Applied Mathematics 3 - Dec 2015

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 Laplace of {t⁵ cosht}(5 marks) 1 (b) Find Fourier series for f(x)=1-x² in (-1, 1)(5 marks) 1 (c) Find a, b, c, d, e if, $$ f(z)= (ax^4 + bx^2y^2+ cy^4+dx^2 - 2y^2)+ (4x^3y - exy^3 + 4xy) $$ is analytic.(5 marks) 1 (d) Prove that $$ \nabla \left ( \dfrac {1}{r} \right ) = \dfrac {f}{r^3} $$(5 marks) 2 (a) If f(z)=u+iv is analytic and $$ u+v = \dfrac {2 \sin 2x}{e^{2y}+e^{-2y}-2\cos 2x}, \text {find f(z)} $$(6 marks) 2 (b) Find inverse Z-transform of $$ f(z)= \dfrac {z+2}{z^2 -27 +1 } \ \text {for} \ |z|>1 $$(6 marks) 2 (c) Find Fourier series for $$ f(x)= \sqrt{1 \ominus \cos x} \ \text {in} \ (0,2\pi) $$ Hence, deduce that $$ \displaystyle \dfrac {1}{2} = \sum^{\infty}_{1} \dfrac {1}{-4n^2 - 1} $$(8 marks) 3 (a) Find $$ L^{-1} \left \{ \dfrac {1}{(s-2)(s+3)} \right \} $$ using Convolution theorem.(6 marks) 3 (b) prove that f₁(x)=1, f₂(x)=x, f₃(x)=(3x²-1)/2 are orthogonal over (-1,1)(6 marks) 3 (c) Verify Green's theorem for $$ \displaystyle \int_c \overline {F}\cdot \overline {dr}\ \text{where} \ \overline {F}= (x^2 - y^2)i + (x+y)j $$ and c is the triangle with vertices (0,0), (1,1), (2,1).(8 marks) 4 (a) Find Laplace Transform of f(t)=|sinpt|, t≥0.(6 marks) 4 (b) Show that F= (ysinz-sinx)i + (xsinz+2yz)j+(xycosz+y²) k is irrotational. Hence, find its scalar potential.(6 marks) 4 (c) Obtain Fourier expansion of $$ \begin {align*} f(x) &=x+\dfrac {\pi}{2} \ \text {where} \ -\pi\ltx\lt0 \\="" &="\dfrac" {\pi}{2}-x="" \="" \text="" {where}="" 0="" \lt="" x\lt\pi="" \end{align*}="" $$="" hence,="" deduce="" that="" i)="" \dfrac="" {\pi^2}="" {8}="\dfrac" {1}{1^2}="" +="" {1}{3^2}="" {1}{5^2}+="" \cdots="" ii)="" {\pi^4}{96}="\dfrac" {1}{1^4}+="" {1}{3^4}="" {1}{5^4}+="" $$\lt="" a=""\gt\ltbr\gt\ltbr\gt \lt/x\lt0\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Using Gauss Divergence theorem to evaluate $$ \iint_s \overline{N}\cdot \overline{F}ds \text{where} \overline {F}=4xi-2y^2j+z^2k$$ and S is the region bounded by x\ltsup\gt2\lt/sup\gt+y\ltsup\gt2\lt/sup\gt=4, z=0, z=3\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Find Z{2\ltsup\gtk\lt/sup\gt cos (3k+2)}, k≥0\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Solve (D\ltsup\gt2\lt/sup\gt+2D+5)y=e\ltsup\gt-t\lt/sup\gt sint, with y(0) and y'(0)=1.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt Find $$ L^{-1}\left { \tan ^{-1} \left ( \dfrac {2}{s^2} \right ) \right } $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Find the bilinear transformation which maps the points 2, i, -2 onto point 1, j, -1 by using cross-ratio property.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Find Fourier Sine integral representation for $$f(x) = \dfrac {e^{-ax}}{x} $$(8 marks)