1(a)
Express the following matrix as the sum of symmetric and skew symmetric matrices where

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1(b)
Obtain the Fourier series for the function f(x) = 2x - 1 . . . 0 < x < 3

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1(c)
Evaluate the following:

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1(d)
If f(z) = u + iv is an analytic function of z = x + iy and u + v = cosx.coshy - sinx.sinhy. Find f(z) in terms of z.

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2(a)
Find the Laplace transform of the following:

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2(b)
Reduce to normal form and find the rank of the matrix:

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2(c)
Find the Fourier series of the function

f(x)= x --- 0 ? x ? ?

= 2? - x --- ? ? x ? 2?

Hence deduce that

f(x)= x --- 0 ? x ? ?

= 2? - x --- ? ? x ? 2?

Hence deduce that

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3(a)
Construct an analytic function f(z) if its real part is:

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3(b)
Find adj A, A

^{-1}and also find B such that:
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3(c)
Find inverse Laplace transform of the following:

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4(a)
Obtain Taylor's and Laurent's expansion of f(z) indicating regions of convergence

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4(b)
Find the half range sine series for the function

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4(c)
Find the Laplace transform of the following functions:

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5(a)
Evaluate the expression that follows. Take C as (i) |z| = 1 (ii) |z + 1 -i| = 2 (iii) |z + 1 + i| = 2

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5(b)
Find non singular matrices P and Q such that PAQ is in the normal form. Hence find the rank of A where:

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5(c)
Solve y'' + 2y' + 5y = e

where y(0) = 0, y'(0) = 1

^{-t}sintwhere y(0) = 0, y'(0) = 1

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6(a)
Evaluate the following along the path (i)y = x (ii) y= x

^{2}:
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6(b)
Use residue theorem to evaluate

where C is |z| = 3

where C is |z| = 3

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6(c)
Investigate for what values of a, b the following equations

x + 2y + 3z = 4

x + 3y + 4z = 5

x + 3y + az = b

have (i)no solution (ii)a unique solution (iii) an infinite no. of solutions

x + 2y + 3z = 4

x + 3y + 4z = 5

x + 3y + az = b

have (i)no solution (ii)a unique solution (iii) an infinite no. of solutions

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7(a)
Show that the set S={sinx,sin3x,sin5x,...} is orthogonal over [0, ?/2]. Find the corresponding orthonormal set.

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7(b)
Find the Fourier series of the function

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7(c)
(i) If u, v are harmonic conjugate functions, show that uv is a harmonic function

(ii) Find the Laplace transform

(ii) Find the Laplace transform

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