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8~ 'S~ C5".--r~ S~~\,\ C?)- (REVISED COURSE)

(3 Hours)

A'?f\\e~ .M(},~e..~oJ1c- 3\\~Question NO.1 is compulsory.Attempt any four questions out of remaining six questions.

Make suitable assumptions if required and justify the same.figures to the right indicate full marks.

O&~o6( 03VR-3390

[Total Marks: 100

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Con. 2642-09.

1. (a) Show that every square matrix can be uniquely expres$ed as the sum of a.symmetric matrix and skew symmetric matrix.

(b) Obtain the complex form of Fourier Series for f(x) = e-x in (-1, 1).(c) Find the Laplace transforms of the following: .

(i) V1 + sin t (ii). te3t sin t.(d) Construct the analytic function whose real part is e-X(xcos y - y sin y).

5

55

5

00 .2 t 1J

-t sin2. (a) Prove that e ~ dt = 4 log 5.a

(b) Find the Fourier expansion of cos px in (0, 2n). Hence, deduce that

100

[

1 1

]ncosecnx = ~ + I (-1)n - + -.

p n=1 p+n p-n

(c) Show that u =cos x cosh y isa harmonic function. Find its harmonic conjugateand the corresponding analytic.function.

6

8

, ,

.6

3.x

(a) Find the analytic function f(z) =u + iv in terms of z if u + v = x2 + y2 .(b) Obtain Fourierseries for

6

8

nf(x)= x + "2 '

-n

-..... -~ - "Con. -2642-VR-3390-09. '-2

(b) Reduce A to norma.U!