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SPPU Mechanical Engineering (Semester 3)
Engineering Mathematics 3
December 2013
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

Answer any one question from Q1 and Q2
1 (a) Solve any two of the following: $i) \ \dfrac {d^2 y}{dx^2} - 3 \dfrac {dy}{dx} + 2y = x^2 + \sin x \\ ii) \ \dfrac {d^3 y}{dx^3}-7 \dfrac {dy}{dx} - 6y = e^{2x} (1+x) \\ iii) \ \dfrac {d^2 y}{dx^2}+ \dfrac {1}{x} \dfrac {dy}{dx} = A + B \log x$
8 M
1 (b) Find the Fourier transform of f(x)=e-|x|.
4 M

2 (a) A body of weight W=3N stretches a spring to 15 cm. If the weight is pulled down 10 cm below the equilibrium position and then given a downward velocity 60 cm/sec. Find the subsequent motion of the weight.
5 M
2 (b) Find $i) \ \ L^{-1} \left \{ \dfrac {1}{s^3 (s^2+1)} \right \} \ \ OR \\ ii) \ \ L^{-1} \left \{ \dfrac {1}{(s+1)(s^2+1)} \right \}$
3 M
2 (c) Using Laplace transform method, solve the differential equation $\dfrac {d^2 y}{dt^3} - 3 \dfrac {dy}{dt}+2y = 12 e^{-2t}$ with conditions y(0)=2, y'(0)=6.
4 M

Answer any one question from Q3 and Q4
3 (a) Find the coefficient of correlation between x and y from the table.
 x 1 3 4 6 8 9 11 14 y 1 2 4 4 5 7 8 9
4 M
3 (b) Prove that (any one): $i) \ \nabla ^4 e^r =e^r + \dfrac {4}{r}e^r \\ ii) \ \nabla \cdot \left [ r\nabla \dfrac {1}{r^n} \right ] = \dfrac {n(n-2)}{r^{n+1}}$
4 M
3 (c) Show that the vector field $\overline F = (y^2 \cos x + z^2) i + 2y \sin x \ j+2xzk$ is irrotational and find scalar potential ϕ such that F=∇ϕ.
4 M

4 (a) If the probability that an individual suffers a bad reaction from a certain injection is 0.001, determine the probability that out of 2000 individual, more than 2 individuals will suffers a bad reaction.
4 M
4 (b) Find the directional derivative of div (x5i+y5j+z5k) at (1,2,3) in the direction of outward normal to surface x2+y2+z2=9 at (1,2,3).
4 M
4 (c) Calculate the first four moments about Mean for the following data:
 x 1 3 4 6 8 9 11 14 y 1 2 4 4 5 7 8 9
4 M

Answer any one question from Q5 and Q6
5 (a) Evaluate: $int_c \overline F \cdot d \ \overline r$ where 'c' is any square with sides of length 5 and $\overline F (2x^2-y) \overline i + (\tan y-e^y + 4x) \overline j.$
4 M
5 (b) Evaluate: $\iint_s (z^2-x)dydz - xydxdz + 3zdxdy$ where 's' is the surface of closed region bounded by z=4 -y2 and planes x=0, x=3, z=0.
5 M
5 (c) Evaluate: $\iint_s (\nabla \times \overline F) \cdot d\overline s \ for \ \overline F = y \overline i + z \overline j + x \overline k$ over the surface x2+y2=1 z, z≥0.
4 M

6 (a) Evaluate: $\int_c \overline F. d \overline r, \ where \ \overline F = y\overline i + xz^3 \overline j - zy^2 \overline k$ and 'c' is the circle x2+y2=4, z=1.
4 M
6 (b) Evaluate: $\int_c \overline F. d \overline$ by using Stokes theorem for $\overline F=4y \overline i - 4x\overline j + 3\overline k,$ where 's' is a disk of radius 1 lying on the plane z=1 and 'c' is its boundary.
5 M
6 (c) If n is the unit outward drawn normal to any closed surface 's' of area s then evaluate $\iiint_v \nabla \cdot \overline n dv.$
4 M

Answer any one question from Q7 and Q8
7 (a) If a string of length 4 cm is initially at rest in its equilibrium position and each of its point is given velocity v(x) such that $v(x) = \left\{\begin{matrix} 3x, & 0\le x \le 2 \\ 3(4-x),& 2 \le x \le 4 \end{matrix}\right.$ Obtain the displacement y(x,t) at any time t.
7 M
7 (b) Solve $\dfrac {\partial u}{\partial t} = \dfrac {\partial ^2 u}{\partial x^2} \ if \\ i) \ \ u \ is \ finite \ for \ all \ t, \\ ii) \ u(0,t)=0 \\ iii) \ u(l,t)=0 \\ iv) \ u(x,0) = \dfrac {3x}{l}, \ 0\le x\le l.$
6 M

8 (a) Use Fourier transform to solve $\dfrac {\partial u}{\partial t} = \dfrac {\partial ^2u}{\partial x^2}, \ 0< x < \infty , \ t>0$ where u(x,t) satisfies the conditions: $i) \ \ \left ( \dfrac {\partial u}{\partial x} \right )_{x=0} = 0 , \ t>0 \\ ii) \ u(x,0)=\left\{\begin{matrix} x, &01 \end{matrix}\right. \\ iii) \ \big |u(x,t)\big | < M.$
7 M
8 (b) An infinitely long uniform metal plate is enclosed between lines y=0 and y=1 for x>0. The temperature is zero along the edges y=0, y=l and at infinity. If the edge x=0 is kept at a constant temperature uo, find the temperature distribution u(x, y).
6 M

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