Applied Mathematics 3 - Dec 2014

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 sint cos2t cosht.(5 marks) 1 (b) Find the Fourier series expansion of f(x)=x² (-?, ?)(5 marks) 1 (c) Find the z-transform of $$ \left ( \dfrac {1}{3} \right )^{1K1} $$(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 e^x (xcosy-ysiny)(6 marks) 2 (b) Find inverse Laplace Transform by using convolution theorem $$ \dfrac {1}{(s-3)(s+4)^2} $$(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 $$ \int_c \dfrac {1}{y} dx + \dfrac {1}{x} dy where c is the boundary of the region bounded by x=1, x=4, y=1, y=?x\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Find the Fourier integral for $$ \begin {align} f(x) & = 1-x^2. 0 \le x \le 1 \ & = 0 \ \ \ x>1 \end{align} $$ Hence Evaluate $$ \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} $$ find the value of ?.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt $$ \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)