STUPIDSID
1 (a)
Prove that:
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1 (b)
Show that matrix
is derogatory
is derogatory
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1 (c)
Evaluate the following:
where C is |z-1| = 1
where C is |z-1| = 1
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1 (d)
Evaluate A∫B(3x2y - 2xy)dx + (x3 - x2)dy along y2 = 2x3 from A(0,0) and B(2,4)
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2 (a)
Prove that: xJ'n(x) = -nJn(x) + xJn-1(x)
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2 (b)
Show that the matrix:
is diagonalizable.
Also find the transforming and diagonal matrix.
is diagonalizable. Also find the transforming and diagonal matrix.
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2 (c)
Evaluate ∬(∇×F). ds where F = (2x - y + z)i + (x + y - z2)j + (3x - 2y + 4z)k and 'S' is the surface of the cylinder x2 + y2 = 4 bounded by the plane z = 9 and open at the other end.
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3 (a)
Evaluate the following:
where C is:
(i)|z - 2 - i| = 2
(ii)|z - 1 - 2i| = 2
where C is:(i)|z - 2 - i| = 2
(ii)|z - 1 - 2i| = 2
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3 (b)
Show that F = (yexy cos z)i + (xexycos z)j - (exysin z)k is irrotational and find the scalar potential for F and evaluate ∫F.ds along the curve joining the points (0, 0, 0) and (-1, 2, π)
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3 (c)
Prove that:
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4 (a)
Define analytic function. State and prove Cauchy-Reimann equation in polar co-ordinates.
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4 (b)
Verify Divergence Theorem; evaluate for F = 2xi + xyj - zk over the region bounded by the cylinder x2 + y2 = 4, z = 0, z = 6
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4 (c)
find A100
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5 (a)
Define conformal mapping. Find Bilinear transformation which maps the points z = 0, i, -1 onto w = i, 1, 0.
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5 (b)
Evaluate the following:
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5 (c)
Find the characteristic roots and characteristic vectors of A3+1
Find the characteristic roots and characteristic vectors of A3+1
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6 (a)
Expand the following:
about z = 0 for
(i) |z| < 1; (ii) 1 < |z| < 2; (iii) |z| > 2
about z = 0 for (i) |z| < 1; (ii) 1 < |z| < 2; (iii) |z| > 2
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6 (b)
if f(z) = u + iv is analytic and 
find f(z)
find f(z)
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6 (c)
Verify Cayley Hamilton Theorem for
and hence find the matrix 2A5 - 3A4 + A2 - 4I
and hence find the matrix 2A5 - 3A4 + A2 - 4I
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7 (a)
Prove that the circle |z| = 1 in the z-plane is mapped onto the cardiode in the w-plane under the transformation w=z2+2z
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7 (b)
Reduce the following quadratic form to canonical form and find its rank and signature:
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7 (c)
Verify Green's Theorem for
where c is boundary of the region defined by x = 1, x = 4, y = 1, y = √x
where c is boundary of the region defined by x = 1, x = 4, y = 1, y = √x
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