1. I'SN 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. Ansb'el 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(I'3 JI(I, €o=8.854 xllJ12 Fhtt'
E PART_A
f9 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 zero
f^r
ii) Rayleigh-Jean's theory ofmdiations agrce with expe metrtal results for
A) all wavelengths B) shorter wavelergths oDly
EY C) longer wavelengths only D) middle order wavelengths only
Pi 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 by
E3 A) photoelectuic effect B) comptor effect
C) difftaction by crystals D) Ramon effect (04 Mrrks)
!5 b_ State and explain de-Broglie's 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, the
z 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. I0PEY12'
2b. State and explain Heiselberg's 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.10.4ms'r 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)
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3. )
tjpnvrz
PART _ B
5 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 num'ler 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. ih'e 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)
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4. 1OPEY12
7b. 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 ms'r 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)
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5. USN 10PHY12t22
First/Second Semester B.E. Degree Examination, June/July 2011
_;
Engineering Physics
Time:3 hrs. Max. Marks:100
Notei l. Aksi'et any FIW fut questions, choosing at leasl ttoo from eoch pqrt.
2. Ahsy'et all objectire qtpe questions only in OMR sheet page S of tie answer bookla.
3. Ansn'et 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:Jxldms't, 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)-
2
fi 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) 0.1
z
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) zero
E 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. 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 ofwidth'a' 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 atT'0kis
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)
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7. 1oPHY12l22
PART - B
5 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 won't 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) infir'1r 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. I0PIIYt2l22
1 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. 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"I'OO
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 Maclaurin's 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)^*' t'l) 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, <l
g
z ' 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. 1OMATl1
3 a. Choose the conect answer :
i) If u = ax'z + bf + ab-ry, tt ={} i,
"n Ax'Ay
A) Zerc B) a+b + ab C)ab D) None ofthese.
ii) The Talor's 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*d'dv
'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
=
Ax'd
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 =y'z r+z"x J +x'yk 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=r'i+y'j+r'f , then frnd gradg, divF, curlF. (06 m'rk)
c. Prove thal divCurlF=V.VxF=0. (05 Mrrk)
d. Prove that the cylindrical coodinate system is orthogonal. (05 Mrrks)
PART. B
5 a. Choose the corect answer:
i) The value of [sir'xcosuxdx 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. 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
'
l'1 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: (y'e'' +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
,f'e*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. 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 9't 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 Marko
8a- 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 (X'AX ) 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. 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 page't*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$=# ,,
=.9
aE 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 (xr'1)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 then
5.c
.. X-+V-=n -- X-+V-=n-
A)
au au B) au au au au
x_+v_.=rnu t))
au Au
x_-v_=nu
' Ax'Ay ax'Ay C)
Ax'Ay Ax'Ay
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, then
z ffi,r.0*ro
A)1 B)r C) 1/r .D) Zerc (04 Marks)
E
b. If u=x'zlan r(y/x)-y)lan-r(x/y).sho*trru, -a'l = *1-yl (04 Mnrk)
oxdy x'+y'
l of 4

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) Jsin'xdx 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. 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-B
5 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
dx'dxdx'dx-' ct $-ey=r Dr 9+ey=e
iii) What is the value 01 !Y. 1o, 1r.61l6r"orial equation
(t r- zxy cos x' * zxy)d* * (.io *' - r')ay = o
A; 2x cos x2 - 2x B) 2y cos x2 - 2x C1 2x cos x2 - 2y D; .2x cos x2 - 2x
iv) The differenlial equarion ofthe family f - 4a(x r aJ is
u'-dY[*r1,dv)
o, -
d* 2'dx) u', r' -,- dv f* *1u dY )
' dxt 2' dx)
c) y,_2vdyfx+l,dy)
- 'dx 2'dx) or ,'=zut(**ulll
'd*( 'd*) (04 Marks)
b. Solve dyldx = e3*
2Y
+1'zg-rr (04 M'rks)
dv
c. Solve cos'y (06 Marks)
d;+xsrnry=x
d. Find the orthogonal trajectories of the family of confocal conics- { * J- - t , *n"." l,
a' b'+). "
the paiameter. (06 Mark)
6 a. Select the correct answer :
.l1l
i) I he series . converges if
f; ,J* T*
A)P>0 B)P<1 C)P> I D)P<1.
ii) h a positive tem series Eu. , if = , *"" the series diverges for
A)i"> I B)1"<1
"t]-.y ^
c)?"=1 D)1,<r.
iil)rhe itermo.he"r_ .-,.
[i_?)'.[i_i)'.[$.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. 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)
246I
7 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 2
8 a. Selecl the cofiect answer :
i) The velociry of the moving particle along the cun'e x = t3 + 1'y =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 Dr'a.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. 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 is
llv 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 ofthese
2E b-a b+a b+a
iv) Maclaurin's series expansion of 1og(l + x) is
8e
E-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 4'n degree tero. (06 Marko
9E
6E
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 is
E 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. 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 {r'aa**Noy= [(9-9]*0, or
; 'R' dy dx ) {r,aox-Nay=(# #)".,
2of4

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 Green's 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 -B
5 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 (D'z- D -2)y = 1 - 2x *
- 9e by the method of mdetermined coefficients. (06 Marks)
6a. Select the corlect answer ilr each ofthe follovr'i[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. vr'ith 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" 1'I * 5!"
* 6x = 0. Give that x(0) = o, = rs. (06 Marks)
f;tol
3 of 4

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 Je'llrtdr B) Je"f(r)dt c) F'f(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
L'l--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 ^* le''IrLrdr (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-l't
.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. 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 also
E, A) B) continuous C) Both A and B
differentiable D) None ofthese
x-dv
iv) lf y--:,then ia atx=0is
slnx dx
Ye 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 Cauchy's mean value theorem. (06Maft)
}E d. Obtain the Maclaurin's 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 f'lv
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 aro
oi 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 called
g A) stationary point B) stationary value C) maximum value D) minimum value
z (04 Msrk)
Lim tanx-x
9 b_ Evaluale: (04 Marks)
x --r 0 x'tan 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
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22. O6MAT21
3 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 f'dxrtf. (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=x'1i+xyj,then JF.di , fion (0,0) to (1, l) alorg rhe liee y = x is
orl ")i c)2 Dt4
ii) Green's 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 Gieeo's theorem ol the plane. (06 Msrks)
d. Express divergence of F, where F = xi - yj + z k in spherical polar coordinates. (06 Mrks)
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