Applied Mathematics 3 - May 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 Transform of $$ \dfrac {\sin t} {t} $$ (5 marks)

1 (b) Prove that f(z)=sinh z is analytic and find its derivative. (5 marks)

1 (c) Find Fourier Series for f(x)=9-x² over (-3,3). (5 marks)

1 (d) Find Z{f(k)*g(k)} if $$ f(k) = \dfrac {1}{3^k}, \ g(k)=\dfrac {1}{5^k} $$ (5 marks)

2 (a) Prove that F =ye^xy cos z i + xe^xy cos z j-e^xy sin z k is irrotational. Find Scalar potential for F Hence evaluate $$ \int_c \overline {F}\cdot d\overline {r} $$ along the curve C joining the points (0, 0, 0) and (-1, 2, π). (6 marks) 2 (b) Find the Fourier series for $$ f(x) = \dfrac {\pi - x}{2}; 0\le x \le 2\pi $$(6 marks)

2 (c) Find Inverse Laplace Transform $$ i) \ \dfrac {s+29}{(s+4)(s^2+9)} \\ ii) \ \dfrac {e^{-2x}}{s^2+8s+25} $$ (8 marks)

3 (a) Find the Analytic function f(z)=u+iv if $$ u+v= \dfrac {x}{x^2 + y^2} .$$ (6 marks)

3 (b) Find inverse Z transform of $$ \dfrac {1}{(z-1/2)(z-1/3), 1/3<|z|<1/2 $$ (6 marks)

3 (c) Solve the Differential Equation $$ \dfrac {d^2y}{dt^2} + y=t, \ y(0)=1, y'(0)=0, $$ using Laplace Transform. (8 marks)

4 (a) Find the Orthogonal Trajectory of 3x²y-y³=k. (6 marks)

4 (b) Using Greens theorem evaluate $$ \int_c (xy+y^2) dx+x\ltsup\gt2\lt/sup\gt dy,$$ C is closed path formed by y=x, y=x². (6 marks)

4 (c) "Find Fourier Integral$$ f(x)= \begin{Bmatrix} \sin x &0 \le x \le \pi \\0 & x> \pi \end{matrix} $$ Hence show that $$ \int^\infty_0 \dfrac {\cos (\lambda \pi /2)}{1-\lambda^2} d\lambda = \dfrac {\pi}{2} $$" (8 marks)

5 (a) Find Inverse Laplace Transform using Convolution theorem $$ \dfrac {s} {(s^4 + 8s^2 + 16) } $$ (6 marks)

5 (b) Find the Bilinear Transformation that maps the point z=1, i, -1 into w=i, 0, -i. (6 marks)

5 (c) Evaluate $$ \int_C \overline {F} \cdot d\overline{r} $$ where C is the boundary of the plane 2x+y+z=2 cut off by co-ordinate planes and F=(x+y)i + (y+z)j-xk. (8 marks)

6 (a) Find the directional derivative of ϕ=x²+y²+z² in the direction of the line $$ \dfrac {x}{3} = \dfrac {y}{4} = \dfrac {z} {5} \ at \ (1, 2, 3). \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Find Complex Form of Fourier Series for e\ltsup\gt2x\lt/sup\gt; 0\ltx\lt2. \lt="" a=""\gt\ltbr\gt\ltbr\gt \lt/x\lt2.\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Find Half Range Cosine Series for $$ f(x) = \left{\begin{matrix} kx &; 0\le x \le l/2 \k(l-x) & ; l/2 \le x \le l \end{matrix}\right.$$ hence find $$ \dfrac {1}{l^2} + \dfrac {1}{3^2} + \dfrac {1}{5^2} + \cdots \ \cdots $$ (8 marks)