Applied Mathematics 3 - Dec 2016

Information Technology (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (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 te^3t sin 4t.(5 marks) 1(b) Find half-range cosine series for f(x)=e^x,
0<x<1.< a="">

</x<1.<></span>(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 $\lt/span\gt/ is irrotational and hence, find φ such that \bar{F}=∇φ\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(c)\lt/b\gt Find Fourier series for f(x) in (0, \ltbr\gt2π), \ltspan class="math-tex"\gt$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 x^{2<\sup>+y2=4,
z=0,
z=6.}(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,
i,
-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)