QP Code 1 6 0 0 6 0 Register Number
VELALAR COLLEGE OF ENGINEERING AND TECHNOLOGY
(An Autonomous Institution, Affiliated to Anna University, Chennai)
Semester Examinations Apr May 2017 Regulations-2016
Programme: B.E./B.Tech. Semester: 2 Max. Marks: 100 Duration 3 Hrs
Course Code Title: 16MAT21 MATHEMATICS-II
Knowledge
Levels
K1 Remembering K3 Applying K5 Evaluating
K2 Understanding K4 Analysing K6 Creating
Part A Answer ALL Questions. 10 x 2 20 Marks
No. Question KL
1. If find grad at K2
2. Evaluate divF at the point given F x yzi xy z j xyz k 2 2 2 . K2
3.
Eliminate from the system 2y sin
dt
dx
x t
dt
dy
2 cos .
K5
4. Find the Wronskian Co-efficient of y y y e x x log . K2
5. State initial value theorem on Laplace theorem. K1
6. Define periodic function. K1
7. Write necessary conditions for a complex function to be analytic. K1
8. Show that the function is nowhere differentiable. K3
9.
Compute the residue of
2 1
2




z z
z
f z at its simple pole z 2.
K2
10. Define removable singularity with an example. K1
Part B Answer ALL Questions. 5 x 16 80 Marks
No Question Marks KL
11. a i. Show that F y xz i xy z j x z y z k

2 2 2 2 2
is irrotational and hence find its scalar potential function.
8 K2
ii. Show that 2 2 n n r n n r . 8 K3
OR
b Verify Green's theorem in the XY plane for

C
3x 8y dx 4y 6xy dy 2 2 where C is the boundary of the region
given by x y2 and y x2.
16 K3
12. a i. Solve D 4xD log x 2 2 . 8 K5
ii. Solve x D D y x 2 2 2 5 . 8 K5
OR
b i.
Solve x t
dt
dy
y t
dt
dx
2 sin 2 2 cos2 .
8 K5
ii. Using the method of variation of parameters, solve
tan2x 2 .
8 K5
13. a i. Find the Laplace transform of the periodic function





a t a t a
t t a
f t
2 2
0
where
8 K4
ii. Using Convolution theorem, find






2 2
1 1
s s a
L
8 K4
OR
b Solve: t y y y 4 2 given 1and 1 by using
Laplace transform method.
16 K4
14. a i. Find the analytic function whose imaginary part is
e y y x cos sin .
8 K3
ii. Show that an analytic function with constant real part is constant and
an analytic function with constant modulus is also constant
8 K4
OR
b i. Find the bilinear transformation that maps their points into the
points .
8 K3
ii. Find the image of the infinite strip 1 x 2 under the transformation
z
1
.
8 K4
15. a i.
Using Cauchy's Integral formula, evaluate
c z z
dz
2 2 1
where
C is
2
3
z .
8 K3
ii.
Expand

1
z z
in Laurent's series valid for 1 z 3.
8 K3
OR
b
Evaluate


2
0 13 5sin
d
using contour integration.
16 K3