1(a)
If ∫∞0e−2tsin(t+α)cos(t−α)dt=14,find α
5 M
1(b)
Find half range Fourier cosine series for f(x) = x, 0 < x < 2
5 M
1(c)
If u(x,y) is a harmonic function then prove that f(z)=ux - iuy is an analytic function.
5 M
1(d)
prove that ∇f(r)=f′(r)ˉrr
5 M
2(a)
If v = ex siny, prove that v is a harmonic function. Also find the corresponding analytic function.
6 M
2(b)
Find Z-transform of f(k) = bk, k≥0
6 M
2(c)
Obtain Fourier series for f(x)=3x2−6xπ+2π212 in(0,2π),
where f(x+2π)=f(x), hence deduce that π26=112+122+132+⋯
where f(x+2π)=f(x), hence deduce that π26=112+122+132+⋯
8 M
3(a)
Find inverse Laplace of (s+3)2(s2+6s+5)2 using Convolution theorem
6 M
3(b)
Show that the set of functions { sinx, sin3x, sin5x,....} is orthogonal over [0, π/2]. Hence construct orthonormal set of functions.
6 M
3(c)
Verify Green theorem for ∫c1ydx+1xdy where C is the boundary of region defined by x=1, x=4, y=1 and y=√x
8 M
4(a)
Find Z{ k2 ak-1 U(k-1)}
6 M
4(b)
Show that the map of the real axis of the z-plane is a circle under the transformation w=2z+i. Find its centre and the radius.
6 M
4(c)
Express the function f(x)={sinx|x|<πo|x|>π as Fourier sine Integral.
8 M
5(a)
Using Gauss Divergence theorem evaluate ∬ and S is the cube bounded by x=0, x=1, y=0, y=1, z=0, z=1
6 M
5(b)
Find inverse Z-transform of F(z)=\dfrac{z}{(z-1)(z-2)}, \ |z|>2
6 M
5(c)
Solve (D2+3D+2)y = e-2t sint, with y(0)=0 and y'(0)=0
8 M
6(a)
Find Fourier expansion of f(x) = 4-x2 in the interval (0, 2)
6 M
6(b)
A vector field is given by \bar{F}=(x^2+xy^2)i+(y^2+x^2y)j. . Show that \bar{F} is irrotational and find its scalar potential.
6 M
6(c)(i)
L^{-1}\left \{ \tan^{-1}\left ( \dfrac{a}{s} \right ) \right \}
4 M
6(c)(ii)
L^{-1}\left ( \dfrac{e^{-\pi s}}{s^2-2s+2} \right )
4 M
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