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L

USN

(A)

x

srn

x

8

Solve

4.,.

u4+ t

l9+6y

=

e* + l.

dx' dx'

dx

Solve

4-ou:

cos h

(2x

-

l)

+

3-.

dx'

1OMAT21

(r)

*ttl"

(04

Marks)

Second Semester

B.E.

Degree

Exan\priffi16pffie

/ July

2014

Engineering

MathemBtlG5-

ll

,

i,i

I

Nbte'L. Answer FIYE

futl

questions choosing at least twofrom

each

part.

i

'"

,P|;finswer

all objective

type

questions

only

in

OMR

sheet

page

5

of

the

Ae*€y

Booklet.

E

,.*ii*"r" to obiective

type

fuestions

on

ih"a,

other

thun OMR

wiT rcnibb"'iatued.

::j...

G;

o_

't,,:,. ,,,,Fime:3

hrs.

Max.

Marko:l@,O

e

E

E l$^**'f /---,.^- Dft/D C.,1,

^..--]i^--

^L^^dien

na lnnsi ttnn ltau onnb antl

d

.E

n

ER {

S=

PART

-A

*i

-?

t

"'*0,,?,,,,,,

S A

I

a. Choose

tlie.gorrect

answer

:

1-. ;,

''.'''

(04

Marks)

=-

I

i) The

g"""*f

solution of the

equation

^'p'

+

3xyp

+

2f

:

0

i*'-'

:[i

'

(e)"tv'*-.lixv-c):o

tsi'tx-y-c)(x2ip-.;:o

H

;

(C)

(*y

-

")

(i2y-c):

0

(D)

(y-N,+)'dx2

+

y'+

c):0

x4)

E

E

ii)

The given

differe#fal

equation is solvable for

y;i,f itiS

possible to

express

y

in

terms

of

''t:"'.,,,""''....;.,'.=

:-l*:I

'(ffixandp

{Q),xandy

(D)

Vandx

es

AA

(A)

yandp

trx

E

.E

iii) The singular solution of

Clpiraut's

equatrb't'S

_

g+

1f

yl*e(*)+r[g(x)]

tsl

v:Jlt+(c)

E

(c)

cy

+

(c)

**r,,,,

(D)

v

g2(x)

+

f

[g(x)]

$;

iv)

The singular solution of

the

.goeia4_y=

px

-

log

p

is

_

  a

(A)

y2:4ax

(B)

*t'{F

lofi

x'

*,(C)

y:

1- rog

(1')

(D)

*':

y

log

x

3E

b. Solve

p'

-

2p

sin h x.- I

:0.

(04

Marks)

 

s c.

soire

i:ip*ituntl

1rB.?1,-t

'imr

(06Marks)

E

E

d.

Obtain the

general

soHfioil=ard singular solution

of Clajraut's

equation

is

(y

-

px)

(p-1)

:

p.

E

S

(06

Marks)

oi

fr

,3

2

a.

Choose the correqi$answer

:

(04

Marks)

i

E

i) The

complemertary

function

of

[Da

+

4] x

:

0

is

-

 

E

(A)

xI.-'[c,

cos t

*czsint]+e'Icrcost+c4sint]

9.i

(B)

x=

[cr

cos

t

*

c2

sin t]

+

[ca

cos

t

+

ca

sin

t]

E'$

(c)

x:[cr

+czt]e-'

ef

(D)

x:

[cr

*

cz

t] e'.

fr

E

ii1 rina the

particular integral

of

(D3

-

3D2

+

4\y:

e'*

is

-

: t

--r'-r*

-.r^r-

2

x

o

i (A)

x'e'*

(B)

x'e'*

(C)

x'e*

(D)

*':o-

;oi

6 b 6

G

4Trr

{i

...:..:.::

E

iii) Roots

"f

g*4+*5v

:

o are

-

dx- dx

''iir.,,:::::'

E (A)

2+i

(B)

3+i

^

(c)

2+2i

(D)

-2+i

"'ir;,

E

iv)

Find

the

particular integral

of

(D3

+

4D)

y:

sin 2x is

-

(B)

-xsinx

8

(c)

:r#a

b.

c.

I of4

(06

Marks)

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d.

Solve

I*u

=re*,

fu

*r:y+e*.

dx"dx

Choose the correct answer

:

a. Choose the

correct answer

:

i) The value of

i

[.'"Xaya*

i,

JJ

o0

(A)

0

(B)

I

1OMAT21

(06

Marks)

(04

Marks)

i) The Wronskian of

x

and

x

e*,is

\

ii)

The complementary function of x2y"

-

xy'-

3y:

*'

log x

is--

(A)

cr

c-os

(log x)

+

c2 sin

(log

x)

(B)

c1

x-r

*

c2x3.

ici

crx

+

czx3-

(D)

crcos

x

*

cz sin

x.

{:'f",,"

(l+x;:.t

(B) (1+x):g-

(C)

(1

+x;2=.'

(D)

(ll*)':e'.

:quation ao(ax

+

b)2y"

.|

a1(ax

+

b)

y'+

azy:

0(x)

is

-

*"

'#q

Lv

,

&r+ryy.lv@Lrvu

q.u\4^ t

u,,,

J

*zJ

_

,su

u,

"

(A)?S{p"ltaneous

equation

(B)

Cauchy's

linereftation

(C)

dre linear

equation

(D)

Euler's

gqua

i

b.

Using

the vaffifon

ofparameters

method to solve the

equati@'t

+

2y'*

y:

e-*

log x.

.

flu=,,

-

.,5t'

(o4Marks)

c.

Solve

*' I-rzJ-illjr+(m2+n2)y:n2x'log*."

t'lt:='

(06Marks)

dx' '& -.d:

\ n i rr

d. Obtain the Frobenius

mdffi solve the

equation

\q:yff"t

d\

(U) CIX

-f

CzX

-

(r-,/

(;lcos

x

-r

s2 slrl

x.

....

,,,"

{,,1,,"'

jii)To

transform(1+x)2y"

+

(1+x)y'+

y

=2

sin

log

(1

+

x)

into a

l$

:dtfferential

$01

gquation

with

constant

coefficient

,,,,,.,

*'

F'

^

*

d'I

*

dY

-,

=

o.

'''S*

* .

",--

(06

Marks)

dx, dx

,

"ptu

#

o*'

a.

Choose the

corect

answer :

\

/13

a

(04

Marks)

g

a

and b from the relation

z

-

(*-u)'

+

(y

-b)'

fu

----------.',."E

qqi

(A) p'q'

:

4z

(B)

pq;

az

(c)

r

=

4z

(D)

t

:

4

ii) The Lagranges's

linear

pffdial differential

eqlihtion Pp

+

Qq

:

R the subsidiary equation

is_

{"k"

(A)

ax=dv=dz,;],$fu)

"=E=d'

(c)

$*Jr=a'

(D)

g(+q+q

R P,e}" P

a

R

Q'".:-'"ft

p P

a

R

iii) By

the

meth@f separation

of variable we seek a solutibn

I)

the form is

tAl

,-

ry'

(B)

z:x2+y2

(C)

x:z+i

(D)

*:<*lvtvl

*"''-.'","

iv) The.sdkhio

n

of

o,-1

:

sin

(xy)

is

-

'

,.-#.,-

*ft )

,:

-*'

sin

(xy)

+

y

(x)

+

0(x)

(B)

#.

x f(y)

+

$(y;

....,.

n'.

'

^..

{.}*,

'-.,,,.,,,,,,

,-

(c)

z:

-smf,r

+

y

(x)

+

0(x)

(D)

None

ofthese. '

.,..

,r..o

{

q*rb.

Form the

partial

differential

equation of all

sphere

of radius

3

units having their cenffe iu the

'

xy

-

plane.

(04

Marks)

c. sbtvl x

(y2

+

z)

p-y (x2

+

z)

q=

z

$2-f).

(06

Markq)

,

c,q,=

c. Solve x

(y'

+

z)

p-y

(x'

+

z)

q=

z

(x"-f).

(06

Mark-r}

#,1*

U u::jn.5jn"U

of separation

ofvariables

to solve

*''

y3

-.u

x'

-

-

o

.

(06

Marks)

'"

'

1vL avL

v--+x--=

"Ax

Ay

PART.

B

(c)

3

2

of4

(D)

%.

(04

Marks)

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t..i.,iv)

The value of

B(3,

%)

is

_

.

-:

."

(A)

ls

(B)

l

(c)

ry

1n)

19

"a.:.-;,

G

\

'

15

au^Ji

\-/

t

,

h**

b. chan

,.order

of

integration

in

{

I

,

or*

and hence walu11;,,,,.,fsbme.

(04

Marks)

4{i,

a

'

i..

c.

Evaluate

I

j-f

(I'*

y' +z'1dx

dy

dz.

,,\

.:.::.

(06

Marks)

d.

Prove

that

i *' ,* x f' I

u.

-

7t

{.rlr-r* trq.f*-qJ,

,"*'

(o6Marks)

-{

a.

Choose the correct

answer :

*

*

(04

Marks)

i)

Let

S be the closed boundarflsyq@ce

ofd@gion

of volume

V then

for

a

vector field

f

defined in V and in S

Jf.nds

is

6

:,,,'=-="=

ii)

If

Jr.ar

where

t=:*yijy']

andCirtfrd theparabola

y=2x2 fromtheregion

(0, 0) to

the

point

(1,2)

is_

(A)

7/-

..#

(B)

-7/.

(C)

3 3y

(D)

-3s

/6

_r

/O

iii)

h

the Green's tlteorem in

the

plane

f

uax +Ndy

=

.

"c

(A)

ffT@*a*)*.,

(B)

t(av

aN).

-

'

'

{tl

a,

-a*]*"

'-'

{[r,

-

*

f*'

._

{c)

ff[+-*lo.o,

to)

11[a.S]a.av

'd\a

Ay)

*[d*

q)

,,

,j,=i}i

A

necessary and sufficient

condition that the

line integral

1i

o;

for

any.'[4

"

C

('

'1""

15 .

p'*

ii)

The

value off

(%)is

_

1OMAT21

Q)

J2"

.

A)

2G

(B)

nJj

\

iii)

The integral

{"

f";;O

Oxdy after

changing the

fiitegration

is

_

b.Usingthedivergencetheorem,evaluateJr.,aswheref=4xzi_y,j*y,iandSisthe

s

surface

ofthe

cube

bounded by

x:0, X

:

l,y

=

0,

y:1,

z:0,2: l.

(04

Marks)

c.

Use the Green's

theorem, evaluate

{z*,

-vr)ax+(x2

+y2)dy

where

C is the

triangle

formed

bythe lines x:0,

Y:0

and

^

*

y:tl.

(06Marks)

d. Veriffthestoke'stheoremfor

f

=-y'i+x'j

wheresisthecircledisc *'+y'1l,z:0.

(06

Marks)

'l't.''""t1.

t

tt"'''='

3

of4

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a. Choose the correct answer

:

D

L{sffi at}

:

1OMAT21

(04

Marks)

(06

Marks)

d.

fcost ifO<t{he" rl^"'

Express

rrtt=.1*rzt

;^";";;

{.[erms

of up$i4 tunction

and

hence nno

r-{r1t1}.

l/dq:a'

[cos3t

if t>2n

,o"'

t

'-"

]

choose the correct answer

(06

Marks)

F

.:,

:,

(04

Marks)

i)

r'l*,,[:)]= *i*

.-t

t

(u/i

----ffi

'-ffm..

(A)

sint

-,..=

) ti'ut

(&ff

tq

u' .,)

sinh

t

t

-j"

t

'*

#ryh

t

(

t I ok*

{l

ii)

r,{+}=.n5

|.+.'

-

36J

r.

"-=.-

.

-iSL\j

(A)

e*

19;

sin3t

(C;

sinh:t"

4m

(D)

cosh3t

t2

iiD

l-cosat

'----'-r-

a-

(B)

l+cosat

a'

(c)

i+. a

' ,i'"1

'n i

'*,{L

a

".,,q

s[

',{{+.r}=

(A)

t-:t+zt3

,,{#=}

(B)

r+.i

(C)

t+1t,+r

32

b. Find

c.

Using Convolution theorem evaluate

r,{----t-}.

l(s+l)(s'+4)J

d. Solve

4*s9*oy=5e2,

given

that y(0)

=

2,

dY,i0)

=1

byusing Laplace transform method.

dt' dt

'

dt

,06

Marks)

*****

4

of4

,.

,r1)

{a,

\D/

-1;i{Q

.rnoF$;-ldz,

*

_u,

\u)

ffi'

1,.:,i

ii)

If L{f(t)}

=

F(s)

then

L{e"'f(t)}

is

=.

ii)

If L{f(|}

=

F(s) then L{e"'f(t)}

is

,::,,,

.

t"

i,fu#,4

,

(A)

F(s+a)

(B)

F(s-a)

(C)

F(9

(D)

None ofjhese

(,

1

-

*-dd

"

ttJ

-

-{"{^",$1,

l+tan-'(s-l1

(B)

I*tun,,

(C)

}-*,'.

1o)

cot-'(${rU*

ir)

ttamsform

of unit step

function L{u(t-a;}

is,

. [-4"

(A)

g:.

tg)

L

(c)

{

'*

p)

{

St;;:=ssr%\,+s

b.

Evaluate

i{r*toszt-cosrt*triot}.

.

rdi=-

(O4Marks)

t

:-t

J

-*""''

c.

Find

the Laplace

ffifrf.orm

of

the

triangular

wave,

giver{ffi

It

o<tcc""0.t-ld

f(t+2c)=f(t).

dffi,

*

(06Marks)

f(f)=l \.'fLv):r\t,,

\.u*ry

l2C-t

C<t<2C

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