## Engineering Mathematics 3 - Dec 2015

### Computer Science Engg. (Semester 3)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** For the function: $$ f(x)= \left\{\begin{matrix}
x \ \ \ in &0< x< \pi \\ x-2\pi n
& \pi < x< 2\pi
\end{matrix}\right. $$ Find the Fourier series expansion and hence deduce the result $ \dfrac {\pi}{4}= 1 - \dfrac {1}{3} + \dfrac {1}{5} - \cdots \ \cdots $(7 marks)
**1 (b)** Obtain the half range Fourier cosine series of the function f(x)=x(l-x) in 0 ≤ x ≤ l.(6 marks)
**1 (c)** Find the constant term and first harmonic term in the Fourier expansion of y from the following table:

x |
0 | 1 | 2 | 3 | 4 | 5 |

y |
9 | 18 | 24 | 28 | 26 | 20 |

**2 (a)**Find the Fourier transform of the function: $\left\{\begin{matrix} 1 \text{ for}&|x|\lt a \\\\ 0 \text{ for}&|x| \gt a \end{matrix}\right.$ and hence evaluate: $ \int^{\infty}_0 \dfrac {\sin x}{x}dx $ (7 marks)

**2 (b)**Obtain the Fourier sine transform of f(x)=e

^{-|x|}and hence evaluate $$ \int^{\infty}_0 \dfrac {x \sin mx}{1+x^2} dx, \ m>0 $$(7 marks)

**2 (c)**Solve the integral equation: $$ \int^{\infty}_0 f(x) \cos pxdx = \left\{\begin{matrix} 1-p, &0\le p\le l \\0, & p>1 \end{matrix}\right. $$ and hence deduce the value $$ \int^{\infty}_0 \dfrac {\sin^2 t}{t^2}dt. $$(7 marks)

**3 (a)**Obtain the various possible solutions of the two dimensional Laplace's equation u

_{xx}+u

_{yy}=0 by the method of separation of variables.(7 marks)

**3 (b)**A string is stretched and fastened to two points 'l' apart. Motion is started by displacing the string in the form $ y=a \sin \left ( \dfrac {\pi x}{l} \right ) $ from which it is released at time t=0. Show that the displacement of any point at a distance 'x' from one end at time 't' is given by $ y(x, t)=a \sin \left ( \dfrac {\pi x}{l} \right )\cos \left ( \dfrac {\pi ct}{l} \right ). $(6 marks)

**3 (c)**Obtain the D'Alembert's solution of the wave equation u

_{tt}=c

^{2}u

_{xx}subjected to the conditions u(x, 0)= f(x) and $ u(x, 0) = f(x) \text { and } \dfrac {\partial u}{\partial t}(x, 0)=a. $(7 marks)

**4 (a)**For the following data fit an exponential curve of the form y=a e

^{bx}by the method of least squares:

x |
5 | 6 | 7 | 8 | 9 | 10 |

y |
133 | 55 | 23 | 7 | 2 | 2 |

**4 (b)**Solve the following LPP graphically:

Minimize Z=20x + 10y

Subject to the constraints: x+2y≤40

3x+y≥30

4x+3y≥60

x≥0 and y≥0(6 marks)

**4 (c)**Using Simplex method, solve the following LPP:

Maximize: Z=2x+4y

Subject to the constraint

3x+y≤22

2x+3y≤24

x≥0 and y≥0.(7 marks)

**5 (a)**Using the Regula-Falsi method to find the fourth root of 12 correct to three decimal places.(7 marks)

**5 (b)**Apply Gauss-Seidel method to solve the following of equations correct to three decimal places:

6x+15y+2z=72

x+y+54z=110

27x+6y-z=8.5

(Carry out 3 iterations).(6 marks)

**5 (c)**Using Rayleigh power method, determine the largest Eigen value and the corresponding Eigen vector of the matrix A in six iterations. Choose [1 1 1]

^{T}as the initial Eigen vector: $$ A=\begin{bmatrix} 2 &-1 &0 \\-1 &2 &-1 \\0 &-1 &2 \end{bmatrix} $$(7 marks)

**6 (a)**Using suitable interpolation formulae, find y(38) and y(85) for the following data:

x |
40 | 50 | 60 | 70 | 80 | 90 |

y |
184 | 204 | 226 | 250 | 276 | 304 |

**6 (b)**If y(0)=-12, y(1)=0, y(3)=6 and y(4)=12, find the Lagrange's interpolation polynomial and estimate y at x=2.(6 marks)

**6 (c)**By applying Weddle's rule, evaluate: $ \int^1_0 \dfrac {xdx}{1+x^2} $ by considering seven ordinates. Hence find the value of log

_{e}2.(7 marks)

**7 (a)**Using finite difference equation, solve $ \dfrac {\partial^2 u}{\partial t^2 } = 4 \dfrac {\partial ^2 u}{\partial x^2} $ subject to u(0, t) = u(4, t)=0 u

_{t}(x, 0) and u(x, 0) = x(4-x) upto four time steps. Choose h=1 and k=0.5.(7 marks)

**7 (b)**Solve the equation u

_{t}=u

_{xx subject to the conditions u(0, t)=0, u(1, t)=0, u(x,0)= sin (π x) for 0≤t≤0.1 by taking h=0.2.}(6 marks)

**7 (c)**Solve the elliptic equation u

_{xx}+y

_{yy}=0 for the following square mesh with boundary values as shown. Find the first iterative values of u(i=1-9) to the nearest integer.

(7 marks)
**8 (a)** Find the z-transform of 2n+sin (nπ/4)+1.(7 marks)
**8 (b)** Obtain the inverse z-transform of $ \dfrac {2z^2 + 3x}{(z+2)(z-4)} $(6 marks)
**8 (c)** Using z-transform, solve the following difference equation:

u_{n+2} + 2u_{n+1} + u_{n}=n with n_{0}=u_{1} =0.(7 marks)