**T.E. (Mechanical/Auto. Engg./Sandwich/Automobile)NUMERICAL METHODS AND OPTIMIZATION(2015 Pattern) (Semester – II)**

**Time : 2Â½ HoursMax. Marks : 70Instructions to the candidates:1) Solve Q.1 or Q.2, Q.3 or Q.4, Q.5 or Q.6., Q.7 or Q.8, Q.9 or Q.10. and Q.11 or Q.12.2) Neat diagrams must be drawn whenever necessary.3) Figures to the right side indicate full marks.4) Use of scientific calculator is allowed.5) Assume suitable data if necessary.**

**Q1)** Evaluate the error in the volume of a tank V\;=\frac\pi4d^{2l} = at d = 1.5 m and I = 2.5 if error in the measurement of diameter is Â± 0.010 m and length is Â± 0.020 m. **[6M]**

**Q2)** A chip-tool interface temperature model is expressed as below : **[6M]**

T = 314 Ã— V^{ 0.42} Ã— f ^{1.5}

Where, T-Average chip-tool interface temperature (Â°C), V-cutting speed (m/min) and f – feed (mm/rev).

Find maximum feed (f) in mm/rev to which the temperature of the tool will not increase above 900 Â°C for cutting speed of 340 m/min. Take initial guesses as [0, 0.5]. Solve for 5 iteration.

**Q3)** Use the Jacobi method to approximate the solution of the following system of linear equations **[6M]**

5x_{1} â€“ 2x_{2} + 3x^{3} = â€“1

â€“3 x_{1} + 9x_{2} + x_{3} = 2

2x_{1} â€“ x_{2} â€“ 7x_{3} = 3

Continue the iterations until two successive approximations are identical when rounded to two significant digits. Take initial approximation as x1 = 0, x2 = 0, x3 = 0.

**OR**

**Q4)** Solve following set of equations using Thomas Algorithm **[6M]**

2.04 X_{1} – X_{2} = 48.8

-X_{1} + 2.04 X_{2} + X_{3} = 0.8

-X_{2} + 2.04 X_{3} = 0.8

**Q5)** a) Write a note on following with example **[4M]**

i) Slack Variable

ii) Surplus Variable

b) Write a note on simulated annealing. **[4M]**

**OR**

**Q6)** a) Solve following LP problem using graphical method : **[5M]**

Minimize Z = 5X_{1} + 6X_{2}

Subject to 2X_{1 }+ 5X_{2} â‰¥ 1500;

3X_{1} + X_{2} â‰¥ 1200

Where X1, X2 â‰¥ 0.

b) Write a short note on Genetic Algorithm. **[3M]**

**Q7)** a) Draw a flowchart for Runge-Kutta 4th order method. **[6M]**

b) Solve, \frac{\partial u}{\partial t}\;=\;\frac{\partial^2u}{\partial x^2} for the following condition. **[12M]**

At x = 0 and x = 0.5, u = 1 for all values of t. At t = 0, u = 2x + 1 for 0 < x < 0.5. Take increment in x as 0.1 and increment in t as 0.01. Find all values of u for t = 0 to t = 0.03.

**OR**

**Q8)** a) Given that dy/dx = yz, dz/dx = xy, y(0) = 1, z(0) = 1, h = 0.1 **[8M]**

Use R-K 2^{nd} order method to find value of y and z at x = 0.1

b) The temperature inside a slab of thickness 16 cm is given by **[10M]**

\frac{dT}{dx}\;=\;-\frac qAx\frac1{0.5\;\ast(1+0.01\ast T)} (T is in deg.C deg.C)

Find the temperature of other surface by taking step size = 4cm, if heat flux (q/A) is 1000 W/m^{2} and temperature at one surface, 500 deg.C. Use R-K 4^{th} order method.

**Q9)** a) A set of x values and respective y values are given below. Using appropriate interpolation method, find the value of y at x = 11.5.**[8M]**

x | 2 | 5 | 10 | 12 | 15 |

y | 45 | 68 | 75 | 90 | 98 |

b) The values of Nusselt numbers (Nu) and Reynold numbers (Re) found experimentally are given below. If the relation between Nu and Re is of the type Nu = a.Re^{b}, find the values of a and b for the given values of Nu and Re. **[8M]**

2000 | 2400 | 2800 | 3200 | 3600 | 4000 |

13.0102 | 13.5091 | 14.0789 | 14.4192 | 15.1297 | 16.7535 |

**OR**

**Q10)** a) A set of x values and respective y values are given below. Using Lagrange inverse interpolation method, find the value of x at y = 0.42** [8M]**

x | 10 | 20 | 30 | 40 | 50 |

y | 0.1105 | 0.1985 | 0.2727 | 0.4101 | 0.123 |

b) Fit a quadratic equation of the form y = a_{0} + a_{1} x + a_{2} x^{2} for a set of given values : **[8M]**

x | 2 | 5 | 8 | 1 | 15 | 20 |

y | 0.2841 | 2.8631 | 12.082 | 23.2612 | 11.6725 | 1.2792 |

**Q11)** a) The Velocity v (m/min) of moped which start from rests is given at fixed interval of time t(min) as follows : **[10M]**

t (min) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

v(m/min) | 0 | 10 | 18 | 25 | 29 | 32 | 20 | 11 | 5 | 2 | 0 |

b) Use Gauss-Legendre three-point formula to evaluate I\;=\int_{-1}^1e^xdx **[6M]**

**OR**

**Q12)** a) Draw a combined flowchart for Simpson’s 1/3 rule and Simpson’s 3/8 rule. **[8M]**

b) Evaluate I\;=\int_0^1\left[\int_0^1e^{(x+y)}dx\right]dy **[8M]**

using trapezoidal rule, Take strip size for x and y axis as 0.5.

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