SHORT ANSWER QUESTIONS) S. No Queston Blooms Tonomy outomes UNIT I Defne grdent? Remember Defne url? Remember 3 Fnd te dretonl dervtve y+ z t (,-,3) n te dreton of +j+3k? Fnd unt norml vetor to te gven surfe y+z= t te pont (,-,3)? 5 Defne rrottonl nd solenodl vetors? Remember 6 Prove tt F=yz+zj+yk s rrottonl? Anlyze 7 Sow tt url(r n )=? Anlyze 8 Prove tt dv url =? Anlyze 9 Defne surfe ntegrl? Remember g Stte Green s teorem? Remember UNIT-II Fourer seres nd Fourer trnsforms Defne perod funton Anlyze d
S. No Queston Blooms Tonomy outomes Defne even nd odd funton Anlyze d 3 Fnd te funtons re even or odd () sn+os+ os Evlute d ()os+ 3 sn Fnd te prmtve perods of te funtons sn3,tn5,se Evlute d 5 Defne Euler s formule Anlyze d 6 Epln bout lf rnge Fourer seres Understnd d 7 Wrte bout Fourer sne nd osne ntegrl Understnd d 8 Fnd te Fourer sne trnsform of Evlute d 9 Defne nfnte Fourer trnsform Anlyze d Wrte te propertes of Fourer trnsform Understnd d UNIT-III Interpolton nd Curve Fttng Defne Interpolton nd etrpolton Remember Epln bkwrd dfferene nterpolton Understnd 3 Defne verge opertor nd sft opertor Remember Prove tt = E - Anlyze 5 Construt forwrd dfferene tble for f()= 3 +5-7 f Crete b =-,,,,3,,5 6 Evlute log f() Evlute 7 Evlute os Evlute 8 Form te dfferene equton orrespondng to te urve y=+b Crete b 9 Derve te norml equtons for strgt lne Anlyze b Epln errors n nterpolton Understnd UNIT-IV Numerl Tenques Defne lgebr nd trnsendentl equton nd gve emple Understnd Wrte bout bseton metod Anlyze 3 Wrte sort note on tertve metod Understnd Derve Newton s Rpson formul Anlyze 5 Epln onvergene of Newton s Rpson Understnd 6 Fnd te squre root of number6 by usng Newton s Rpson metod evlute 7 Derve te formul to fnd te reprol of number Anlyze 8 Epln LU deomposton metod Understnd 9 Epln te proedure to fnd te nverse of te mtr by usng LU deomposton metod Understnd Wrte sort note on Guss Sedel tertve metod Understnd UNIT-V Numerl Integrton nd Numerl solutons of dfferentl equtons Derve te Newton-ote s qudrture formul Anlyze Epln Smpson s /3 nd 3/8 rule Anlyze 3 Epln two pont nd tree pont Gussn qudrture Understnd Compute usng Guss ntegrl d, n 3 Understnd 5 Defne boundry vlue problem Remember 6 Epln Tylor s seres metod Anlyze 7 Epln Euler s metod Anlyze 8 Gve te dfferene between Euler s metod nd Euler s modfed metod Evlute
S. No Queston Blooms Tonomy outomes 9 Epln Runge-Kutt seond nd lssl fourt order Evlute Epln power metod to fnd te lrgest Egen vlue of mtr Evlute GROUP - B (LONG ANSWER QUESTIONS) S. No Queston Blooms Tonomy UNIT-I Vetor Clulus Fnd te onstnts nd b so tt te Surfe wll 3 5 6 byz ( z) 3 be ortogonl to te Surfe y z t te pont (-,,). Prove tt f r s te poston vetor of ny pont n te spe ten rrottonl nd s solenodl f n 3. C r n. r If F (5y 6 ) (y ) jevlute F. dr long te urve C n y plne y= 3 from (,) to (,8). Evlute A. nds were A Z j 3y zk nd S s te surfe of S te ylnder +y =6 nluded n te frst otnt between Z= nd Z=5 Evlute f. dr were f 3 y y j nd C s te prbol y= from (,) to (,). Evlute ( yz d z dy y dz) over r of el ost, y sn t, z kt s t vres from to. 7 Verfy guss dvergene teorem for te vetor pont funton F=( 3 -yz)- yj+zk over te ube bounded by =y=z=nd =y=z= 8 3 Verfy Green s teorem n te plne for ( y ) d ( y y) dy were C s squre wt vertes (,),(,),(,),(,). 9 Verfy Green s Teorem n te plne for 3 ( y ) d ( y y) dy were C s squre wt vertes (,),(,),)(,),(,) f ( y ) yj Verfy Stokes teorem for over te bo bounded by te plnes =,=,y=,y=b,z= UNIT-II Fourer seres nd Fourer trnsforms Obtn te Fourer seres epnson of f() gven tt C f ( ) ( ) s n Anlyze Anlyze Evlute Evlute Evlute Anlyze Anlyze Anlyze Anlyze Evlute Outome d
S. No Queston Blooms Tonomy nd dedue te vlue of.... 3 6 ( ) sn Outome Fnd te Fourer Seres to represent te funton f n - d <<. 3 Epress f()= s Fourer seres n,. Anlyze d Crete d Epnd te funton f ( ) s Fourer seres n,. 5 Fnd te Fourer seres to represent te funton f() gven by: d 6 Epnd f()=os for n lf rnge sne seres Crete d 7 Fnd te Fourer trnsform of f() defned by d f ( ) f f > 8 Fnd te Fourer sne trnsform for te funton f() gven by 9 f ( ) sn, Fnd te fnte Fourer sne nd osne trnsforms of f() = Fnd te nverse Fourer trnsform f() of d UNIT-III Interpolton nd Curve Fttng d d Fnd te nterpolton polynoml for te followng dt usng Newton s forwrd nterpolton formul.. 3...8 5.6 f() 7. 8. 38. 3 5. 7 5 3 35 5 Fnd f(), from te followng dt usng y 35 33 9 6 3 Newton s Bkwrd formul. Understnd 3 Te populton of town n te deml ensus ws gven below. Estmte te populton for te yer 895 Yer () 89 9 9 9 93 Populton (y) 6 66 8 93 Fnd by Guss s bkwrd nterpoltng formul te vlue of y t = 936 usng te followng tble X 9 9 9 93 9 95 Y 5 7 39 5 5 Usng Lgrnge s formul fnd y(6) gven 3 5 7 9 y 6 58 8 7 6 Fnd y(5) gven tt y()=, y()=3, y(3)=3 nd y(8) =3 usng Lgrnge s formul Understnd rete
S. No Queston Blooms Tonomy 7 A urve psses troug te ponts (, 8),(,), (3,-8) nd (6,9). Fnd nlyze te slope of te urve t =. 8 Ft strgt lne y= +b from te followng dt: evlute 3 y.8 3.3.5 6.3 9 By te metod of lest squres, ft seond degree polynoml y=+b+ to te followng dt. 6 8 y 3.7.85 3.7 57.38 Usng te metod of lest squres fnd te onstnts nd b su tt y=e b fts te followng dt:.5.5.5 y..5.5 9.5.35 8.75 evlute evlute Outome UNIT-IV Numerl tenques Fnd te rel root of te equton 3 --= by bseton metod. Fnd te squre root of 5 up to deml ple s by usng bseton metod 3 Solvee = by tertve metod Fnd rel root of te equton, log os usng Regulfls metod 5 Fnd rel root of te equton 3-os-= usng Newton Rpson metod 6 Usng Newton s tertve metod fnd te rel root of orret to four deml ples 7 Fnd te squre root of 8 by Newton Rpson metod. 8 Solve by LU deomposton metod +y+z=9,-3y+z=3,3+y+5z= 9 Solve 5-y+3z=,3+6y=8,+y+5z=- wt ntl ppromtons (3,,-) by Job s terton metod Solve +y-z=7,3+y-z=-8,-3y+z=5 by Guss-Sedel tertve metod UNIT-V Numerl Integrton nd Numerl solutons of dfferentl equtons Derve te Newton-ote s qudrture formul Anlyze Epln Smpson s /3 nd 3/8 rule Anlyze 3 Epln two pont nd tree pont Gussn qudrture Understnd Compute usng Guss ntegrl d, n 3 Understnd 5 Defne boundry vlue problem Remember 6 Epln Tylor s seres metod Anlyze 7 Epln Euler s metod Anlyze 8 Gve te dfferene between Euler s metod nd Euler s modfed metod Evlute 9 Epln Runge-Kutt seond nd lssl fourt order Evlute Epln power metod to fnd te lrgest Egen vlue of mtr Evlute b b b
Group - III (Anlytl Questons) S. No Questons Blooms Tonomy Outomes UNIT-I Vetor Clulus If ten wt s? If url ten wt s Evlute 3 If re rrottonl vetors ten wt s Crete Wt s te pysl nterpretton of Anlyze 5 If ten wt s lled? I 6 Wt s? Evlute 7 Wt s te neessry nd suffent ondton for te lne ntegrl for every losed urve? Anlyze H 8 Wt s? Evlute H 9 If were, b, re onstnts ten wt s Evlute were s s te surfe of te unt spere? If ten wt s? Evlute I Unt-II Fourer Seres If s n even funton n te ntervl ten wt s te vlue Anlyze d of? If n ten wt s te Fourer oeffent? Evlute d 3 Wt re te ondtons for epnson of funton n Fourer seres? Anlyze d If s n odd funton n te ntervl ten wt re te vlue Crete d of? 5 If n ten wt s? Evlute d 6 Wt s te Fourer sne seres for n? Evlute d 7 Wt s te lf rnge sne seres for n? Evlute e 8 Wt s te Fourer sne trnsform of? Evlute e 9 Wt s te Fourer osne trnsform of? Evlute e Wt s te? Evlute e Unt-III Interpolton nd Curve fttng If by Bseton teorem metod frst two ppromtons Evlute b re nd ten wt s? For wt vlues of te Guss forwrd nterpolton formul s used to Anlyze nterpolte? 3 Wt s te order of onvergene n Newton Rpson metod? Anlyze b Wt s te onvergene n Bseton metod? Evlute b 5 Wt s te order dfferene of polynoml of degree? Crete 6 For wt vlues of te Guss bkwrd nterpolton formul s used to Crete nterpolte? 7 If ten wt s te trd norml equton of Crete by lest squres metod?
8 If ten wt s te frst norml equton of? Crete 9 If ten wt s te frst norml equton of? Evlute If s te best ft for 6 prs of vlues by te metod of Evlute lest-squres, fnd f? Unt-IV Numerl Tenques Wt s dfferene between polynoml nd lgebr funton? Anlyze Wt s Trnsendentl equton Anlyze 3 Defne root of n equton Anlyze Wt re te merts nd demerts of Newton-Rpson Metod Anlyze 5 Epln bout order of onvergene? Evlute 6 Defne lner,qudrt nd ub onvergene? Evlute 7 Epln bout Flse-poston metod? Crete 8 Epln bout Regul-Fls metod Anlyze 9 Wt s Crout s metod n LU deomposton Evlute Wt s Dolttle s metod n LU deomposton Remember Unt-V Numerl Integrton & Numerl Solutons of dfferentl equtons How mny number of subntervls re requred to get ury, wle evlutng defnte ntegrl by trpezodl rule? Anlyze Wt s te ntervl for loser pplton, n Smpson s rule? Anlyze 3 Wt s te dsdvntge of prd s metod? Anlyze Wt s te metod of Runge-Kutt metod? Anlyze 5 If ten by usng Euler s metod wt s Evlute te vlue of? 6 If ten by usng Euler s metod wt Evlute s te vlue of? 7 wt s te tertve formul of Euler s metod for solvng Crete wt y( )=? 8 Wt s te dfferene of polynoml of degree? Anlyze 9 If nd y()= ten by prds metod wt s te vlue of Evlute ()? Wt s te Adms-bsfort orretor formul? Remember Prepred by: Mr J. Sures Goud, Assote Professor HOD, FRESHMAN ENGINEERING