Kerala University 2009-2nd Sem B.Tech Combined First and , - Question Paper

III Semester B.Tech. Degree Examination, June 2009
(2003 Scheme)
03 – 301 : ENGINEERING MATHEMATICS – II
(CMNPHETARUFB)
Time : three Hours Max. Marks : 100
PART – A
ans all ques.. (10×4=40 Marks)
1. Solve P(p + y) = x(x + y)
2. obtain the orthogonal trajectory of the family of parabolas y = ax2.
3. State the Dirichlets condition for the convergence of a Fourier series.
4. find the Fourier series of
? ? ?
= = p
- p = =
=
sin x, 0 x
0, x 0
f (x)
5. Evaluate ? ? ?
p
?
?
dr d
2
0
aec
0
r2 cos2
6. obtain the angle ranging from the tangents to the curve x = t, y = t2, z = t3 at t = ± 1.
7. obtain the directional derivative of u = xy + yz + zx at the point (1, 2, 3) along the X-axis.
8. Show that ?( + ) + +
B
A
2xy z3 dx x2dy 3xz2dz is independent of path joining the
points A and B.
9. Evaluate F.nds
S
?? where F = (x + y2 )i - 2xj + 2yzk and S is the surface of the plane
2x + y + 2z = six in the 1st octant.
10. Solve (D2 – 4D + 4)y = e2x.
PART – B
ans 1 ques. from every Module. (3×20=60 Marks)
Module – 1
11. a) Solve P2 + x2 = 4y.
b) Solve (D2 – 4D + 4)y = x2 + ex + cos 2x.
c) Solve by the method of variation of parameters 4y sec 2x
dx
d y
2
2
+ =
12. a) Solve ( ) ( ) y four cos log(1 x)
dx
dy
1 x
dx
d y
1 x 2
2
+ two + + + = +
b) Solve 2x 0
dt
dy
3x y e
dt
d x t
2
2
- - = - =
Module – 2
13. a) find the Fourier series of
? ? ?
p - = =
p = =
=
(2 x), one x 2
x, 0 x 1
f (x) in the interval (0, 2)
b) Evaluate ? ?
-
+
1
0
2 x
x two 2
2
x y
x
dy dx by changing the order of integration.
14. a) find the half range cosine series for
? ? ?
- < <
= < < l l l
l
k( x), /2 x
f(x) kx, 0 x /2
b) Evaluate ?? r3 dr d? over the area bounded by r = two sin ? and r = four sin ? .
Module – 3
15. a) Using Stokes theorem evaluate ?
c
F.dr where F (2x y) i yz2 j y2zk
_
= - - -
-
and S
is the upper half surface of the sphere x2 + y2 + z2 = one and C is its boundary.
b) Using Stokes theorem prove that div curl F = 0 .
16. a) Show that div grad ??
?
?
?
?
r
1
= 0.
b) If
_
F = x2y j+ x2z k + x3 i
-
, evaluate ??
s
F.n ds by using divergence theorem where
S is the surface bounding the region x2 + y2 = a2 z = 0 and z = b.
———————

Earning:   Approval pending.