Subject Code: 2140706
GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER- IV (New) EXAMINATION - WINTER 2019Date: 12/12/2019
Subject Name: Numerical and Statistical Methods for Computer Engineering
Time: 10:30 AM TO 01:00 PM
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1
(a) If X = 3.1416, find the absolute and relative errors if :
- X is truncated to three decimal places.
- X is rounded off to three decimal places.
(b) Construct an Interpolating polynomial which takes the following values :
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| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | -10 | -8 | -8 | -4 | 10 | 40 |
MARKS
03
04
(c) By using Method of least squares , fit a second degree parabola y=a +bx+cx’ to the following data:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Y | 1 | 1.8 | 1.3 | 2.5 | 23 |
07
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Q.2
(a) Write an algorithm for Bisection Method.
03
(b) If P is the pull required to lift a load W by means of pulley block, find a linear law of the form P =m W + c connecting P and W using following data:
| P | 12 | 15 | 21 | 25 |
|---|---|---|---|---|
| W | 50 | 70 | 100 | 120 |
04
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(c) Obtain Cubic spline for any of given subinterval from the following data:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| S(x) | 1 | 2 | 5 | 11 |
07
OR
(c) Using Lagrange’s interpolating polynomial, find Interpolating polynomial from the given data:
| X | 2 | 3 | 5 | 7 |
|---|---|---|---|---|
| S(x) | 0.1506 | 0.3001 | 0.4517 | 0.6259 |
Q.3
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(a) Use Secant method to find the real root of equation x3-5x-7=0.
03
(b) Find a real root of x3—x—1=0, correct to three decimal places using Newton-Raphson method.
04
(c) Use Gauss-Seidel method to obtain the solution of the system
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83x+11y—4z =95, 7x+52y+132z =104, 3x+8y+29z=7107
OR
(c) Apply Budan’s theorem to the equation x4—7x2+6x—1=0 to draw the inference about the roots in the interval (-2,—1).
Q.3
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(a) Solve the given System of Linear equations by using Gauss Elimination method:
x+y+z=7, 3x+3y+4z=24, 2x+y+3z=16
03
(b) Given that dy/dx =y2+x2y2 , y(0)=1, y(0.1)=1.06, y(0.2) =1.12, y(0.3) =1.21
Estimate y(0.4) using Milne’s predictor-corrector method.
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04
Q.4
(a) Considering following tabular values, Determine the area bounded by the given curve and X-axis between x =10 to x =16 by Trapezoidal rule.
| x | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|
| y | 1.02 | 0.94 | 0.89 | 0.79 | 0.71 | 0.62 | 0.55 |
03
(b) Using Simpson’s 1/3 rule, evaluate ?01 1/(1+x) dx by taking 4 sub intervals.
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04
(c) Use Fourth order Runge-Kutta method to find y(0.2) with h=0.1, given that dy/dx =2x+y, y(0) =1
07
OR
Q.4
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(a) Use Euler’s Method to find y(0.10) in five steps from the differential equation dy/dx =x+y+xy ,y(0)=1
03
(b) Use Modified Euler’s method to solve dy/dx =x+3y , y(0)=1. Hence find y(0.5) with h=0.1.
04
(c) (i) Write an algorithm for Newton’s Forward Interpolation Formula
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(ii) Using Newton’s divided difference formula, calculate the value of f(6)
| x | 2 | 7 | 8 |
|---|---|---|---|
| S(x) | 1 | 5 | 4 |
07
Q.5
(a) Compute the Median from the data:
| Class | 0-30 | 30-60 | 60-90 | 90-120 | 120-150 | 150-180 |
|---|---|---|---|---|---|---|
| Frequency | 8 | 13 | 22 | 27 | 18 | 7 |
03
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(b) Find the correlation coefficient between the sales and expenses of the following 10 firms:
| Firms | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sales | 50 | 50 | 55 | 60 | 65 | 65 | 65 | 60 | 60 | 50 |
| Expenses | 11 | 13 | 14 | 16 | 16 | 15 | 15 | 14 | 13 | 13 |
04
(c) In a state, data shows the demand of towers for the sufficient network for each of the last 7 weeks.
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Demand | 23 | 29 | 33 | 40 | 41 | 43 | 49 |
(a) Calculate a two week moving average for weeks two to seven
(b) Calculate mean square error (M. S. E)
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07
OR
Q.5
(a) Find the standard deviation of a group of data points:
101.8, 103.2, 104.0, 102.5, 103.5
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03
(b) 10 Participants in a musical test were ranked by the three judges in the following order. Using Spearman’s Rank Correlation Co-efficient method, determine which pair of judges has the nearest approach to common liking music.
| 1st Judge | 1 | 6 | 5 | 10 | 3 | 2 | 4 | 9 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2nd Judge | 3 | 5 | 8 | 4 | 7 | 10 | 2 | 1 | 6 | 9 |
| 3rd Judge | 6 | 4 | 9 | 8 | 1 | 2 | 3 | 10 | 5 | 7 |
04
(c) Obtain the two lines of regression for the following data :
| X | 190 | 240 | 250 | 300 | 310 | 335 | 300 |
|---|---|---|---|---|---|---|---|
| Y | 5 | 10 | 15 | 20 | 20 | 30 | 30 |
07
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