1(a)
State Dirichlet conditions for the expansion of f(x) as Fourier series. Examine whether f(x)=sin(1/x) can be expanded in Fourier series in [-?,?]

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1(b)
Find Laplace transform of:

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1(c)
Find Z {f(k)} where f(k) is given by:

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1(d)
Express the function f(x) as a Fourier integral hence evaluate the integral that follows:

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2(a)
Define linear dependence and independence of vectors. If the vectors (0,1,a),(1,a,1) and (a,1,0) are linearly dependent then find the value of 'a'

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2(b)
Find Laplace transform of:

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2(c)
Find {f(k)} if F(z) is as given below and if ROC of F(Z) is:

(i)|z|<2 (ii)2<|z|<3 (iii) |z|>3

(i)|z|<2 (ii)2<|z|<3 (iii) |z|>3

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3(a)
Determine the value of a and b for which the system:

x + 2y + z = 6

x + 3y + 5z = 9

2x + 5y + az = b

has (i)no solution (ii)unique solution (iii)infinite solutions.

Find the solutions in case of (ii) and (iii)

x + 2y + z = 6

x + 3y + 5z = 9

2x + 5y + az = b

has (i)no solution (ii)unique solution (iii)infinite solutions.

Find the solutions in case of (ii) and (iii)

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3(b)
Evaluate the following:

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3(c)
Find the Fourier series for f(x) in (0,2?) -

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

= 2? - x ... (? ? x < 2?)

Hence deduce that

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

= 2? - x ... (? ? x < 2?)

Hence deduce that

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4(a)
Find two non singular matrices P and Q such that PAQ is in the normal form where

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4(b)
Find L(|cost|)

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4(c)
(i) If A, B are Hermitian prove that AB-BA is skew Hermitian

(ii) Show that A is Hermitian and iA is skew Hermitian if:

(ii) Show that A is Hermitian and iA is skew Hermitian if:

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5(a)
Solve y'' + 2y = r(t); y(0) = 0, y'(0) = 0

Using Laplace Transform where

r(t) = 1 ... (0 ? t ? 1)

= 0 ... (t > 1)

Using Laplace Transform where

r(t) = 1 ... (0 ? t ? 1)

= 0 ... (t > 1)

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5(b)
Find the complex form of the Fourier series of the function

f(x) = x

f(x) = x

^{2}+ x ... (-? < x < ?)
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5(c)
Find z(a

^{n}), z(cos n?), z(sin n?)
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6(a)
Show that e

^{x}is equal to:
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6(b)
Find the inverse Laplace Transform of

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6(c)
Obtain the Fourier series for the function

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

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

Hence deduce that:

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

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

Hence deduce that:

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7(a)
Find the Fourier Transform of:

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7(b)
Using Laplace Transform evaluate

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7(c)
Show that the system of equations

ax + by + cz = 0

bx + cy + az = 0

cx + ay + bz = 0

has a non Trivial solution if a+b+c=0 or if a=b=c.

Find the non Trivial solution when the condition is satisfied

ax + by + cz = 0

bx + cy + az = 0

cx + ay + bz = 0

has a non Trivial solution if a+b+c=0 or if a=b=c.

Find the non Trivial solution when the condition is satisfied

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