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MU Information Technology (Semester 3)
Applied Mathematics 3
December 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) Find the Laplace transform of te3t sin 4t.
5 M
1(b) Find half-range cosine series for f(x)=ex,
0
5 M
1(c) Is $$f(z)=\frac{z}{z}$$/ analytic?
5 M
1(d) Prove that $$\nabla x\left ( \bar{a}x \nabla \log r\right )=2\frac{(\bar{a}.\bar{r})\bar{r}}{r^4}$$/, where \bar{a} is a constant vector.
5 M

2(a) Find the Z- transform of $$\frac{1}{\left ( z-5 \right )^3}$$/ if |z|<5.
6 M
2(b) If V=3x2y+6xy-y3, show that V is harmonic & find the corresponding analytic function.
6 M
2(c) Obtain Fourier series for the function $$f(x)=\left\{\begin{matrix} 1+\frac{2x}{\pi }, -\pi\leq x\leq 0 & \\ \\ 1-\frac{2x}{\pi },0\leq x\leq \pi & \end{matrix}\right.$$/ hence deduce that $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+.........$$/
8 M

3(a) Find $$L^{-1}\left [ \frac{(s+2)^2}{(s^2+4s+8)^2} \right ]$$/ using convolution theorem.
6 M
3(b) Show that the set of functions $$1,\sin \left ( \frac{\pi x}{L} \right ),\cos\left ( \frac{\pi x}{L} \right ),\sin \left ( \frac{2\pi x}{L} \right ),\cos \left ( \frac{2\pi x}{L} \right ),..........$$/ Form an orthogonal set in (-L,
L) and construct an orthonormal set.
6 M
3(c) Verify Green's theorem for $$\int \left ( e^{2x}-xy^2 \right )dx+\left ( ye^x+y^2 \right )dy$$/ Where C is the closed curve bounded by y2=x&x2=y.
8 M

4(a) Find Laplace transform of $$f(t)=K\frac{t}{T}for 0/ 6 M 4(b) Show that the vector, \(\bar{F}=\left ( x^2-yz \right )i+\left ( y^2-zx \right )j+\left ( z^2-xy \right )k$$/ is irrotational and hence, find φ such that \bar{F}=∇φ
6 M
4(c) Find Found series for f(x) in (0,
2π), $$f(x)\left\{\begin{matrix} x,& 0\leq x\leq \pi \\ 2\pi -x, & \pi \leq x\leq 2\pi \end{matrix}\right.$$/ hence deduce that $$\frac{\pi ^4}{96}=\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+..........$$/
8 M

5(a) Use Gauss's Divergence theorem to evaluate $$\iint_{s}\bar{N}.\bar{F} ds$$/ where$\bar{F}=2xi+xyj+zk$ over the region bounded by the cylinder x2<\sup>+y2=4,
z=0,
z=6.
6 M
5(b) Find inverse Z- transform of $$f(x)=\frac{z}{\left ( z-1 \right )\left ( z-2 \right )}, |z|>2$$/
6 M
5(c) i) Find $$L^{-1}\left [ log\left ( \frac{s+1}{s-1} \right ) \right ]$$/
ii) $$L^{-1}\left [ \frac{s+2}{s^2-4s+13} \right ]$$/
8 M

6(a) Solve (D2+3D+2)y=2(t2+t+1) with y(0)=2 & y'(0)=0.
6 M
6(b) Find the bilinear transformation which maps the points 0,
i,
-2i of z-plane onto the points -4i,
∞,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
6 M
6(c) Find Fourier sine integral of $$\left\{\begin{matrix} x, &02 \end{matrix}\right.$$/
8 M

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