STUPIDSID
1(a)
If \( \int ^{\infty}_0e^{-2t}\sin(t+\alpha)\cos(t-\alpha)dt=\dfrac{1}{4},\text{find}\ \alpha \)
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 \( \nabla f(r)=f'(r)\dfrac{\bar{r}}{r} \)
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 \( f(x)=\dfrac{3x^2-6x \pi+ 2\pi^2}{12}\ \text{in}(0,2\pi), \)
where f(x+2π)=f(x), hence deduce that \( \dfrac{\pi^2}{6}=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots \)
where f(x+2π)=f(x), hence deduce that \( \dfrac{\pi^2}{6}=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots \)
8 M
3(a)
Find inverse Laplace of \( \dfrac{(s+3)^2}{(s^2+6s+5)^2} \) 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 \( \int _c \dfrac{1}{y}dx+\dfrac{1}{x}dy \) where C is the boundary of region defined by x=1, x=4, y=1 and \( y=\sqrt{x} \)
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 \( w=\dfrac{2}{z+i} \). Find its centre and the radius.
6 M
4(c)
Express the function \( f(x)=\left\{\begin{matrix}
\sin x & |x|<\pi\\
o & |x|>\pi
\end{matrix}\right. \) as Fourier sine Integral.
8 M
5(a)
Using Gauss Divergence theorem evaluate \( \iint _s \bar{N}.\bar{F}ds \ \text{where}\ \bar{F}=x^2i+zj+yzk \) 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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