1 (a)
Determine the constants a,b,c,d if f(z)=x

^{2}+2axy+by^{2}+i(dx^{2}+2cky+y^{2}) is analytic.
5 M

1 (b)
Find a cosine series of period 2π to represent sin x in 0≤x≤π.

5 M

1 (c)
Evaluate by using Laplace Transformation \[ \int^\infty_0 e^{-3x} t \cos t \ dt. \]

5 M

1 (d)
A vector field is given by \[ \overline {F} = (x^2 + xy^2)i + (y^2 + x^2 y)j. Show that F is irrotational and find its potential. Such that F=∇ϕ.

5 M

2 (a)
Solve by using Laplace Transform.

(D

(D

^{2}+2D+5)y=e^{-t}sin t, when y(0)=0, y(0)=1.
6 M

2 (b)
Find the total work done in moving a particle in the force field. F=3xy i-5z j+10x k along x=t

^{2}+1, y=2t^{2}, z=t^{3}from t=1 and t=2.
6 M

2 (c)
Find the Fourier series of the function f(x)=e

^{-x}. 0
8 M

3 (a)
Prove that \[ J_{1/2}(x) = \sqrt{ \dfrac {2} {\pi x } }\cdot \sin x \]

6 M

3 (b)
Verify Green's theorem in the plane for ∮(x

^{2}-y)dx+(2y^{2}+x)dy Around the boundary of region defined by y=x^{2}and y=4.
6 M

3 (c)
Find the Laplace transforms of the following. \[ i) \ e^{-t} \int^t_0 \dfrac {\sin u} {u} du \\ ii) \ t \sqrt{1+\sin t} \]

8 M

4 (a)
f f(x)=C

_{1}Q_{1}(x) + C_{2}Q_{2}(x) + C_{3}Q_{3}(x)_{t}where C_{1}, C_{2}, C_{3}constants and Q_{1}, Q_{2}, Q_{3}are orthonormal sets on (a,b), show that. \[ \int^b_a [f(x)]^2 dx = c^2_1 + c^2_2 + c^2_3. \]
6 M

4 (b)
If v=e

^{x}sin y, prove that v is a Harmonic function. Also find the corresponding harmonic conjugate function and analytic function.
6 M

4 (c)
Find inverse Laplace transform of the following: \[ i) \ \dfrac {S^2} {(S^2 + a^2) (S^2+b^2)} \\ ii) \ \dfrac {S+2}{S^2 -4S+13} \]

8 M

5 (a)
Find the Fourier series if f(x)=|x|, -k

6 M

5 (b)
Define solenoidal vector. Hence prove that \( \overline {F} = \dfrac{\overline{a}\times\overline{r}}{r^n} \) is a solenoidal vector.

6 M

5 (c)
Find the bilinear transformation under which 1, i, -1 from the z-plane are mapped onto 0, 1, ∞ of w-plane. Further show that under this transformation the unit circle in w-plane is mapped onto a straight line in the z-plane. Write the name of this line.

8 M

6 (a)
Using Gauss's Divergence theorem \( \iint_s \ \overline {F} .d\overline {s} \) where F2x

^{2}yi-y^{2}j+4xz^{2}k and s is the region bounded by y^{2}+z^{2}=9 and x=2 in the first octant.
6 M

6 (b)
Define billinear transformation, And prove that in a general, a bilinear transformation maps a circle into a circle.

6 M

6 (c)
Prove that \[ \int xJ_{2/3} (x^{3/2})dx = - \dfrac {2}{3} x^{-1/2}J_{-1/3}(x^{3/2}) .\]

8 M

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