This document appears to be an exam for an engineering chemistry course. It contains 8 multiple choice questions across two parts (Part A and Part B). The questions cover topics like fuels and combustion, electrochemistry, corrosion, polymers, water analysis, and more. The maximum marks for the exam is 100. Students are instructed to answer 5 full questions by choosing at least 2 from each part, and to answer all objective type questions on the OMR sheet provided.

1. 06cHEt2t22
Time: 3 hrs- Max. Marks:100
Notei I. Answer any FtVEfutl qaestions, choosing at least twofrom each parL
2. Answer all objective trye q eslions oilt! on OMR sheet page S of ihe answer booklet.
3. Ansper to objecrive type questions on sheets other than O-MR ilt not be valued.
PART_A
I a. Selecl lhe correcl aDswer:
i) The rendency of knocking is more in ......... hydrocarbons K=
A) Aromaric B, SEaighl chain Olefins
C) D) Cy"l,y'.ini-
ii) Catalyst used in fluidized-bed caralyic cracl,irg is-
us:dl cracl,ing is
'r7-rn"";L t,'
lat:lyjr fl' _-"otl, ',j
.,
A)Zrol+Ceo) B)Al2Or rFe2Or :t tEr.,. ";r /
C ) Fluidi,.ed At2O, andSiO2 DrAt2Oj CeOr :.. "'" /
iii) Petr_ol obtained liom petroleum oil is subjected 1c rcforming because ti.f 7/
A) To remove impurities and water B) For structural modifrcation
C) For degradation ofhigh mol. weight hydrocarbons D) To reduce weighr.
iv)...........jrmctionisusedintheconversio[ofsoiarenergyintoelectricalenergy.
A)p-n-p B) -p-n C)p-n D) none ofthese. (04 Mark)
b. DeIine net and gross calorific value of a fuel. Describe how calorific value ofa solid sample
is determined- (07 Marks)
'c. Explain the terms 'octane' and 'cetaae' numbe$. (05 Mark)
d. Calculate the gross calorifi-c ofa sample ofcoke frolr the following data:
Mass ofcoke: 0.73x10-rkg Water equivalent ofthe calorimeter: 0.32g kg
Mass ofwater: 1.25 kg Specific heat of water: 4.187 kJ kg"LK1
Rise in temperature: 1.9K. (0a Mark)
a. Select the corect answer:
i) Glass electode can not be used in the presence of fluoride ions because
A) alkalile eror B) loss its attivity
C) glass membnne dissolves D) leads to as),rnmetric potential.
a-! ii) The value of EMF is , if AG : -212.3 klll.rcI, T = 298 K and
Faraday constart: 9.65 kJ,nr'lmol.
E3 A) 11.0 V B) r.r v c) 2.2 V D) 22.0 V
iii) EMF ofa cell mainly depends on
oi A) Size ofthe cell B) Quantity oithe electrolyte
C) Weight of the cell D) Difference between Ec,rn,a. and E*.a..
z iv) Use of secondriry reference electrode is prefefied over primary reference electrode
becasue
A) it is light weight B) it is compact
E C) it is reversible with electrol]te D) it gives constant and reproducible potential.
(04 Marks)
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2. 06cHEt2/22
2 b What are ion-selective electodes? Explain the principle, construction and working of a glass
elecbode. (06 Mark!)-
c Derive the Nernst equation for a single elecrode. Wdte Nemst equation for the give! cell:
Cu(s) Cu (aql Ae (aq) lAg (0s Mark)
d. Calculate E:",./2, , ifthe emfofzn I Zn"(a=l) | lNl*(a=l) Ni is 0.51 and
Ni I Nl.(a=r) II sCE is 0.49 V. Given E:cu = 0.24v. (05 Mark)
3 a. Select the collect aaswer:
i) The electrolye used in Li-MnO2 battery is
' B) 6M KOH
A) l0% Hrso4
C) Li-halide a1ld organic solvents D) NLIqCI + ZnCI,
ii) Hot KOH solution should be used in H2O2 fuel cell
A) to incrcase conductiviry B) to enhance the absorption ofhydrogen on anode
C) for better reduction ofO2 D) to avoid dilution ofKOH.
iii) In some batteries, graphite is used as an additive, because
A) it reduces the total weight B) it absorbs the moistue
C) it improves the conductivity D) to increase the sulface area ofthe electrode.
iv) Lithium metal is used as anodic mate al in Lithium batte es because
A) it readily iooses the electrons B) its rcduction potential is very high
C) its size is very small D) it rcadily reacts with ivater. (04 Marks)
b. What are the modern batteries? Explain the construction, reactions and applications of a
Ni-MH battery. (06 Mark)
c. Explain the following chamcte stics ofa battery:
i) Capacity ii) Cycle life. (0s Mark)
d. How is a fuel ceil, in which chemical energy is coltverted into electrical enetgyj more
advantageous over the cotrventional batteries? Jusdry - "fuel air is only an energy
conversion device and not an energy storage dcvice". (05 Marks)
1a. Select the corect answer:
i) Appropdate corosion control for a buried pipeline is
A) Painting B) Cathodic protection C) Calhodic coating D) Meta]lic coating
ii) Cathodic coating is
A) coating oftin on iron B) coatiag ofzinc on iron
C) conversion of whole specimen into cathode D) immemion ilto ho1 ZnC12 soln.
iii) A bolt and a nut made ofthe same metal is p.efered in practice to avoid
A) stress comosion B) difGre[tiai aeration corosion
C) differential metal cooosion D) caustic emb ttlement.
iv) The rcaclion at an anode duringcotosion is
A) Oxidation B) Reduction . C) Redox D) Conversion. (04 Marks)
b. Define metallic colrosion. Discuss tbe following factoN which inlluence the rate of
corosion: i) Natue ofcorrosion products ii) Hydrogen over-voltage (0s Marks)
c. What is arodizing? Explain the anodizing ofaluminium. (06 Mark)
d. Write a note on phosphating. (0s Marks)
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3. 06cHEt2t22
PART _ B
5 a. Select the cofect answer:
i) 'Pickling'is
A) alkali cleaning B) acid cleaning C) organic solvcnt cleaning D) water cleaning
. ii) As lemperature increases the elecbo-deposition
A) decreases B) increases
C) no effect D) first decreases and then increases
iii) The experimentally detemined discharge potentiai oi an electrode is 2.58V and its
theoretical discharge potential is 1.53V; theIl over-voltage is
A) 3.345 V B) -r.0s v c) 4.11 V
D) 1.05 V
iv) As over-voltage increases, electroplating
A) decreases B) increases C) no change D) increases and decreases.
(04 Mark)
Explain the significance ofthe following electroplating:
i) Polarization ii) Over-voltage. (08 Mark)
Differentiate between electroplating and electroless plating. (0,t Mark)
d. Explain the sudace propagation and electroplating olchromiurn on a steel material.
(01Markj)
6a. Select the corect answer:
i) The molecular ordering in nematic phase is
A) No positional order and orientational order
B) Perfect positional alld orientational order
C) No positional but orientational order
D) Total loss in molecular ordering.
ii) The layered structure is found in ........,..... phase.
A) nematic B) discotic C) meso
iii) ln porenriometric acid-base rilralion, the indicalor elecrrode is
A) calomel B) Ag-AsCl C) Glass-electrode D) platinum
iv) Mathematical expression ofBeer-Lambert law is
A)A:-logT=C/ B)A=1ogT=C/
C) log (Io / [) = eC/ D)lo9(11116)= eC/ (04 Mark)
b- What are the liquid crystals? Discuss the classification, with examples. (07 Mark)
Eraiain the liquid crystalline behaviour in PAA homologous series. (01 Mark)
d. Explain the theory behind the conductometric titation. Draw aod explain the
conductometric titration curve for the following fitations:
i) A strong acid and a strong base ii) A strong acid and a weak base. (05 Mark)
Select the corect answer:
i) A poll.rner of high optical clarity used in preparation of lelses is
A) Teflon B) Phenol-fonnaldehyde C) Neoprene D) PMMA
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4. 06cHEt2t22
7 a. ii) Tte glass transirion rempemture (TJ depends on
A) t,?e ofmonomer used B) t)?e ofpol),rnerizarion
C) chemical resistance D) molecular weight
iii) Monomer used in Teflon is
A) fluorine B) bisphenol C) tetrafluoroethylene D) epicblotohydrin
iv) The moiecular weight of addition polymer is integral multiple of rhe motecuiar weighl of
monomer used because
A) it is a linear pollmer B) no solvent in added
C) it involves a catalyst D) no elimination of byproducts (04 Mrrks)
b. Wlat is polymerization? Explain the solution polymerization. (0s Marks)
c. Justiry rhe following :
i) Crystalline pollmers possess higher streogdr tharl amorphous pol1mers,
ii) Themal control is mther difficult in buik polymedzation. (0s Mrrk)
d. What are lesins? Give the synthesis, properties and uses of phenol-formaldehyde. (06 Marls)
8 a. Select the correct ans er:
r) tne puresl ioral oj tdte[ Is
A) dver B) Borewell C) Rain D) Spring
ii) In chlbride analysis (Argentometric), the end point is bdck-red, due to
A)AgCr2O7 B) AsNOI C) AeCl D) Ag2CrOa
iii) Complexing agent for spectophotometric analysis oftitrate is
A) SPADNS B) Ammonia C) Phenol suiphonic acid D) phenol disuiphonic acid
iv) The method removing temporary hardness is
A) soda process B) boiling C) distillation D) reverse osmosis (04 Mark)
b. Define alkalinity. lf 'P' represents the phenolphthaiei! alkaiinity and .M, represents rhe
methyl orange alkalinity, for a given sample of water, how do you pledict the rypes of
alkalinity, when (i) P = M (irP= %M (iiDp>%M (iv)p<%M? (osrrlarrsr
c. A river courses firsr *uough a tenain rich ir limesrone and througb a terrain rich rn gypsum.
Identify the tlpe of hardness it accumulates in its path. Explain the need for maintaining a
constant pH ard the colour changes that occur, when the hardness of this water is beiflg
determined by titmlion against EDTA, using Eriochrome black-T as the indicator. (05 Marks)
d. Gve an account o[the following:
i) COD is higher lhan BOD.
ii) IlgSOa and Ag2SOa are to be added during COD analysis.
iiD Chlorine is a powerful disinfectant only at lower pH values. (06 Marks)
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5. USN 06cHEt2t22
First/Second Semester B,E. Degree Examination, Jr*ne/July 2011
Engineering Ghemistry
Time: 3 hrs. Max. Marks:l00
Note,l.Akswef a ! FIWfu questions, choosing at least t ro from eorh part,
2-4nswer all objeclhe 4)pe questions otul! in Otr IR sheel page S af.t*e Answer Booklet.
3.Answer to objective 4)pe qaestions on sheets othel tht i OMn ffi ,rat be yalaed-
PART - A
1. a. Choose the corect answer (04 Marks)
i) Ignition point of fuel depends on
A) Hydroger content B)
Oxygen cootelt
C) Sulf,r content D) Nituogen corterit
i, Mention the catalyst and working temperature required for fllidized bed catalltic
cracking process.
A) Clays and 3500C B) Clays and 55O9C
C) Zeolites and 55OoC D) Zeolites aoil.3500C
iii) Reformation is a process of
A) Structulal rearangement of hydrocarbon
B) Breaking of heavier hydrocarbon to lower hydrocarbon
C) Stuchral rearangement without altering the number of qsfbqr
D) None ofrhese
iv) Mentior the catalyst used to convert caxbon monoxide and,hldrogen at 1g0 _ 2500C
to form a mixture of aliphatic hydrocarbons
A) Copper B) Tin C)
Zirrc D) Iron
b. Defiae calorific value offuel. Explair the determination of c.il.sific value of solid fuel
sample. (06 Marks)
Define octalle number. Exptain the methods of improving octane fnmrber. (05 Marks)
d. Explain tie consrrucion and r.rorkirg ofphotovoliaic cell.
(0s Marks)
2, a. Choose the correct answer (04 Mark)
D Galvanlc cells can convert
A) Chemical energy into electical e[ergy
B) Electrical energy into chemical energy
C) Solar energy into chemical energy
D) None ofthese
ii) Electochemical reactiols axe spontaneous when change in froeenrlgy is
... A) Positive B) Negative
iii)
C1 Zero , t Non"
Cell porenlial ofa concenrrarion cell is positive only if
.. 4) cr.c, exchanges
iv)Glass electrode
B) Cr=Cr c) a2>cr .D) Nore
A)Fluoride ions B)
Hydrogen iuns
C)
Chloride ions D)
None
b. Define single electrode potential. Derive Nemst equation for si+ile.6lectrode.
(06 Marl$)
c. What are reference electrodes. Explain the construction arfl iwoiking of coloured
electode. (05ll{arl(!)
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6. L
06cEm2D2
d. For the cell, Fe,Ec'z.(O.OtStv0 llag*(0.t3M) / Bg, wdte the cell reaatior and calculate
the emf of the cell at STP, if standard electrode potentials of iron and silver arc -0.44v
and 0.80V rcspectively. (05 Marb)
3. a. Choose the corect answer (04 Mrrks)
i) Shelf life ofa dry cell is limited to 2 years because
A) Zinc rnetal always contacts rvith ammonium hydroxide.
B) Zinc metal always contacts with ammonium chloride.
C) Zinc metal always contacts with potassium ahloride
D)None oflhese,
ii) Mention the electrolyte used itr Nicad battery
A) Sodiumhy&oxide B) Sutfiuic acid
C) Hydrochloric acid D) Potassium chlodde
iii) Which bau,ery has more power density?
A) Zn - MnO2 B) Zr- Ag;.O C) Zn-HgO D) Zn-O2
iv) Which electrolyte is used in hydrogen - oxygen fue1 cell
A) Potassium chloride B) Potassium nitate
C) Potassim hydrodde D) Potassium sulfate
b. What are primary batteries? Explain the co[struction and working ofa dry cell.
(06 Marks)
c. Explain the construction and working oflead - Acid batiery. (05 Marks)
d. What are fuel cells? Explain the construction and working of hydrogen - oxygen fuel
cel[. (05 Marks)
4. a. Choose the corect answer (04 Marks)
i) Rate of corosion is more if
A) Cathodic area is larger than anodic area
B) Cathodic area is smaller than anodic area
C) Cathodic and anodic areas are equal
D) None ofthese
ii) Water line conosion is an example of
corosior
A) Differenlial metal B) Differential aeratio!! corosiotr
C) Stress conosion D) None ofthese
iii) Caustic embdttlemetrt is a classical example of
A) Stress conosion B)
Differential metal conosion
conosion
C) Differential aeration D)
None ofthese.
iv) Galvanising is the process of coatitrg a base metal
by :
A) Tin B) Copper C) Zirrc D) Nickel
b. Define the term conosion. Explain the electrochemical theory of corosion with respect
to ircn, (06 Marks)
c, What is cathodic protection? How a metal is cathodically Eotected by sacrificial anode
method. (06 Marks)
d. Write a note otr Galvanisation. (04 Marks)
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7. 06cH812t22
PART -B
5, a. Choose the corect answet (04 Msrks)
i) Poladzation effect will be minimum during metal finishing if
A)
Rate ofanode reaction is equal to cathode reactioL
B)
Rate of anode reaction is smaller than that of cathode.
C) Rate ofanode reaction is greater than that ofcathode
D) None ofthese.
ii) Which anode is used in chromium electroplating?
A) Soluble chromium anode B) Insoluble anodes
C) Inert anodes D) None ofthese
iii) When the metal structure to be plated is iregular, the process employed is :
A) Electroplating B) Electroless plating
C) Electopolishing D) None oftheie
iv) Reduchg agent used in electroless plating ofcopper is :
A)Sodiwr hnophosphire B) Formaldehyde
C) Sodium acetate D) Sodium Succinate
b. Detrne the term metal finishing. Mention any four technological importa[ce of metal
frnishing. (06 Marks)
c. Explain the process ofelectroplating ofgold. (04 Mark)
d. What is electroless plating? Explain the process of electoless plating of nickil.
(06 M.rks)
6, a. Choose the correct answer (04 Marks)
,) The mesophases which are formed by heating and cooling are called
A) Lyotropic phases B) Thermotropic phases
C) Monotropic phases D) None ofthese
ii) Mention thg electodes used in conductivitv cell
A) Platinum elechode B) Glass eiectrode
C) Both are platinum electrode
D) Glass electode and platinum electode
iii) In the estimation of FAS by potediometry, the indicator electuode used is
A) Calomel electrode B) Glass electrode
C) Platinum electrode D) None ofthese
iv) An ioll selective electrode used in the determination ofpH ofa solution is
A) Calomel electrode B) Glass electrode
C) Ag - AgCl electrode D) None ofthese
b. thermotrophic and tyotrophic liquid crystal with suitable example. (06 Marks)
,Define
c. What axe homologous series? Explain the liquid crystalline behaviour in pAA
series.
(05 Marks)
d. Explain the nature ofconductometdc gaph for the folJowing tihations
:
i) Saong acid with strong base ii) Strcng acid with w;ak base. (0s Marks)
7. a. Choose the correct answer
r) Temporary hardness ofwater is caused due to the presence of dissol.,r"a ,ao(31*'*)
A) ?lcium and magnesium carbonates B) Calcium and magnesium bicarbonates
C) Calcium and magnesium sulfate D) Calcium nitate
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8. USN O6MAT11
X'irst Semester B.E. Degree Examination, June/July 2011
Engineering Mathematics - I
Time: 3 hrs. Max. Marks:100
Note|l.Answer FIW full q estiohs choosihg at least t to from each part
. 2-4nswer all objectire q)pe questiohs only it OMR sheet pqge S olftheAnswer Baoklet.
i.Anstter to objective tlpe queslions on sheets other than OMR i'ill t ot be talued-
PART-A
I a. select the correct answer :
i) Ify: a" then y, is
41m loga.amx 8)1m loga.1'.a^" C) loga.a'* D) (m loga)2.a'*
€ ii) The nr derivative ofsis(cx + b) is
A) a" sin(ax + b + I! ) B)-a'sin(ax+b+
c,2 2 T)
C)aosintax+b+Ir
- D) a' sin(a + bx + 11)
5.i 2'
iii) If { be the angle betwe€n the radius vector and the tangent at any point of the curve
r = f(0) then,
'dr.!9
A) cotq=!9 Bt trnd = Quna=f D) None of these.
iv) The Pedal equatiol in polar coorditate system is
A) l0r - 0: l:-t B)r:(1 - cosO) ^
tt d0 - I I f_t /dr']
t))_= I _l
e! taDo
' =_
dr p' r' r'[d0]
(04 Marks)
b. Fi-od the n'hderivaLive ol ) e"'sin(bx r- c).
. (04 Mark)
c. lfy|a+y' r'm 2x. prove thar (xl ltyn,z, (2n+ l)xyn-r t(n2 - m')y, = 0 (06 Mrrk)
d. Fin<l the angle between the cuwes . = *O . = (06 Mark)
, _** ;..,aO
2a. Select the corect answer :
i) Ifu=x,, + is €quatto
theD
9E dx.N
A) x*l(ylogx + 1) B)xv l(ylogx+ 1) i) xv-tlxlogx + 11
..
ii) If u be a -1)
homogeneous function of degree n in a and y then
ar
^4+vff=n sr-$-r.fr -n, 6,1 *A-r4_n, *. x__ v_ = nrr
D)
au Au
&'ay
iii)Ifu=x'z+2xy+f+x+y then ,@*y4 i. rqoul to
oi A) 2u B)u c)zerc D) None ofthese.
iv) Ifx=rcoso, y : r sino, then lG:I) is equal to
z o(r,u)
A)1 B)r C) llr D) Zerc (04 Mark)
E
IJ u - x) tan-r() /x) - y) tan r(x/)). show lf,a i" - *i -y' (04 Marks)
dxfo x2 +y2
I of 4

9. 06cHEl2/22
The products that are fomed urder anaerobic oxidation in BOD experimeat are
A) Carbon dioxide and water B) Ammonia and hydrogen sulfide
C) Both A and B ' D) None ofthese
iii) Role of oxidizing agent in detemination of COD of waste water
A) It helps to reduce wastes present in water
B) It helps to oxidize wastes preselt in water
C) Both A and B
D) None ofthese
iv) Which indicator is used in determination of chloride ions present in water usiog silver
nitate solution?
A) Potassium dichromate B) Potassium nitrate
C) Potassium sulfate D) Potassium chomate
b. Explain the determination oftotal hardness ofwater sample. (06 Mark)
c. What^is meant by desalination? Explain the process of reverse osmosis. (05 uark)
d. 25cm of seuage sample lorCOD is reacfed with I0cmrof K:CrzOr solurion and rhe
uneacted K2C.2O7 requires 8.5cm3 of0.025N FAS solution. Under similar conditioos,
in blank titmtion 15cm3 ofsame FAS is used up. Calculate the COD ofthe sample.
(05 Marks)
8. a. Choose the correct answer (01Marks)
i) Phenol - fomaldehyde resin is commercialiy called as
A) PVC B) Nylon C) Bakelite D) Teflon
ii) Hexamethylene diamine and adipic acid gives the following polymer.
A) Nyton 66 B) PVC C) Nylon D) None
iii.) Sulfur is used particularly in
A) Compoundingofresins B) Vulcanization of rubber
C) Both A and B D) None ofthese
iv) Wl ch tlpes ofpol-vmer is used in smart window material.
A) Teflon B) PVC C) Polyaniline D) Buna S
b. Explain the ftee radical mechanism of addition pollmerization. (06 Marks)
Explain the Feparation and mention the application ofthe following pollmers :
i) Pollmethyl methacrylate ii) Teflon (06 Marks)
d. Define the te1m glass transition temperature of a polymer. Mention the factors which
affect the sarne. (04 Mark)
4 of4

10. O6MAT11
c. lf u=x2 -1. v=2xy and x - r cos0, y = rsin0, frnd 4I*.
d(r,0)
(06 Marks)
d. ,n estimating the cost of a pile of bricks measued as 2mx15mx1.2E, the tape is stetched
1% beyond the standard length. If the count is 450 bricks to 1 cu.m and bricks cost Rs.530
per 1000, find the approximate error in the gost. (06 Mark!)
3 a., Select the corect answer :
it lsinn xdx is eoual to
nrl n+l
r1,., ^ D-1. -1
A) "-l-11., B) C) i"-, Dl -iI",
nn =- tr
... f. r
hI
,) lsln'xcos- x ox $eouallo
A)l 16
B)!
32
c)a
324
D)I
iii) The cuwe is s,,rDmetrical about the
f(Za-x)=x1
A) y - axis B) x - axis C) x and y axis D) None ofthese.
iv) The asymptote for the cuwe r = a sin30 is equal to
A)0=a B)0=30 C)0=a D) No aslmptotes.
(04 Marks)
b. Using the reduction formula, evaluate Jtanu x ax . (04 Mark)
" ------. o*- (2n+I)l a" ,
z
c. If n is a posirive inreger. show thal J*'Jz* - *' a-,*o ?
(06 Mrrks)
d. Trace the Leminiscale if = * G -x') (06 Marks)
4a. Select the corect answer :
i) Area bounded by the curve r - f(0) and the mdii vectors 0 : o, 0 = p is
A) B)
10-
cr -: lr'do
.,J o1 :18-
Jfdo
; Jr,do ; Ir'de
ii) The length of the arc of the cuwe y - f(x) between the points where x = a and x = b is
a;"dr.(fl)'
l,"
,i,F*fl* ",'ilF.[tJ'l* ,,'1ffi*
iii) The surface area of the solid generuted by the revolution about x-axis of tlle arc of the
cuIvey= l(x) lrom x. a to x=b is
A) f2rwds gr )'
fzrvdx C) J2Trxds
D) I2?rds
.. d [0.-. ].
rv) ll(x,(I)dxl ls equal to
-l Li
do l
,cr igrtx,qtdx s; iarl*,.,'t a* c)i1rt*.ota*
jd(t
D) t+r(x.o)dx (04 M,rk)
iAq ;dx :oq
b. x"t + y'" a'''.
Find the entire length ofthe asteroid = using tlrc value of
fl = #.
(04 Marks)
2.,f4

11. O6MAT11
c. Find the volume of the sotid geoeiated by the revolution of the cardioid r=a(1 + cosO)
about the initial line. (06 Marks)
Evaluarc
jljjdx. cr> 0. (06 Marks)
log x
PART -B
5 a. Selectthe colrect answer :
i) rte order orthe ,
eo*,i", l,.rf+)''l =al*l'
dx/.1
,.
L ax,/
A)1 B)2 c)3
D) None ofthese.
ii) The standard form ofa linear differe*ial equalion ofthe fust order is
'-'
A) 9+v=p
dx ' 9+Py=a cr
sl'd 'dx 9-pv=p
' o) 9+av=a
'dx '"
dY,
'tu
iii) what is the value of for the differenlial equation
(t + zxycos x' - zxylx r (sin x' - x')ay = o
A) 2x cos x2 - 2x B) 2y cos x2 - 2x C.12x cos x2 - 2y D) -2x cos x2 - 2x
iv) The differcntial equation ofthe family l= 4a(x + a) ls
. dv( I dv)
A) v' = --:1 x +-v--1 | ,, ,, -rdy[** tudy)
dx 2' dx ) 'dx( 2" dx)
cr ,, =zudYl**-!rdY l
'dx[ 2' dx o, u' = zr9[,.* u dY)
" dxt 'dx)
(04 Msrks)
)
b. Solve dy/dx = el"-'?Y + x'?s-'zv (04 Merks)
c. solus !I -.1s;12y = yr 6632 y (06 Marks)
Fid the orthogonal trajectories ofthe family ofconfocal conics
' "'*-y'-=1,*6"."1i"
a' b'+)"
the parameter. (06 Marks)
6a. Select the corect answer :
i) The series
.111, , r........ converges if
,, ,, ,.,
A)P>0 B)P<1 c)P>l D)P<1.
u) Ir a posrhve term senes }un , --
lf u
Lim a
=
- , then llre senes drverges lor
-
tl)@ un ^
A)1,>1 B)l<1 c)),=1 D)r<1.
iii)rhe nb termorthe *,i* [4-?] '*[4-1) '*f4, 4)
1) 2) (3" 3] *...-.-..."o i.
[r' 2'
"
o,ftn rt)'_n+tl-' *n+ll 6lf rn-rr,,_n*r l' plf rn+t)"'_n+rl"
L n"-' nl I n'"
".,ftn-r)"" nl 1," "l 1"" nl
.2345
iv) The series
f -;ri-:. r......... is
A) Conditiomlly convergenl B) Absolutely coNergent
C) Divergent D) None ofthe above. (04 Marks)
3 of4

12. O6MAT11
b. Testtheconvergenceoftheseries 1 * 3 * 5 *.-.. 2n -l +..-. co (04 Marks)
t.2.3 2.i.4
t.4.5 n(n+lxr + 2)
c. Discuss rhe nanre ofthe series
' ,'^*l,t)'^,*l.1ix,,...-..o (x>0i (06 Mark)
2 j 4./ 5,/
d. Discuss the absolute convetgence 'and conditional convetgence of the sedes
579tl
---+---
246I +......-.- (06 Marks)
7 a. Select the corect answet i
i) If l, m, n be the dircction cosine of the nomal to the plare, then the normal form of the
equation ofthe plare is
'A)ln+my+oz:0 B)ln+my-nz=p
C)ln+my+IIz=p D) None of these.
ii) Sfnmetrical form ofthe equations ofthe straight line through the poirt A(xr, yr, zr) and
having direction cosines l, m, n are
,., x xr_ y-Yr _ z-z
A Rl x+xr y+yr z+zl
| ,-n , lm
at x-xt y yt z zl D)lx+my+nz=0.
'' l,. *y -;
iii) The equation of any plane through the lt" *
:t =
? =
?
A) a(x xl) + b(y y1) + c(z - z1):0 where al+ bm + cn = 0
:
B) a(x + xr) + b(y + yr) + c(z + zl) 0 where al + bltr + cn = 0
C) (x + xr) + (y + y, + (z + zr) = 0 where al+ bm + cn = 0 D) None ofthese.
i) x+l 1
A point on tle line :j--: -
v-l --
- z is
' 2 I -t =
A) (1, 6, 1) B) (-1, 6, -1) c) (1, -6, 1) D) (1, 6, -1) (04 Marks)
b. Find the equation of the plane which passes through the poiat (3, -3, 1) aod is parauel to the
plane2-+}y+52+6=O (04 Mrrks)
c. Show that flrc thes ]j=J:f z+3. x 8 v-4 5
.445713 -z are coplanar. Find their
common point and the equalion of the plane on which they lie. (06 Msrks)
d. Find the magnitude and the equations ofthe shortest distance between the lines
xyz.-Ax2y-tz+2 (06 Marks)
2 3t 3 5 2
8 a. Select the corect answer :
i) Tle velociry of rhemovingpaniclealongtfrecunex-rl .l.y t2.z 2r. 3 is
A)(r'r lI'11 -(21 -l)k B)fl'- tli 2rj-(2r rl)k
c)3lir 1j | (2r , l,k D)3l,ir2ri I ik
ii) The divergence of a continuously differentiable vector point function F is denoted by
divF and is dehned by
a,i9-;9F,r4 B,iT.ig-k9t c);!l -a)
dx 4 ,Jz a). i9l-pdF D) r.aF r "fu
d( .) cz Az d idj+r.91 Az
iii) div curl F is equal to
A)i+j+k B)1 C)0 D)2.
iv, Il F. x2 ' y2 - z2.lhen curl grad F is
A)-l B)0 c)l D)2. (04 Marks)
b. Find dir F. r.rhere F = grad {xr + y' zr - 3x1z) (04 Marks)
c. Prove that curl (g.ad O) = 0. (06 Mark!)
d. Show thal roR is any irrotational vectot for any value of c. but is solenoidal if o + 3 : O
where R: xi + yj + zk and r is the magnitude ofR. (06 Mark)
*****
4 of 4

13. l
I
USN lOMATI1
I
t." First Semester B.E. Degree Examination, January 2011
Engineering Mathematics - t
Time: 3 hrs. Max. Marks:100
Note'. l. Answer any FIYE full questions, choosing ot least twa each palL
from
2, Answer all objectite t)pe questions onb) on OMR sheei page S of the answer baokleL
3. Answer to objectit'e Drpe questiot s on sheets other than OtrlIR ii not be ralued,
I a. 'Choose the correct answer :
E i) Iff(x) is continuous in [a, b], differenriable in (A b) and f(a): f(b), then there exists
C e(a, b) such that f(c):0.
A) unique B) inlinite C) al least one D) no such
E
i, The Maclaurin's series off(x) - k(constant) is,
EO: A)(x)=k B) f(x) = 0 C) does not exist D) f(x) = k!
ijil The nrh derivative of i.
(x r
-L- 2)'
^. (-lf (n+2)l ^. I
.
^., 7i* * 2f -
o, c) zERo
t* , ,r
D) None ofthese,
Etl
iv) The 126 derivative of y -err r;"[, 1,
$@ary B) -4096y c) (32)y D) None ofthese.
!e (04 Marks)
b. Ifx=tan(logy), prove that (l+x)y,1r +(2Ix(- 1)yn + n(Il l)ya r=0 (06 Marks)
bd c, Expand log(sec x) by using the Maclaudn's series expansioa
EE (05 Marks)
.sd d. (0s Marks)
2d. Choose ihe collect ans$er :
i) Which stateme[t is true?
A)
i ,3, - -, O*- are nor indererminale
.o B) 00,
C) I - is not indelerminate D) None ofthese.
ii) The angle between r = asinO and r = bcosg, is
tsE
A 1tl2 B ) ,t C) - n/2 D) None ofthese.
iii) The radius of a curvahre it the polar fo.m is,
o, +,i']t"
'-' 12[t' 2r,, _rr, B) 1r,' + r2l3t2
c, r'It'n l'1"'
+ tj + 2r' - fr) ' + 2r1, -r.,
D) None ofthese.
E. Lim )r -1r
IS,
x-+0 5'-6'
o,log(213) B) tosl :-i
-13 I c) r"J4l D) None of these.
z loc( 5/ 6) 6l LY6 ] (04 Marks)
E
Evaruare:i)::, '*+t l)::o
E
[rll:{]' (06 Marks)
c,
d.
Derive an expression for the mdius ofcurvirrule in the pedal form. (05 Marks)
Find the mdius ofcurvature of a2y at the loint ;here the curve cuts x_axis.(05 Mark)
=3,ar
1of4

14. 10MAT11
3 a. choose the conectaoswer: .}
i) Ifu= a* + bf + auxy, th". 0't ,^
'aq tt
A) Zerc B) a + b + ab C) ab D) None ofthese.
ii) The Talor's series of{x, y) = xy at (1, l) is
A)l+(x-1)+(y-1)l + [(x - l) + (y- l)] + [(x- 1Xy - l)l
B) 1
C)(x-lXv-1) D) None ofthese.
iii.; The Jacobian of transformalion from the Ca(esiaD to polar coordinate system is,
. A)l B) lcosg C) lsin0 D)None ofrhese.
iv) Ifu = f(x, y), x: O(t), y = iy(t), then du/dt is,
n; !I!I* d'$ et
gl*9I c)
to & & dY
'dxdt dy dt dt dt 'a(dr * ay dt -'
D)None orrhese.
(04 Mirks)
b. 1J.1, u =It. .6o*,t,u, *$*y$ =3ruou
x+y ox dy
(06 Mrrks)
c, 16u: II,, = Il .16 * = I1, 9,6, - 6(u,v,w) (05 Mrrks)
z' x y' d(x,y,z)'
.
d. If the H.P. required by the steamer varies as the cube of the velocity aad the square of the
length, firrd the percentage change in H.P. fot 3Vo aurrd, 4yo increase ia velocity and length
respectively. (0s Marks)
4 a. Choose the corlgct answer :
i) The gadient, divergence, cwl are respectively
A) scalar, scalar, vectot B) vector, scalar, vector
C) scalar, vector, vector D) vector, vector, scalar
ii; V=y'zi+z'xj+x'y[ is
A) constant vector B) solenoidal vector C) scalar D) None ofthese.
iii)Curl grad f
is,
A) gad curl f B) curl giad f+ $ad curl C) f
D) does not exist. zero
iv) Ifthe curvilinear system is spherical polar coordinate system then the radius vector R is
A) rsin gcos rsin gsinQj +
{i+ rcos0[
B) rsin0i + rcos0j + d
j+k
C) i+ D) None ofthese. (0{ Marks)
b. Iti=x2+'f +* ana F-x'i+y'j+z'i,thenfiod gradQ, divF, crulF. (06 MsrLs)
c. Provethat div Curl F = V.Vx F:0. (05 Ma.ks)
d. Prove that the cylindrical coordinate system is orthogonal. (0s Marks)
PART _ B
5 a. Choose the correct answer:
i) The value of Jsin5
xcos6 x dx is
0
- 5x3xl o, 4x2 'n
o'rr*s, ^, 2x4x2
c),"9- D) None oftbese'
'llx9x1
ii) x'z+f =*f is symmehic about
A) x-axis B) y-axis C) the line y :x D) All of these
iii) Surface area ofa solid ofrevolution ofthe curve y - f(x), if rotated about x-axis, is:
A) J2ry
dx z1 lzm dy c) Jzry as D) J2mds
2of4

15. IOMAT1l
iy) Asymptote to the curve f(a - x) : x3 is
A)v=o B)x=0 C)x=a D) None ofthese.
(01MrrlG)
'l--o r
b. Evaluate l1----id{, q>0. (06 Mrrks)
; log x
rlz
c. Derive the reduclion formula for J[sin' x dx . (0s Mrrts)
0
d. Compute the perimeter ofthe car<liod r = a (l + cose). (0s Msrks)
6 a. Chqose the correct answ€r :
2 / - 6 _ -
Id'vl Id'vl
i) For the differential eouahon I ---r. I +l ---:- I
' +y=x', the order and degee
I dx'./ ldx'J
respectively are
A)2,6 B)3,2 c) 2,4
ll I
dv v
..- --i-+a={l rs
'dx x
A) Variabte separable and homogeneous B) Linear
C) Homogeneous and exact D) All of
iii) ydx - xdv 0 can be reduced to exact, ifdivided by
i.1*'*f B)l c)xy
iv) Olthogonal trajectory ofl = 4aG + a) is
' '
e; x'?= +a 1y i a; nj x'? + f = uu C) Selforthogonal D) None ofthese.
(01Marlc)
b. Solve:'(1 + f)dx + (x -e-d 'Y)dy = O (06 Marl(s)
c, Solve: (y'ze*v' + 4x3 )dx + (2xye.' - 3y'?)dy = 0 (0s Marks)
d. Find the ofihogonal trajectory of the cardiods r: a(l - coso) , using the differential
equation method. (05 Mrrls)
7 a. Choose the conect answer :
i) Ilhich ofthe following is not an elementary transformation?
A) Adding two ro1s B) Adding two columrls
C) Multiplying a rcw by a non-zero number D) Squaring all the eiements of the matrix.
fr 23'l
ii) RaDt( of rhe matrixA=
l2 4 6lis
3
l:I o o.l
A) B) C)2
D) None ofthese.
iii) The solution of the simultaneous equations x + y = 0, x
-2y = O is
A) orly trivial B) onty unique
C) unique aad trivial D) None ofthese.
iv) Which of rhe following is in rhe normal form?
1000
lr ool lr ool 0100
ara=lo r ol nrs=l orol c) c= 0010 D) All ofthese.
Lorol Lor,.] 0001
0000
(04 Mrrk)
3 of4

16. f
lOMATlI
91 92 93 94 95
-r
92 93 94 95 96
b. Fitrd the mnk of the matrix 93 94 95 96 97 (06 Msrk)
94 9s 96 97 98
95 96 97 98 99
Fqr what Blu€s of l" and p , the following simultaneous equations have i) No solutiofl ii)
a mique solrtion iii) an inlinite number of solutions?
x+.y+z:6 t x+2y+32=10. x+2y+Xz=lt. (os Mirks)
d. Solve. usiug tbe Causs-Jordan meLhod.
,t+y+z:g; x-2y+32=8; 2x+y-z= 3. (0s Marks)
8a. Choose the collect a.nswer :
i) The eigen values ofthe matrix A exist, if
A) A is a square malrix B) A is singular matrix
C) A is aay matrix D) A is a null matrix.
ii) A square Darrix A oforder .n' is similar ro a square mahix B ofthe order .n. if
....
Al A: P-'BP B) AB = Null matll C) AB = Unit matrix D) None of these.
iii) Which ofthese is in quadratic form?
A1x2 t f +) -2xy t yz- rx B)x3.l yl-22
C) (x y - z)' D.l None ofthese.
iv) Quadmtic form rX'AX) is positive detuire, if
A) All the eigen values ofA are > 0 B) At least ore eigen value ofA is > 0
C) All eigen values ) 0 and at least one eigen value = 0 D) No such condition.
b, Find the eigetr values and eigen vector coresponding to ttre largest eiu"rr rufo"
of#JHiH
11 r3l
a:Ir s r]
'' l - 'l (o6Mark)
p I 1l
[-l r rl
tt
IfP:10 1 2 I is a modal mahi.( of the ryiatdx A in e.No.8(b)raad ioverse of p is
Lr 1 1l
[-3 o ]l
I-t = I 2 -2 2 J, then transform A in to diagoral folm and hence find Aa.
lr z
(0s Marks)
tl
d. Fird the trature of the quadratic forms for which coresponding eigen values of the
conesponding mahices are given as
Matrix Eigen values
2, 3,4
B
C 0, 1,6
D 0,3,-4
E 1, 3,4
(0s Mark!)
4 ol4

17. O6MAT21
Second Semester B.E. Degree Examination, June/July 2011
. Engineering Mathematics - Il
Time: 3 hrs. Max. Marks:100
Notet L Answer at y FIYE full questions, choosing at least t)o fiot , each part.
2. Answet all objectiye rype quesrions orrbt ih OMR sheet page j of the dhsu'et bookleL
3. Anst'er lo obJective lrpe qaestiohs on sheets othel than OMR uill not be ralued.
E
e
PART*A
3
1 a. Select the conect answer :
a i) 'An expression for the radius ofcurvature in parametric fonn is
1l(3
/],.,2,}
t,t p=,''.,,, p=ll-tJ;
", o=i,li,-{lj}
A) B) D)Norcor$ese
I: ); |.*Y-rx]
Ea ii) The curvature ofa circle is a
5h
A) constant B) variable C) 1 D)0
iii) Ifa firnction {x) is continuous in [a, b] ther ril(x)=f(x)-kr is also
A) differentiable B) continuous C) Both A and B D) None ofthese
iv) ll xdv
--:-,,11.n
Y= - atx=0is
rlr
Aq
A)l B)0 C) Both A and B D) 2 (04 Mrrkl
=.9
sE 4a'zQa - x
3d Find the radius of cuivahne for the cuwe y2 = , where the curve meets the
x-axis- (04 Marks)
EE c. State and prove Cauchy's mean value theorem. (06 Mark)
}E d, (06 Mrrks)
">6 2 a. Select 1ft9 cortect atsw€r I
i1 Theralueof
Lh logl
E+ 1,
x -r 0 cosec x
l6
A)0 B)1
ii) If f '(a) = O and g'(a) = O, th"o *" hurre
Lh lG)*=
:g x -+ a c(x)
A)
Li- -t]u B)
Li' t-l*l a, t'' {Q D) Non" orrh"r"
x-)a g(x) x--+a g (x) xra g"(x)
5g iii) The ni:cessary conditions for f(x, y) = 0 to have extuemum arc
:>
A) f-r=6=1* B) f-=o=f,y C) fx=o=fy D) None ofthese
.i 6i 1v) The point (a, b) is called a statiomry point and the value (a, b) is called
g
A) stationary point B) stationary value
C) m4ximum value D) minimum value
z (04 Marks)
E
Lim tanx-x
b. Evaluate :
x --) 0 x'tan x
(04 Marks)
c. Examine the function f(x,y) = xa 1ya - 2(x - y)'? for exheme values. (06 Marks)
d. 5.xyz,
Ifxyz = 8, find the values ofx, y, z for whicb u=x+2y+42 is a maximum. (06 Marks)
1of4

18. O6MAT21
PART _ B
5a. Select the correct answer :
The-differential equation !I = y' is
A) Linear B) Quasi linear C) NonJinear D) )lone ofthese
The P.I of y'+y = cosxis
I
1"or* _.
"si ,
A) -sitrx B) c) I U)
I
x
2 -xsrn
2
iii) The P.l. of(D2+ 3D + 2)y = I + 3x + xr is
A) *2 B)!
) a) 2x2 D) 4x'z
iv) Tie general solution of 2a qb 61d91 6ilferential equaiior, rontains
A) Atleast 'n' ind:oendent constrnts n; Airaos ,n.
inaefendent cc'stants
C) Exactly n' ind;pendent constants D) Exactly ,n,
dependent consraal:
b. Solve ; (;r - 252 ..4p - gy, = O. (04 Marks)
(o1M'rks)
c. Scivt : y'- 2y'+ y = xe' 5ii; .
c.. Sotve : y' - 4y' + 3y = 20 aos :{ ,by the methcd of undetermined cosffic,iejs. [: #::]
6 a" Select the correct answer :
i) The homogeqeous linear Citbrential equatili y,,665e auxiliary equation
has roots
l, l, and -2 is
AJ (Dr - Dz - :ti - 2)y - 0 B)(D3+3D-2)y=0
C) @'-3D+2)y=0 D) None ofthese
ii) The general solutioa of (x-2D2 _ 1p;, = 6 i,
A) y=C,+Cre* B) y=C,+Cr* C) y=C,+Czx, D)y=C,x+Crx,
.. The
iii) equarion + b), y, + a,(ax + b)y, + ary
ao 1ax
= O(x) is
4 ) Legencire s linear equation 81 Cauchl.s ijrrcar equauor,
C) Borh A and B
iv) The differential equatior y,+ 5y,+ 61 ,
= , ,idr1T"J:;1;;; = , ,. *
A)
_Initial
value ryoblem B) Boundary val-ue problems
C) Both A aird B D) All ofthese (04 Msrks)
b. Solve by the method ofvariation ofparaoeters
y,+ury =.gsuj.. (05 Marks)
Solve {x + 1)'y' + (x + 1)y'+ y 4cos log(l + x)
. = (0d Marks)
.i. Solvey'+4y'+4y=8x'],giveny(0)=landy(l)=1.
(05 Marks)
7a. Select the corect answer ;
i) L[ry]=
LtJ
A) cot-rs B);+ C) tan-r s D) cof I
1s - I.1
ii) L[3sin h 2t]=
.q)-6 B) 6
c,) i6,
s'-4 s'+4 s'+4 D) None of these
3 of4

19. O6MAT2I
7 a. 1ll)
.lt-"-l
LI-J
a1 rogfr]
s+a/s,/s,/
. n; r"rf:ll cl rrfr-l D) Noae ofthese
iv) Ifu(t * a) is a unit step filnction then Laplace transform ofu(t - a) is
A)-
s
B)e s
C)L
s
D)
-s (04 Mrrks)
nl
b. Prcve that Llt"l = + (04 Mrrk)
c, tf f(r) = t'7. 0 < t < 2. is a periodic tunction witb period 2, then find L[ (r)1. (06 Mrrk)
fsint 0<t<4
d. Find Laplace transform
- of f(0 = .{ using unit step function. (06 M{rks)
[cosr r> i
8a, Select rhe correct lnswer :
l r,l-1-l=
+5J
Ls-
$* B) -T-
sinJ5t
.,* D)sin.G.t
--- - ,[s3 +6s':+l2s+8.ll=
'Ls"l
fl) l,'l.......................-
A)
t',tott+-+- -2
B) i-+t'+ tt + t'
'21 2t 15
-+t- '2
c) t'+t'+tn+f D) None ofthese
iii) Convolution of(t) and g(t) is given by (0 ,i g(t) is equal to
a; Jrlu; 96 -u;au
B) F(u) s(t+u)du
q lr6)du D) Jg(u)
du
iv) L [y"(t) ] is equal to
A) s'2 Lty(t)l * sy(o) - y'(0) B) s'? - sy(0) - y'(0)
c) s' Lty(t)l - sy'(o) - y(0) D) None ofthese (04 Mukr)
s+2,
Fiod , L'f ) (0{ Mrrks)
G+l).,
c. rl.a l.'l' - l-), by using convotution theorem. (06 Mrrl(s)
s(s" + a'),
d. Solve by Laplace traosfom method, given y" + k'y = 6 and y(0)
: 2, y'(0) = 0. (06 M&ks)
4of4

20. USN lOMAT?1
Second Semester B.E. Degree Examination, June/July 2011
Time: 3 hrs.
Engineering Mathematics - tl Max- Marks: l0O
Notei l. Answer an! FfWfull questions, choosing dt leoat two lrom each pdtL
2. Answer all obJective lype questions onl! in OMR sheet page 5 oJ lhe ans$'er booklet,
3. Afis$,er to objective q,pe quations oa sheels othet lhan OMR teill not be valued.
PART-A
s
1 a. Choose your answem for ttre following :
i) A differential equation of the first order but of second degree (solvable for P) has the
E general solutioo as,
A) 4 (x,y, c) + F, (x, y,c) = 0 B) (x, y,c) x (x,y,c) = 0 I
C) (x, y,c) - F, (x, y,c) = 0
F, D) F, (x, y,c) /F. (x, y,c) = 0
ii) Ifthe given differential equation is solving for x then it is ofthe form,
a1 x=f(y'1) B) y=f(x,P) c) x=f(i) D) x=f(y,P)
iii) Clairaut's equation of P = sin(y - xP) is,
A) y=llrio-rp B) y=Px+siqP c) y=Px+sinrP D) Y=x+sin-'P
x
iv) The di.fferential equation lor R, L series circuit is.
Ar {+tu=E
'dt BtL{+i=e cr di+Ri=E o)r-$+ru=r
=.9 dr dr L
(04 Mrrk)
bE b. solve P(P+ y) = x(x + y) by solving for P. (0s Marks)
c. Solve P3 - 4xyP + 8y'z = 0 by solving for x. Mark)
d. Solve (Px - y)(Py +x) = a'P, use the substitutionX = x2,
q=x (05
(06 Marks)
nB 2 a. Choose )our answers for the following :
6C i) Roots of y'-6y'+ 13y = 6 ale,
AR A)2t3i ,- B) 2+i C) 31
ii) The value of
*(f(x))
is, -8 e *rC,r2
'U A) f'(x) Br I
'f'(x) c) F(x)dx or []-a*
' 1( x.)
iii) The pa icutar integat of (D'? - 6D + 9)y = log 2 is,
A) 2 gt ltosz C) 91o92 or llosz
61o9
9' '6 -
:>
-.i 6i
iv) The displacement in the simpte harmoni" rno,ior, {=-p'* ir,
c A) Cj cos Ft + C, sinpt B) C, cos pt - C, sin prt
z
C) Cr cospt t C, sinpt D) cosfrtt sirpt (04 Mrrks)
b. Solve (D3 -D)y = 2e* +4cos x. (0s Marks)
c. Solve (D'? +2)y = x2er* +aos2x (05 Mark)
d. Solve the simultaneous dilfcrential equations,.dt
' $+5x-2y' = r, 9+2*+y =g. 196 vu.l",
dt
1of 4

21. -
10MAT21
3 a. Choose your answels fol the following :
u and v
i) If y, and yrare the solutions of second ordq differentiat equation and are
vadation of paBmeters of yo = uy, + ry, then v =
.. r (y,x)dx ., ti,i',tviv, .,., t X&
I fulx)dx -' rv,v',-viv,n f dx
"'t :*,-viv, '' ')v,Y',.'viv'
iD lt y' + 4xy' + 2y =e' if x = e' then we get for x2y' as,
x1
A) (D-l)y B) D(D-l)y C) D@+l)v D) D@+2)v
ii0 ro tu iot (ax+b)'?y'+K,(ax+b)y'+Kry=x iito Legendre's linear equation we
put ax+b =
A) e-' B)l C) 1+ e' D) 1-et
Series solution is a regular singutarity of the equation Poy' + P,y '+PrY = 0
when
iv)
A) x<0 B) x> 0 C)x=0 D)x *0 (o4Mark)
(05 Marks)
b. Solve v'-6y'+9v=1.
-xt usine variation ofparameters.
c. solve x2 y'+ xy'+ y = 2cos'](log x ) (05 Marb)
(05 Mrrks)
d. solve 2xy'+3y'- y = 0 by Frobenius medtod'
a.
- Choose your aoswers for the following :
b ftom the relation
1) iattil Aferentia equation by eliminating a and
+b) is,
Z = (x'?+ a)(y'z
A) Z*2, =xYz B) 2,, = xY2 Ct Z = 4xvz D) Z,Z, = 4aY2
The solution of Z}y = sinxy is Z = -
A) sinxy+f(x) + g(y) B) -1cos xy + f(x) + g(Y)
x
I
C) - x'. sitr Ky+ f(x) + g(y) D) -sinxy+ f(x) + xg(Y)
iii) For the Lagraage's linear partial differential equation, Pp+Qq - R' &e subsidiary
equations are
..
A' dx -dv dz -. -dx=-dY dz
t'.) p =R
PQR
-=-=- a
^. dx dy
,P Q
dz g=q=E
,P,
D)
R Q, R,
iv) In the method of separation of variables to solve un-2ux+q = 0, the trial solution is
A) x(x)r@ srX(*)
' T(t) o llE
r(t)
o) x(x)",8(0
(04 Mrrks)
b. Solve Z*y = sinxsiny for ruhich Z, = -2siny when x = 0 and z = 0 when y is an odd
_,
multrple ol-1t (0S Mark)
,.
c Solve (xr - yr -Z'z)P +zx-yq= 2xz. (05 Mrrks)
d. Solve 3u* + 2uy = 0, u(x, 0) = 4e-" by the separation ofvariables. (06 Mrrks)
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22. 1OMAT21
PART-B
Choose your answers for the following :
i) The value of ij
ij-ra-o
h
A) 6 B)1 c)8 D)9
,lTi
ii) The ntegral
f j1* * yyoyo* after changing the order of integration is
,,fi ,l-' ,.Fi ,"/iT
A) J ilx + v)dxdv B) i lr**vu*av c) iit"*vx*avD) J lx'v)dxdv
iii) The value of 'Je idx is
l,) r,D qz../; q J2"
rv) rhe value of r(][(;)
--
=
")+
$ 2Ji il+
-
c1 rJi Dr !a rol nr".tst
'2
b"-Y"
b
b. Evaluate
J Jvaxdrtr chans,as the order ofintegmtion. (05 M.rks)
c. Evaliate JJ J("+r+zXra"az (05 Mrrk)
-l0 r-z
d. show that
I(l+ x)'-'(l - x)*'dx = 2-."-'p(m,r). (06 Mrrk)
I
a. Choose your answers for the following :
1 r cJr.i=o then F is calted
A) Rotational B) Solenoidal
C) Irotational
D) Depend€nt
ii) If f is the vector field over a region of volume V in tbree dimensional space then F.dv
is called
A) Scalar volume integral B) Vector volume integral
C) Scalar surface integral D) Vector surface integral
iii) In oreen's theorem in rhe plane
J'|f+- +I-dyir
^4, C)al
A) - Ndy) B) I (Mdx) x(Ndy) IOdx-Mdy) D) + uay)
J(Mdx [tuax
iv) IfC be a simple closed curve in space and S be the open surface, - fbe the vector field
oen Jr.ar =
A) f(cwlfl.nds B) Ioxf).ds C) lN2DJlas D) Jo.0.nds (04 Mrrks)
b. f of rhe cyiinder x1 +y2 =9
lJt."a" *irer" = tzi+2y'j+xzlk
Evaluale ar,d S rs the surface
s
contained in the fEst octant between z = 0 and z = 2. (0s Mrrk)
c. Veri& Greeu's theorem for f1ry+y2px+x2dywhere C is the closed curve made up ofthe line
y = x and the parabola y = x2. (05 Marks)
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