1(a)
Find L(te

^{3t}sin t)
5 M

1(b)
Use the adjoint method to find the inverse of

5 M

1(c)
Find p if f(z) = r

^{2}cos 2? + ir^{2}sin p? is analytic
5 M

1(d)
Find the Fourier Series for f(x)=x

^{2}in (-1,1)
5 M

2(a)
Show that u=cosx.coshy is a harmonic function. Find its harmonic conjugate and corresponding analytic function.

8 M

2(b)
Show that the set of functions cosx, cos2x, cos3x,... form a orthonormal set in the interval (?, -?)

6 M

2(c)
For a matrix A verify that A(adj.A)=|A|I

6 M

3(a)
Find the Laplace Transform of each of the following:

6 M

3(b)
Find half-range series for the function:

6 M

3(c)
Find non singular matrics P and Q such that PAQ is normal form. Hence find its rank where A is given by

8 M

4(a)
Solve the system of equations x-y+2z=9, 2x-5y+3z=18, 6x+7y+10z=35

6 M

4(b)
Find the inverse Laplace Transform of the following:

6 M

4(c)
Expand the function f(x) with the period 'a' into a fourier series
f(x)=x

^{2}--- 0 ? x ? a
8 M

5(a)
Using Convolution Theorem, find the inverse Laplace transform of the following:

8 M

5(b)
Find the analytic function and its imaginary part if real part is:

6 M

5(c)
Find the Fourier series of the function

6 M

6(a)
Using Laplace Transformation, solve the following equation:

(D

(D

^{2}- 3D + 2)y = 4e^{2t}, with y(0) = -3 & y'(0) = 5
8 M

6(b)
Find the Fourier series of the function

f(x) = ?x ... 0 < x < 1

= 0 ... 1 < x < 2

f(x) = ?x ... 0 < x < 1

= 0 ... 1 < x < 2

6 M

6(c)
Determine l, m, n and find A

^{-1}if A is orthogonal
6 M

7(a)
Evaluate the following integral by using Laplace Transform

6 M

7(b)
Find the values of f(1), f(i), f'(-1), f''(-i) if:

8 M

7(c)
Reduce the following matrix to normal form and find its rank:

6 M

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