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

(b) Evaluate: \[\int \int _s\left ( x\bar{i}+y\bar{j} +z\bar{k} \right ).d\bar{s}\] over the surface of a sphere \[x^2+y^2+z^2=1\].
4 M
(c) Evaluate: \[\int \int _5 cur1\bar{F}.\hat{n} ds\] for the surface of the paraboloid
\[z=9\left ( x^2+y^2 \right )\] and \[\bar{F}=\left ( x^2 +y-4\right )\bar{i}+3xy\bar{j}+(2xz+z^2)\bar{k}\].
5 M

Solve any Two question.Q1(a)(i, ii, iii)
1(a)(i) ) \[\left ( D^{3}-7D-6 \right )y= e^{2x}\left ( 1+x \right )\]
4 M
1(a)(ii) ii) \[\left (D ^{2} +1\right )y = 3x-8\cot x\] (by variation of parameter method).
4 M
1(a)(iii) iii) \[\left ( 2x+1\right )^2 \frac{d^2y}{dx^2} -2\left ( 2x +1 \right )\frac{dy}{dx}- 12y=12x\]
4 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 spring. A pull of 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 displacement of the body from its equilibrium position in time t seconds, the maximum velocity and period of oscillation.
4 M
Solve any one question.Q2(b)(i, ii)
2(b) i) Find Laplace transform of:
\[F(s)=\frac{s^{2}2s+3}{\left ( s-1 \right )^2(s+1)} \].
4 M
2(c) Using Laplace transform, solve the differential eqation:
\[\frac{dy}{dt}+2y(t)+\int_{0}^{t}y(t)\ dt=\sin t\] given y(0) = 1.
4 M

Solve any one question.Q3(a,b,c) Q4(a,b,c)
3(a) The first four moment of a distribution about the value 2 are -2,
-20 and 100. Find the first four central moments and B1, B2.
4 M
3(b) Number of absent student in 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 \[\phi =x^{3}yz^{3}\] at (2,
-1) along the vector
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
\[\bar{F}=\bigtriangledown \phi\].
4 M
Prove the following any one question.Q4(c)(i),(ii)
4(c)(i) \[\bigtriangledown ^4\left ( r^2\log r \right )=\frac{6}{r^2}\]
4 M
4(c)(ii) \[\bar{a}.\bigtriangledown \left [ \bar{b}.\bigtriangledown \left ( \frac{1}{r} \right ) \right ]=\frac{3(\bar{a}.\bar{r})(\bar{b}.\bar{r})}{r^5}-\frac{(\bar{a}.\bar{b})}{r^3}\]
4 M

Solve any two question no.Q7(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:
i) u(0,
ii) u(L,
iii) \[\(u\left ( x,0 \right )=x,0\leq x\leq \frac{L}{2} \\ =L-x, \frac{1}{2}\leq x\leq L \)\]
iv) u(x,
∞) is finite.
6 M
7(b) If
\[\frac{\partial^2y }{\partial x^2}= c^2\frac{\partial^2y }{\partial x^2}\] represents the variation of string of Length L fixed at both ends, find the solution with boundry conditions:
i) y (0,
ii) y (L,
iii)\[\left ( \frac{\partial y}{\partial t} \right ) =0\]t=0
iv) \[y(x,0)=k(Lx-x^2),0\leq x\leq L\]
7 M
7(c) An infinitely long plane uniform plate is bunded by two parallel edges x=0 and x=π and an end at right angles to them. The breadth of the plate is π. This edge is maintained at tempreature u0 at all points and other edges at zero tempreature. Find the steady state temperature function u(x,
y). Also use y→∞,
6 M

Solve any two question no.Q8(a,b,c)
8(a) Solve;
\[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}\] if
i) u is finite for all t
ii) u(0,
iii) u (π,
iv) u (x,
6 M
8(b) A string streched and fastened to two point '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 diplacement y(x,
t) from one end.
7 M
8(c) A rectangular palte with insulated surface is 10 cm wide and so long to its width that it may be considered infinite in length without introducing an appreciable error. If the tempreature of short edge y=0 is given by
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,
6 M

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