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MU Information Technology (Semester 3)
Applied Mathematics 3
May 2014
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[seπss2+2s+2]
5 M
1 (b) State true or false with proper justification "There does not exist ar. Analytic function whose real part is x3 - 3x3y-y3.
5 M
1 (c) prove that f1(x)=1, f2(x)=x, f3(x)=(3x21)2 are orthogonal over (-1,1)
5 M
1 (d) Using Green's theorem in the plane, evaluate  c(x2y)dx+(2y2+x)  by around the boundry of the region defined by y=x2 and y=4.
5 M

2 (a) Find the fourier cosine integral representation of the function f(x)=e-ax, x>0 and hence show that
0cosws1+w2dw=π2ex, x0
6 M
2 (b) Verify laplace equations for U=(r+a2r)cosθ Also find V and f(z).
6 M
2 (c) Solve the following equation by using laplace transform dydt+2y+t0ydt=sint given that y(0)=1
8 M

3 (a) Expland f(x)={πx, 0<x<10, 1<x<2 with period 2 into a fourier series.
6 M
3 (b) A vector field is given by ˉF=(x2+xy2)i+(y2+x2y)j show that ˉF is irrotational and find its scalar potential.
6 M
3 (c) Find the inverse z-transform of- f(Z)=z+2z22z+1,|z|>1
8 M

4 (a) Find the constants 'a' and 'b' so that the surface ax2-byz=(a+2) x will be orthogonal to the surface 4x2y+z3=4 at (1, -1, 2)
6 M
4 (b) Given  L(erft)=1SS+1, evaluate 0t eterf(t)dt
6 M
4 (c) Obtain the expansion of f(x)=x(π - x), 0 (ii)  11n4=π490
8 M

5 (a) If the imaginary part of the analytic function W=f(z)is V=x2y2+xx2+y2 find the real part U.
6 M
5 (b) If f(k)=4k U(K) and g(k)=5k U(K), then find the z-transform of f(k)·g(k)
6 M
5 (c) Use Gauss's Divergence theorem to evaluate and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4
8 M

6 (a) Obtain complex form of Fourier series for f(x)= \cos h\ 3x + \sin h\ 3x in (-3,3).
6 M
6 (b) Find the inverse Laplace transform of \frac{{\left(S-1\right)}^2}{{\left(s^2-2s+5\right)}^2}
6 M
6 (c) Find the bilinear transformation under which 1, I, -1 from the z-plane are mapped onto 0,1,&infin of w-plane. Also show that under this transformation the unit circle in the w-plane is mapped onto a straight line in the z-plane. Write the name of this line.
8 M



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