University of Mumbai 2008-4th Sem B.E Electrical and Electronics Engineering Applied Maths IV - Question Paper
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CD-5793
Con. 5250-07.
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(REVISED COURSE)
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(3 Hours)
N.B.(1) Question No. 1 which is compulsory.
(2) Answer any four questions from the remaining six questions.
(3) If in doubt make suitable assumption, justify your assumptions and proceed.
(4) Figures to the right indicate full marks.
1. (a) State and prove Cauchys-lntegral theorem.
(b) Evaluate / (z~2 ) where C is the upper half of the, circl9 lz-21 =3.
4 6 6 1 3 2 -1 -4 -3
1 2 -3
(c) Determine A , A and A . If A
(d) Prove thal V x j = a - n r (n,2> ( a r )r where 1 is constant vector.
2. (a) What is :he directional derivative of 1 xy2 + yz3 at the point (2,-1, 1) in the direction of the normal to the surface x logz - y2 - 4 at (- 1, 2, 1).
(b) Find the eigenvalues and eigenvectors of the matrix
A =
(c) Expand f(z) = (z + 1)(z+4) " the region
, show that An = An + A? -1 for every integer n 3 and hence find Aso 6
1 0 0 1 0 1 0 1 0
3. (a) If A *
(b) Evaluate J z2 - iz 2
c
(i) I z + i 1 = 1
(ii) the rectangle with vertices at (1. 0). (1, 3), (-1,3) and (-1,0).
(c) Verify Greens theorem in plane for
I (x2 - 2xy) dx + (x2y + 3) dy c
where C is the boundry of the region defined by y2 a* 8x and x 2.
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4. (a) Find b such that the force field 7Y//o -
F = (exz - bxy) I + (1 - bx2) j+ (e* + bz)k is conservative. Find the scalar potential <J> of Ff when F is conservative.
5 -6 -6 -14 2 3-6-4
is derogatory.
<b) Test whether the matrix A =
ML/
0 ( a2 + xz
f de
J 1 + sin?o
dx
(D
(c) Evaluate
(2)
3 1 1 3
then find (1) 4A (2) eA.
5. (a) II A =
(b) Find the sum of the residues of the function
sinz
f (z) = at its poles inside the circle I z I = 2.
' 1 z cosz r
(c) Verify Divergence theorem for F = 4Xj - 2y2j + z2 k taken over the region bounded by the cylinder
x2 + y2 s 4, z = 0, z = 3.
8 -8 -2 4 -3 -2 3 -4 1
is diagonalisable. If yes, find the transforming
6. (a) Test whether the matrix A *
matrix p and the diagonal matrix D.
(b) Define; Singular point, Essential singularity and Removable singularity with one example.
(c) Verify Stoke theorem for F = (x2 + y2) i - 2xyj taken round the rectangle bounded by the lines
x = a, y = 0, y = b.
7. (a) Evaluate jj F- nds where F = (x + y2) i - 2xj + 2yzk and S is the surface of the plane
2x + y + 2z = 6 in the first octant.
sinz
and z = it.
(b) (i) Expand the function f(z) =
n
(ii) Expand cos z in a Taylors series about z
(c) Reduce Ihe given quadratic form to a canonical form by orthogonal transformation and hence find rank index and signature.
Q = 3x2 + 5y2 + 3z2 - 2yz + 2xz - 2xy.
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| Earning: Approval pending. |
