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MU Information Technology (Semester 3)
Applied Mathematics 3
December 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


1(a) Find the Laplace transform of te3t sin 4t.
5 M
1(b) Find half-range cosine series for f(x)=ex,
0
5 M
1(c) Is f(z)=zz/ analytic?
5 M
1(d) Prove that x(ˉaxlogr)=2(ˉa.ˉr)ˉrr4/, where \bar{a} is a constant vector.
5 M

2(a) Find the Z- transform of 1(z5)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,πx012xπ,0xπ/ hence deduce that π28=112+132+152+........./
8 M

3(a) Find L1[(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.
6 M
3(c) Verify Green's theorem for (e2xxy2)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=(x2yz)i+(y2zx)j+(z2xy)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,0xπ2πx,πx2π/ 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.
6 M
5(b) Find inverse Z- transform of f(x)=z(z1)(z2),|z|>2/
6 M
5(c) i) Find L1[log(s+1s1)]/
ii) L1[s+2s24s+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.
6 M
6(c) Find Fourier sine integral of \(\left\{\begin{matrix} x, &02 \end{matrix}\right. \)/
8 M



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