r

tl#IW
. ...--L
G- • -.~
-A...J~
Of' l'ee hno lo

n. /FU / f.\T l/-11

\'l'rn11d Sr•\\i1111a/ f ,,,r f ,_-,., .,, S, •111,
•,ft•r !fl!- 1-!5 )
Time: 3:fl fll/n l:'11~•i111'1',j!!.;;_J /!1[/1_, •11wrin I/( t:
I.\'! fl.I
lla \". 1/orl, ,: 7fJ
Note: (I) No atudent will be allowed to
leave the e11mln d :
· (II) Be precise In your 1n1wer. • on room be"-
,ure en d of e11m.
.
(Ill) All queatlou ■re compul1ory.

Course
F, 0 IIow In1t ■re Outcomes:
rr C0'1Codes ', \ ',
- the coune outcoma of the 1ablect: -
! COURSE OUTCOMES Bloo mLc vd
BAS203.1 Rem~ber the concept,diff'er~ntiation to
eval
' coefficient and LOE with vanable coefficie uate LOE of nth order with constant Kl&KS
nt of 2nd order. ·
BAS203.2 · Understand and apply the concept of Lapl
to evaluate differential equations ace Transform and inverse Laplace aansfonn
K2,IO&.KS
BAS2033 . Un de~ d the conce~t of convergence to ·
expansion of the function for Fourier serie analy ze the convergence of series and
s. . :· .K2&K4
BAS203.4 Apply the concept of analyticity, harmonic -
· aoolyjniz conformal transformation. function and cr~ e·the image of the func
tion Kl&K4
Apply the CQ~cept of Cauchy integral theo : · ·
BAS203.5 rem, Cauchy integral fonnula, TaylQr's and
Laurent's Senes and calculus of residue KJ&KS
to evaluate inteura.1s · .
-

0 .N.-1 \'E CT ION A

( ) . Find ~he differential equation which repre
se~ts the family of straight line passing thro
M■ rkl CO'•
ugh the origin 2 CO.I
.h} X 'l _ :c. o
.:::0
J;
.Find the Laplace transform of e.-t cos
t dt L{_Coot).= _&___ e_- ~ ~ t-ColtJ
If f(x) = lcosxl in -1r < x < 1r, then
find the value of a
-(.-d)
:C --+ -., - - - - - - - -
0 ::1_
- - - - - - . . , . . 1---'-
Fin dth eim age sof lz-2 il=2 underth 71
"-.. .l.!,_ _-- ---' ---- -+- ----
()- - - + - - - - - - - - - - - -
emappingw=; · lf\J...t-1-== D <bf' '1=:--- +-: -:-- -:-+
C0.4
-:,K:,:--
3
- , • u = - - -.- ' - - - - - - -
e Evaluate the complex integration - - - - - - - - - - - - 1 -~
le (.t+Os dz, where C is the circle lzl = 2. 4 Tf.f
---+--CO-.S--+-K-4-

·.N. -2
\'£C T/O N B
-3
Solve by method of variation of paramete 2 d2y dy
r x - dJIZ
+ x- -
dJr 9y· = 48 x S CO.I K4
I · State Convolution theorem and find i- 1
[,sz+i:csz+♦)] using.Convolution theorem. =. C0.2 KS
:J Find the Fourier series for the function
f(x) .= x -x 2
in the interval -1 S x S 1 .
C0.3 K4
I) Find the bilinear transfonnation which map
the poin
w = -1, -i, 1'show that under this maP. i , t ts z = -1, 0, 1 onto the point
e upper half ofth.e z. plan maps opto the inter C0.4 · KS
the unit circle lwl = 1 · yJ.e;:, 1 ior of
, .W =- /., Z. ..
Find the Laurent Expansion for f (z) =
(i) lzl < 1 ii lz - 11> 1
Cr-t) (Z-z ) in the region
z.+1 6 CO.5 KS
(iit) o < lz- 21 < 1
 ' .. ., ' ...
• ._-. I 1 1\ - 'G'
1 J - ~: • ". • ••• , i. ;' •
\'E C7 10 N<
CO'S BL
M• rb KJ
e:_ . ,T . _.e~ I . 6
co.1
. '
' ii' .,z·". ··
::::.
Solve b ch . . . 1
.. •· : ·' . ·,,.'sn
dJf -·"
den t , .. .:~1 . ·
e,x a---• -- 11x . - -"y ·=•--8¥"
d .,, · - K3
Y anging the inde ' pen .'~., · ~ - · · CO.1•..
.· u
ya11
"' ... ... '· -' •. , · ... 6 . ·'1
Sol ve· thes1•multan = dy + 2x + Sv = e't •
. .,.;.. ·;- .··-· . . . . eous equ
·~,. ,·. -('· -r~.D+ l 4 X ==- t~
atio n;;-
5t- .Je
+ 4x·+ '3y t,
x: c,~~-<~c~e--, Marki CO'S BL
K5
,3 ~. . 6· CO.2
Find the i La ·, r= L--b = ,e.-3ts1,,.'{t K3

F'
ind
.
nverse p ace transfonn of cot
. .
-1

the _Laplace Transfonn of F(t ) _=

l 1, 0 ~ t:<J · · . ..
t, 1 <t s 2 ·r=-t&J =---1- ' f
~g i-1
. 'C,..,. -;g- . i= ::L
t -t
.i _
2-
t 2 2 <;.t < 00 .
.. . . BL
Marki
· · 6
·· or t!.../
-Test' the convergen1♦ce of the-seri
es
. u.., :::c.
· ·
..
2- "'
~
,,. 1L_-""' ~.-f __ r-. - ·.-1.
:x
: ,' · ,,C_oi,,-, ·
~ ._=
. 2 6 2
x 3+• ·· ~':'+1 · A..
. , .. -ti . ' . . . .-,._.. .u.. .-~.
1 +- x+ -x +- .J
· . a:t X. -=-;f - · Yt) =- .. -J'
...... , .. . .... . ,. ... ..
~ -.
9 17
·
2 Dhence deduce that co; .. 6 ,+ ..
f (t) = ~xi:- x in the int e~I
l
~~nd the half nui~e,sin~ series,o_ r(: t)- 011 oo , , ·., 'l')riX . · , .- J:,;.
-1-- . Sit\ ,--, --; -
. 11..=.
•

-= 1- 2.. +2
3
..-
3
2..8 + ... ·:· ..... .... ._..... ··- T - 0 )(. ·

Marks CO'S BL
('rJ 150cld ' K4
6 C0.4

ow that the function/ defined (z) by = f(z) :c~!;;\ z

-:i:,~Q1{J~) = O, is not analyti~ at the
'.;. ,:;;;:~~ ,'-- -
_ .. _
-·
' t,'

gh it-satisfies Cau chy -Rieman n equation at origin. ·

origin_evc;n thou C0.4 . K4
= + .and u = e-r[x ;iny - cosyJ
Find 6 y
-ff · f(z ) = ·u + iv is an ~al yti c fun
.
ction ofz x iy,
;({?2~}J+:J-~1-,.,~) c_. -,. - ,..f z.)=
\t-::: e_- i ~-z..
f(z ) _intermsofz.
Marks CO'S BL
6 co.s . .K3

- y - ix2 ) dz along real axis from z = 0 to z f 1 and then

Find the value of the integral fot+t(x
alon a line · arallel to ima in
axis from z = 1 to z =1 +i
. -l = + ·
6 00. S KS
,1n
.
+2 dz, -where C is circ
le lzl =
Evaiuate by'Cauchy's integral formula le 2112-511

--
1% & )·
K5, 30, K2, 2,
3 K3, 32,
· . . ' . 34'6
10 - -- ~ ·
60 . ~ ' '

so K4, 32,
40 ?3% ,
30 .
zo
I Blooms Level,Dlstrlbutlon I
10 .
. I
0
coi COJ cos ·,
C01

J
 .. 11111
mititil

H. Tl:'< ·11
.\'/:' ll-11
(Mme&LMbii&Mi& iiNii&i&,mm,j
Time: 3:0(Jllr., tln4DMM4ibth&iiM,W ;m1,w
lla x. .llar/,'i: 711
Note: (I) No student will be allowe
d to leave the e11mlnatlon roo
· (II) Be preclle In your m before end of e11m.
an1wcr.
(Iii) All questions are com
pulsory.
Course Outcomes:
FoIIow In2 are the cou
! CO'sCoda ne outcomes of the 1ubject:
-
COURSE OUTCOMES
: BAS203.1 . Remember the concept differ Blooms Level
entiation to evaluate LDE of
coefficient and LOE with var
iable coefficient of 2nd order.
nth order with co!l5t8nt I'
I
BAS203.2 Understand and apply the con Kl &KS
I to evaluate differential eQuat
cept of Laplace Transform and
inverse Laplace transform !
ions lI
BAS203.3 · Understand the concept of con K2 ,K3 &KS
vergence to analyze the conver l
exoansion of the function for gence of series and
Fourier series.
BAS203.4 Apply the concept of analyti K2&K4
city,
' annlvimi conformal transform hart]lonic function and create the image of the function
i ation.
I'. BAS203.5 . _Apply the concept of Cauch
y integral theorem, Cauchy inte . K3&K6
i
Laurent's Series and calculus gral formula, Taylor's and
of residue to evaluate integra
ls . K3&K5
\

Q. N. -1 ",£CT/ON A
. ''
renti'al equatio
. dzy + 2 dy
•• • J •a:: Marki • CO's
For a dtue BL
n tb:2 a c1z +y = 0, Fm•
d the val.ue of a for which the
equation characteristic equatio differential 2 CO.I K2
n has equal number of roots.
(b)
~:::: ±I
Find ~e Laplace transform 'F st
(t) = ; t
If f (x) = 1 + lxl in -3 <
~ rt c.- J¼ I
1
4_ .:::
~ :& ~ ¾/cf - · C0.2 IO
x < 3 , then find the value o~
,:-d:::--)- --; --F-in_d_th_e-im
Clo -=: b 2 CO J IO
--
ag_es_ of_c_ir_cl_e_l_z_-_- l_l_=_l_u--,--
n.d-er_th_e-.m
- ap
- pi-ng_ w_=_;: - - - -Lt
- - - ,-/ -'.2-
---- - - ' - -2
- -J -_
C_0 -.4-l --IO
~
Evaluate
• the· complex integration
1
",
Y, - (
Z Z+ff I) dz , where C.is
C0.5 IO

---.--
:- N.-2 SECTION B
A=-
. . ~ t / · Marks CO's
Solve by method of variation ~ d 1 BI.
of parameter x 2
dx~ + 2x ~ : 12y = x 3
logx 6 CO.I K4 \
.I
State Convolution theorem.
Find L- 1 lcs2: ) 2] using by Co
4 nvolution theorem 6 00 .2
') Obtain the Fourier series to rep KS
resent f (x) = !. (rr - x) 2 in
the interval O S x S 2rr
• C0.3 · K4
·l Find the bilinear transformatio
n which p the points z = 0, ~ +r ~~
· es ofthe uni· . I Ima
also find the 1ma -:-1, i onto the points w =. _t, ~• 6 .
t c1rc e zI = 1 oo C0.4 1,,,'l.
,,
Find the Taylor's and Laure ~ ,- :z+ 1, / 'Z..c::Jt.O
nt's series which represents the
(i) lzl < 1 fµnction f (z) = zz+ :z+ l in the 6 C0.5
(ii) 1 < lzl < 3 region
(iii t < lz+ 11 < 2

~ ld) l!..l=- 1 )'(\Q p-& cx:t-

. '\l=:::U.
 .,,,
BL
1(.3
21
6 co.t
Solve by changing the independent variable +x2
dx1
(4x 2 - 1) dy+ 4x 3 y
= 2x 3
1(.3
d.r
6 CO.I
- + 2x + 4v = 1 + 4t dy +x ~ y = ! tz
Solve the simultaneous equation 1
· dt ., ' dt Z

Marte, CO'S BL
KS
6 co.2
rto,,21;-.e-~~
Find .t he.·inverse of Laplace transfonn of log G-s:-::-::-:) Fli) =
(b)-~ -. : . _ - ~ - ~ ---=--'----_:_::::_ __!_~:b_ ____,.....---4-.i; ~~rcl(.3
0 s t ~ .2
2
. . { t /:,. .l. -~ .
~ t :S 3 -n:&J-== ,J~ -.,g~ -l,.),-r_.3-&-r_.3 & /.
' '1- '
Find the .Laplace Transfonn of F(t) = t - 1, 2
7 t>3

CO'S BL '
· -·- -4--::
Marks
KS
-
Test the convergence of the series 6 C0.3
:2..
. sx z +'iox
X + ·3 8 3 15 4
+11x 't-1
+ ... ......... ......... ...... ... ...)1 ...... (_t)Y) ;( LI !- div x~ I
C0.3 1(4
~ind the half range cosine series of f(x) =x sinx in the interval (0, 11') 6

--f at,))L

DL
I
K4
. ,) If f (z) = u + iv is an analytic function of z = x + iy ana.v, =e-x(x cosy + ystn y) Find /(z) in
tenns ofz. U = ..:.,c XS I
Show th.at the function f(z) defined by/ (z) = . x: 2
2
~;
1
> , z =I:- 0, f (0) = ~• is not analytic at the .
K4

ori in even thou it satisfies Cauch -Riemann uation at ori in. Wx..~ O ~ tl~ = \9..~ c:: \Q

Marks CO'S BL
Eval~Jt'(z)2dz ~long (i) the real axis from z = 0 to z =;. and then along a line.P.arallel toy- 6 C0.5
axis from z = 2 toz=2+t (ii)alongthe line2y=x ,. (.JJ Qi'; ~ .... -2.t:Y-J
2 .
3 ...3
Evaluate by Cauchy's integral fonnula ~c z
z+t 2- z
~
+4
dz, where C ts circle lzl = 3 6 co.s

60 -----
70

50
40
30
20
,...,.._,..,..........,.,,,..,,,..,,,,.."'_,.....,,....,..;;,;.a,;<
0

10
0
! Blooms tevel'Dlstrlbutlon i
COl CO2 C03 C03 COS