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 f

_{1}(x)=1, f_{2}(x)=x and f_{3}(x)=-1+ax+bx^{2}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) = sin

^{5}t
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)=e

^{x}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-x

^{2}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)=a

(iii) f(k)=1/2

(i) f(k)=1 ... (k?0, |z|>1)

(ii) f(k)=a

^{k}... (k?0, |z|>a)(iii) f(k)=1/2

^{k}... (k?0, |2z|>1)
6 M

7(c)
Solve the following:

8 M

More question papers from Applied Mathematics 3