STUPIDSID
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
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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
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
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)
(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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