STUPIDSID
1(a)
Determine the constants a, b, c, d, e if \( f(z)=\left ( ax^4+bx^2 y^2 +cy^4+dx^2-2y^2\right )+\left ( 4x^3 y-exy^3+4xy \right ) \)/ is analytic.
5 M
1(b)
Find half range Fourier sine series for f(x)=x2, 0
5 M
1(c)
Find the directional derivative of \( \varphi \left ( x,y,z \right )=xy^2+yz^3 \)/ at the point (2,-1,1) in the direction of the vector i + 2j + 2k.
5 M
1(d)
Evaluate \[\int_{0}^{\infty }e^{-2t}t^{5}\cosh t\ dt.\]
5 M
2(a)
Prove that \[\jmath_ \frac{3}{2}(x)=\sqrt{\frac{2}{\pi x}}\left ( \frac{\sin x}{x}-\cos x \right )\]
6 M
2(b)
If f(z) = u + iv is analytic and \( u-v=e^x\left ( \cos y-\sin y \right )\)/, fin f(z) in terms of z.
6 M
2(c)
Obtain Fourier series for \(\begin{align*} \begin{matrix} f(x)&=
x+\frac{\pi }{2} &-\pi / Hence deduce that \[\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{s^2}+.....\]
8 M
3(a)
Show that \(F=\left ( 2xy+z^3 \right )i+x^2j+3xz^2k \)/, is a conservative field. Find its scalar potential and also find the work done by the force F in moving a praticle from (1,-2,1) to (3, 1, 4).
6 M
3(b)
Show that the set of functions \( \left \{ \sin \left ( 2n+1 \right )x \right \},n=0, 1, 2,...\)/ is orthogonal over [0,π/2}. Hence consturct orthonormal set of fucntions.
6 M
3(c)
i)\[L^{-1}\left \{ \cot ^{-1}\left ( s+1 \right ) \right \}\]
ii)\[L^{-1}\left ( \frac{e^{-2s}}{s^2+8s+25} \right )\]
ii)\[L^{-1}\left ( \frac{e^{-2s}}{s^2+8s+25} \right )\]
8 M
4(a)
Prove that \[\int \jmath _3(x)dx=\frac{2\jmath _1(x)}{x}-\jmath _2(x)\]
6 M
4(b)
Find inverse Laplace of \( \frac{s}{\left ( s^2+a^2 \right )\left ( s^2+b^2 \right )}\left ( a\neq b \right ) \)/ using Convolution theorem.
6 M
4(c)
Expand f(x) = xsinx in the interval 0≤x≤2π as a Fourier series. Hence, deduce that \[\sum_{{n=2}}^{\infty }\ \ \frac{1}{n^2-1}=\frac{3}{4}\]
8 M
5(a)
Using Gauss Diveragence theorem evaluate \(\int \int _s\bar{N.}\bar{F}ds\ \ \text{where} \bar{F}=x^2i+zj+yzk \)/ and S is the cube bounded by x=0, x=1, y=0, y=1, z=0, z-1
6 M
5(b)
Prove that \[\jmath ^'_2(x)=\left ( 1-\frac{4}{x^2} \right )\jmath _1(x)+\frac{2}{x}\jmath 0(x)\]
6 M
5(c)
Solve \( \left ( D^23D+2 \right )y=2\left ( t^2+t+1 \right )\)/, with y(0)=2 and y(0)=0 by using Laplace transform
8 M
6(a)
Evaluate by Green's theorem for \(\int _c\left ( e^{-x}\sin dx+e^{-x} \cos y dy\right ) \)/ where C is the rectangle whose vertices are (0,0), (π, 0), (π, π/2)
6 M
6(b)
Show that under the transformation \( w=\frac{z-i}{z+i} \)/ real axis in the z-plane is mapped onto the circle |w|=1
6 M
6(c)
Find Fourier integral representation for \[f(x)\frac{e^{-ax}}{x}\]
8 M
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