Engineering Mathematics 2 Jan 2014

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USN 1OMAT21

Second Semester B.E. Degree Examination, Dec.201"3lJan.20l4Engineering Mathematics - ll

Time: 3 hrs. Max. Marks:100

Note: l. Answer any FIW full questions, choosing at least two from each part. ,tiy:

:: ,...,, 3. Answer to objective type questions on sheets other than OMR will not be vuluedii'','",

. .',gef equation of the form,

a; d[*,0,9) = o B) of v,p.9] = o\ cly/ \ dx)\ J./

:::,:::'

ii) The generalsolution of the equation, p2

A) (y+x-,.c:JQ+2x-c)C) (-y-x-cXy-2x-c)=0

iii) Clairaut's equation is of the form.

,,.,,,,,.

,, ,

(04 Marks)y = f (x,0) then difft-rentiating x we

...:

A) x=py+f(p) ts)iy=p'^+f(p) C) y=px+f(p) D)Noneoftheseiv) Singular solution of y: pf,* 2p2 is, ,n ".."""'::'::"'

A) y2+8y=g B) i2*8y=g C)r'+8y-c=0 D)x'+8y=gb. Solve p2 +2pcoshx+1=0. (o4Marks)c. Find singular solution of p = sin(y -xp). (06 Marks)

d. Solvetheequation, y2(y-xp)=rop' ,ringru tion X=l and V=]. (06Marks)

z a. choose the correct J;i";. tn rouo*ing ' ,i',,,,

x v (04 Marks)

i) A second order linear differential "quuiio,

hur, . ""' :,::::,,,

A) two arbift*y solution B) One afbifiary solution

PART _ AI a. , Choose the correct answers for the following :

i)'r Suppose the equation to be solved is of the form,

ciC)

oL .,,,Q :,::,

()

I3soo#

.F

_-o

oo ll

troo.=N

6J Y3

()tr-c0)-'F

a-^

a::

bU

o0trEb>!/66-9G

6.y

rdo!eo.J

O:

@tE

La)

>\ +ioo-tro0o=9Otr>=OCJ:

U<-No

a!oa.

c) 0(*, y, p, =r.,, , D) g(x, y,0) = 0

-3p+2=0is,,B) (vr'x-c)(y-2x-c)=QD) (y-x-cxy*x-c)=Q

C) np arhitrary solution D) None of theseii)

[r,j:f:::: -4i are the roots of A.E of a homogeneous linea differential eWaffiits. solution is,

'lrrr'ri'r

',..' ' C) C,e2' + Cre* cos4x + C,e' sin 4x D; C,e2* cos4x + Cre2'sin.

iii) P.I. of (D + 1)2 y = e-^*3

y'seB; C,e:t + C, cos 4x + C, th !*Y

x'A)-2

B) x3e* 6', X3

"-**:JD) {"-}

2

iv) Particular integral of f(D)y = e*V(x) is,

A) 9H B)eu'frlur,.ll C)eu*ffi[u,,,1

Sorve g'v

-39'v.*:9l-v =0.dx' dx' dxSolve y' -3y' -t2y = 2sin xcosx

Solve the system of equation, # - 2y = cos2t, S * 2x = sin 2t .

D)#.r) [t"'vt*l]

b.

c.

d.

(04 Marks)

(06 Marks)

(06 Marks)

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3 a. Choose the correct answers for the following :

i) In xzyo + xy'- Y = 0 if e' = x then we get x2y" as,

A) (D-l)v B) (D+1)y C) D(D+l)yii) In second order homogeneous differential equation,

x: a is a singular point if

1OMAT21

(04 Marks)

D) None ofthesePo (x)y'+ P, (x)y'+ P, (x)y = S ,

(04 Marks)

(06 Marks)

(06 Marks)

(04 Marks)

A) Po(a) > 0 B) Po(a) + 0 C) &(a):0

iii) The general solution or r' 1'{ +r+-y =0 is.dx' dx"llA) )'=C,x-C,- B) Crx*cr, C) C,x+Crx

D) P.(a) < 0

D) C,x - Crx

6

C) Ia,x'ir ' '''' D) None ofthese

iv) Fiobenius series solution of second order linear differential equation is of the form,,:e

A) xt!a.x'r=0

B) f a,x'r=0

b. Solve y' +a'y = secax by the method of variation of parameters.

c. Solve *,Q+4x9+2y=e^.dx' dx l

obtain the series solution of "ldI', 2*n = 0 .

Choose the correct answers for the following :

i) PDE of M+b = a2x+ y is" '

d.

4a.

x 9.9=tdx (N

B\ q 9=o'dx av

ii) The solution of PDE 2,, =2y' is"

A) z=x2*#ril"ig(v) B) z"';*2 y2 + xf (y) + g(y)

D) ,=y'+x{(y)+g(y)

c) x(t)Y(t)

c\ g+Z=o D) Z*4=,'^4 ox oy dx dy

''1i, ::

C) z=x2y'iflx;+g(x)iii) The,,,pulsidiary equations of (y2

.dxdvdzal _=j_=_R,,..'"t"'l'..t"' P q

^\ dx dy dz) )-y- +z' xy xz

+22)p+x(yq -z)=0 arb, ,

dx _dy _d,u) --------=y- +z- x xz

D) None of these

ir) In the method of separation of variables to solve xz" *zr=0 the assumed solution is

.: ofthe form,A) x(x)Y(x) B) x(y)Y(y)

11.l-

b. Solve +=cos(2x+3y).Ax'Ay

c.

d.

Solve xp-yq =y' -x'Solve 3u* + 2r, = 0 by the separation of variable method given

D) X(x)T(t) "":::',,,,

(04 Marks)

(06 Marks)

that u:4e-* when y: 0.

(06 Marks)





PART - B5 a. Choose the correct answers for the following :

lx2D JJ.xaya" =00

A)1

.

iv)"

A) j*'-'.-.d* B) i*'-'.-'o*00

The Beta and Gamma functions relation for B(m,n) =

c.

1OMAT21

(04 Marks)

B) - 1t2 c) u2 D) None ofthese

o; ll41r,e)rdrdo

I 'r't '

DJ Three

the surface o,f the sphere

D) 4n

ii) The integrat l[ f 1x, y;dxdy by changing to polar form becomes,

A) l[Otr,e)a.ae B) llr(r,0)drd0 c) l[rlr,eyraraeiir) For a real positive number n, the Gamma function f(n) =

O1'[x'e-.ax

,, - , o

4, Pril s, +:+(,:) C) r(m)r(n) D)I (m-n) l (m + n)

^ "lY^By changing the order of integration evaluate , ^!t'j:O+ y,;dydx. a > 0.

Hence evaluate 'f d*lw(06 Marks)(04 Marks)

D The scalar surface integral of i ore, s, where s is a surface in a three-dimensional

region R is given by, f?."ar :

-

Uy s divergence theorem.-*

A) IIIv iav ' s) lJv ?a*oy cr ffJv.iav D) None orthesev S -v-

iit If all the surface are closed in a region containing volume V then the followingtheorem !,s applicable.A) Stoke's theo.rem B)Green's theorem C)Gauss divergence theorem D)None of these

iii) The value of [(Z"y-*'h*+(x' +y'])dy] by using Green's theorem is.J

. J ) J -- -Q -

C) Jx'e-.dx0

d.

6a.

a x x+y ::, ,:,

Evaluate II I.-.'''O zdydx.

t,Express the integral I+ in terms of the Gamma function.

o!l-X

Choose the correct answers for the following :

A) Zero B) One C) Twoiv) fff.nas : _, where f : xi+yj+2k and S is

JS

*' +y' +22 =a2

A) 4na B) 4na2 g7 4xa3

b.

c.

Find the work done by a force f =(2y-x')i +6yzj-Bxz2k fromthe point (0,0,0) to thepoint (1, 1, 1) along the straight-line joining these points. (04 Marks)If C is a simple closed curve in the xy-plane, prove by using Green's theorem that the

integral llt"Or-ydx) representstheareaAenclosedbyC. Henceevaluate 4*-L =,.rs/\ | - -r- ---.--- ,J ". - A- b,

_, .

(06 Marks)

d. Verify Stoke's theorem fo. i =(2x-y)i-yz' j-y'zk for the upper half of the sphere

*' + y' +22 =I. (06 Marks)For More Question Papers Visit - www.pediawikiblog.com




7 a. Choose the correct answers for the following :

D L[t"1=.nA)-' sn+l

ii) L[e-3'1 =a-lA)

-- s-3

nB) "_,S

11

B)s+3

is equal to,

B) 3!

' (s -2)o

1OMAT21

(04 Marks)

p1 A' qn+l

1

D)s-3

,''a1,

D)(s -2)

....

c) n!' n-1

S

c)

c)

I

s+3

aJM

iii)

iv)

r{r1t-a)H(t-a)}3!A).--_

(s + 2;o

't{ 6rt - 1)} =

ry,,1,t,p;t B) et C) e^ ::,,,,,, . D) e-^

b. Evaluate Lisin'Zt) ,,, ,'.,. (06 Marks)

lz.. o<t<3c. Find L{(t)}" given rhar f1r; : J-, ;:,,,r [,, t > 3

(06 Marks)

li o<t<2I

d. Express f(t)= j4t 2<t<4 in terms of unit s.tep function and hence find their Laplace

[a t>4transform. ,,i...',',.,, ,,,

"""':: (04 Marks)".!j'".

8 a. Choose the correct answers for the foilowing : (04 Marks)

ai --!- sy --l- C) --f-- pl .--l ,S- +a- ,',,;;;.. S- -a- ,i S'+a' S'-a'

.l /r\liii) L-r]cot-rl+ ll=L \s'll,,!1, ,i!

L lr\D-a)J- _ _ .,.,..,,,1:

A) e'f(t) ,,,:,... B) e"tf(t) C) e--'f(t) D) None of these

:

sin t sinh at ^ sin atA) ts) C;.-_-'::.'.""":"tt"'"""" t

For the function (t) : 1, convolution theorem condition, ,,i:,;

A) Not satisfied B) Satisfied with some conditionC) Satisfied D) None of these

h at sin at ,_. sinh ttc)t''?

^ 2sr -6s+ 5

(s - 1)(s - 2Xs -3)

1i,

b. Find the inverse Laplace trans

() .

c. Find l, I

]+- ^ | .rsing convolution theorem. (06 Marks)l(s-lxs'+4)J

d. Solve differential equation y'(t)+y=F(t), where af,l={? 0<t<1. Given that

12 r>ly(0) = 0 = y'(0). (06Marks)

*****

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Engineering Mathematics 2 Jan 2014

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