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RGPV Civil Engineering (Semester 3)
Engineering Mathematics 2
December 2015
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

1(a) Expand π-x2 in a half range since series in the interval (0,π) up to the first terms.
2 M
1(b) Find the Fourier transform of Dirac Delta Function δ(t-a).
2 M
1(c) Find the function whose sine transform is $$\dfrac{e^{-s}}{s}$$
3 M
Solve any one question from Q.1(d) & Q.1(e)
1(d) Find the Fourier series to represent the function f(x) given by
$f(x)=\left\{\begin{matrix} 0\ \ \ for\ \ -\pi\leq x\leq 0\\ \sin x \ \ \ for \ \ 0 \leq x \leq \pi \end{matrix}\right.$
Deduce that $\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-\cdots\cdots=\dfrac{\pi-2 }{4}$
7 M
1(e) Develop $$\displaystyle \sin\left ( \dfrac{\pi x}{l} \right )$$ in a half-range cosine series in the range 0<x<1.
7 M

2(a)

Find L{F(t)} if $$\displaystyle F(t)=\left\{\begin{matrix} \sin \left ( t-\dfrac{\pi }{3} \right ),& t> \dfrac{\pi }{3} \\\\ 0, & t<\dfrac{\pi }{3}\\ \end{matrix}\right.$$

2 M
2(b) Define Unit Step Function and Find its Laplace Transform.
2 M
2(c) Prove that :$$L^{-1}\left ( \dfrac{s}{s^{4}+s^{2}+1} \right )=\dfrac{2}{\sqrt{3}}sinh\dfrac{t}{2}sin\dfrac{\sqrt{3}}{2}t$$
3 M
Solve any one question from Q.2(d) & Q.2(e)
2(d) State convolution theorem and hence evaluate
$L^{-1}\left \{ \dfrac{p}{(p^2+1)(p^2+4)} \right \}.$
7 M
2(e) Solve the simultaneous equations using Laplace Transform:
$$\dfrac{dx}{dt}-y=e^t,\dfrac{dy}{dt}+x=sint,given x(0)=1,y(0)=0$$
7 M

3(a) Solve the differential equation by Removal of first derivative method.
$\dfrac{d^{2}y}{dx^{2}}-4x\dfrac{dy}{dx}+(4x^2-1)y=-3e^{x^2}\sin2x$
2 M
3(b) Solve by changing the independent variable
$(1+x^2)^2\dfrac{d^{2}y}{dx^{2}}+2x(1+x^2)\dfrac{dy}{dx}+4y=0$
2 M
3(c) Solve by the method of variation of parameters:
$\dfrac{d^{2}y}{dx^{2}}-y=\dfrac{2}{1+e^x}$
3 M
Solve any one question from Q.3(d) & Q.3(e)
3(d) Solve in series the equation
$2x(1-x)\frac{d^2y}{dx^2}+(5-7x)\frac{dy}{dx}-3y=0$
7 M
3(e) Solve in series the equation
$(1-x^2)\dfrac{d^2y}{dx^2}-2x\dfrac{dy}{dx}+2y=0$
7 M

4(a) Solve the differertial equation
$(z^2-2yz-y^2)p+(xy+zx)q=xy-zx$
2 M
4(b) Solve $p^2-q^2=x-y$
2 M
4(c) Solve $(D^2+5DD'+6D'^2)z=\dfrac{1}{y-2^x}$
3 M
Solve any one question from Q.4(d) & Q.4(e)
4(d) If a string of length l is initially at rest in equilibrium position and each of its points is given the velocity
$\left ( \dfrac{dy}{dt} \right )_{t=0}=bsin^{3}\dfrac{\pi x}{l}$,find the displacement y(z,t).
7 M
4(e) A bar with insulated sides is initially at a temperature 0°C,throughout. The end x=0 is kept at 0°C,and heat is suddenly applied at the end x=l so that $$\displaystyle \dfrac{\partial u}{\partial x}=A$$ for x=1, where A is a constant. Find the temperature u(x,t).
7 M

5(a)

Find the directional derivative of$$\displaystyle \phi =5x^2y-5y^2z+\dfrac{5}{2}z^2x$$at the point P(1,1,1) in the direction of the line$$\displaystyle \dfrac{x-1}{2}=\dfrac{y-3}{-2}=\dfrac{z}{1}$$

2 M
5(b) Prove that vector $f(r)\vec{r}$ is irrotational
2 M
5(c)

A Vector field is given by$\overline{F}=(\sin y)\hat{i}+x(1+\cos y)\hat{j}$.Evaluate the line integral over the circular path given by
$x^2+y^2=a^2,z=0.$

3 M
Solve any one question from Q.5(d) & Q.5(e)
5(d) Verify Stoke's theorem for the vector $$\overline{F}=z\hat{i}+x\hat{j}+y\hat{k}$$taken over the half of the sphere $$x^{2}+y^{2}=a^{2},z=0$$ lying above the xy-plane.
7 M
5(e) Evaluate$\iint_s \overline {A}\cdot\hat{n}\ ds\ \text{where}\ \overline{A}=z\hat{i}+x\hat{j}-3y^{2}z\hat{k}$and S is the surface of the cylinder x2+y2=16 included in the frst octant between z=0 and z=5.
7 M

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