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Applied Mathematics 3 : Question Paper Dec 2014 - Computer Engineering (Semester 3) | Mumbai University (MU)

## Applied Mathematics 3 - Dec 2014

### Computer Engineering (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 sint cos2t cosht.(5 marks) 1 (b) Find the Fourier series expansion of f(x)=x2 (-π, π)(5 marks) 1 (c) Find the z-transform of $\left ( \dfrac {1}{3} \right )^{|K|}$ (5 marks) 1 (d) Find the directional derivative of 4xz2+x2yz at (1, -2, -1) in the direction of 2i-j-2k(5 marks) 2 (a) Find an analytic function f(z) whose real part is ex (xcosy-ysiny)(6 marks) 2 (b) Find inverse Laplace Transform by using convolution theorem $$\dfrac {1}{(s-3)(s+4)^3}$$(6 marks) 2 (c) Prove that $$\overline{F} = (6xy^2 - 2z^3) \overline{i} + (6x^2 y +2yz) \overline{j}+ (y^2 - 6z^2 x) \overline {k}$$ is a conservative field. Find the scalar potential ? such that ??=F. Hence find the workdone by F in displacing a particle from A(1,0,2) to B(0,1,1) along AB.(8 marks) 3 (a) Find the inverse z-transform of $$f(z)= \dfrac {z^3}{(z-3)(z-2)^2}$$
i) 2<|z|<3 ii) |z|>3
(6 marks)
3 (b) Find the image of the real axis under the transformation $$w= \dfrac {2}{z+i}$$(6 marks) 3 (c) Obtain the Fourier series expansion of \begin {align*}f(x)&=\pi x;0\le x \le 1 \\ &= \pi (2-x); 1 \le x \le 2 \end{align*} Here deduce That $$\dfrac {1}{1^2} + \dfrac {1}{3^2}+ \cdots \ \cdots = \dfrac {\pi^2}{8}$$(8 marks) 4 (a) Find the Laplace Transform of \begin {align*}f(t) & = E; 0 \le t \le p/2 \\ & = E; p/2 \le t \le p, \end{align*} f(t+p)= f(t)(6 marks) 4 (b) Using Green's theorem evaluate $$\int_c \dfrac {1}{y} dx + \dfrac {1}{x} dy where c is the boundary of the region bounded by x=1, x=4, y=1, y=√x\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Find the Fourier integral for$$ f(x)=\left{\begin{matrix} 1-x^2 &0 \le x \le 1 \0 & x \ge 1 \end{matrix}\right. $$Hence Evaluate$$ \int^\infty_0 \dfrac {\lambda \cos \lambda - \sin \lambda}{\lambda^3} \cos \left ( \dfrac {\lambda}{2} \right )d \lambda $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt If \ltspan style="text-decoration:overline"\gtF \lt/span\gt=x\ltsup\gt2\lt/sup\gt \ltspan style="text-decoration:overline"\gti \lt/span\gt+ (x-y)\ltspan style="text-decoration:overline"\gtj\lt/span\gt+ (y+z)\ltspan style="text-decoration:overline"\gtk \lt/span\gt moves a particular from A(1,0,1) to B(2,1,2) along line AB. Find the work done.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Find the complex form of fourier series f(x)= sinhax(-l,l).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Solve the differential equation using Laplace Transform. (D\ltsup\gt2\lt/sup\gt+2D+5)y=e\ltsup\gt-t\lt/sup\gt sint y(0)=0 y'(0)=1\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt$$ If \ \int^\infty_{0} e^{-2t} sn(t+\alpha)\cos (t-\alpha) dt = \dfrac {3}{8} $$find the value of α.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt$$ \iint_s (y^2 z^2 \overline{i} + z^2 x^2 \overline {j}+ z^2 y^2 \overline{k})\cdot \overline n ds $$where is the hemisphere x\ltsup\gt2\lt/sup\gt+y\ltsup\gt2\lt/sup\gt+z\ltsup\gt2\lt/sup\gt=1 above xy-plane and bounded by this plane.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Find Half range sine series for f(x)=lx-x\ltsup\gt2\lt/sup\gt (0, l) Hence prove that$$ \dfrac {1}{1^6}+ \dfrac {1}{3^6}+ \cdots \cdots = \dfrac {\pi ^6}{960} (8 marks)