(4) Total No. of~stlons : 8) {Total No.

ofPrlnred Pages : 4

e" RoU No ...........- ••,.................

7. a) Find lhc poles Bild residues ot each pole of - 1- . 7
z +I BT-202(GS)
t
z +I
~~qi!' tR qi!'~ am)'q m ~• B.Tccb., I & II Semester
Examinalioa, December 2024
b) Find the din:ctionnl derivative of 0 - z2yz + 4.zr at
{I, - 2, - 1) in the din:ction of 2T - ] - if. 7 Grading System (GS)
27 -7-2f .fil ~ if (1,-2,- 1) ~ 0 -x2y.t+ 4.zril'iT Mathematics - II
~~ll@~I Ca,epr.-~.C&.:: 11me: Three Hours
Maximum Marks : 70
8. VcrifyGn:cn'sthcorcmfor J [(xy+y1 )ctr+.r1~] whcrcC Now i) Attempt any five questions.
C
~qf.rJ!t'tftll)~~I
is the boundruy by y - x andy - xl. 14
ii) All questions carry equal mlllb.
1
J[ (xy+ /)ctr + x
C
~ ] ~~Jlr-f~JJI~~~~~ ri Rt'tf ~ ~ lAR I f
iii) In case of any doubt or dispute the English version
C,y - x ~y • xl<fil~!I
question should be treated as final.
•••••• ~ ,n Jlffl~~ 3MT ~ <fi'I M if~ 'qllf(
lW lfA 1ffl ~ 'll'RT ~ I
Co.w, -fuw.coot.
I. a) Solvc(t+/)ctr = (Ulll- 1 y-x)~. 7

(t+ /)ctr =( tan-•y - x)~ lfi)~ ll'RI

b) Solve (1)2 + 3D + 2)y • sink 7
(D1 + 3D+2)y • sin3xlffl ~ ~

BT-202(0S) BT-202{GS) PTO

 +
Ill
S. a) Construct a pllttiAI diffcrcolinl equation &om lhc relation
2. n) Sol\'c lhe simulllUICOus equations dx -1x + y"' 0 1111d 1
dJ

dy 2x- Sy=0
f(r+ y2 u2,z2 -2.xy)"'0
dt . 1
1
~11 J(.r2+y +z2.z2-2xy) =O ll 31iftlc1; 3ftR
~~ dx - 1x+y• 0 am dy -2x-Sy =0 lR
dJ dt ~.flt~~,
~~·
b) Solve by I.be method of variation of parameter
b) Show lhat u "' ,:-• (.uin y - ycos y) is Hannonic. 7

(D2 + 1}y=x. 7 ~f.f; u,,,e-• (xsiny-ycosy) ~!1

~ (Dl + J}y• x ~ ~ ; f i l ~ ~ ~ ~ l
6. a) Dctcnninc P such that the function

l d 1y dy /(z) = ½log(x2 +y 2 )+itan-•(;) be an analytic

3. 11) SolYC (l+x) dx2 +(l+x)dx +y,,,cos log(l+x). 7
function. 7
2d 1 y dy
(l +z) dx2 +(l+x) dx + Y"'coslog(l+x) ~~iffll P'i/il~~ll<li'R~~ti;ifffl

b) Show lhat 1.(-z) ={- I)" J,.{r) when n is positive Of /{z) .. ½log(x2 + l)+1t11n-•(;) ~ fhc\11011c11i/i
.f.);
negative integer.

Pt,
J.(- x) ={-1)" J {r) ;)l'lf n~ l f l ~
·
0

C!mJr ..fr,i.-r.rr;,,t
7
lJicf1R it I

b) EvalUB!CusingCauchy'slheorem
1
z· e-:
J--
,(z-1)
dz when:cis
1

4. a) Sol\'cbyCharpit'smclhodpr+qy• pq. 7
"1fil? ~ pr+qy• pq l) ~ <ffll 1-tl=½-
1 Catee,-imn'.eoot 1
b) Sol\'C lhe Partin! difTcrcntl11I equation
12
(IY-401 01 +4DD )z,,,cos(2x+ Y). 1
ili'f;ft~~'i/il~~ I r>e-',dz~~~
,(z-1)

• 31ifffl;3ion,ifllfRVT (CY-4o>o' +4Dif)z=cos(2x+y)

er;)~~,
BT-202(0S) Con1d••• BT-202(0S) PTO
 (4) Total No. o/Quulio,is : 8) (Toto/ No. ofPri111td Pages : ./

b) Write short note on: 7 RoUNo ..................................

i) Cauchy's intc:grul fonnuln BT-202 (GS)
ii) Solenoidal and lrrototioruil 8 .Tcch., I & IT Semester
~:ire~, Examination, June 2024
i) <ffl-.ft~~ G rading System (GS)
ii) 11M-1~~ 3iR~ffi'! Mnthemntics - II
1imc: T/trce flours
•••••• Max/11111111 JI.larks: 10
Note: i) Attempt nny five qucsdons.
~qfvRAilf>1~~1
ii) All questions CQrT)' cqWII marks.
fl'1l R,:i'fi)r3ic!;l!'tl'ffl
iii) In cue of any doubt or dispute the English version
question should be treated llS tinnl.
~ ,n llffl ~ ~ W i n ~ if!I ~ i f ~ "tfl'IT
ti Rt'f lffl 3ifctif "!RT~ I

I. a) Solve x dy + y .. x'y 6 using Bernoulli's. 7

dx

~lfflffl1T~ x! +y =xJy6 lm~<ml

b) Solve lhedifrcrcntilll cqWllion {xe""' + 2y): + ~..,. = 0

using Exact method. 7
Exact Fclffl ~ ffl1T m 3llll@ Wflili·M

BT-202(GS) BT-202(GS) PTO

 12] [3)

2. n) Solvc{D2 -60+13)y=Sc'"'sin2.r. 7 6. n) ,1u>(

Evnlunte Jco.o) 3.r2 + 4.ry + t• r' ) d:: aloogy • .r2. 7

(n2-6D+13)y=8clxsin2r <R~1'RI
y • .r2~~ J(~;i(3.r2 +4.ry + i.r )dz
2
<j;f~ ~I
b) Showlhal ![.r"J.(.r)]=.r"J.-i(.r) . 7 b) Find the Poles 11J1d Residues 111 c11ch pole or
• 2
/(z): sin z 2
~~~ ![.r•J.(.r)]=.r"J._,(.r) I
(z-:) . 7

3. Solve (0 2 +1)y • .rsin.rusingvnria1ionofp:1rnmctcrs. 14

~lflt~tfi'l~fb1J;1~M (02 +1)y • .rsln.r ll1l

~?!RI
7. Vc.-ify0r,:,:n•$ thc::orcm In the pl1U1c for
4. n) Form the pllltial dUfcrcn1illl equation (By eliminating lhe
nrbl1rnryfunclions)from Z • {.r+y);{.r - y
2 2
) . 7
Jc{.r2-..y)ttr+(y2- 2.ry)dy where C is II squnrc with
venices (0. 0), (2, 0), (2, 2), (0, 2). 14
z .. {.r + y); (.r2 - y2) ~ 3ITTfflll 3i"!R ~ <ll"fllA fc(.r2- .tY')ttr+(y2 - 2.ry)dy ~ ~ <f'lira 11 J!R 1); w)ti
~1li)wm:r~)~I ~ ~~ ~C t1llil (0, 0), (2, 0), (2, 2), lO, 2) ~ ~
b) Solvc(D1 -DD1-6011Z• .xy. 7 ;pf i,
(02 -DD 1-6D11Z• .xy1li) ~ ~I
S. 11) Find the directional derivative of /(.r,y,z) = .xy2 + yz'
S. n) Solve the partial diffcrcntio.l cquallonyp-.rp • z. 7 a1 point (2, - 1, I) In the direction of the vector
3liftJq; ~ ~ yp- xp • z cm~~, T+2] + 2f . 7
b) Showlh01 u •c-•(.niny - y<'Osy) isHrumonic. 7 t~ T +2] +U tPl .f~tTT il ~ (2, - 1, I) ~
~ u"' e-'(.rsiny-ycosy) ~ !1 I (x. y. %) =.xyl + yz' <l>T ~ ~ ~ I
BT-202(OS) ConuL BT•202(GS) PTO
 (41 Total No. of Questions : 8/ /Total No. of Primed l'ageJ : 4

Roll No ... - ............................ .

8. n) Prove tlinl curt( r"r) = 0 7
BT-202 (GS)
Im:~~~ curl(r"r) =O B.Tccl1., I & n Semester
b) Wri1c short nolc on: 7 Exwnination, Dcccmbcr2023
I) Cauchy Riemann equations Grading System (GS)
ii) S1okcs theorem
Mathematics - fl
Tif!m'lre~I 7ime: Tlm:a /lours
l) tr,fil) til:! ~•frq,<111 Marlmum Marks: 70
Ii) «!f<l~I ST~ Nora: i) Attempt nny five questions.
~ qt.J flt'TI Ir,! l!t'I ;f.l~,
•••••• ii) All questions cnny equal mnrks .
~Ost~~~~i!I
lll)ln case of nny doubt or dispute the English version
question i.hould be tn:ntcd os final .
A;;U ,n AlliR ~ ~ ~l!lQT ~ lfil ~ ,, 3ill-..n '11'11
~ ln"I tr,) 3ir.til 'lFII v!Tl)-tn1

I. a) Solve (t+ y1)tit=(lnn-• y-x)dy usingLcibnil.7.lincnr

method. 7
t'll~ 1!wn f.lfq <lil QIMTJ ~
(1+ l)tit .. (u1n- 1y-x)dy lb'~ <!RI

b) Solvc{e1 +t)cosxt.1.t+111 sinxdy=O. 7

(e' + I)c:osxt.1.t+111sin x dy = 0 ir,l l!t'I <!RI

2. a) Solve (n2-4D+3)y=cos2... 7
1
(0 -4D+3)y=cos2x ctlt~l'!RI
UT-202 (OS) BT-202 (GS) !'TO
 121 131

b) Dc1cnmnc p so that the function

b) Showtha1Ji{,r)= 12sinx . 7
"i -.J~ /(z) =½los(x2 -, y 1)+il3n- 1( ';)

~~ Ji(x)= innalytic funclion. 7

; v-;;
(2sinx
p f.lufft., ifR mf.), $1"""

/(z)=½los(.r1 + i)+itan- 1( ';)

J. Solve (D-2 +9),, = tanJx by using method of vllrinlion of
p:irnmeters. 14 ~,;i11011te1u. y;lJV•nil 1 .. .~Jul
~ I
~ .li\ f!r-;rnJ iii) f.ttb lli1 ~ m (01+9)1= lllnJ.r 6. 11) Show lhllt lhe function II (.r, )') = e' cos)' is Hllflllonic.
tlil~~I DelCtlllinc h's 1i11JTI1onic conjugate. 7
lklJte.&. -fuen.e.o;•• ~ Rf \i;1ttA 11(r.y) :o ~•cosy 1!Pl'ff.r.l"i ti {fllffl
4. a) Solve the p:u-t.ial dlffercotinl equation
~~ r.iumc, ~I
(x-y)p+(.r+ y)q•2xz. 7
Ze' .. I
b) F'mdlhcrcslducor (z-1)3 lllllSpOe. 7
~~~ (.r-y)p+(.r+ y)q=2.rt <NF!
!!RI
b) Soh•e (p1 + ql)y =qr.by using ChllrJlh 's method. 7
u'
~ qlf 1R - - , iii! 3f!n°\t( S'I@ ti~ I
(Z-1)
~~ ' I l l ~ ~ (pl+<r)y=qzlffl~l!RI

7. Verify Gauss divcsgcnec theorem for F=..-37 + y3]+t3k

llllccn over the cube bounded by x =0 • .r =a, )' = 0, y = a,
12
5. ll) Solve ( D2 +4 OD1-5D ) Z=sin(2x+Jy). 7 t=O.z=a. 14
F=..-37 + il +t3"f ~ fclil ttm ~ ~ <fil·~ ~.
;;ii x=O,x=a,y= 0,y=u,z=O, z =a~ RR tl9'R ~~
t, .
BT-202 (GS) Contd... BT-202 (GSJ PTO
 l~I 1btol No. of Q11c,t/011s : 8) (Total No. ufPrlntt:d Pages : 4

Roll No _.....·-·--···................
8. a) Fmd lhc dircction:11 dcriv:mvc of /(x,y, :) = ,:1-' cosy:
01 (0, 0, 0) in the dircc1ion of 1111: 1MJ1cn1 ro lhc curve BT-202 (GS)
B.Tcch ., J & II Semes ter
r =asint ,y=aco st, :=at 1111=!: . Eltaminntioo,Junc2023
4

/ (r,y,:) • e,_, cosy: ~ (0, 0, 0) ~ u;

Gracting Syste m (GS)
M:ithc m:itics - Il
r•asin t,>•=o cost,:a at at,.!: Timt! : Thrt!ll /lours
4 !rlaxl11111n1 Marks: 70
l)r mm 1fi'I ~"' il ~ ~ m.t, Note: i) A1tcmpt1111y five question s.
b) Usina Green's theorem, find the area of!ht region in lhc ~ 111'1 flAl cffl Wi=f ~ I
I X
u) AU questions cmy equal masks.
tirst quadran t bounded by the CUl'\'1: y =x, y = - , y = - .
X 4 "'1!J1 Al$3i lf;~i,
ul)ln case of any doubt or diJpwc the English YCBion
ll'fm 1111 ~~~. u y•x, y•.!.,
X
y.~ ~AA
4
~ 'fl1l'/lt1 rf ffif 111T ii~ m;, 1l'R I
...........
~ !ft SllllR ti,,~ 3l'JllT
$11A1"'3iffl,JlJAl;;mml
-uh
question should be treated u final

r
~ i l 3ibvft 'llTIIT
Iv

I. a) Solve : : =cos(x +y) +sin(x +y).

CoJUJ -fua-TT.COUl
i•cos( x+y)+ sin(x+ y) 1"'Wi =f~I

b) Solve: (l+ y 1)dt=(IM _, y-x)dy

(l+ y 1)d\'•(lan-l y-x)dy lffl~ ~I

dly dy
2. a) Solve: - + - = (l +e-tr'
dr1 dx
dldy
~+ dz •(l +e'r' 11!1~ ~1

BT-202 (GS) BT-202 (GS) PTO

 (21 Ill
6. a) Use Cauchy lntcgr:il fonnulo 10 solve
b) Soh·c : dr-y=c', dy +x=si111;x(0)=l,y(0)=0
d1 d1 • .2 . l •
~ SlllJl'Z + cosnz dz whcrcC isthecirclcl~ = 3.
c (z-l)(z-2)
f!=.. - y=c'. dy +x=si111;x(0)=l,y(0) = 0
d1 dt
• 2 2
ili't iic1 clll~ I <mil~ l f i l ~ ~ ~ J, smnz + cos,n dz
<Jzl !; 'fc (z - l)(z-2)
3. Solve the difTcrcntinl cquntion
tr.I~ lffi~I lifiii C ~ 1:1 =Jt
x(l-x)y• +2(1 - 2x)y'- 2y .. o
b) Using complex intcgr:ition method,
using Frobcnius method.
~f.nm ~ ~ 3!ilili<'r~. · Ji. cos40 dO
solve: 0 5+4cos0
x(l-x)y• +2(1 - 2x)y'- 2y = 0
tr.I !<'I cfil~ I ,,,. U A A 21 cos40
VJJ'fl'Jt _f;;ro;i- 1>nr11 ~ 'Mlcti<'l'l Fclft1 ~ ~ ~ 11! Jo S+◄cosO dO

4. 11) Prove that J½(x) .. [f .r/1vr ~~~·

7. a) Solve: J!"(x - y+l.r2 )dr nlongthercnloxisfromz • O

Im:~ J½(x) a /f.r111x
10 : • I nnd then nlong II llne pnrnllel to !mnglnruy nxls
b) Soh·c by Ou11pi1's method, the P.D.8 (Jr+ ql)y• q:. from :r • I to :r • I+ I .
tfRfqe ~ it P.D.E(,r+,r)y• q: ili't !<'I ~ I
J!•' (x-y+l.r2)tb ili't ~~ a1«1mih 31~c)nm1: • O
~ z• I~~~ @t,,qfi)ih;Jttit/;'WIAiffi1!lliWTc);~
5. 11) Solve : (D 2 -6DD'+9Dq)z= 12t'+36xy.
: = l~:c- 1+ /
(D 2 -6DD' + 9 D.,)z = I 2x2 + 36.ry ifi't !<'I~ I
2
b) Pro\'C tlult lUl nnolytic function with constlUlt modulus is b) Prove that .· V2/(r) = j(r)+-
r J(r)
ConstnnL
~~ fcl;@R'IJQf<li(llffi ~ f.htl\ttOll<4\ih lfx'l'I @ff 2
~~ V 2/(r)=;(r) + -/'(r)
ii@l ii r

BT-202 (GS) Coatd.... BT-202 (GS) PTO

 Total No. a/Questions: 8] {Total Na. afPrimed Pages : J

Roll No .................- ....·-··--···

BT-202 (GS)
B.Tccli., l & II Semester
Examination, November 2022
Grading System (GS)
MaU1cmatics- Il
Time: Three Honrs
t}f aximum !,!arks: 70
Nate: i) Am:mpt nny five questions.
~~$11:U~~~I
ii) All questions cnrry cqunl marks.
~!Wfli);3rcp'flX!'Agl
iii) In c:isc of nny doubt or dispute the English version
question should be trentcd ns fimtl.
~ ,ft J!i!j'Ri);~ 3iwn f.rcllc:cfj) ~~ q 3W;;ft 'll11ll
~ Rt=r lfi1 ~ tJA'T ;:;rrlPITI
.-
I. a) Sol\'C cos,.rdy = y(sin.r- y)dx using Bernoulli's.
~ ifil ~ ~ cos.rdy=y(sin.r-y)dx lfi1 i:cJ
ifiil
b) Solve lhe Linear diffcrcnlilll equntion

sin2.r! - y= ton.r. Cmwi- -fuc.1 r.CiJa~

~~~ sin2.r!-y=tan.r ~i:cl~I

2, a) Soh•c(r+sin8-cos8)dr+r(sinO +cosO)d8=0 .
(r+sin8-cosO)dr+r(sinO+cosO)d0=0 <!l!~~

BT-202 (GS) PTO

 [21 Ill
b) Solve the diffcrc:ntinl equation. S~~v tluu /(Z) = z:z is diffcrcntinblc but not nnnlytic nl
6. :i)
(0 3 - 70? + 14D- 8).f' = e-' cosl.\- ongin .

~ wft<!RCT {D 3 - 7D2 +14D - 8)y• e' cos2x ii;'t

~ f.l; / (Z) "'= 3liflliM ~ ~ ~ ~ RtHqOllc4<h
:rc!f !1
~~I
Show thnt 11(x,y) = c-?.r sin 2y is harmonic nnd
de1cnninc it's Harmonic conjugnle.
3. Solve (0 2 + 'l)y = tlln 2.r by using method of vnrintion or
parameters. ~f.l; 11(x,y)=<1!-lx sin2y ~131R~F-ltlf~
~cfilfl'r.@Jcfil ~ < l i l ~ ~ (D +4)y= tnn2.r
2 cm f.l; ~ ~ ~ i,
iffl ~~I
r:_ , 7. D) By Residue theorem, Ewlunte if, ~nz d::, whcrcC:IZl=2.
r C =- I
4. u) Show that 3,. is solenoidal
31cltN ~ ~. ~ cm f t~ z dr ~ C:IZl=2
c -1
~
Z
~f.l; solcnoidol ~I b) Using Cau~y integral 11\eorem, to cvnlunte the integral
r
.,_
b) Show that the vector
2 ~ 2 - 2 -
J e- dr , where C is the circle IZI = 2.
c(z-1)2 (z-3)
(x -y.:)1 +(y -=x)j +(;: -~)k
2z
is lrrolntionnl. Find it's scalar potential.
~ r.o c..-2 - .v:)i+ (y2 - =.r)J+<=2 - ~>i"
.),re~
lc.-,;l(::-3) dz ~R'milil~~~. Cnuchy
3111r.!<1i1~~l);~c~~ 121 - 2t1
lrrot:itionnl i, ~ ~" eft@l cl'il Q'ffi ~ I

8. n) Solve x2 pl+ y 2q1 = z 2 .

s. VcrifyGrcm'slhcort:mfor L[3x 1
- si] d.r+(4y - 6~)dy , :x2pl + y lq2 .,_ ::2 ,m ~ cfil~I
Where C is the region bounded by x • O,y • 0 nod ..-+y• I. 12
b) Solve (0 2 - 4OOt +40 )Z =- cos(x-2y)
Jfl::rilHIIW<mmmlira~ L[3x 2
-8y2
] d.r+(4y - 6.ry)dy
12
{D 2 - 4DD1 +4D )Z•cos(x- 2y) ilTt if<'! ~ I
;;m C. x • O,y • O 3m x+ y • I 1' ftm ~ ml t1
......
OT-202 (GS) Contd.- BT-202 (GS)