Applied Mathematics - 3 - Dec 2015

Electronics 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) Evaluate $ \int^\infty_0 e^{-l} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt $(5 marks) 1 (b) Obtain the Fourier Series expression for f(x)=2x-1 in (0,3)(5 marks) 1 (c) Find the value of 'p' such that the function $ f(z) = \dfrac {1}{2} \log (x^2 + y^2)+ i \tan^{-1} \left ( \dfrac {py}{x} \right ) $ is analytic.(5 marks) 1 (d) If $ \overline F = (y \sin z-\sin x)\widehat{i} + (x \sin z + 2yz)\widehat{j}+ (xy \cos z + y^2)\widehat{k}. $ Show that $\overline F$ is irrotational. Also find its scalar potential.(5 marks) 2 (a) Solve the differential equation using Laplace Transform $ \dfrac {d^2y}{dt^2}+ 2 \dfrac {dy}{dt}+ y = 3te^{-l}, $ given y(0)=4 and y'(0)=2.(6 marks) 2 (b) Prove that $ J_4 (x) \left ( \dfrac {48}{x^3} - \dfrac {8}{x} \right ) J_1 (x) - \left ( \dfrac {24}{x^2}-1 \right )J_0 (x) $(6 marks) 2 (c) (i) In what direction is the directional derivative of φ x², y², z⁴ at (3, -1, 2) maximum. Find its magnitude.(4 marks) 2 (c) (ii) if $ \overline r = x\widehat i + y \widehat j + z\widehat k $ Prove that ∇rⁿ=m^n-2r(8 marks) 3 (a) Obtain the Fourier Series expansion for the function $$ \begin {align*}f(x) &=1+\dfrac {2x}{\pi}, \ -\pi \le x\le 0 \\ &= 1- \dfrac {2x}{\pi}, \ 0\le x \le \pi \end{align*} $$(6 marks) 3 (b) Find an analytic function f(z)=u+iv where. $$ u-v = \dfrac {x-y}{x^2 + 4xy + y^2} $$(6 marks) 3 (c) Find Laplace transform of $$ i) \ \cosh t\int^{1}_0 e^{u} \sinh u \\ ii) \ t\sqrt{1+\sin t} $$(8 marks) 4 (a) Obtain the complex form of Fourier series for f(x)=e^m in (-L, L)(6 marks) 4 (b) Prove that $ \int x^4 J_1 (x)dx=x^4J_1(x)-2x^3J_3(x)+c $(6 marks) 4 (c) Find $$ i) \ L^{-1} \left [ \dfrac {2s-1}{s^2 + 4s + 29} \right ] \\ ii) \ L^{-1} \left [ \cot^{-1} \left ( \dfrac {s+3}{2} \right ) \right ] $$(8 marks) 5 (a) Find the Bi-linear Transformation which maps the points 1, i, -1 of z plane onto 0, 1, ∞ of w-plane.(6 marks) 5 (b) Using Convolution theorem find $$L^{-1} \left [ \dfrac {s^2}{(s^2 + 4)^2} \right ] $$(6 marks) 5 (c) Verify Green's Theorem for $ \int_c \overline F.\overline {dr} $ where $ \overline {F} = (x^2 - y^2)\widehat{i}+ (x+y)\widehat{j} $ and C is the triangle with vertices (0, 0), (1, 1) and (2, 1).(8 marks) 6 (a) Obtain half range sine series for $$ \begin {align*} f(x) &= x, 0\le x \le 2 \\ &=4-x, 2\le x \le 4 \end{align*} $$(6 marks) 6 (b) Prove that the transformation $ w = \dfrac {1}{z+l} $ transforms the real axis of the z-plane into a circle in the w-plane.(6 marks) 6 (c) (i) Use Stoke's Theorem to evaluate $ \int_c \overline F. \overline {dr} $ where $ \overline {F}=(x^3 - y^2) \widehat{i}+2xy\widehat{j} $ and C is the rectangle in the plane z=0, bounded by x=0, y=0, x=a and y=b.(4 marks) 6 (c) (ii) Use Gauss Divergence Theorem to evaluate. $ \iint_s \overline F. \widehat{n}ds \text{ where }\overline F= 4x\widehat{i} + 3y\widehat{j} - 2z\widehat{k} $ and S is the surface bounded by x=0, y=0 and 2x+2y+z=4.(4 marks)