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University of Mumbai 2007-2nd Sem B.E FE Applied Maths II - Question Paper

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C *1-2688-07.

y E (A (I fy, .jl ( Applied i4+u-ir,w >

(REVISED COURSE)    

3 Hours)

N.B.(1) Question No. 1 is compulsory.

(2)    Attempt any four questions from remaining six.

(3)    Figures to the right indicate marks.

\

1. (a) State Duplication formula for Gamma function and prove, that

1


2"

n +

2s)o|0'}-

, , 1 NO-995

Il

[Total Marks : 100

20

1 3 5 7.............( 2n - 1) =

Jx


Solve: (l + e) dx + eV 1-*Jdy = 0.

(b)


(c)    Evaluate JJJ (x2 + y2 + z2) dx dy dz over the first octant of the sphere x2 + y2 + z2 = a2.

v

(d)    Sketch the region bounded by the curves xy = 16, y = x, x = 8 and y = 0. Express area of this region as a double integral in two ways.

2. Solve the following differential equations.

(a)    (xy2 - ev*3 j dx - x2y dy = 0.

(b)    f -r tan 0 -

COS0 '

(c)    (D2 - 3D + 2) y = 2 ex sin (f).

(d)    (D3 - 7D - 6 ) y = (1 + x2) e2*.

3. (a) Find the length of loop of the curve 3 ay2 = x ( x - a)2.

(b)    Use the rule of D.U.I.S. to prove that

J e +x2dx=e"2a)a>0. o    d

(c)    Change the order of iotegration and Evaluate.

Jz -x2 x


dx dy,.

r~2 2

Vx +y

dx dy


If


4. (a) Evaluate (i+x2+y2) over one loop of Leminiscate

(x2 + y2)2 = x2 - y2 by converting into polar co-ordinates.

(b)    Find the volume bounded by paraboloid x2 + y2 = 4z and the cylinder x2 + y2

(c)    Prove that


00 O)


= 16.


a-X3

Jx


xj y4 e-dy = |.


dx


(i)


T _dx_ =lRf-Q- H'l

i ( ex + e-x)" 4 I 2 ' 2 J 1

(*39    C' 0 l|fv V :

*TT

sfw(<s\ ,S-

5.    (a) Show that the line e = divides the length of astroid x2'3 + y2* = a2* in fir4 quadrant 6

v

in the ratid 1:3.

(b)    Solve by the method of undetermined multipliers.    6

(D2 - 4D + 4) y = x2 + ex + Cos 2x.

(c)    (i) Evaluate JJ xy dx dy. where R is bounded by circle x2 + y2 = 2x, parabola 5

R

y2 = 2 x and the line y = x.

(ii) Find the area of the region common to two circles r = a cos 0 and r = a sin 9 by 3 double integration.

   frc dx    a>0    _

6.    (a) Evaluate Jq a+boos3; b>0 and applying rule of D.U.I.S. deduce that    b

P dx _ a

(i)    (a-i-bcosx)2 (a2-b2)3/2

cosxdx    -b

r*_c___

(ii) Jo (at-bcosx)2 Ja2_tj2j3/2

(b)    Find the mass of Lamina in the form of cardioide. r = a (1 - cos 0) if the density at any e, point varies as it's distance from the pole.

(c)    Solve :    8

x2 y_ 3 dfy dy y _ 4,

* dx dx dx x

7. (a) Prove that    

,3/4

r1 (1~x4) dx =i _l prz i\

Jo (H x4)2 4 2W 4'4

(b) Evaluate

dx dy dz

111

v (1i-x+y+z)3

over the volume of tetrahedron bounded by planes x = 0, y = 0, z = 0 and x + y + z = 1.

(c) Solve by the method of variation of parameters.    flL

d2y    i

y +y_ j -

dx    2 1 1 + sinx


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