University of Mumbai 2008-4th Sem B.E Electrical and Electronics Engineering Applied Maths-IV - Question Paper
Applied Maths-IV Sem IV June 2008
N.B.: (1) Question No. 1 is compulsory
(3 Hours)
(2) Attempt any four questions from question Nos. 2 to 7
(3) If in doubt make suitable assumption. Justify your assumption and proceed.
(4iFiflurM to theriflbt indicate full marks. . > . I * i ___
1. (a) If the angle between the surfaces x2 + axz + byz = 2 and x2z + xy + y+1=zat (0, 1, 2) is cos*1 then find the constants a and b.
(b) Find Ihe eigen values and eigen vectors of the orthogonal matrix
1 2 2
B = 3
2 t -2 2 -2 1
(c) Evaluate J f(z)dz along the parabola y * 2k2 from z*0toz = 3 + 10i where 5
c
f(z) = x2 - 2ixy.
(d) Find unit normal vector to the unit sphere at point - 5
a a
73*73
2, (a) Find the direotional derivative of xy2 + yz3 at the point (2, -1,1) along Ihe tangent to 6
*
the curve x = a sin t, y * a cos t, z = at at t = ~.
(b) Verify Cauchy's integral theorem for I(z) * ez along a circle c : I z I = 1. 6
(c) Reduce Ihe Quadratic form - 8
8x2 + Ty2 + 3z2 + 12xy + 4xz - 8yz to sum of squares and find the corresponding Linear transformation also find the rank, index and signature.
3. (a) Using Caley-Hamilton theorem for -
-3 -I -4 A = 2 2 1 0 1 2 Find A64 + 2A37 - 581.
Prove that V2f(r) = f"(r) -i- f(r) and hence show that V4er = 1+ -j er
(c) Find all possible Laurent's expansion of the function
7z-2
z<z-2)(z+1) about z = -1.
' 2 01
1/2 2 th0D prove that both A and B are not dia90na,iza&le 6 but AB is not diagonali2able.
<b) Verify Green's theorem in plane for 6
|(x2 -2xy) dx f (x2y 3)dy where c is the boundary of the region defined
by y2 Bx and x * 2.
6
(b)
f(z) =
|
'1 2' | ||
|
u < |
0 1_ |
and B = |
ITtlDli rtiWD
.3436-00-9712. -g2
Con
(c) (i) Evaluate _J (,2 + a5) (x2 + b2) dx ' a > - b>0
2n
J
dO
()j) Evaluate J V2 - CO80
f n 4
5. (a) Evaluate J (2 _ n ]$f where c is I z I e 1.
(b) Deline Minimal polynomial and derogatory matrix and Test whether the matrix
5-6-6 -14 2 3-6-4
is derogatory.
(c) Verily Gauss-Divergence theorem lor -
F 4xi + 21 + z2k taken over the region of the cylinder bounded by xz -i y2 = 4.
z = 0 and z = 3.
[1 A~
~ then prove that 3 tan A A tan 3.
(fc) Evaluate J(2 - z2) dz where c is the upper half of circle I z - 2 I * 3. c
(c) (i) Show that F * (ye** coszji + (xe*y cosz)j + (-e*y sinzjk is irrotational and
find the scalar potential $ such that p _ y .
(ii) Rnd div F where F * -4J
x*+y*
7. (a) State and prove Cauchys-Residue theorem and hence -
f 1 + 2 a
Evaluate J Z(2-z) where c is I z I = 1
2
0
(b) Evaluate JJ F*nds where F = (x + y2) i - 2xj + 2yzk and s is the surface of the plane
2x + y + z = 6 in the first octant.
<c) Show that the matrix
-9 4 4 A = -8 3 4 -16 8 7
is diagonalizable. also find the diagonal term and diagonalizing matrix P.
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| Earning: Approval pending. |
