ISN                                                                                          1OPHY12                      ...
I0PEY122b.    State and explain Heiselbergs uncertaiaty   principle,                            (0a Marks)       Find the ...
)                                                                                                   tjpnvrz               ...
1OPEY127b.    Derive an expressiou for inter planar spacing in terms of Miller indices.           (06   Marls)  c.   Defin...
USN                                                                                       10PHY12t22                  Firs...
IOPHY12D2  b. What is a wave filnction? Explain the properties ofa waYa functio&               (04 Marks)  c. Dedve the ex...
1oPHY12l22                                             PART - B5 a.   Choose your aoswers for the following :       i)   W...
I0PIIYt2l221 a. iii)   Which ofthe following has geatest packing fraction            A) simpie cubic                      ...
lOMATl1                                      First Semester B.E. Degree Examination, January 20ll                         ...
1OMATl13 a. Choose the conect answer                     :     i) If u = axz + bf + ab-ry, tt ={} i,                      ...
lOMATlI           iv) Aslmptote to the cuve f(a - x; = )(3 i5                 A)v=o                    B)x=0              ...
IOMAT1         I                                                    9t 92 93 94              95                           ...
O6MAT11                          First Semester B.E. Degree Examination, JunelJuly 2011                                   ...
O6MAT11   c.    lf u=x2-l,r 2xy              and x = r cose,   r = rsino. flJrd                                           ...
O6MAT11        Find the volume of the solid gercrated by the rcvolution of the cardioid                                   ...
O6MAT11  h    Test the converaence of the series                                                                          ...
USN                                                                                                                      O...
06MAT21        .                                                        l.d   b     Evaluare                       u lr-[ ...
O6MAT21             iv) The cylindrical co-ordinate system is                 A) Not orthogonal B) Orthogonal C)          ...
I                                                                                                                    O6MAT...
IJSN                                                                                                               O6MAT21...
O6MAT213 a.   Select the conect answer         :                               lJx       i)     The value    of J Jxy dydx...
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
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Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
Physics Stream (2011-July) Question Papers
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Physics Stream (2011-July) Question Papers
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Physics Stream (2011-July) Question Papers

  1. 1. ISN 1OPHY12 First Semester B.E. Degree Examination, January 2011 Engineering Physics Time: 3 hrs. Max. Marks:100 Notei l. Ansv,er an) FIYEJu questiohs, choosing at leosr twofrom each pan 2. Answet all objeclive type questions only in OMR shee, page 5 ofthe answer booklet. .9 3. Ansbel to objective type queslions on sheels other than OMR will nol be valued- tE 4. Physicat constants : h:6.625 x 11131 J-5, c:3 x ld ms-), m"= 9.1 x LOtl hg, .? k =1.38 x 1(I3 JI(I, €o=8.854 xllJ12 Fhtt E PART_Af9 1 a. Choose yor[ answers for the following :9p: i) - Green light incided on a surface releases photoelecftons fiom the surface. If now blue;, light is incident on the same surface,the velocity of electrons 5d A) inqeases R) decreases C) remairx same D) becomes zerof^r ii) Rayleigh-Jeans theory ofmdiations agrce with expe metrtal results for A) all wavelengths B) shorter wavelergths oDlyEY C) longer wavelengths only D) middle order wavelengths onlyPi iii) The de-Broglie wavelength of an electron accelerated to a potential difference of 100 volts is A)1.24 B)10A C) 100 A D)t2A iv) The wave rratule associated wili electrons in motion was verified byE3 A) photoelectuic effect B) comptor effect C) difftaction by crystals D) Ramon effect (04 Mrrks)!5 b_ State and explain de-Broglies hypothesis. (04 Mark) c. Define phase velocity and group velocity. Obtaia the relation between goup velocity and-bts particle velocity. Obtain rhe exprcssion for de-Broglie w qsing group velocity.H! (08 Marhs) Find the kinetic energy and group velocity of an wavelength of 0.2nm. (04 Mark);# 2 a. Choose your answers for the following : i) The uncertainty in the determhatior of position ($).**,,n" uncertainty in the deterrninatioo ofits momentum is ii) ^)% oflocating probability D% c) % D)3 The a particle is maximum A) at the cente ofthe wave packet B) at the nodes of the wave packel C) catnot be determined D) none ofthese iii) In Davision and Germer experiment, wher 54 volts was applied to electons, thez prcnounced scattering direction was found to be at;E A) 90" B) 120" c) s0" D) none ofthese iv) The giourld state energy of an elechon in an one dimensional infrdte potenlial well of width 2 A is 16 eV. Its energy in third excited state is A) 32 eV B) 64 eV C) 144 eV D)256 eV (04 Ma*s) l of 4
  2. 2. I0PEY122b. State and explain Heiselbergs uncertaiaty principle, (0a Marks) Find the eigin value and eigen functions for an electron in on€ dimeisioml potential well of infmite height. (08 Marks) d. Estimate the time spent by an atom in the excit€d state dudng the excitation and de-excitation processes, when a spectml liBe of wavelenglh 546 nm-and width 10-ra m is emitted- (01Marks)3 a. Choose your arswers for the following : i) The mobility of electrons in a conductor is 4 x 10-3 m2Vrs-l. Then the &ift velocity of the electloo in the presence of applied electric field of strength 1 00 vm-l is .ql+.ri B) loms:r c.10.4msr D)004ms-r ii) The classical value ofmolar specific heat of a co[ductor is A)lR B) ;R c)3R or ln iii) ofa metal at absolute zero temperature is proportional to The Fermi energy A) rl B) n% C) n% D) n where n is number offree electrons per unit volume. i") At 50I! the probability of findiog an electron at Fermi energy is %. Te $obability offindiflg electon at the same energy level at 100 K is A)l B) zero ct Y4 D) U (o4.rrnrks) b. Obtain the expression for electical conductivity on the basis of fiee electron theory of metals. (08 Marks) c. Explain Fermi energy ald Fermi factor. (04 Marks) d. Calculate the probability of ar electron occupying an energy level 0.02 eV above the Fermi level and 0.02 eV below the Fermi level at 200 K. (04 Marks) a. Choose your answers for the following : i) Choose the correct relalion: A) E=eu (e, l)P B) D=eo (e, -l)E C) P=e, (e. -l)E D) e.=1-l ii) Electronicpolarization A) decreases rvith increase in temperature B) increases with temperature C) is independent of temperalure D) may iacrease or decrease with temperatue iii) Hysteresis loss occurs when the mag material is subjected to A) DC voltage B) AC voltage C) both AC and DC voltage D) none ofthese iv) The relative permeability for diamagnetic materials is A) slightly greater than one B) zero C) less than one D) very much greatd than one (04 Marks) b. Obtain the expression for intemal field in solids. (08 Marks) c. Distinguish between hard and soff magnetic materials. (05 Mark) d. Find the polarization produced in a crystal by an electric field of stenglh 500 vmm-r if it has a dielectdc constant of6. (03 Marks) 2of4
  3. 3. ) tjpnvrz PART _ B5 a. Choose your answers for the following: i) Rate ofinduced absorptioo depends on A) nulnber ofatoms in lower eoergy state B) the elergy-d€nsity Cj number ofatoms in higher erergy state D)bothAatrdB ii) Iu semiconductor laser the maredal used is A)any semiconductor B) direct band gap semicondctor i) indirect band gap semiconductor D) no1 a semiconductor iii) Tie required condition to achieve laser action in a system is A) state ofpopulation inversion B) existence ofmetastable Jtate C) a resonant cavity D) allthe thxee object iv) In recording the image on the photographic plate the reference bean and the beam undergo ---- at the photographic plale - - i"tetf"t"oce D) polarization A) difftactio-n --Bfiflectio, C) -. (04 *larks) b. Explain the construction and working of He-Ne laser, with the help of sritable diagrams rks) c. Metrtion the applications ofholography. (04 Marks) .. 633 nm is d. Th" ur"rug" o,rprt po.". of laser-source emitting a laser beam of wavelength S mw. f ;ia tle numler of photons emitted per second by the lascr source (04 Marks)6 a. Choose your answers for the following : i) The crilical temperanfe ofmercury is A) 4.2 K B) 6.2 K c) 7.8 K D)20K ii) Tire temperature of a superconductor kept in a weak magnetic field is reduced below critical temperature, then A) R=0;B +0 B)R*0; B=0 c) D)R:0; B=0 iii) The numerical apertde of an optical fiber in ical ap€rture 1n water (n,, = f) is A) 0.43 B) 0.24 c ) 0.96 iv) Graded index fiber can be A) single mode fiber onlY B) D) medium C) both single mode and multimode (04Mrrks) b. Define the terms : i) angle ofacceptance ii) numedcal aperture. change iii) ftactional index iv) modes ofpropagation (04 Mark) (08 Mark) c, Explain BCS theory of supercotrductivity. Wdle a short note on Maglev.vehicles d. ihe refiacrive indices ofcore and clad<ling are 1.50 ard 1.48 respectively in an optical fiber. (04Marks) Find the numerical aperture and angle of acceptance.7 a. Choose your answers for the following : i) The relation for angles between a-xes of a ticlinic crystal is A)cr=p=v=90" B)o*p*y=90" C)o*p+v+90" D)([=B--y*90" ii) The coordination number for a face centered cubic lattice is A) 12 B)8 c)6 D) 26 iii) The packing factor offcc structue is A) s2% B) 68% c) 92% D) none ofthese rv) The Miller i[dices ofthe platre parallel to t]rc x and y axes are A) (1 00) B)(0 10) c)(001) D)(rll) (04 Marks) 3 of 4
  4. 4. 1OPEY127b. Derive an expressiou for inter planar spacing in terms of Miller indices. (06 Marls) c. Define packing ftaction. Calculate packing ftaction for sc and bcc structures. (06 Msrks) d. Inter planar distance for a crystal is 3 ,4. and the glancing atrgle for second order spectoum was observed to be equal to 10o30. Fitrd the wav€length olthe X-rays used. (04 Marts)8a- Choose your answers for the following : i) In a carbon nano tube, the bond betwee[ the carbon atoms is A) metallic B) iooic C) hydrogen D) covalent ii) Fullerene is A) a sheet of carbon aloms rolled up into long tube B) sixty carbon atoms ananged in the shape of a football C) one dimensional aray of atoms D) three dimensional aray of atoms iii) Ultrasonic waves arc sound waves havilg l A) velocity greater tian 330 msr B) velocity lesser than 330 ms C) frequercy geater than 20000 Hz D) ftequency less than 20000 IIz iv) The t,?ical size ofnano matedal is betweetr A) 1-10om B) 10-50m C) 1-l00nm D) 1 1000nm b. What axe oano matedals? Explain carbon nano tubes and their physic"l ,."r"rri"JoilffiT,l few applications ofcarbo[ nano tubes. (08 Marks) c. Explain the principle and method of nondesauctive method of testing of material using ultrasonics. (oE Ma*s) 4of4
  5. 5. USN 10PHY12t22 First/Second Semester B.E. Degree Examination, June/July 2011 _; Engineering Physics Time:3 hrs. Max. Marks:100 Notei l. Aksiet any FIW fut questions, choosing at leasl ttoo from eoch pqrt. 2. Ahsyet all objectire qtpe questions only in OMR sheet page S of tie answer bookla. 3. Ansnet to objective qrpe questions on sheets other tha; OMR wilt not be vatued. 3 4. Physical constants : h:6.625 xl(lia t-5, c:Jxldmst, m,=9.1 x1O31 kg, ? k =1.38 x t (fri JKt, €o= B.BS4 x tOi, Fnit. E PART-A 1 a. Choose your answers for the following : i) In Complon Effect, the wavelength ofthe x-rays scattered at an angle 0 > 0. Ed A) iocreases B) doesn,t change C) decreases D)none ofthese;h ii) Ke, Kp artd Ko an lEspective kinetic eflergy ofan e, a proton and cr - ptuticle ofsame de-Broglie wavelength, then B)&Kp<K" C)K<Kp<K" D)K=Kp=K, iii) 9S&K ofthe particles has smallest de-Broglie wave length when both ofthem. .... The heavier A) move with same speed B) move wiih same momentum te C) move with same kinetic energy D) none ofthese iv) Matter waves are not electomagletic waves because A) they move with variable velocity B) depead on charge C) move with corstad rclocity D) rcne of these ,__ (04 }Iarlls) b. What are the basic postulates of quantum theory of radiations? Explain how planck,s overcome the drawbacks ofweins law and Rayliegh Jean,s law. (06 Mark!) c. Define gr_oup and phase velocity. Derive the expression for de-Broglie wavelength using group velocity concept.->i (06 Mrrks) d. C^o.mpute the de Broglie wavelength for a neutrou moving with one tenth part ofthe velocity oflight. (04 Marls)66 Za. Choose your ansuers for the following . i) An electron is moving in a box of length a; if y, is the wave function at x 3with = 4 n= I and Vr2 atx =a forn= 2, then I: is A)E a B)11 cr o D)- 2fi ii) For_a particle in an i[Enite poterfial well in its l,t excited state, the probability of finding the pafticle at the center of box is A)0 B) 0.2s c) 0.s D) 0.1z iii) To become a nuclear constituent, the K.R of e must be ofthe order of A) 20 MeV MeV B) 2 C) 20eV D) zeroE iv) An electon has a speed of 100 ur/s accuate to 0.05%. The uncertainty in its position ts A) 0.01m B)0.0115n C)0.024m D) 0.04m (0aMark) I of 4
  6. 6. IOPHY12D2 b. What is a wave filnction? Explain the properties ofa waYa functio& (04 Marks) c. Dedve the expression for energy eigen value for an electron in potential well of infiaite depth. (06 Marl6) d. A quantum particle confiued to one-dimeruional box ofwidtha is in its first exerted state. What is the probability offinding the particle over an interval of marked srmmeticallr (:) at the ceBtre of box. (06 Marks)3 a. Choose your answers for the following : i) Ifthe mobility of E in a metal increases the resistivity A) decreases B) increases C) remains constant D) none ofthese ii) The tempelatwe dependence ofelectrical resistivity ofmetal is A) p"+ B)p"# C) pcrJT D) poT iii) Zero percentage probability is the probability for E to occupy the energy level above the Fermi energy ler el atT0kis A)E+Er B)E=EI C) E>Er D)E<Er tO If the Fermi energy of a metal is L4eV, the Fermi temperatue of the me,tal is apprcximataly A) 1.6 x 103 k B) 1.6 x 104 k C) 1.6 x los k D) 1.6 x 106 k (04 Marks) b. Discuss the various drawbacks of classical ftee e.lscton Aeory of metals. What are the assuilptions made in Quantriu theory to overcoae {ra ws? (06 M!r}r) c. Explai! d@sity of slates? Derive the expression for r,i.rtrt io{l *al}du0tivit--v io temrs of meall d. #"#^lfri;, potassiri. is 2.lev. whar are the . m6ier fnr ,tuch the ,.ro"l1tr"f., oc.up&cy at 300 K are 0.99 and 0.5? (04 Me*s)4 a. Choose your answers for the followiag : i) For fenomagnetic substances, the Curie-Weise law is given by c at r=! T gr ,/ =-L T_E 61 ,=l-o C DrT-0 ii) Clausius-Mossotti equation docs not hold for A) gasses B) liquids C) crystalliue solids D) none ofthese iii) The Ferro electric matedal losses spontaneous polarization at A) room temperatue B) 0 K C) TCK D) 100 K rv) In hysterisis, polarization A) moves with the electric field B) lags behind electric filed C) ahead to the electric field D) oone ofthese. (04 Marks) b. Explain the teflr intemal field. Derive an expression for intemal field in the case of one dimensional allay of atoEls ill di-eleotric solids. (07 Marks) c. Derive Clausius-Mossotti equation. (04 Mark!) d_ Sulphur is elemental solid di-elecaic whose di-electdc cotsta is 3.4. Calculate the t electronic polarizability ifits density is 2.07 x 103 kg/ot3 and atomic weight is 32.07. (0s Marki) 2of4
  7. 7. 1oPHY12l22 PART - B5 a. Choose your aoswers for the following : i) Wavelength of a laser beam can be used as a standard of A) time B) tempemtule C) argle D) length ii) Image is stored on a hologram in the form of A) interference pattem B) diffraotioo pattem C) photograph D) aone of these iii) Which event is likely to takes place, when a photon of energy equal to the difference in energy betwee[ two levels is incident in a system A) absorption B) emission C) absorption and emission D) none of these iv) Quartz plates arc fixed at the ends ofthe discharge tube in a He-Ne laser so that A) there wont be leakage ofgas B) the tube can withstand high eleclric voltage C) the loses light can pass out without any loss D) the emergeucy light is polarized (04 Mrrks) b. Explain the requisites atrd conditio$ of a laser system. (05 Mlrks) c. Describe the principle and working of LIDAR used to measure pollutant in abrosphere. (06 Marks) d. Find the member of mode of standing waves and thefu ftequency sepaxation in the resonant cavity of 1m length of He-Ne operating at a wavelength of6 (05 Marts) -..^6 a, Choose yoDJ ar:*irers fer ti.. folloliDg : d ,i( ") AL i-, i) The cord: ,i1:yiil a! a st,tc.,cri1d.Jrtor is " A) infir1r Li:t.tr!) no ,9 ) none ofthese ii) The rel.riaq bei.aeec superconductirg (Tg) and alomic weight (p) of isotope is A)Tccrp B) Ao1 p C; f.o.,,[ O, f"oI vlt iii) If optic fibre is kept in a medium of R.I. p (> I ) instead of air, the acceptance angle A) increases B) decreases C) remains constant D) none ofthese iv) In graded index fibre, the R.I, ofcladding vades A) exponertially B) linearly C) parabolically D) noae ofthese (04 Mark) b. Discuss t)?es of optical fibres ard uodes ofpropagation using suitable dia$am. (06 Marks) c. Distinguish between q?e- I and rype - II superconductors. (05 Mark!) d. The angle of acceptance of an optical fibre is 30o when kept in air. Find the angle of aoceptarce when it is in a medium ofR.I. 1.33. (05 Marks)7 a. Choose your answers for the following : i) Four types ofBravais lattices are obtained in A) rhombhohedEl system B) orthorhombic system C) triclinic system D) mo[oclinic system ii) In BCC structure, the packing density ofcrystal is equal to or* u,+ C)I ,8 ,)* 3of4
  8. 8. I0PIIYt2l221 a. iii) Which ofthe following has geatest packing fraction A) simpie cubic B) body centred cubic C) face centred cubic D) all have equal packing ftaction iv) The space lattice ofdiamond is A) simple cubic B) body cented cubic C) face ceotr€d cubic with two atoms/unit cell D) face centred cubic with four atoms/unit cell (04 Mark) b. With a neat figul€, explain the stluctrre of diamond ard show that atomic packhg factor of diamcnd is 0.34. (ro M"rks) Calculate the glancing angle of the (1 l0) plane of a simple cubic crystal (a = 2.814 A ) coresponding to secord order diffraction maximum for the x-rays of wavelength 0.710 A. (06 Marks)8a. Choose lour ar:swers for rhe following i) The slate ofmafter around ihe name - size is known as A) solidstate B) liquid state C) plasma state D) rnesoscopic state ii) The ultrasonic waves are delected by A) electromagnetic induction B) tuning fork C) piezo electric effect D) i[verse piezo eleclria eflecl iii) A constant testing ofproduci without causing any damage is called A) milute testing B) destruclive testing C) tron-de structive tssting D) random testing iv) The ftequency ofulhasonic waves is A) < 20 kIIz B) between 20 Hz ad 20 kHz C)>20W12 D) rcne of these (04 Mnrks) b. Describe a method for measurement of velocity of ultrasonic waves in a iiquid and mention how the buik modulus of the liquid could be evalualed. (08 Marks) c. Write a note on carbon nano tube. Discuss the various quantum structures. (08 Marks) 4of4
  9. 9. lOMATl1 First Semester B.E. Degree Examination, January 20ll Engineering Mathematics _ I Time: 3 hrs. Note. L Answer dny FIVE full questiorrs, choosing st least tterTro- no", 2. Anfl,er dlt objective qtpe questions only oi gMR fii.**"O"IOO sheeipag. 5 ofihe aaswer booktet. 3. Answer lo objecrive lype questions on ihee* other than O"MR ilt not be valued- ,9 PART-A E E l a. choose the corect answer: E i) If f(x) is cortinuous in [a, b], differentiable in (a, b) and (a) = tlb), then there exists C €(a, b) such thar f(c) O. : g . A) unique infinite B) C)alleastone D)nosuch t9 E ii) I The Maclaurins series of fix l(conslant) is, 9p: k A) t{x) = 0 B) f(x) = c) does no1 exist D) f(x) = k! !H iiil The nd derivarive of f,;, (x+2) (-l)(n +2)l _. 1 ^. t1r- EY 2l(x+2)^* tl) lx+2),*1 c) zERo D) None ofthese. iv) The I 2t derivative of y = etr* .in i, v.3 tt) (6t)y B) -40e6y "[" - c) (32)y D) None ofthese. (04 Marks) ;1 b. Ifx= tan(log y), prove thar (1+x2)y,+1+ (2nx t)),, + n(Il _ - t)y" r=0 (06 Marks) LY. Expad log(sec x) by usiag the Maclaurin,s series eipansi (rfqry c. containing xa. (05 Marl(s) State and prove the Lagrange,s mean value theorem. .ry5 d. -<,/ _e. (0s Mark) 61 2a. Choose the corect answer : tli- cri.rrru;. lii :f Ee i) Wlich statement is tlue? :-lsrun; ;1{ a 0"o A) - . -. co - co. oco are not irdeterminate B) 00. C) i is rlot irdetemimte D) None ofthese. qe i, The angle between r = asin0 and r = bcos0. is EE ....!"D of B)r c)-nn D) None ofthese. iii) The radius a curvature in the polar form is,;E A) tLf4- r B) fr,+1213/ E1 I" c) r+2rrr,-n, D) None ofthese. + 2rr" - tr., tt + lt -t!a lv) Lim ,t-1* . - - is. x-+0 5-6.-.i .i o, logt 2/3) f, <lgz e) togl i-i i D) Nooe of these. 109(5/6) LJ 6] "r,.rfX] (04 Marks) Lim sinxsin rx .. Lim I z- +:, +q, ) m;]E Eraluare: i) c. d. g;*""*:;1,.",il,,*.","J,*ll Find the tadius ofcurvatue of ::,,:; a2y =x3-al at the point where ttre curre cuts x_axis.(os Marks) 1of4
  10. 10. 1OMATl13 a. Choose the conect answer : i) If u = axz + bf + ab-ry, tt ={} i, "n AxAy A) Zerc B) a+b + ab C)ab D) None ofthese. ii) The Talors series off(x, y) = xy at (1, 1) is B) 1 + (x- l) + (y- 1)l + (x- lxy - 1)l D)None ofthese: iii) The Jacobian of hansformation ftom the Cartesian to polar coordinate system is, A)l B) lcosO C) lsin0 D) Non€ ofthese. iv) Ifu = f(x, y), x : $(t), y = y(t), then du/dt is, ,o,r didi*ddv dxdl dy dr B) 9x+gI dr cr to&*&dY &dt D) Nore ol lhese. dr aydl (01Ma*s) b. rinIli . , (06 Marks) 16 x+y"1ro* 1ru1 ^4*u4=31*u = Axd 11rr= II,, = E 614y7= I3,616r =19,v,w]. z x y- A(x,y,z) (05 Mrrl(!) If the H.P. required by the steamcr varies as the cube of the velocity and the square of the lengtlL find the percentage chaage in H.P. for 3% and 404 increase in velocity ard length rcspectively. (05 Mark)4 a. Choose the correct atrswff : i) The $adienl, divergence, curl are respectively . A) scalai, scalar, vector B) vector, scalar, vector C) scalar, yector, vector D) vector, vector, scalar rr) V =yz r+z"x J +xyk ls A) constant vector B) solenoidal vector C) scalar D) None ofthese. iii) Curl grad f is. A) grad curl f B) curl grad f+ grad curl f C) zero D) does not exist. iv) Ifthe cuvilinear system is spherical polar coordinate system then the radius veclor R is A) rsinOcosOi+ rsin 0singj + rcos0[ B; rsin0i+rcos0j-+r[ C) i+ j+k D) None ofthese. (01Marks) b. lf g=x2+f +*arrd F=ri+yj+rf , then frnd gradg, divF, curlF. (06 mrk) c. Prove thal divCurlF=V.VxF=0. (05 Mrrk) d. Prove that the cylindrical coodinate system is orthogonal. (05 Mrrks) PART. B5 a. Choose the corect answer: i) The value of [sirxcosuxdx is 0 5x3xl B) A.v) + : -.-:: 2 C) -" "- )YAv) D) None orrhese. 11x9x7 llx 9 llx9x7 til * +f =xzf is symmetric about A) x-axis B) y-axis C) the line y = x D) Att ofthese iii) : Surface area ofa solid ofrevolution ofthe curve y f(x), if rotated about x-a,is, is: 1) pry dx B) I2d dy Q J2zrYos D) f2rrx ds ,J 2o{4
  11. 11. lOMATlI iv) Aslmptote to the cuve f(a - x; = )(3 i5 A)v=o B)x=0 C)x=a D) None ofthese. (0{ Marls) l--o t b. Evaluate j l-dx.cr>0. (06 Mrrl(3) log x rl2 Derive the reduction fomula fo! Isin x dx . (05 Mrrk) 0 d. Compute the perimeter ofthe cardiod r = a (1 + cose). (05 Mark!)6a. Choose the corect answer : i) For the differenrial .Ouurion . / d.y l, .,(#)".,="., *" order and degrce [6d.) respectively are A)2,6 B)3,2 c)2.4 D) None ofthese. 11) dv v , -. --: +1=0 ls dxx A) Variable separable and homogeneous B) Linear C) Homogeneous and €xact D) All ofthese. iii) ydx - xdy = 0 can be reduced to exacr. ifdivided by A)x"t B)f C) xy D) All of these. iv) Onhosoral haiectorv oft = 4a(x I a) is l1 xr= la 1y a1i Bjx:+f =a2 C) Selfonhogonal D) None of these. (04 Marks) b. solve: (1 + f)dx + (x -"-* )dy = 0 (06 Marks) c. Solve: (ye +4x3)dx +(2xyev -3y?)dy =0 (0s Mrrks) d. Find the orthogonal trajectory of the cardiods r = a(l - coso) using the differcntial equalion method. (0s Marks)ta, Choose the i) corect answer : Which ofthe following is not an elementary A)Addirgtworows B) Adding ,fe*q luB4gsrr: r"i C) Multiplying a row by a non-zero number D) Squaring 23"l fr 4 6l is a,hil;, ir) Rank of (he matrix A "12 l, u ,] A)3 B) 1 c)2 D) None ofthese. iii) The solution of the simultaneous equations x + y = 0, x -2y = 0 is A) only trivial B) only unique C) unique aad aivial D) None ofthese. iv) Wlich of the foltowing is in the normal form? 1000 [rool [rool 0100 etr=l orol B)B-l orol oc 0010 D) A11 ofthese. Loool Loo,.] 0001 0000 (04 Mrrks) 3 of 4
  12. 12. IOMAT1 I 9t 92 93 94 95 92 93 94 9s 96 b. Find the rank ofthe matrix 93 94 9s 96 97 (06Mark) 94 95 96 97 98 95 96 9t 98 99 c- For what values of i, and p , the following simultaneous equations have i) No,solution ii) a uaiquesolution iii) an infinite number of solutions? x+yt..6t xt2y t3z=lo x+ 2y + ?,2= y. (0s Marks) d. Solve, using the Gauss-Jordan method. x+y+z=9; x-2y + 32- 8; 2x+y -z= 3. (0s Marko8a- Choose the corect answer : i) The eigen values of the mat ix A exist, if A) A is a square matrix B) A is singular matrix C) A is any matrix D) A is a null matdx. ii)A square matrix A oforder n, is similar to a square matrix B ofthe order ,n, if .... 1)+:p^pp B)AB=Nuumarixc)^AB=r;;a;ix;iNi-*Lr,l,"*. iii) Which of these is in quadraric form? a|tx2t f +y2-2yy -yL-n Btxr+l rl ( ) (x y + z) D) None offtese. iv) Quadratic form (XAX ) is posirive definirg it A) All the eigen values ofA are > 0 B) At least orc eigen value ofA is > 0 C) A11 eigen values ) 0 and at least one eigen value :0 D) No such condition. Find the eigeu values and eigen vector coresponding to t}te largest eig"" ,loJX;H [Ir]l II "ulr" "f A=11 5 ll roo marrsl L3 1 rl f-r I rl c. IfP = I 0 -I 2 | is a modal matrix ofthe matrix A in e.No.8(b)rand inve$e ofp is L1 I rl f-3 o 3l ^-,1^^-l- r It -2 I J. therl transfbm A in to diagonal form atrd hence fir1d Aa. lr 2 )l Find the nature of the quadratic forms for which corresponding (05 Marks) eigen va.lues of the corresponding matrices are given as ven Matrix Eigen values 2,3,4 B c 0,J.6 D 0, -3, -4 E 2,3, 4 (0s Ma*s) 4of4
  13. 13. O6MAT11 First Semester B.E. Degree Examination, JunelJuly 2011 Engineering Mathematics. --l Time: 3 hrs, Max, Marks:100 NoteJ.Answet FIVE full queslions choosing at leost two fiom eail+Nrr. 2ulnswet all objectlve tlpe questions onl! ln OMR sheet paget*df the Ansv,er Booklel. 3.An$,et to obJective q)pe questio s on sheels other thqn Anl*rotll fiol be valued- PART-A E I a. answcr: select the conect E i) lfY = u" t t1l"n,n E A)m loga.a* B;(m loga)".atr C) loga.a" D) (m loga)2.a* ii,) The nd derivarive ofsin(ax + b) is - n?r A)a"sintex+h+-) B) a? sin(ax +b,+,l 2 T)9:6"! il C) a sinrax + b + l D) a sir(a+.bxl+5d 2 T) iil) If $ be the angle between the radius vector and lie tangent,at ary point of the curve:ET r = (0) then,ES At coro=40 drdrdr B) rand=r@ c) tano-!9 D) None ofrhese.Ei iv) The Pedal equation in polar coordinate system is -coso) I fdr Atl0, -0zl=-l B)r=(1 c) rarl$=# ,,=.9aE i=i- r (ae] (0a Marks) b. Find the nh derivative of y = e* siu(bx - c). (04 Marks) c. Ifyr y-t^=2*, prove lhat (xr1)yn-2 r(2n-l)xy,*r t- (n2 m2)y" 0 (06 Mrrk) d. Find the angle berween lhe curves r - *a , - -l = . (06 Marks) 1+ cos 0 I -cosg;r 2 a. Select the corlecl answer: -a 3)u js i) Ifu=xY, ther oxoy equalro A) xx-r(ylogx + -l) B) xv-r(yloex + l)C) xrr(xlogx+ l) D) xv-(ylogx - 1) ii) Ifu be a homogeneous function ofdegree n il1 a ard y then5.c .. X-+V-=n -- X-+V-=n- A) au au B) au au au au x_+v_.=rnu t)) au Au x_-v_=nu AxAy axAy C) AxAy AxAy iii) Ifu=x2+ 2;y-f -x +y then lr.Ouu1ro "q*yQ; .r A) 2u B)u C) Zero LD) None ofthese.-c iv) Ifx=rcos0, y=rsinO, thenz ffi,r.0*ro A)1 B)r C) 1/r .D) Zerc (04 Marks)E b. If u=xzlan r(y/x)-y)lan-r(x/y).sho*trru, -al = *1-yl (04 Mnrk) oxdy x+y l of 4
  14. 14. O6MAT11 c. lf u=x2-l,r 2xy and x = r cose, r = rsino. flJrd ffi. (06 Mrrkr) d, In estimating the cost of a pite of bdcks measued as 2mx15Bx1.2m, the tape is stetched l% beyond ihe standard le;$h. If the count is 450 bricks to 1 cu.m and bdcks cost Rs 530 per 1000, iurd the apprcximate enor in the cost, (06 Marks)3 a. Select the correct answer : i) Jsinxdx is equal to e) 4r.-, o s) 4r*, n c) I1r",, n D) !I"_, n ii; Jsina xcos] x d,x is equal to A)* B)--L 1), c)a i2 Dx iii) The curve flza-x1=x3is symmetrical about the A) y - axis B) x - axis C)xandyaxis D) None of these. iv) The asymptote for the curve r = a sh30 is equal to A)e=a B) e :30 c)0=0 D) No asymptotes. (01Mrrks) b. Using the rcduction formula, evaluate Jtan x ax (04 Mrrk) c. tf nisaposirive integer. sho that J*"Jz*-.*=fi5h , " 0 (06 Mrrks) d. Trace the Leminiscate *y? = * <* -*l (06 Marks)4a.
  15. 15. O6MAT11 Find the volume of the solid gercrated by the rcvolution of the cardioid r=a(l + cosO) about the idtial line. (06Marks) l-*o r Evaluate l:----:dx - d> 0- (06 Mrrks) j logx PART-B5 a. Select the correct answer : i) rhe order ofthe eouutio, L*ldY )l =.i4) ," L dxr I d*i A)1 B)2 c)3 D) None of these. ii) The standard form ofa linear differential equation ofthe first order is ar $r y-P Br S+Py=q dxdxdxdx- ct $-ey=r Dr 9+ey=e iii) What is the value 01 !Y. 1o, 1r.61l6r"orial equation (t r- zxy cos x * zxy)d* * (.io * - r)ay = o A; 2x cos x2 - 2x B) 2y cos x2 - 2x C1 2x cos x2 - 2y D; .2x cos x2 - 2x iv) The differenlial equarion ofthe family f - 4a(x r aJ is u-dY[*r1,dv) o, - d* 2dx) u, r -,- dv f* *1u dY ) dxt 2 dx) c) y,_2vdyfx+l,dy) - dx 2dx) or ,=zut(**ulll d*( d*) (04 Marks) b. Solve dyldx = e3* 2Y +1zg-rr (04 Mrks) dv c. Solve cosy (06 Marks) d;+xsrnry=x d. Find the orthogonal trajectories of the family of confocal conics- { * J- - t , *n"." l, a b+). " the paiameter. (06 Mark)6 a. Select the correct answer : .l1l i) I he series . converges if f; ,J* T* A)P>0 B)P<1 C)P> I D)P<1. ii) h a positive tem series Eu. , if = , *"" the series diverges for A)i"> I B)1"<1 "t]-.y ^ c)?"=1 D)1,<r. iil)rhe itermo.he"r_ .-,. [i_?).[i_i).[$.i). "1..#-+] rlg#.+] r[-*]: +] lv) lhe senes , - )71< -. "lq+,+]" l.2,3.4, -+-- r......... is A) Cooditionally convergent B) Absolutely convergent C) Divergent D) None ofthe above. (04 Marks) 3 of4
  16. 16. O6MAT11 h Test the converaence of the series -l 2n --L , -l | --L+.... n(n + 1)(n + 2) +.... co (04 Mark) " t.2.3 2.3.4 3 4.5 c. Discuss the natwe ofthe series 1*lr*i]l**f 1l x3+....."o (x>0) (06 Ma*s) 2 3 i.4l 5, d. Discussthe absolute cotrvelgence and conditional coflvergence of the series 5 7 9 ll (06 Marks) 246I7 a. Select the corect arswer : i) If 1, m, n be the diection cosine of the nomal to the plare, then the nomal folm of the equation ofthe plane is A)ln+my+nz=o B)h+mY-nz=P C) ln + my + nz = p D) None ofthese. ii) Slmmetri;al form of the equations ofthe staight line thrcugh the point A(x1, y1, z1) and having diection cosines 1, m, n are A) -x ryi - /-2, B) !,th = )+)-=3Jl lmnl c1 I:Jr -.I l=z zr D)lx+my l-nz 0. Ix mv nz iii) The equation of any plaoe thrcugh the la" ? = ? = ? * A) a(x xD + b(y - y1) + c(z - 21) = 0 where al + bm + cn = 0 - B) a(x + xr) + b(y + yr) + c(z + zr) = 0 where al + bm + cn 0 : C) (x + x1) + 1y + y,)+ (z + zr) = 0 where al + bm + cn = 0 D) None ofthese. i ) A noinr on rhe lin" **l- Y l= 1 i, 2 3 -l A) 0, 6, 1) B) (-1, 6, -l) c) (1, -6, 1) D) (1, 6, -1) (04 Mark) Find the equaiion of the plane which passes llrough the point (3, -3, 1) atrd is parallel to the plane2x+ 3y + 52+ 6=0 (04 Mark) c. Show that tlrc lines g=-l:] z+3 x-8 or" coplarar. Find their 4 4 -5I =4 -v-4 3 7 cornmon point and the equation of the plane on which they lie. (06 Marks) d. Find the magnitude and the equations of the shortest distance between the lines ).-2 I z+2 (06 Marks) 2-3 I 3 5 28 a. Selecl the cofiect answer : i) The velociry of the moving particle along the cune x = t3 + 1y =C,z=2t+3 is A)(C, l)j +liF(21 r l)i t2d r(2t-3)k B)(t+lti c) 3li - t!j (2r + llk D) 3lri+ 2d | 2k ii) The divergence of a continuously differentiable vector point function F is denoted by divF and is defined by er idF-i9l,ral ax-dy d7 ial-r9 cri9F i9l-rg Dra.ttav**F eritr, "at az 6, dJ a, F rF iii) divcurlF is equal to ^ A)i irk B)l C)0 D) 2. iv1 lf F=x -f l.rhen curl grad F is Ar-l B)0 9)l D)2. (04 Marks) b. Find dir F. wherc F = grad ( | t lxirz) (04 Mark) c, Prove that curl (grad O):0. (06 Marks) d. : Show that r"R is any irotatioml vector for any value of tr but is solehoidal if oc + 3 0 whercR=xi+yj +zk and r is the magnitude ofR. (06 Mrrk) *ri*** 4 of4
  17. 17. USN O6MAT21 Second Semester B.E. Degree Examination, May/June 2010 Engineering Mathematics - Il Timei3 hrs. Max. Marks:100 Note:l.Answet any FIVE full qaestions, choosing al least twoJflom each part. 2-4nswer all objective q/pe qaestions only in oMR sheet pqge 5 oflhe answet booklel 3,Arrs ler to objecth,e Etpe queslions on sheets olher lhdt OMR will not be talaed PART_A 1 a. Select the corect answer in each ofthe foliowing : E i) Curvatue of a staight line is B) zero C) Both A and B D) None ofthese. ii) Radius ofthe curvatwe ofthe curve y: a sin 0 at the pole isllv n , sr !. )) Ct4" D) zero. iii) Iff(x) is continuous ir the closed interval [a, b] differenlial in (a, b) then I at least orc value c of x in (a, b) such that f(c) = A) t{b)-f(a) B) i(b)+f(a) C) f(b)-[ra) D) Nore ofthese2E b-a b+a b+a iv) Maclaurins series expansion of 1og(l + x) is8eE-E nr * I r *t - **........... 2 3 4 Br *-!2,. i3,. *" *........... * - 4l c) ,- ** ** **........... D) x+ * r l - - .. (04 Marks) 234).]t4l ..1E fo"" b- Show that fbr the ellipse in the pedal +=+-:-+,rheradiusof D_ a- h- -b- a the cuwature at the point (p, r) is a2b2lpr. (04 Marks)=t1 c, Verifi the Roller theorem for the function f(x) = (x - a)(x b)", x e (a, b). (06 Marks) d. Expand 4 tantl + x t using lhe Maclaurin s expansion upto the 4n degree tero. (06 Marko9E6E 2a. Select the correct answer in each ofthe foliowing : i) The basic firndamental i[determinate folms are q-o 11 Ar 0 B)- c)0 D) both A and B;: It r) lhe value ol a- x lopsin ls x-->n/2 (n (2-x I )z A) zero B)% c)-% D)-2 iii) The necessary and sufficienl condition for maximum and minimum isE alf.(xy) = 0 B)t(xy)=0 C)fdxr=0=fy(D,) D) None ofthese. iv) In a plane triaogla ABC, the maximum value ofCosa Cos D Cos c is, A) 3/8 B) 1/8 c) 5/8 D)25f8. (04 Msrks) 1 ol4
  18. 18. 06MAT21 . l.d b Evaluare u lr-[ *ll " x-+al a)l (04 Marks) c. Expand tanr(y/x) about the point (1, 1) up to 2id degree tenn. (06 Marks) d. Find the minimum value of x2 +f +* subject to the condition ax + by + cz = p. (06 Mark)3 a. Select the correct answer ill each ofthe following : rJ" i) Value of I jra- i. A) zerc B)a 24 cr a 24 D) 24 ii) R is the region ofxy plale bounded by the curves y = y1(x) , y = y2(x) and line x = and 4 x : b. rhen i" lJrl*yp"ay A) J lf(rr)dydx B) J F(xy)dxdy c) J Jrrx vlava* D) All are correct. rn) Jldxdy repesents A) Area ofthe rcgion in polar form B) Area ofthe region in Cartesian form C) Both A and B D) None ofthese. iv) The value ol f(n r I) is A) nr(n) B) n! C) (n- 1)! D)BothAandB. (04 Marks) b. IfAisthe-areaoftherectangularregionboundedbythelinesx=0,x=1andy=O,y=2 Evaluare - t dA Jtx A 1 . (04 Marks) c. With usual notations, prcve that Jx f1Z-;2,-,f1-)f(2m + 1). (06Mark, rJx d. Evaluate J Jxy dy dx , by changing the order ofirte$ation. (06 Marks) 0^ a. Select the corect answer in each ofthe following : i) If F is the force acted upon by the particle moves fiom one end ofa curve to the other end. fhen the total work done by p is A) JFxd; B) JF.di q jd; D) None ofthese. ii) The line integral of F: x2i + xyj from O(0, 0) ro p(1, 1) along the straight line is A) 1/3 B) t/3 c)2t3 - iio ]f aN/ax , al4/ay arc continuous functions, C is a simple closed curve enclosing the D4i region R in the xy - plane. The Grcen,s theorem states that ar flaa,,+N+=(#-#).*, "r f,a..*a,=(#-#1.*, c1 {raa**Noy= [(9-9]*0, or ; R dy dx ) {r,aox-Nay=(# #)"., 2of4
  19. 19. O6MAT21 iv) The cylindrical co-ordinate system is A) Not orthogonal B) Orthogonal C) Coplanam D) Non_coplaoar. (04 Marks) b. Find the total work particle round lhe circle x done, by tle force represenled fy F=:xyi - y = 4. - yj 2z),k, in moving a (04 tarks) Verifi the Greens theorem for ,)a," c. {V* f **raf , where c is the alosed curve of the region bounded by y = r and ) = y2. (06 Marks) d. Express the vector I = zi - 2xj + yk , in cylindrical coordinates. (06 Mark) PART -B5 a. Select thg corect answer in each ofthe followiag : i) Solution ofthe differential equation (D, - aJy is A) are*+ cze- B) (a + b)ed C) (cr + c2x + cax2)es D) (crx + c#)e* ii) Particular integal ofthe differential equation (D, +5D + 6)y = e, is A) e* B) e/12 /30 c) e" D) e, I 6. iii) Complementary function ofy,,- 2y, + y = x exsin x is A) c1e + c2e B) (c1x + c2)ex C) (c1+ c2x)e-* D) None oflhese. iv) Particular integal of(D2 4)y=sin3x is A) U4 B) - y13 c) t/5 D) None of these. (04 t{rrk) b. Solve (D3 + D2 + 4D + 4)y = 0 (04 Mark) c. Solve y" + l6y = x sin 3x. (06 Marks) d. Solve (Dz- D -2)y = 1 - 2x * - 9e by the method of mdetermined coefficients. (06 Marks)6a. Select the corlect answer ilr each ofthe follovri[g : i) The wronskin ofcos x and sin x is A)0 B)t c)2 D)4 ii) To transfonn 1t + x;Q 161*y!I + y = sin 2[og(1+ n)] into a L.D.E. vrith constant coeJfrcients put (1+x) = A) B) 1og x c) e "t D) t. iii) The solution ofthe differential equation y,, + 6y = 0 satisfies the condition y(0) = 1 and y(n/2) - 2 is A) cosx + 2sinx B) 2cosx + sio< C) cosx sinx - D) None ofthese. iv) crcosax + czsinax - a cosax is the general solution of 2a A ) fD^2 + a21y = "in * B) (D2 - a?)y = sin ax C) (D+ a)y = cos ax D)@+a)y=sinx (0a Mark) b. solr. * dl .*dY.-u-lor*. dx dx (04 Marks) c. Solve y"-3y+2y =;!;, bf vadatlon otpamneter merhod. (06 Marks) go1.o" 1I * 5!" * 6x = 0. Give that x(0) = o, = rs. (06 Marks) f;tol 3 of 4
  20. 20. I O6MAT21 a. Select the corfect answer in each of the following : i) Laplace transform of f(r, t > 0 is defined by ..-. D) fe of(rldr a1 Jellrtdr B) Je"f(r)dt c) Ff(1)dr ii) Laplace transform ofcos at is a s_ +a B)-j+a s" , C)-l , s+a- D_ t ^2 ^2 rr) ,lir.rl Ll--f rs lsl A) F(t)dr e1 [!t)6, c) t" (t) lr) Norc ofthese. ;t iv) Laplace tratrsfo1m of f(t) is A) s f(s)-f(0) B)sf(s)-f(0) c) fG) f(0) D) None ofthese. (04 Mark) b. Find | {e"t + 2f - 3 sin 3t + 4 cosh 2t } (04 Mark) c. If f(t) is a periodic function ofperiod w, then show that LtttL)t= y ^* leIrLrdr (06 Marks) lsint o<t<n/2 d. *e function f{l)={ t>nt2 , Express in lerms ofunjt step nrnclion and lind il.s Icost Lallace tansform. (06 Marks) 8 a. Select the collect affwer in each of the following : i) Inverse l,aptace transfom of s-a tt (. -uf +bt A) ercostt eorcosbt C) edcosbt B) D) e"tsinbt [., -r.-al ii) lnerse Laplace lranslorm of l-lt .A) 1. 3t+2f B) 10-3t+21 C)4-31+4f D) None ofthese. iii) L {u(t (t a)}, where u a) is a lmit step tunction is A)L a Btl c) e* D)se^ iv) L {5 (t - a)}, where 5(t - a) is n unit impulse firnclion A)e^ Bre^ C.)e Dtl (04 Mark) s +2 Find the inverse Laplace translbm of Js (04 Marks) s-s-2 tbeorem obtain ,{ (s+a,t, ., } , )(s+b) c. Lsing the coovolution (06 Mark) L J d. Solve the differential equation y"(1) + 4y(t) + 4y(t) = e{ with y(0) = 0 y(0), using the Laplace transform me&od. (06 Mrrl$) 4of4
  21. 21. IJSN O6MAT21 Second Semester B.E. Degree Examination, June/July 2011 Engineering Mathematics - lI Time: 3 hrs. Max. Marks:100 Notai l, Answet an! FfvEfall questions, choosing at least twofiom edch pu . 2. Ans qll obJecdve type queslions onl! in OMR sheet page 5 ofthe answet booklet. et ri 3. Answet lo objectlye type questions on sheets other than OMR will not be volued. PART.A 3 1 a. Selegt the correct answer : I i) An expression for the radius of curvature in parametric form is 1! ar p= !L:iL e) p= !t-!)l c; o= {tli,.Yllj i D) None orthese lz Yi Ixv-vx ] ii) The curvature of a circle is a;r A) constant B) variable c)1 D)0 iii) Ifa tunction (x) is continuous in [a, b] then O(x)=f(x)-lc< is alsoE, A) B) continuous C) Both A and B differentiable D) None ofthese x-dv iv) lf y--:,then ia atx=0is slnx dxYe A) 1 B) 0 C) Both A and B D) 2 (04 Mark!)!E b. Fhd rhe radius of curvature for rhe curve y -4a}l2a-xl ,lrhere the curve meets the x x-axis. (04 Marl(s)!65 c. State and plove Cauchys mean value theorem. (06Maft)}E d. Obtain the Maclaurins series expansion oflog (l + e*), upto 46 degree terms. (06 Marks)Ef..r ,E .!t i) ii) The value A)0 of Lim x --t U lop r __::ej_ is cosec x B)l *. huu. Lit IEI i" c) *l $ffi Drrlii{,}r--f.; /.+ lff(a) = o and g(u) = 0. ,h.n x -+a g(x) "qru1 ,o ,tl Lim flv A)x-)a g(x) B)xra -:::! Lim f(x) a, Lim flx) D) None ofthese+U S(x) x-)a C(x) iii) The necessary conditions for f(x, y) = 0 to have extremum arooi A) fv=o=fy, B)f*=o=fyy c) f"=o=fy D) None ofthese iv) The point (a, b) is called a stationary point and the value f(a, b) is calledg A) stationary point B) stationary value C) maximum value D) minimum valuez (04 Msrk) Lim tanx-x9 b_ Evaluale: (04 Marks) x --r 0 xtan x c. Examine the flrnclion f(x, y) = xa 1 ya - 2(x - y)? for extreme values. (06 Mark) -, d. Ifxyz = 8, find the values ofx, y, z for which u =--LUL isamaxinum. (06 Marks) x+2y+42 1 of4
  22. 22. O6MAT213 a. Select the conect answer : lJx i) The value of J Jxy dydx is 0x A) -L 24 sr -l 48 c)a 25 D)l -50 iD I= J Idx dy represents the area oftriangle with vertic€s. A) (0. 0) (0. r ) (r. 0) B) (0, 0) (0, r) C) Both A and B D) None ofthese iii) The funcrion Jn+l is defined forall A) Positive integels B) Real numbers C) Both A ard B D) Real numbers except for rcgativ€ ftactions iv) Thevalueof 0i1-11 is l))l 4)3.1416 B) 1.1416 C) 2.1416 D) None ofthese (0{ Msrks) r ,4-* Change the order ofintegration and hence evatuate J J fdxrtf. (04 Marks) Prcve that B( m, n) = 14j11 . (06 Markr) {E+n d. "** JJr-f !lt* rhai i=l-, i-- show = :,- cJr . (06 Mark)4 a. Select the corect answer : i) If F=x1i+xyj,then JF.di , fion (0,0) to (1, l) alorg rhe liee y = x is orl ")i c)2 Dt4 ii) Greens theorcm in the plane is applicabie to A) xy - plane B) yz - plane C) xz - plane D) All ofthese iii) With usual notatiom Causs-divergence theorem state that JlJaiv F Ov is equal to A) flF. fids n) f.lFxfids C) fJF,a.as D) Nore ofthese ss iv) Cylindrical polar coordinates (p, $, z) are given by A)x=pcos$ y=psing z=1 B) x = cos$ y=psino z=p C) x = pcos$ y=psino z=z D) None oftbese (04 Mark) b. Find the total work done by the force represented ty F =Zxyi-yj+Zxzk in moving a particle around the circl e x2 + y2 = 4 . (04 Marks) c. State and prcve Gieeos theorem ol the plane. (06 Msrks) d. Express divergence of F, where F = xi - yj + z k in spherical polar coordinates. (06 Mrks) 2of4

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