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MU Information Technology (Semester 3)
Applied Mathematics 3
December 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Find L(te3t sin t)
5 M
1(b) Use the adjoint method to find the inverse of
5 M
1(c) Find p if f(z) = r2 cos 2? + ir2 sin p? is analytic
5 M
1(d) Find the Fourier Series for f(x)=x2 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)=x2 --- 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:
(D2 - 3D + 2)y = 4e2t, 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
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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