### MA 2161 – MATHEMATICS - II Second Semester B.E./B.Tech. DEGREE EXAMINATIONS, MAY/JUNE 2010 Question paper

Reg. No.
Question Paper Code: E3120
B.E./B.Tech. DEGREE EXAMINATIONS, MAY/JUNE 2010
Regulations 2008
Second Semester
Common to all branches
MA2161 Mathematics II
Time: Three Hours Maximum: 100 Marks
Part A - (10 x 2 = 20 Marks)
1. Transform the equation x2y00 + xy0 = x into a linear di®erential equation with constant
coe±cients.
2. Find the particular integral of (D2 + 1) y = sin x:
3. Is the position vector ~r = x~i + y~j + z ~k irrotational? Justify.
4. State Gauss divergence theorem.
5. Verify whether the function u = x3 ¡ 3xy2 + 3x2 ¡ 3y2 + 1 is harmonic.
6. Find the constants a; b; c if f(z) = x + ay + i(bx + cy) is analytic.
7. What is the value of the integral
Z
C
µ
3z2 + 7z + 1
z + 1

dz where C i s jzj =
1
2
?
8. If f(z) =
¡1
z ¡ 1
¡ 2
£
1 + (z ¡ 1) + (z ¡ 1)2 + ¢ ¢ ¢
¤
, ¯nd the residue of f(z) at z = 1:
9. Find the Laplace transform of unit step function.
10. Find L¡1 fcot¡1(s)g.  229  229  229
Part B - (5 x 16 = 80 Marks)
11. (a) (i) Solve the equation (D2 + 4D + 3) y = e¡x sin x. (8)
(ii) Solve the equation (D2+1) y = x sin x by the method of variation of parameters.
(8)
OR
11. (b) (i) Solve (x2D2 ¡ 2xD ¡ 4) y = x2 + 2 log x. (8)
(ii) Solve : (8)
dx
dt
+ 2x + 3y = 2e2t;
dy
dt
+ 3x + 2y = 0:
12. (a) (i) Prove that ~F = (6xy +z3)~i +(3x2 ¡z)~j +(3xz2 ¡y)~k i s irrotational vector and
¯nd the scalar potential such that ~F = r'. (8)
(ii) Verify Green's theorem for
Z
C
(3x2¡8y2) dx+(4y¡6xy) dy where C is the boundary
of the region de¯ned by x = y2, y = x2. (8)
OR
12. (b) Verify Gauss-divergence theorem for the vector function ~ f = (x3 ¡ yz)^i ¡ 2x2y^j + 2^k
over the cube bounded by x = 0, y = 0, z = 0 and x = a, y = a, z = a. (16)
13. (a) (i) Prove that every analytic function w = u + iv can be expressed as a function of
z alone, not as a function of ¹z. (8)
(ii) Find the bilinear transformation which maps the points z = 0; 1;1 into w =
i; 1;¡i respectively. (8)
OR
13. (b) (i) Find the image of the hyperbola x2¡y2 = 1 under the transformation w =
1
z
: (8)
(ii) Prove that the transformation w =
z
1 ¡ z
maps the upper half of z-plane on to the
upper half of w-plane. What i s the image of jzj = 1 under this transformation?
(8)
14. (a) (i) Find the Laurent's series of f(z) =
7z ¡ 2
z(z + 1)(z + 2)
in 1 < jz + 1j < 3. (8)
(ii) Using Cauchy's integral formula, evaluate
Z
C
4 ¡ 3z
z(z ¡ 1)(z ¡ 2)
dz, where `C' i s the
circle jzj =
3
2
: (8)
OR
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229  229  229
14. (b) (i) Evaluate
Z1
¡1
x2 ¡ x + 2
x4 + 10x2 + 9
dx using contour integration. (8)
(ii) Evaluate
Z2¼
0

2 + cos µ
using contour integration. (8)
15. (a) (i) Apply convolution theorem to evaluate L¡1
·
s
(s2 + a2)2
¸
. (8)
(ii) Find the Laplace transform of the following triangular wave function given by
f(t) =
(
t; 0 · t · ¼
2¼ ¡ t; ¼ · t · 2¼
and f(t + 2¼) = f(t). (8)
OR
15. (b) (i) Verify initial and ¯nal value theorems for the function f(t) = 1+e¡t(sin t+cos t).
(8)
(ii) Using Laplace transform solve the di®erential equation y00 ¡3y0 ¡4y = 2e¡t with
y(0) = 1 = y0(0). (8)
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