1(a)
Find adj A, A

^{-1}if A is a matrix as given below. Also find B.
5 M

1(b)
Find Laplace transform of

5 M

1(c)
A regular function of constant magnitude is constant.

5 M

1(d)
Find the Fourier series f(x) =1-x

^{2}in (-1,1)
5 M

2(a)
Expand f(x) with period 2 into a Fourier series.

6 M

2(b)
Find the orthogonal trajectories of the family of curves e

^{-x}(x siny - y cosy) = c
7 M

2(c)
Using convolution theorem, prove that,

7 M

3(a)
Show that every square matrix A can be uniquely expressed as P+iQ where P and Q are Hermitian matrices.

6 M

3(b)
Using Cauchy's residue theorem evaluate the following where C is the circle (i) |z| = 1/2 (ii) |z + i| = 3:

7 M

3(c)
Solve the following equation by using Laplace transform. Given that y(0) = 1

7 M

4(a)
State Laplace equation in polar form and verify it for u = r

^{2}cos 2? and also find V and f(z).
6 M

4(b)
Find the Fourier expansion for

f(x)= ?(1-cosx) ... 0 < x < 2? and hence show that

f(x)= ?(1-cosx) ... 0 < x < 2? and hence show that

7 M

4(c)
Evaluate the following:

7 M

5(a)
Using residue theorem evaluate:

6 M

5(b)
Reduce the following matrix to normal form and find its rank

7 M

5(c)
(i) Express the function as Heaviside's unit step function and find their Laplace Transforms

f(t) = 0 ... 0 < t < 1

= t

= 0 ... t > 3

(ii) Find L {f(t)} where

f(t) = t ... 0 < t < 1

= 0 ... 1 < t < 2

f(t) = 0 ... 0 < t < 1

= t

^{2}... 1 < t < 3= 0 ... t > 3

(ii) Find L {f(t)} where

f(t) = t ... 0 < t < 1

= 0 ... 1 < t < 2

7 M

6(a)
Investigate for what values of ? and ? the equations:

x + 2y + 3z = 4

x + 3y +4z = 5

x + 3y + ?z = ?

have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

x + 2y + 3z = 4

x + 3y +4z = 5

x + 3y + ?z = ?

have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

6 M

6(b)
Show that the set of functions sin(2n + 1)x where n=0, 1, 2, ... is orthogonal over [0, ?/2]. Hence construct the orthogonal set of functions.

7 M

6(c)
Find all Laurent's expansion of the function f(z)

7 M

7(a)
Find L{cost cos2t cos3t}

6 M

7(b)
Show that the vectors [1, 0, 2, 1], [3, 1, 2, 1], [4, 6, 2, -4], [-6, 0, -3, -4] are linearly dependent and find the relation between them

7 M

7(c)
Obtain half range sine series for f(x) where

7 M

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