Question Paper: Applied Mathematics - 3 : Question Paper Dec 2015 - Electronics & Telecomm. (Semester 3) | Mumbai University (MU)
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## Applied Mathematics - 3 - Dec 2015

### Electronics & Telecomm. (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) $$\text {Evaluate }\int^\infty_0 e^{-t} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt$$(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 $$f(x)=\dfrac {1}{2} \log (x^2 + y^2)+t\tan^{-1}\left ( \dfrac {py}{x} \right )$$(5 marks) 1 (d) $$\text {If} \overline {F}(y \sin z-\sin x)\widehat{i}+ (x\sin z+2yz)\widehat{j}+ (xy\cos z+y^2)\widehat{k}$$ Show that $\overline{F}$ is irrotational. Also find its scalar potential.(5 marks) 2 (a) Solve the differential equations using Laplace Transform $$\dfrac {d^2y}{dt^2} + 2 \dfrac {dy}{dt}+ y = 3te^{-t}$$ given y(0)=4 and y'(0)=2.(6 marks) 2 (b) Prove that $$J_4 (x) \left ( \dfrac {48}{x^2} - \dfrac {8}{x} \right )J(x)- \left ( \dfrac {24}{x^2}-1 \right )J_0(x)$$(6 marks) 2 (c) (i) In what direction is the directional derivative of ϕ=x2y2z4 at (3, -1, 2) maximum. Find its magnitude.(4 marks) 2 (c) (ii) $$\text{If } \overline{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ \text{prove that, } \nabla r^n=nr^{n-2}\overline{r}$$(4 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 from of Fourier series for f(x)=em in (-I, L)(6 marks) 4 (b) Prove that $$\int x^4 J_1 (x)dx = x^4 J_2(x)-2x^3J_3 (x) +c$$(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 $\int_c \overline{F} \cdot \overline {dr}$ 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 $w=\dfrac {1}{z+1}$ transforms the real axis of the z-plane into a circle in the w-plane.(6 marks) 6 (c) (i) Use Stroke's theorem to evaluate $\int_c \overline {F} \cdot \overline {dr}$ where $\overline {F} = (x^2 - y^2)\widehat{i} + 2xy \widehat{j}$ 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 $\iint_s \overline {F} \cdot \widehat {n}ds$ where $\overline {F} 4x\widehat{i} + 3y\widehat{j},$ 2zk and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4.(4 marks)