STUPIDSID
1 (a)
Prove that:
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1 (b)
Show that:
is derogatory
is derogatory
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1 (c)
Evaluate the following:
where S is the surface of the plane
in the first octant and
where S is the surface of the plane in the first octant and
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1 (d)
Evaluate the following:
where C is the curve |z-2| + |z+2| = 6.
where C is the curve |z-2| + |z+2| = 6.
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2 (a)
Prove that for any positive integer n, J-n(x) = (-1)n Jn(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)
Show that the area bounded by simple closed curve C is given by 
Find the area of ellipse
Find the area of ellipse
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3 (a)
Verify Cauchy's theorem for function f(z) = 3z2 + iz - 4 if 'C' is the perimeter of square with vertices at 1±i, -1±i
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3 (b)
Prove that
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3 (c)
Show that F = (2xy+z3)i+(x2)j+3xz2k is irrotaional and find its field from (1, -2, 1) to (3, 1, 4)
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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 = xi - 3y2j + zk over the region bounded by the cylinder x2 + y2 = 16, z = 0, z = 5
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4 (c)
Find eAt If
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5 (a)
Define conformal mapping. Find bilinear transofrmation which maps the points z = 2, I, -2 onto w = 1, I, -1
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5 (b)
Evaluate the following:
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5 (c)
Show that cos(x sinθ)=J0(x) + 2cos2θJ2(x) + 2cos4θJ4(x)-…
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6 (a)
Find all the possible Laurent's expansion of the function
about z = -1 indicating the region of convergence
about z = -1 indicating the region of convergence
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6 (b)
Prove that there does not exist any analytic function whose real part is 3x2-2x2 y+y2
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6 (c)
Verify Cayley Hamilton Theorem for
\(A=\begin{bmatrix} 3 &1 \\-1 &2 \end{bmatrix}\)
and hence find the matrix 2A5 - 3A4 + A2 - 4I
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7 (a)
Explain removable singularity with example.
Evaluate ∫C tan z dz, where C is the circle |z| = 2, using residue theorem
Evaluate ∫C tan z dz, where C is the circle |z| = 2, using residue theorem
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7 (b)
Find the analytic funstion whose real part is: e-x {(x2 - y2)cosy + 2xy siny}
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7 (c)
Reduce the following quadratic equation to canonical form and find its rank and signature x2 + 4y2 + 9z2 + t2 - 4xy + 6zx - 12yz - 2xt - 6zt
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