I B.Tech - Regular Examinations, June 2009
First Year
MATHEMATICS-I - SET NO: 1
(Regulation 2007)
Time : 3 hours |
Maximam Marks:80 |
Answer any FIVE Questions
1. (a) Solve ( xy sin xy + cos xy ) y dx + (xy sinxy - cos xy ) x dy = 0.
(b) The world population at the beginning of 1970 was 3.6 billion. The weight
of the earth is 6.586 × 1021 tons. If the population continues to increase
exponentially, with a growth constant k = 0.02 and with time measured in
years, in what year will the weight of all people equal the weight of the earth,
if we assume that the average person weighs 120 pounds?
2. Solve d4y/
dx4 - y = cosx coshx + 2x4 + x - 1.
3. (a) Verify Rolle’s theorem for f(x) = x2 - 2x -3 in the interval (1,-3 ).
(b) Prove that u = x2-y2/x2+y2 , v = 2xy/
x2+y2 are functionally dependent and find the
relation between them.
4.(a) Find the radius of curvature of x = aeθ[sinθ-cosθ],y = aeθ[cosθ-sinθ] at θ=0.
(b) Trace the curve y2 (a-x) = x3, (a>0)
5.(a) Find the volume of the solid that results when the region enclosed by the
curves xy = 1, x-axis and x = 1 rotated about x-axis.
(b) Evaluate 0 ∫ 1 x ∫ √x x2y2(x + y) dydx
6.(a) Test the convergence of Σ (1+1/√n)-n2
(b) Test the convergence of Σ (Xn/n) ,(X > 0)
7.Verify Stoke’s theorem for over the box bounded by the
planes x = 0, x = a, y = 0, y = b, z = c.
8.8. (a) Find L [ f ( t ) ] where f ( t ) is a periodic function of 2 π and it is given by
f ( t ) = sint, 0 < t < π ,
f ( t ) = 0, π < t < 2 π
(b) Find L - 1 [ s/ (s2 +4s +5 )].
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