Subject Code:2140606
GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER-1V (NEW) EXAMINATION - WINTER 2018
Subject Name:Numerical and Statistical Methods for Civil Engineering
Date:22/11/2018
Time: 02:30 PM TO 05:00 PM
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q.1
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(a) In usual notations show that ? +? = ? - ?
(b) Find the cubic polynomial which takes on the values f(0)=4, f(1)=1, f(2)=2, f(3)=11, f(4)=32, f(5)=71. Also find f(6) and f(2.5).
(c) Obtain by Power method the numerically largest eigen value of the matrix
A=
Q.2
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(a) In how many different ways can the director of a research laboratory choose 2 chemists from among 7 applicants and 3 physicists from among 9 applicants?
(b) A class consists of 6 girls and 10 boys. If a committee of three is chosen at random from the class, find the probability that, (i) three boys are selected; (ii) exactly two girls are selected.
(c) Solve the following system of equations using Gauss Jacobi iteration method:
4x1 + x2 + x3 = 2; x1 + 5x2 + 2x3 = 6; x1 + 2x2 + 3x3 = -4
OR
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(c) At checkout counter customers arrive at an average of 2.0 per minute. Find the probabilities that
1) At most 3 will arrive in any given minute
(i) At least 3 will arrive during an interval of 4 minutes
(iii) At most 10 will arrive during an interval of 6 minutes.
Q.3
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(a) Using Regula — Falsi method determine the root of the equation x log x = 1.2.
(b) Use Euler’s method to solve the initial value problem dy/dx = x on the interval [0,3] with y(0) = 1. Compare the numerical solution with exact solution for the step size h=0.25.
(c) Using Runge — Kutta fourth order method solve dy/dx = y - 2x/y; y(0)=1. Evaluate the value of y when x = 0.2, x = 0.4, take step size 0.2.
OR
Q.3
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(a) Using Taylor’s series method, find y(1.1) correct to four decimal place, given by dy/dx = x2 + y2, y(1) = 1.
(b) Evaluate ?1/(1+x2) dx from 0 to 1 by Simpson’s 1/3 rule taking eleven ordinates and hence find the value of loge 2 correct to five significant digits.
(c) Use Newton’s divided difference method to evaluate f(4) from the below data:
X: 0 1 2 3
f(x): 2 3 12 147
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Q.4
(a) The runs scored by two batsmen A and B in 9 consecutive matches are given below. Find which batsman is more consistent?
A: 85 | 20 | 62 | 28 | 74 | 5 | 69 | 4 | 13
B: 72 | 4 | 15 | 30 | 59 | 15 | 49 | 27 | 26
(b) Derive an iteration formula for vN and hence find 3v58.
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(c) Solve the following system of equation using Gauss — Seidel method:
5x + y - z = 10; 2x + 4y + z = 14; x + y + 8z = 20
OR
Q.4
(a) Find the mean and standard deviation for the following data:
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Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70
Frequency | 6 | 14 | 10 | 8 | 1 | 3 | 8
(b) Find the equation of the cubic curve which passes through the points (0, -5), (1, -10), (2, -9), (3, 4), and (4, 35).
(c) Solve the following system of equations using Gauss elimination method with partial pivoting. x + y + z = 7, 3x + 3y + 4z = 24; 2x + y + 3z = 16
Q.5
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(a) Find the median from the following data.
Class limits | 0-30 | 30-60 | 60-90 | 90-120 | 120-150 | 150-180
Frequency | 8 | 13 | 22 | 27 | 18 | 7
(b) Compute the correlation coefficient between X and Y using the following data:
X | 2 | 4 | 5 | 6 | 8 | 11
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Y | 18 | 12 | 10 | 8 | 7 | 5
(c) Following table gives the data on rainfall and discharge in a certain river. Obtain the line of regression of Y on X.
Rainfall(inch) X: 1.53 | 1.78 | 2.60 | 2.95 | 3.42
Discharge(1000cc) Y: 33.5 | 36.3 | 40.0 | 45.8 | 53.5
OR
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Q.5
(a) A train is moving at the speed of 30 m/s suddenly brakes are applied. The speed of the train per second after t seconds is given by the following table:
Time(t) | 0 | 5 | 10 | 15 | 20 | 25 | 30
Speed(v) | 30 | 24 | 19 | 16 | 13 | 11 | 10
Apply Simpson’s 3/8th rule to determine the distance moved by the train 30 sec.
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(b) An unbiased coin is tossed 6 times. Find the probability of getting (i) exactly 4 heads, (ii) at least 4 heads.
(c) At constant temperature, the pressure P and the volume V of a gas are connected by the relation PV? = constant. Find the best fitting equation of this form to the following data and estimate V when P=4.
P(Kg. Sq. cm) | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0
V(cc) | 1620 | 1000 | 750 | 620 | 520 | 460
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