Guru Gobind Singh Indraprastha Vishwavidyalaya 2009 B.E APPLIED MATHEMATICS - 2 - Question Paper
END TERM exam
SECOND SEMESTER [B.TECH] – MAY-JUNE 2009
APPLIED MATHEMATICS - 2
(Pltaae t&rtteyoxir&xcMi Rott No.) x<zm Roll No.; \&3jkM<SY'
SEXXMfDSEMESTER [B,TEGH.}t- MXY-JUKB 2009
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Paper Code: BTBLA-102 V 'T*' faperlD: 99102 ;- |
/; Subject: AppUed Mathematics-II (Batch: 2004-2008) |
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Time : 3 Hours 'V -* |
Maximum Marks :! |
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rT. Note: Attempt one question from each unit, Q.No.l is compulsory. | |
x? ,-.y2 z2 If ; + -T- + -=? 1, prove that
. a + u b + u c +u 3 (du*'2
fuY-fdu V f du) nf du du
__ +---+ x. + y-+:
\dz) '\dy) \dzj \ dx * dy
: >
du
dz
(3)
For the function, (p (x, y) = -
, find the magnitude of the x + y
|
vV> |  |
directional derivative along a line making an angle 30 with the positive x-axis at (0, 2).
fS)
Find the inverse Laplace transform of cot1It r
Apply Greens thm. in plane to evaluate j[(3x2 -8y2)dx +(4y-6xy)dy],
; - c
where C is the boundary of the region defined by y = Jx & y = x2.
QJ
Show that when |z+l[< l, Z'2 =I + ](n + l)(Z + I)0 .
(3)
(3)
* (c) (d)
5
?
3
?
*5'
?'"
3
?
*\
O'
(3)
(4)
(3)
(3)
(3)
t2 ; 0< t<2 t-I ; 2 < t < 3 . 7 : t > 3
(Fi
Find Laplace transform of f(t) =
y
(g) Expand tan-.1 . in the neighbourhood of (1, 1}.
X ... ___________ . :
(h) Prove that j>(z-a)udz = 0 (n is an integer / -1), where C is the circle
Z-al = r.
UNIT-I
Q.2 (a) X If u =si-H X + y
A
sin prove-tnal
vx+Vy.
d2u sin u cos 2 u 4cos3u
> <?2u d2u 2
x";~ + 2xy-+ y -
dx dx'dy dy
(6)
. i-
(3.5)
(3)
|6.5)
/ fob# If w f(x, y), x = rcos9, y
* f -n \2 . \2 / \ 2
rsinO, show that
I, dr ) r2 V J (.fix J
v\><
dy
dx
dz
If x* + y2 + z2 - 2xyz = 1, show the -
\l K* JT-y2
52u 3:u
Transform the equation - -J---- ~ 0 into poJar coordinates.
dx' d\~
= 0.
Q.3 (a} (b)
Examine the following function for extreme values x4+y - 2x2 v 4xy -2y*. . (6)
: " " --J- ' .\l :/ /?
Q.4 Show that the function f{z) = e~z , z 0 and f\0) - 0 is notanalytic at
2 = 0, although CaudHy-Rieriiaxin equationsare satisfied at this point- (6-5)
{hyX If w * 7, + prove that, 'when z describes the circle + yl = aVw > < . ; 7
Cj /*/ z :t v ' .- o*-...-..../. ' ' v*n;
/ describes a straight line and find its length. Also prove that if z
describes the circle x2 + y2 = b2, where b > a, w describe? an ellipse. (6)
r x2 dx CT'
Q.5 (a) Evaluate by the method of complex variables, the integral J : . (6)
- (l *1' X - y CL_
(b) State and prove Cauchys Integral formula. Hence evaluate
t 7-rr dz, where C is the circle | Z | = 2* (6.5)
Jc (z-ij)
UNIT-III
Q.6 (a) AppJy Stokers theorem to evaluate f (ydx + zdy + xdz)where C is the
curve of intersection of x'2 + y2 + z2 = a2 and x + z = a. (6.5)
(b) Using Divergence theorem, evaluate |f. ds where
f* 52-u xi-~*2y2j-i-zzk & S is the surface bounding the region xa + y* = 4,
v, - 0 and % = 3. (6)
Q.7 Find the work done in moving a particle oncc round the circle Q-
x.+y>.=9 in the xy-plane if the field of force is
j ' F ~ (2x - y - z)i + (x + y - zz )j + (3x ~ 2y + 4z)k - if possible, find its scalar W.
, // potential. (6)
*wi Find the values of a, b, c for which the , vector.
V = (x -i- y az)i + (bx + 3y z)j + (3x + cy + z)k is irrotational. (3.5)
Evaluate Jr.n dS, where S is a closed surface. (3)
UNIT IV C
da dv 17
Q.8 (a) Solve by using Laplace Transform, + 2*---h5y = sin2t,given v =
x df dt 2 " -
d
..... 2 and -4, when t = 0. (6.5)
(b) Find the Laplace transform of the triangular wave function of period y
/ t, 0 < t < c
2c given by f(t) - . (6)
2c -1, c < t < 2c >.
ft \ j g i Q.9 I (al) Apply convolution theorem to evaluate L i -3--- I . (4)
Vs a J
Evaluate l| dl j . (4.5)
Find the inverse Laplace Transform of * . (4)
SJ 4-4al
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Attachment: |
| Earning: Approval pending. |
