GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-1V (NEW) EXAMINATION - WINTER 2018
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Subject Code:2140706 Date:01/12/2018Subject Name:Numerical and Statistical Methods for Computer Engineering
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 (a) Discuss about mathematical modeling. 03
(b) Discuss various types of errors used for numerical calculations. 04
(c) Obtain cubic spline approximation for the function defined by the data given below for the first two subintervals. Take M1 =Mn =0. 07
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| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| f(x) | 1 | 2 | 33 | 244 |
Q.2 (a) Write an algorithm for Simpson’s rule. 03
(b) Using Simpson’s rule, find ?00.6 e-x2 dx taking seven ordinates. Show the calculations up to four decimal places. 04
(c) Define divided difference. Using Newton’s divided difference interpolation, find f(6) from the following table: 07
| X | 1 | 2 | 7 | 8 |
|---|---|---|---|---|
| f(x) | 1 | 5 | 5 | 4 |
OR
(c) Define interpolation. Using Lagrange interpolation, fit a second degree polynomial passing through the points (0,0),(1,1) and (2,20). 07
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Q.3 (a) State Budan’s theorem. Define diagonally dominant system with example. 03
(b) Use Newton-Raphson method to find a positive root of x3 + x2 —1=0 correct up to four decimal places taking x0 =1. 04
(c) What do you mean by diagonally dominant system? Solve the following system of linear equations using Gauss-Seidel method: 19x+y+z=10, 2x+10y+3z=19, 3x+4y+11z=0 . 07
OR
Q.3 (a) Explain geometrically the method of false position. 03
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(b) Using Euler’s method, find y(1) if dy/dx =x+y and y(0)=1. Take n=10. 04
(c) Perform one iteration of the Bairstow method to extract a quadratic factor from the polynomial x4 + x3 +2x2 + x + 1 with initial factor x2 —0.5x —0.5. 07
Q.4 (a) Write the steps for engineering problem solving. 03
(b) Determine the condition number of the matrix | 4 9 | 04
| 16 25 |
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(c) State direct and iterative methods to solve system of linear equations. Solve the following system of linear equations using Gauss elimination method: X+y+z=9, 2x-3y+4z=13, 3x+4y+5z=40 . 07
OR
Q.5 (a) Write the formula for Runge-Kutta fourth order method. 03
(b) Fit a second degree polynomial to the following data using least square method. 04
| X | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| Y | 12 | 4 | 1 | 2 | 7 | 15 | 30 |
(c) Calculate the first four moments of the following distribution about the mean. 07
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| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| f(x) | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 |
Q.5 (a) Develop a C program to fit regression line of y on x through given set of points using the least square method. 03
(b) The probability distribution of a commodity is given below. 04
| Demand | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|
| Probability | 0.05 | 0.10 | 0.30 | 0.40 | 0.10 | 0.05 |
Find expected demand.
(c) For the following data, obtain trend values using five years moving average. 07
| Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Value | 3 | 7 | 14 | 8 | 10 | 11 | 14 | 12 | 16 | 20 | 25 |
OR
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(a) Discuss the pitfalls of Gauss elimination. 03
(b) Define the following terms with examples: 04
- Ill-conditioned system
- Significant figure
(c) Obtain the correlation coefficient for the following data: 07
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| x | 100 | 98 | 78 | 85 | 110 | 93 | 80 |
|---|---|---|---|---|---|---|---|
| y | 85 | 90 | 70 | 72 | 98 | 81 | 74 |
This download link is referred from the post: GTU BE/B.Tech 2018 Winter Question Papers || Gujarat Technological University
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