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MU Information Technology (Semester 3)
Applied Mathematics 3
May 2015
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 Laplace Transform of sintt
5 M
1 (b) Prove that f(z)=sinhz is analytic and find its derivative.
5 M
1 (c) Find Fourier Series for f(x)=9-x2 over (-3,3).
5 M
1 (d) Find Z[f(k)*g(k)] if f(k)=13kg(k)=15k
5 M

2 (a) Prove that F =yexy cos zi+°xtxy cos x j ?xy sin r k ° is irrotational and Scalar Potential for F . Hence evaluate c¯Fd¯r along the curve C joining the points (0,0,0) and (-1,2,?).
6 M
2 (b) Find the Fourier series for f(x)=πx2,0x2x.
6 M
2 (c) Find inverse Laplace Transform of i)s+29(s+4)(s2+9)ii)e2ss2+8s+25
8 M

3 (a) Find the Analytic function f(z)=u+iv if u+v=xx2+y2
6 M
3 (b) Find Inverse Z transform of 1(z1/2)(z1/31/3<|z|<1/2
6 M
3 (c) Solve the Differential equation d2ydt2+y=i,y(0)=1,y(0)=0 using Laplace Transform.
8 M

4 (a) Find the Orthogonal Trajectory of 3x2y-y3=k.
6 M
4 (b) Using Green's theorem evaluate c(xy+y2)dx+x2dyC is closed path formed by y=x.y=x2
6 M
4 (c) Find Fourier Integral of f(x)={sinx0xπ0x>π Hence show that ccot(lπ/2)1λ2dλ=π2
8 M

5 (a) Find Inverse Laplace Transform using Convolution theorem 3(s4+8s2+16)
6 M
5 (b) Find the Bilinear Transformation that maps the points z=1,i,-1 into w=i,0,-i.
6 M
5 (c) Evaluate c¯Fd¯r where C is the boundary of the plane 2x+y+r=2 cut off by co-ordinates planes and F=(x+y)f+(y+z)j-xk.
8 M

6 (a) Find the Directional derivative of ?=x2+y2+z2 in the direction of the line x3=y4=z5 at (1,2,3,).
6 M
6 (b) Find complex Form of Fourier Series for e2x; 0
6 M
6 (c) Find Half Range Cosine Series for f(x)={kx;=;0x1/2k(1x);1/2x1 hence that 112+132+152+ 
8 M



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