Anna University Chennai 2012-2nd Sem B.E /B.tech /e , ester, , subject code:MA2161, MATHEMATICS ll (regulation 2008) - Question Paper
B.E./B.Tech. DEGREE EXAMINATION, MAY/JUNE 2012.
Second Semester Common to all branches MA 2161/181202/MA 22/080030004 MATHEMATICS - II
(Regulation 2008)
Time : Three hours Maximum : 100 marks
Answer ALL questions.
PART A (10 x 2 = 20 marks)
12 j
1. Transform the equation (2x + 3)2 - - 2(2: + 3)12 = 6x into a
dx dx
differential equation with constant coefficients.
2. Find the particular integral of (D - if y = ex sin x .
3. Find X such that F = (3x-2y + zj + (4x + Ay-z)j + (x-y + 2z)k is solenoidal.
4. State Gauss divergence theorem.
5. State the basic difference between the limit of a function of a real variable and that of a complex variable.
6. Prove that a bilinear transformation has atmost two fixed points.
7. Define singularpoint. ~ -
( \ 4
8. Find the residue of the function f\z) = -r at a simple pole.
z (z - 2)
9. State the first shifting theorem on Laplace transforms.
10. Verify initial value theorem for f(t) = 1 + e~l (sin t + cos t).
PART B (5 x 16 = 80 marks)
11. (a) (i) Solve (D2 + a2)y = sec ax using the method of variation of
parameters. ' (8)
(ii) Solve : (D2 - 4D + 3)y = ex cos 2x. (8)
Or
(b) (i) Solve the differential equation (x2D2 - xD + 4)y = x2 sin(log x). (8)
dx
(ii) Solve the simultaneous differential equations + 2y = sin2t,
- 2x = cos 21. (8) dt
12. (a) (i) Show that F = (y2 +2xz2}i +(2xy-z)j + i{2x2z - y + 2zz is
irrotational and hence find its scalar potential. (8)
(ii) Verify Greens theorem in a plane for
J[(3x2 -8y2)d x + {4y -Qxy)dy\, where C is the boundary of the
c
region defined by jc = 0, y = 0 and x + y = 1. (8)
Or
(b) (i) Using Stokes theorem, evaluate J F -dr, where
c
F = yzi + x2j - (jc + z)k and C is the boundary of the triangle with
vertices at (0, 0, 0), (1, 0, 0), (1, 1, 0). (8)
(ii) Find the wrork done in moving a particle in the force field given by
F = 3x2i + (2xz - y)j + zk along the straight line from (0,0,0) to (2,1, 3). (8)
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 z . (8)
(ii) Find the bilinear transformation which maps the points z = 0,l,co into w = i,l, -i respectively. (8)
Or '__
( d2 d2 1 r (b) (i) If f[z) is an analytic function of zy prove that =-+r- log /M = 0. (8)
ydx dy J 1 1
(ii) Show that the image of the hyperbola x2 - y2 = 1 under the
transformation w = is the lemniscate r2 = cos 26. (8)
z
14. (a) (i) Evaluate [ --rr- where C is \z - 2| = by using Cauchys
c(s-W-2) 2
integral formula. (8)
(ii) Evaluate f(z) = i-r in Laurent series valid for the regions
(z +1 ){z + 3)
|| > 3 and 1 < \z\ < 3 . (8)
Or
f z 1 I -I (b) (i) Evaluate ---- dz, where C is the circle \z-i\ = 2 using
l(z + lf(z-2)
Cauchys residue theorem. (8)
00
C COS TYIX
(ii) Evaluate =-r- dx, using contour integration. (8)
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15. (a) (i) Apply convolution theorem to evaluate L 1 (ii) Find the Laplace transform of the fo' |
/ >2 {s2+a2f J lowing triangular wave |
function given by fit) = J and fit + 2?r) = fit). (8)
[2n -t, 7i < t < 2k
Or
eat _ e~bt
(b) (i) Find the Laplace transform of-. (4)
t
co
(ii) Evaluate j te~2t cos tdt using Laplace transform. (4)
o
(iii) Solve the differential equation ---S~ + 2y = e~t with (o) = l
dt dt
and /(o) = 0, using Laplace transform. (8)
2 10394
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Attachment: |
| Earning: Approval pending. |
