SPPU Mechanical Engineering (Semester 3)
Engineering Mathematics 3
December 2016
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any two Q.1a(i, ii, iii) & Solve any two question. Q.1(a,b) &Q.2(a, b, c)
1(a) \( \begin{align*}i)&\left ( D^3-7D-6 \right )y=e^{2x}\left ( 1+x \right )\\ ii)&\left ( D^2+1 \right )y=3x-8\cot x\text{(by variation of parameter method)}.\\ iii)&\left ( 2x+1 \right )^2\frac{d^2y}{dx^2}-2\left ( 2x+1 \right )\frac{dy}{dx}-12y=12x.\end{align*} \)/
8 M
1(b) Find the Fourier cosine transform of:
\[f(x)=e^{-x}+e^{-2x}, x>0.\]
4 M

2(a) A body weighing W = 20 N is hung from a spring =. A pull 40 N will strech the spring to 10 cm. The body is pulled down to 20 cm below the static equilibrium position and then released. Find the the displacedment of the body from its quilibrium position in time t seconds, the maximum velocity and period of oscillation.
4 M
Solve any one question from Q.2(b)(i, ii)
2(b) i) Find Laplace transform of:
\[t e^{-2t}\left ( 2\cosh3t-4\sin h2t \right ).\] ii) Find inverse Laplace transform of:
\[F(s)=\frac{s^2-2s+3}{\left ( s-1 \right )^2\left ( s+1 \right )}\]
4 M
2(c) Using Laplace transform, solve the differential equation:\( \frac{dy}{dt}+2y(t)\int_{0}^{t}y(t)dt=\sin t \)/ given y(0)=1.
4 M

3(a) The first four moments of a distribution about the value 2 are -2, 12, -20 and 100. Find the first four central moments and B1, B2.
4 M
Solve any one question from Q.3(a, b,c) &Q.4(a, b,c)
3(b) Number of absent student in a class follow Poisson distribution with mean 5. Find the probability that in a certain month number of absent student in a class will be :
i) More than 3
ii) Between 3 and 5
4 M
3(c) Find the directional derivative of \( \phi =x^2yz^3 \text{at} (2, 1, -1) \)/ along the vector \[-4\bar{i}-4\bar{j}+12\bar{k}.\]
4 M

4(a) Find coefficient of correlation for the following data:
x y
6 9
2 11
10 5
4 8
8 7
4 M
4(b) Show that vector field: \( \bar{F}=\left ( x^2-yz \right )\bar{i}+\left ( y^2-zx \right )\bar{j}+\left ( z^2-xy \right )\bar{k}\)/ is irrotational. Hence find scalar potential φ such that \[\bar{F}=\nabla\phi.\]
4 M
Solve any one question from Q.4(c)(i, ii)
4(c) \( \begin{align*} i)&\nabla^4\left ( r^2\log r \right )=\frac{6}{r^2}\\ ii)&\bar{a}.\nabla\left [ \bar{b}.\nabla\left ( \frac{1}{r} \right ) \right ]=\frac{3\left ( \bar{a}.\bar{r} \right )\left ( \bar{b}.\bar{r} \right )}{r^5}-\frac{\left ( \bar{a}.\bar{b} \right )}{r^3} \end{align*}\)/
4 M

Solve any one question from Q.5(a, b,c) &Q.6(a, b,c)
5(a) Evaluate: \(\int _c\bar{F}.d\bar{r} \)/ where \( \bar{F}=x^2\bar{i}+xy\bar{j} \)/ and 'c' is the arc of the parabola joining (0, 0) and (1, 1). Equation of parabola is y=x2.
4 M
5(b) Show that: \[\iint_s\frac{\bar{r}}{r^3} .\hat{n} ds=0.\]
4 M
5(c) Verify Stoke's theorem for:\( \bar{F}=xy^2\bar{i}+y\bar{j}+z^2x\bar{k}\)/ for the surface of a reactangular lamina, bounded by:
x=0, y=0, x=1, y=2, z=0.
5 M

6(a) Evaluate\( \oint _c\bar{F}.d\bar{r}\)/ where \( \bar{F}=\left ( 2x-y \right )\bar{i}+\left ( x-2y \right )\bar{j}+z\bar{k} \)/ and c is the arc of the ellipse \[\frac{x^2}{9}+\frac{y^2}{4}=1, z=0.\]
4 M
6(b) Evaluate : \( \iint_s\left ( x\bar{i}+y\bar{j}+z\bar{k} \right ).d\bar{s} \)/ over the surface of shere \[x^2+y^2+z^2=1.\]
4 M
6(c) Evaluate : \( \iint_s \text{cu r1}\bar{F}.\hat{n}ds\)/ for the surface of the paraboloid \( z=9-\left ( x^2+y^2 \right )\text{and}\\ \bar{F}=\left ( x^2+y-4 \right )\bar{i}+3xy\bar{j}\left ( 2xz+z^2 \right )\bar{k}. \)/
5 M

Solve any one question from Q.7(a, b,c) &Q.8(a, b,c)
7(a) Solve the equation: \(\frac{\partial u}{\partial t}=a^2\frac{\partial^2u }{\partial x^2} \)/ where u (x, t) satisfies the following conditions: \( \begin{align*} i)&\ u\left ( 0,t \right )=0\\ ii)&\ u\left ( L,t \right )=0\\ iii)&\ u\left ( x,0 \right )=x,0\leq x\leq \frac{1}{2}\\ & \ \ \ \ \ \ \ \ \ \ =L-x,\frac{1}{2}\leq x\leq L\\ iv)&\ u\left ( x,\infty \right )\text{is finite}.\end{align*} \)/
6 M
7(b) If \( \frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2y }{\partial x^2}\)/ represents the vibrations of a string of Length L fixed at both ends, find the solution with boundary conditions: \[\begin{align*}i)y\left ( 0,t \right )&=0\\ ii)y\left ( L, t \right )&=0\\ iii)\left ( \frac{\partial y}{\partial t} \right )_{t=0}&=0 \\ iv)y\left ( x,0 \right )&=k\left ( Lx-x^2 \right ),0\leq x\leq L.\end{align*}\]
7 M
7(c) An infinitely long plane uniform plate is bounded by two parallel edges x= 0 and x=π and an end at right angles to them. The breadthof the the plate is π. This edge is maintained at temperature u0 at all points and other edges at zero temperature. Find the steady state temperature function u(x, y). Also use y→∞, u=0.
6 M

Solve any two questionQ.8(a)(i, ii, iii, iv)
8(a) Solve: \( \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} \)/ if \[\begin{align*} i) &u\ \text{is finite for all t } \\ ii)u\left ( 0, t \right )&=0\\ iii)u\left ( \pi , t \right )&=0\\ iv)u\left ( x,0 \right )&=\pi x-x^2,0\leq x\leq \pi \end{align*}\]
6 M
8(b) A string streched and fastended to two points 'L' apart. Motion is started by displacing the string in the form \(y=a\sin \frac{\pi x}{L} \)/ from which it is released at time t =0. Find displacement y(x, t) from one end.
7 M
8(c) A reactangular plate with insulated surfaced is 10 cm wide and so long its width that it may be considered infinite in length without introducing an appreciable error. If the temperature of short endge y=0 is given by: u=20x, 0≤x≤5
=20(10-x), 5≤x≤10
and two edges x =0 and x= 10 as well as other short edge are kept at 0°C, then find u(x,y).
6 M

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