MORE IN Engineering Mathematics 3
SPPU Mechanical Engineering (Semester 3)
Engineering Mathematics 3
May 2014
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

Solve any Two of the following:
1 (a) (i) (D2+D+1)y=x sin x
4 M
Answer any one question from Q1 and Q2
1 (a) (ii) (D2-4D+4) y=ex cos2 x
4 M
1 (a) (iii) $(3x +2)^2 \dfrac {d^2y}{dx^2} + 3 (3x+2) \dfrac {dy}{dx}-36 y = 3x^2 + 4x +1$
4 M
1 (b) Find the Fourier transform of: \begin{align*} f(x)& =1 , &|x|\le a \\ &=0, &|x|>a \end{align*}
4 M

2 (a) A body weighing W=20 N is hung from the spring. A pull of 40 N will stretch the spring to 10 cm. The body is pulled down to 20 cm bellow the static equilibrium position and then released. Find the displacement of the body from its equilibrium position in time 't' seconds. Also find the maximum velocity and Period of oscillation.
4 M
2 (b) Solve any one of the following:
i) Find the Laplace Transform of: f(x)=te3t sin 2t
ii) Find inverse Laplace Transform of:
$F(s) = \dfrac {1} {(s-2)^4 (s+3)}$
4 M
2 (c) Solve the following Differential equation by Laplace Transform method. $\dfrac {d^2y}{dt^2} + 2 \dfrac {dy}{dt}+ 5y =e^{-t}\sin t$ Given that: y(0)=0, y'(0)=1.
4 M

Answer any one question from Q3 and Q4
3 (a) Following are the values of import of raw material and export of finished product in suitable units.
 Export 10 11 14 14 20 22 16 12 15 13 Import 12 14 15 16 21 26 21 15 16 14

Calculate the coefficient of correlation between the import and export values.
4 M
3 (b) Find curl F at the point (1,1,2) where $\overline F = x^2 y\overline i + xyz \overline j + z^2 y \overline k$
4 M
3 (c) Prove the following (any one): $i) \ \ \nabla \cdot \left ( \dfrac {\overline a \times \overline r}{r} \right )=0 \\ ii) \ \overline a \cdot \nabla \left [ \overline b \cdot \nabla \left ( \dfrac {1}{r} \right ) \right ] = \dfrac {3 (\overline a \cdot \overline r)(\overline b \cdot \overline r)}{x^5}- \dfrac {(\overline a \cdot \overline b)}{x^3}$
4 M

4 (a) In a distribution, exactly normal, 7% of the items are under 35 and 89% are under 63. Find the mean and standard deviation of the distribution.
[A1=0.43, z1=1.48, A2=0.39, z2=1.23]
4 M
4 (b) Number of road accidents on a highway during a month follows a Poisson distribution with mean 5. Find the probability that in a certain month, number of accidents on the highway will be:
i) Less than 3
ii) Between 3 and 5
4 M
4 (c) Find the constant a and b, so that the surface ax2-byz=(a+2)x will be orthogonal to the surface 4x2y+z3=4 at the point (1, -1, 2).
4 M

Answer any one question from Q5 and Q6
5 (a) Find the work done in moving a particle along the path x=2 t2 =t, z=t3, from t=0 to t=1 in a force field $\overline F = (2y+3) \overline i + xz \overline j + (yz-x)\overline k$
4 M
5 (b) Evaluate: $iint_s ( x\overline i + y \overline j + z^2 \overline k) \cdot d\overline s,$ where s is curved surface of cylinder x2+y2=4 bounded by the planes z=0 and z=2.
5 M
5 (c) Apply Stoke's theorem to calculate ∫c (y dx+z dy+ x dz), C being intersection of x2+y2+z2=a2, x+z=a.
4 M

6 (a) Sing Green's lemma evaluate $\oint_c x^2 dx + x \ y \ dy,$ where C is the boundary of region R which is enclosed by y=x2, and y=x.
4 M
6 (b) Evaluate: $\iint_s (\nabla \times \overline F) \cdot \widehat n \ dS,$ where S is the curved surface of the parabola x2+y2=2z bounded by the plane z=2 and $\overline F =3 (x-y) \overline i + 2xz \overline j + xy \overline k$
5 M
6 (c) Prove that: $\iint_s (\phi \nabla \psi - \psi \nabla \phi ) \cdot d\overline S = \iiint_v (\phi \nabla^2 \psi - \psi \nabla^2 \phi) dv,$ where S is closed surface enclosing volume V.
4 M

Answer any one question from Q7 and Q8
7 (a) A string of length 1 is stretched and fastened to two ends. Motion is started by displacing the string in the form $u(x)= a \sin \left ( \dfrac {\pi x}{l} \right )$ from which it is released at t=0, Find the displacement u at any time 't', if it satisfies the equation. $\dfrac {\partial^2 y}{\partial t^2} = C^2 \dfrac {\partial ^2 y}{\partial x^2}$
7 M
7 (b) $solve \ \dfrac {\partial u}{\partial t} = a^2 \dfrac {\partial ^2 u}{\partial x^2} \ if$ i) u(x,∞) is finite
ii) u(0,t)=0
iii) u(l,t)=0
iv) u(x,0)=x, 0
6 M

8 (a) An infinitely long plane uniform plate is bounded by two parallel edges in the y direction and an end at right angles to them. The breadth of the plate is π. The end is maintained at temperature u0 at all points and other edges at zero temperature. Find steady state temperature u(x,y), if it satisfies $\dfrac {\partial ^2 u}{\partial x^2} = \dfrac {\partial ^2 u}{\partial y^2} = 0.$
7 M
8 (b) Use Fourier Transform to solve $\dfrac {\partial u}{\partial t}= \dfrac {\partial ^2 u}{\partial x^2}; \ 0< x < \infty, \ t>0 ,$ where u(x,t) satisfies the conditions: i) \ \ u(x,t)0 \\ iii) \ \begin {align*} u(x,0) & = x, &01 \ \ \ \ \ \ \ \end{align*}
6 M

More question papers from Engineering Mathematics 3