Applied Mathematics 3 - Dec 2015

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 Laplace of {t⁵ cosht}.(5 marks) 1 (b) Find Fourier series for f(x)=t-x² in (-1, 1).(5 marks) 1 (c) Find a, b, c, d, e, if,
f(z)=(ax⁴+bx²y²+cy⁴+dx²-2y²)+i (4x³-exy³+4xy) is analytic.(5 marks) 1 (d) Prove that $ \Delta = \left ( \dfrac {1}{r} \right ) = \dfrac {r}{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} $ Find f(z).(6 marks) 2 (b) Find inverse Z-transform of $ f(z)= \dfrac {z+2}{z^2 - 2z+1} $ for |z|>1.(6 marks) 2 (c) Find Fourier series for $ f(x) = \sqrt{1-\cos x }\text{ in } (0, 2\pi) $ Hence, deduce that $ \dfrac {1}{2} = \sum^\infty_1 \dfrac {1}{4n^2 - 1} $(8 marks) 3 (a) Find $ L^{-1} \left { \dfrac {1}{(s-3)+(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 $ \int_c \overline {F} \cdot \overline{dr} \text { where } \overline {F} = (x2-y2)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 < x < 0 \\\\ &= \dfrac{\pi}{2} - x \text { where }0< x<\pi \end{align*} $
Hence, deduce that $ i) \ \dfrac {\pi^2} {8} = \dfrac {1}{1^2} + \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots \\\\ ii) \ \dfrac {\pi^4}{96} = \dfrac {1}{1^4} + \dfrac {1}{3^4} + \dfrac {1}{5^4} + \cdots \ \cdots $(8 marks) 5 (a) Using Gauss Divergence theorem to evaluate $ \iint_s \ \overline{N} \cdot \overline {F}ds \text{ where } \overline {F} = 4xi - 2y^2 j+ z^k $ and S is the region bounded by x² + y² = 4, z=0, z=3.(6 marks) 5 (b) Find Z{2^k cos (3k+2)}, k≥0.(6 marks) 5 (c) Solve (D²+2D+5)y=e^-t sint, with y(0)=0 and y'(0)=1.(8 marks) 6 (a) Find $ L^{-1} \left { \tan^{-1} \left ( \dfrac {2}{s^2} \right ) \right } $(6 marks) 6 (b) Find the bilinear transformation which maps the points 2, i, -2 onto points 1, i, -1 by using cross ratio property.(6 marks) 6 (c) Find Fourier Sine integral representation for $ f(x) = \dfrac {e^{-ax}}{x} $(8 marks)