Karunya University 2009 B.Sc Mathematics Differential Equations, Numerical Analysis and vectors - Question Paper
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Illllllllllllllllll (Pages : 3) 1049
Name :.........................................
Final Year B.Sc. Degree Examination, March 2009 Part - III : Group - I : MATHEMATICS Paper IV - Differential Equations, Numerical Analysis and Vectors
(Perior to 2006 Admission)
Time : 3 Hours Max. Marks : 65
Instruction x Maximum of 13 marks can be earned from each Unit.
UNIT - I
dy y - x +1
1. Solve ~7~- 7. 4
dx y- x + 5
2. Show that the equation (x2 - 4xy - 2y2) dx + (y2 - 4xy - 2x2) dy = 0 is exact and hence solve it. 4
3. Find the orthogonal trajectories of the circles x2 + (y - c)2 = c2 4
4. Solve : (D2 - I) y = 2x2. 4
5. Solve : (D2 - 2D +2) y = ex cos 2 x. 4
UNIT - II
2 d2y dy _ ,
6. Solve : x T - x + 2y = xlog x. 5
dx
dx
7. Solve the system - x + y
dy
dt
8. Find the Laplace transforms of
i) e-t Cos 2t ii) 4e5t + 6t3 - 3 Sin 4t.
9. Solve the equation y" (t) + y(t) = t, y(0) = I, y! (0) = -2, using Laplace transforms.
UNIT - III
10. Prove that A*1 Sin (ax + h) = (2Sin-]T Sin
n
ax+h+ (ah+ 7t)
11. Prove that i) 1 + A = E
12. The following data gives the melting point of an alloy of lead and zinc, where t is the temperature and p is the percentage of lead in the alloy.
p : 60 70 80 90
t : 226 250 276 304
Applying Newtons interpolation formula, find the melting point of the alloy containing 84 percent of lead. 5
13. Apply Lagranges formula to find f (5) given that f (1) = 2, f(2) = 4, f (3) = 8,
f (4) = 16, f(7) = 128. 5
UNIT - IV
14. Prove that (ax b) (cx d) = (a c)(b d)-(a d)(b c). 4
_ r
15. If r = r , where r = xi + yj + zk, prove that Vr-
i
16. Find the directional derivative of the function 2xy + z2 in the direction of the vector
i + 2j+2k at the point (1, -1, 3). 4
17. Show that F = (2xy + z3)i + x2 j+ 3xz2 k is a conservative force field . Find the scalar potential. 4
18. If F is any vector point function, prove that div (Curl F ) = 0. 4
UNIT - V
19. Evaluate J F dr , where F = x2i + y2j and c is the arc of the parabola y = x2 in
c
the xy - plane from (0, 0) to (1, 1). 5
20. Evaluate jj F n ds , where F = 6zi - 4j + yk, where s is the portion of the plane
s
2x + 3y + 6z = 12 in the first octant. 6
21. State Greens Theorem. 3
22. Apply Stokes theorem to evaluate J" (ydx+zdy+xdz) where C is the curve of
c
intersection of x2 + y2 + z2 = a2 and x + y = a. 6
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