1 (a)
Find L−1{e4−3s(s+4)52}
5 M
1 (b)
Find the constant a,b,c,d and e If
f(z)=(ax4+bx2y2+cy4+dx2−2y2)+i (4x3y−exy3+4xy) is analytic.
f(z)=(ax4+bx2y2+cy4+dx2−2y2)+i (4x3y−exy3+4xy) is analytic.
5 M
1 (c)
Obtain half range Fourier cosine series for f(x)=sin x, x ∈ (0, π).
5 M
1 (d)
If r and r have their usual meaning and a is constant vector, prove that
ablax[a xbarrrn]=(2−n)rna+n(a⋅ˉr)ˉrrn+2
ablax[a xbarrrn]=(2−n)rna+n(a⋅ˉr)ˉrrn+2
5 M
2 (a)
Find the analytic function f(c) =u+iv if 3u+2v=y2- x2 + 16 xy.
6 M
2 (b)
Find the z-transform of {a|k|} and hence find the z-transform of {(12)|k|}
6 M
2 (c)
Obtain Fourier series expansion for f(x)=√1−cosx, x∈(0, 2π) and hence deduce that ∞∑n=114n2−1=12.
8 M
3 (a)
(i) L−1{s(2 s+1)2}
(ii) L−1{logs2+a2√s+b }
(ii) L−1{logs2+a2√s+b }
6 M
3 (b)
Find the orthogonal trajectories of the family of curves e-x cos y+xy=? where ? is the real constant in xy - plane.
6 M
3 (c)
Show that ˉF=(y exycosz)i+(x exycosz) j−(exysinz) k is irrotational and find the scalar potential for ˉF and evaluate ∫ cˉF
dr along the curve joining the points (0,0,0) and (-1,2,π).
8 M
4 (a)
Evaluate by Green's theorem ∫ e-x sin y dx+e-x cos y dy where c is the rectangle whose vertices are (0,0) (π,0) (π, π/2) and (0, π/2)
6 M
4 (b)
Find the half range sine series for the function.
f(x)=2 k xl, 0≤x≤l2
f(x)=2kl(l−x), l2≤z≤l
f(x)=2 k xl, 0≤x≤l2
f(x)=2kl(l−x), l2≤z≤l
6 M
4 (c)
Find the inverse z-transform of 1(z−3)(z−2)
(i) |z|<2
(ii) 2<|z|<3
(iii) |z|>3.
(i) |z|<2
(ii) 2<|z|<3
(iii) |z|>3.
8 M
5 (a)
Solve using Laplace transform.
d2ydx2+4dydx+3y=e−x, y(0)=1, y′(0)=1.
d2ydx2+4dydx+3y=e−x, y(0)=1, y′(0)=1.
6 M
5 (b)
Express f(x)= π/2 e-x cos x for x>0 as Fourier sine integral and show that
∫∞0w3sinwxw4+4 dw=π2 e−xcosx.
∫∞0w3sinwxw4+4 dw=π2 e−xcosx.
6 M
5 (c)
Evaluate ∬ and s s=is the cylinder formed by the surface z=0, z=1, x2+y2=4, using the Gauss-Divergence theorem.
8 M
6 (a)
Find the inverse Laplace transform by using convolution theorem.
L^{-1}\left\{\frac{s^2+2s+3}{\left(s^2+2s+5\right)\left(s^2+2s+2\right)}\right\}
L^{-1}\left\{\frac{s^2+2s+3}{\left(s^2+2s+5\right)\left(s^2+2s+2\right)}\right\}
6 M
6 (b)
Find the directional derivative of ? = 4e2x-y+z at the point (1, 1, -1) in the direction towards the point (-3,5,6).
6 M
6 (c)
Find the image of the circle x2+y2=1, under the transformation w=\frac{5-4z}{4z-2}
8 M
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