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)=13k⋅g(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¯F∗d¯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)=π−x2,0≤x≤2x.
6 M
2 (c)
Find inverse Laplace Transform of i)s+29(s+4)(s2+9)ii)e−2ss2+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(z−1/2)(z−1/3−1/3<|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+x2dy⋅C is closed path formed by y=x.y=x2
6 M
4 (c)
Find Fourier Integral of f(x)={sinx0≤x≤π0x>π Hence show that ∫∞ccot(lπ/2)1−λ2dλ=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¯F∗d¯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;=;0≤x≤1/2k(1−x);1/2≤x≤1 hence that 112+132+152+⋯ ⋯
8 M
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