RGPV Engineering Mathematics 2 - December 2015 Exam Question Paper

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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