MORE IN Applied Mathematics - 3
MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Evaluate $$\int^\infty_0 e^{-l} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt$$
5 M
1 (b) Obtain the Fourier Series expression for f(x)=2x-1 in (0,3)
5 M
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 M
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 M

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 M
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 M
2 (c) (i) In what direction is the directional derivative of φ x2, y2, z4 at (3, -1, 2) maximum. Find its magnitude.
4 M
2 (c) (ii) if $$\overline r = x\widehat i + y \widehat j + z\widehat k$$ Prove that ∇rn=mn-2r
8 M

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 M
3 (b) Find an analytic function f(z)=u+iv where. $u-v = \dfrac {x-y}{x^2 + 4xy + y^2}$
6 M
3 (c) Find Laplace transform of $i) \ \cosh t\int^{1}_0 e^{u} \sinh u \\ ii) \ t\sqrt{1+\sin t}$
8 M

4 (a) Obtain the complex form of Fourier series for f(x)=em in (-L, L)
6 M
4 (b) Prove that $$\int x^4 J_1 (x)dx=x^4J_1(x)-2x^3J_3(x)+c$$
6 M
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 M

5 (a) Find the Bi-linear Transformation which maps the points 1, i, -1 of z plane onto 0, 1, ∞ of w-plane.
6 M
5 (b) Using Convolution theorem find $L^{-1} \left [ \dfrac {s^2}{(s^2 + 4)^2} \right ]$
6 M
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 M

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 M
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 M
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 M
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 M

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