STUPIDSID
1 (a)
Determine the constants a,b,c,d if f(z)=x2+2axy+by2+i(dx2+2cky+y2) 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.
(D2+2D+5)y=e-t sin t, when y(0)=0, y(0)=1.
(D2+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=t2+1, y=2t2, z=t3 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 ∮(x2-y)dx+(2y2+x)dy Around the boundary of region defined by y=x2 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)=C1Q1(x) + C2Q2(x) + C3Q3(x)t where C1, C2, C3 constants and Q1, Q2, Q3 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=ex 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 F2x2yi-y2j+4xz2 k and s is the region bounded by y2+z2=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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