1(a)
Find the Laplace transform of te3t sin 4t.
5 M
1(b)
Find half-range cosine series for f(x)=ex,
0
0
5 M
1(c)
Is f(z)=zz/ analytic?
5 M
1(d)
Prove that ∇x(ˉax∇logr)=2(ˉa.ˉr)ˉrr4/, where \bar{a} is a constant vector.
5 M
2(a)
Find the Z- transform of 1(z−5)3/ if |z|<5.
6 M
2(b)
If V=3x2y+6xy-y3, show that V is harmonic & find the corresponding analytic function.
6 M
2(c)
Obtain Fourier series for the function f(x)={1+2xπ,−π≤x≤01−2xπ,0≤x≤π/ hence deduce that π28=112+132+152+........./
8 M
3(a)
Find L−1[(s+2)2(s2+4s+8)2]/ using convolution theorem.
6 M
3(b)
Show that the set of functions 1,sin(πxL),cos(πxL),sin(2πxL),cos(2πxL),........../ Form an orthogonal set in (-L,
L) and construct an orthonormal set.
L) and construct an orthonormal set.
6 M
3(c)
Verify Green's theorem for ∫(e2x−xy2)dx+(yex+y2)dy/ Where C is the closed curve bounded by y2=x&x2=y.
8 M
4(a)
Find Laplace transform of \( f(t)=K\frac{t}{T}for 0/
6 M
4(b)
Show that the vector, ˉF=(x2−yz)i+(y2−zx)j+(z2−xy)k/ is irrotational and hence, find φ such that \bar{F}=∇φ
6 M
4(c)
Find Found series for f(x) in (0,
2π), f(x){x,0≤x≤π2π−x,π≤x≤2π/ hence deduce that π496=114+134+154+........../
2π), f(x){x,0≤x≤π2π−x,π≤x≤2π/ hence deduce that π496=114+134+154+........../
8 M
5(a)
Use Gauss's Divergence theorem to evaluate ∬sˉN.ˉFds/ whereˉF=2xi+xyj+zk over the region bounded by the cylinder x2<\sup>+y2=4,
z=0,
z=6.
z=0,
z=6.
6 M
5(b)
Find inverse Z- transform of f(x)=z(z−1)(z−2),|z|>2/
6 M
5(c)
i) Find L−1[log(s+1s−1)]/
ii) L−1[s+2s2−4s+13]/
ii) L−1[s+2s2−4s+13]/
8 M
6(a)
Solve (D2+3D+2)y=2(t2+t+1) with y(0)=2 & y'(0)=0.
6 M
6(b)
Find the bilinear transformation which maps the points 0,
i,
-2i of z-plane onto the points -4i,
∞,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
i,
-2i of z-plane onto the points -4i,
∞,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
6 M
6(c)
Find Fourier sine integral of \(\left\{\begin{matrix}
x, &02
\end{matrix}\right. \) /
8 M
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