MORE IN Applied Mathematics 3
MU Computer Engineering (Semester 3)
Applied Mathematics 3
May 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 the Laplace transform of f(t) = e-4t sinht sint
5 M
1(b) Express the matrix A as the sum of a symmetric and a skew symmetric matrix
5 M
1(c) If the functions f1(x)=1, f2(x)=x and f3(x)=-1+ax+bx2 are orthogonal in [-1,1] then determine the constants a and b.
5 M
1(d) Find the Fourier transform of f(x) = e-|x|
5 M

2(a) Find the Laplace transform of f(t) = sin5t
6 M
2(b) For the matrix A find the non-singular matrices P and Q such that PAQ is in normal form
6 M
2(c) Find the Fourier series for f(x) = x in (0, 2?)
8 M

3(a) Find the Laplace transform of
6 M
3(b) Reduce the following matrix to normal form and find its rank:
6 M
3(c) Find Fourier expansion of f(x) = [(?-x)/2]2 in (0,2?) and hence prove that:
8 M

4(a) Find inverse Laplace transform of
6 M
4(b) Is the matrix A unitary. If yes, find A-1
6 M
4(c) Obtain the half range sine series in (0,2?) for f(x) = x(?-x) and hence find the value of ?(-1)n/(2n-1)3
8 M

5(a) Find inverse Laplace transform of
6 M
5(b) Find the complex form of Fourier series for f(x)=ex in (-?,?)
6 M
5(c) Express the function
f(x) = 1 ... (|x| < 1)
= 0 ... (|x| > 1)
as a Fourier integral. Hence evaluate
8 M

6(a) Using convolution theorem find Laplace inverse of
6 M
6(b) Find the Fourier series for f(x) = 1-x2 in (-1,1)
6 M
6(c) Solve the following system of equations
x + 2y + 3z = 14
3x + y + 2z = 11
2x + 3y + z = 11
8 M

7(a) Evaluate the following:
6 M
7(b) Find the z-transform of
(i) f(k)=1 ... (k?0, |z|>1)
(ii) f(k)=ak ... (k?0, |z|>a)
(iii) f(k)=1/2k ... (k?0, |2z|>1)
6 M
7(c) Solve the following:
8 M

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