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Applied Mathematics - 3 : Question Paper Dec 2016 - Mechanical Engineering (Semester 3) | Mumbai University (MU)

## Applied Mathematics - 3 - Dec 2016

### Mechanical 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 $$\int _c(\bar{z}+2z)dz$$ along the circle x2+y2=1(5 marks) 1(b) Evaluate the integral using Laplace Transform $$\int ^\infty _0 e^{-t}\left ( t\sqrt{1+\sin t} \right )dt$$(5 marks) 1(c) Determine the analytic function whose real part is u = r3 sin 30.(5 marks) 1(d) A rod length l has its ends A and B kept at 0°C and 100 respectively until steady state conditions prevail. If the tempreature at B is reduced sufddenly to 0°C and kept so while that of A is maintained. Find the tempreature u(x,t) at a distance from A and at time t.(5 marks) 2(a) Find complex from of Fourier series of f(x)=e2x in (0,2)(6 marks) 2(b) Find the orthogonal trajectory of the family of curves given by 2x-x3+3xy2=a(6 marks) 2(c) Using Bender Schmidt method solve
$$\frac{\partial^2u }{\partial x^2}-\frac{\partial u}{\partial t}$$ = 0 subject to the conditions u (o,t)=0,
u(1,t) =0,
u(x,0) = sinπx,
0≤x≤1. Assume h=0.2
(8 marks)
3(a) Find k such that $$\frac{1}{2}\log \left ( x^2+y^2 \right )+i\tan^{-1}\left ( \frac{kx}{y} \right )$$ is analytic(6 marks) 3(b) Evaluate $$\int \frac{1}{\left ( z^3-1 \right )^2}$$dz where C is the circle |z-1|=1(6 marks) 3(c) Show that the set of function
$$\left \{ Sin\left ( \frac{\pi x}{2L} \right ),Sin\left ( \frac{3\pi x}{2L} \right ),Sin\left ( \frac{5\pi x}{2L} \right )...... \right \}$$ form an orthogonal set over the interval [0, L]. Construct corresponding orthonormal set.
(8 marks)
4(a) Find Laplace Transform of the periodic function
$$\begin{Bmatrix} sin2t,0<1 &\frac{\pi }{2} \\ \\0,\frac{\pi }{2}\ltt &="" \lt\pi="" \end{bmatrix}f(t)="\left" (="" t+\pi="" \right="" )$$<="" a="">

</t>
(6 marks)
4(b) Find half range sine series for x sin x in (o,π)(6 marks) 4(c) Expand
$$f(z)=\frac{z^2-1}{z^2+5z+6}$$ around z=1
(8 marks)
5(a) Using residue theorem evaluate $$\oint _c\frac{e}{\left ( z^2+\pi ^2 \right )^2}dz$$ where C is |z|=4(6 marks) 5(b) Find Fourier expansion of f(x)=x+x2 in (-π,
π) and f(x+2π)=f(x)
(6 marks)
5(c) Find $i)\ L\left ( e^{-4t}\int_{0}^{t} u\sin 3udu\right )\ \ ii) L^{-1}\left ( \frac{1}{s} log\left ( 1+\frac{1}{s^2} \right )\right )$(8 marks) 6(a) Show that the function $w=\frac{4}{z}$/ transform the straight lines x=c in the z-plane into circles in the W-plane.(6 marks) 6(b) Solve using Laplace Transform$R\frac{dQ}{dt}+\frac{Q}{C}=V$/, Q=0 when t=0(6 marks) 6(c) Solve the Laplace equation $\frac{\partial^2u }{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$/ for the following data by sucessive interations (Calculate first two interations)
!mage
(8 marks)

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