MORE IN Applied Mathematics 3
MU Information Technology (Semester 3)
Applied Mathematics 3
May 2016
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) 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 $$\dfrac{\pi^2}{6}=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots$$
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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