STUPIDSID
GTU Civil Engineering (Semester 4)
Numerical & Statistical Methods for Civil Engineering
December 2015
Numerical & Statistical Methods for Civil Engineering
December 2015
1(a)i
Two students x and y work independently on a problem. The probability
that x will solve it is 3/4 and the probability that y will solve it is 2/3. What is the probability that problem will be solved?
2 M
1(a)ii
Evaluate the integral \[\int ^1_{-1}\dfrac{dx}{1+x^2}\] by Gaussian integration two point formula.
2 M
1(a)iii
Using Taylor's series method, find y(1.1) correct to four decimal places, given that \[\dfrac{dy}{dx}=xy^{1/3}\], y(1) = 1.
3 M
1(b)i
Obtain the binomial distribution for which mean is 10 and variance is 5.
2 M
1(b)ii
With the usual notation show that Δ = ehD-1.
2 M
1(b)iii
Find the \[\sqrt{10}\] correct to three decimal places by using Newton-Raphson iterative method.
3 M
2(a)
A book contains 100 misprints distributed randomly throughout its 100 pages. What is the probability that a page observed at random contains at least two misprints. Assume Poisson Distribution.
7 M
Solved any one question from Q.2(b) & Q.2(c)
2(b)
State Bayes's theorem. In a bolt factory, three machines A, B and C
manufacture 25%, 35% and 40% of the total product respectively. Of these outputs 5%, 4% and 2% respectively, are defective bolts. A bolt is picked up at random and found to be defective. What are the probabilities that it was
manufactured by machines A, B and C?
7 M
2(c)
What are the properties of Binomial Distribution? The average percentage of failure in a certain examination is 40. What is the probability that out of a group of 6 candidates, at least 4 passed in examination?
7 M
Solved any one question from Q.3 & Q.4
3(a)
Calculate the Mean, Median and Mode for the following data:
|
Class interval |
50-53 | 53-56 | 56-59 | 59-62 | 62-65 | 65-68 |
| Frequency | 3 | 8 | 14 | 30 | 36 | 28 |
|
Class interval |
68-71 | 71-74 | 74-77 | |||
| Frequesncy | 16 | 10 | 5 |
7 M
3(b)
Calculate the coefficient of correlation and obtain the lines of regression for the following:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Y | 9 | 8 | 10 | 12 | 11 | 13 | 14 | 16 | 15 |
7 M
4(a)
From the following regression equations
8x-10y=-66, 40x-18y=214 and variance of x=9
Find (i) Average values of x and y.
(ii) Correlation Coefficient between the two variables.
(iii) Standard Deviation of y.
8x-10y=-66, 40x-18y=214 and variance of x=9
Find (i) Average values of x and y.
(ii) Correlation Coefficient between the two variables.
(iii) Standard Deviation of y.
7 M
4(b)
Fit a second degree parabola y = ax2 + bx + c in least square sense for the following data:
| x | 1 | 2 | 3 | 4 | 5 |
| y | 10 | 12 | 13 | 16 | 19 |
7 M
Solved any one question from Q.5 & Q.6
5(a)
Explain False position method for finding the root of the equation f(x) = 0.Use this method to find the root of an equation x = e-x correct to up to three decimal places.
7 M
5(b)
Explain Euler's method for solving first order ordinary differential equation. Hence use this method, find y (2) for \[\dfrac{dy}{dx}=x+2y\] with y (1) = 1.
7 M
6(a)
Determine the interpolating polynomial of degree three by using Lagrange's interpolation for the following data. Also find f(2)
| x | -1 | 0 | 1 | 3 |
| f(x) | 2 | 1 | 0 | -1 |
7 M
6(b)
Explain Bisection method for solving an equation f(x) = 0. Find the real root of equation x2 - 4x - 10 = 0 by using this method correct to three decimal places
7 M
Solved any one question from Q.7 & Q.8
7(a)
Apply Runge-Kutta fourth order method to calculate y(0.2) and y(0.4) given
\[\dfrac{dy}{dx}=y-\dfrac{2x}{y}\] , y(0) = 1.
\[\dfrac{dy}{dx}=y-\dfrac{2x}{y}\] , y(0) = 1.
7 M
7(b)
Solve the following system of equations by Gauss elimination method with partial pivoting. 2x1+2x2+x3=6, 4x1+2x2+3x3=4, x1+x2+x3=0
7 M
8(a)
Compute values of f(0.12) and f(0.40) using suitable interpolation formula for the following data:
| x | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 |
| f(x) | 0.1003 | 0.1511 | 0.2027 | 0.2553 | 0.3093 |
7 M
8(b)
Derive Trapezoidal rule and Evaluate \[\int ^{1.3}_{0.5}{e^{x^2}}dx\] by using Simpson's 1/3rd rule.
7 M
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