Processing math: 84%




MU Information Technology (Semester 3)
Applied Mathematics 3
December 2013
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 L1{e43s(s+4)52}
5 M
1 (b) Find the constant a,b,c,d and e If
f(z)=(ax4+bx2y2+cy4+dx22y2)+i (4x3yexy3+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]=(2n)rna+n(aˉr)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)=1cosx,  x(0, 2π) and hence deduce that n=114n21=12.
8 M

3 (a) (i)  L1{s(2 s+1)2}
(ii)  L1{logs2+a2s+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,  0xl2
f(x)=2kl(lx), l2zl
6 M
4 (c) Find the inverse z-transform of 1(z3)(z2)
(i) |z|<2
(ii) 2<|z|<3
(iii) |z|>3.
8 M

5 (a) Solve using Laplace transform.
d2ydx2+4dydx+3y=ex, 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 excosx.
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\}
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



More question papers from Applied Mathematics 3
SPONSORED ADVERTISEMENTS