1(a)
Find the Z-transform of f(k)=(3

^{k})/k where k?1.
5 M

1(b)
Prove that every skew Hermitian Matrix A can be expressed as B+iC where B is real skew symmetric and C is real skew symmetric

5 M

1(c)
Find the complex form of Fourier series of f(x) = cosh 2x + sinh 2x in (-5, 5)

5 M

1(d)
Show that L{f(t) }=e

f(t) = g(t-a) ... (t > a)

= 0 ... (t < a)

And hence find L{f(t)} for f(t)=e

g(t) = (t-4)

= 0 ... (t < 4)

^{-as}g(s) wheref(t) = g(t-a) ... (t > a)

= 0 ... (t < a)

And hence find L{f(t)} for f(t)=e

^{3t}.g(t) whereg(t) = (t-4)

^{2}... (t > 4)= 0 ... (t < 4)

5 M

2(a)
Find the Fourier series of f(x) where

6 M

2(b)
Find all the possible values of k for which rank of A is 1,2,3 where

6 M

2(c)
(i) Find L{J

(ii) Find L{(1+te

_{0}(t)} where(ii) Find L{(1+te

^{-t})^{3}}
8 M

3(a)
Define orthogonal matrix. If A is an orthogonal matrix prove that |A| = +/- 1.

Also find whether A is an orthogonal matrix or not where

Also find whether A is an orthogonal matrix or not where

6 M

3(b)
Find Fourier expansion of f(x) = [(?-x)/2]

^{2}in (0,2?) and hence prove that:
6 M

3(c)
Find the Laplace inverse of

(i) cot

(ii) (s+1)e

(i) cot

^{-1}s(ii) (s+1)e

^{-s}/(s^{2}+s+1)
8 M

4(a)
Find the Z-inverse transform of z/(z-a) for |z| < a and |z| > a

Given a>0

Given a>0

6 M

4(b)
Using Convolution theorem find and verify

6 M

4(c)
Find the values of k for which the following equations have a solution:

x + y + z = 1

x + 2y + 3z = k

x + 5y + 9z = k

Also find the solutions for these values of k.

x + y + z = 1

x + 2y + 3z = k

x + 5y + 9z = k

^{2}Also find the solutions for these values of k.

8 M

5(a)
Examine whether the vectors [1,0,2,1], [3,1,2,1], [4,6,2,-4], [-6,0,-3,-4] are linearly independent or dependent.

6 M

5(b)
Find the Laplace transformation of

6 M

5(c)
Express the function

f(x) = 1 ... (|x| < 1)

= 0 ... (|x| > 1)

as a Fourier integral and hence evaluate:

f(x) = 1 ... (|x| < 1)

= 0 ... (|x| > 1)

as a Fourier integral and hence evaluate:

8 M

6(a)
Show that the fourier transform of f(x)=e

^{-x2/2}is given by F(s)=e^{-s2/2}
6 M

6(b)
Find Z{f(k)} where

6 M

6(c)
Find the fourier series of

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

= sinx ... (0 < x < ?)

Hence deduce that

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

= sinx ... (0 < x < ?)

Hence deduce that

8 M

7(a)
Test for the consistency of the following equations and solve them if possible

x + 2y -z = 1

x + y + 2z = 9

2x + y - z = 2

x + 2y -z = 1

x + y + 2z = 9

2x + y - z = 2

6 M

7(b)
Solve the equation if y=1 at t=0

6 M

7(c)
Show that the set of functions:

1, sinx, cosx, sin2x, cos2x,...is orthogonal on (0,2?) but not on (0,?)

1, sinx, cosx, sin2x, cos2x,...is orthogonal on (0,2?) but not on (0,?)

8 M

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