Applied Mathematics 3 : Question Paper Dec 2016 - Computer Engineering (Semester 3) | Mumbai University (MU)

Applied Mathematics 3 - Dec 2016

Computer Engineering (Semester 3)

(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a) Find the Laplace transform of te3t sin 4t.(5 marks) 1(b) Find half-range cosine series for f(x)=ex,
0<x&lt;1.&lt; a="">

</x&lt;1.&lt;&gt;<>(5 marks)
1(c) Is $ f(z)=\frac{z}{z} $/ analytic?(5 marks) 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 marks) 2(a) Find the Z- transform of $\frac{1}{\left ( z-5 \right )^3} $/ if |z|<5.(6 marks) 2(b) If V=3x2y+6xy-y3, show that V is harmonic & find the corresponding analytic function.(6 marks) 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 marks) 3(a) Find $ L^{-1}\left [ \frac{(s+2)^2}{(s^2+4s+8)^2} \right ] $/ using convolution theorem.(6 marks) 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 marks)
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 marks) 4(a) Find Laplace transform of $ f(t)=K\frac{t}{T}for 0/\lt/span\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(b)\lt/b\gt Show that the vector, \ltspan class="math-tex"\gt$\bar{F}=\left ( x^2-yz \right )i+\left ( y^2-zx \right )j+\left ( z^2-xy \right )k $\lt/span\gt/ is irrotational and hence, find φ such that \bar{F}=∇φ\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(c)\lt/b\gt Find Found series for f(x) in (0, \ltbr\gt2π), \ltspan class="math-tex"\gt$f(x)\left{\begin{matrix} x,& 0\leq x\leq \pi \\ 2\pi -x, & \pi \leq x\leq 2\pi \end{matrix}\right. $\lt/span\gt/ hence deduce that \ltspan class="math-tex"\gt$ \frac{\pi ^4}{96}=\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+.......... $\lt/span\gt/\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(a)\lt/b\gt Use Gauss's Divergence theorem to evaluate \ltspan class="math-tex"\gt$\iint_{s}\bar{N}.\bar{F} ds $\lt/span\gt/ where$$ \bar{F}=2xi+xyj+zk$$ over the region bounded by the cylinder x2<\sup>+y2=4,
(6 marks)
5(b) Find inverse Z- transform of $ f(x)=\frac{z}{\left ( z-1 \right )\left ( z-2 \right )}, |z|>2 $/(6 marks) 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 marks)
6(a) Solve (D2+3D+2)y=2(t2+t+1) with y(0)=2 & y'(0)=0.(6 marks) 6(b) Find the bilinear transformation which maps the points 0,
-2i of z-plane onto the points -4i,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
(6 marks)
6(c) Find Fourier sine integral of $\left\{\begin{matrix} x, &02 \end{matrix}\right. $/(8 marks)


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