GUJARAT TECHNOLOGICAL UNIVERSITY
SEMESTER-IV(NEW) — EXAMINATION - SUMMER 2019
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Subject Code:2141703
Subject Name: Numerical Techniques & Statistical Methods
Date:09/05/2019
Time:02:30 PM TO 05:30 PM
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q1
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(a) If X =3.4327, find the absolute and relative errors if :
(a) X is truncated to three decimal places.
(b) X is rounded off to three decimal places. [03]
(b) Calculate mean and mode for the following data: [04]
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Frequency | 10 | 14 | 19 | 17 | 13 |
(c) Use Fourth order Runge-Kutta method to find y(0.2) with h=0.1, given that
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dy/dx = 2x+y, y(0)=1 [07]Q2
(a) Using the power method, find the largest Eigen value for A:{1 2; 3 4} [03]
(b) Apply Gauss — Seidel iteration method to solve :
20x+y-2z=17, 3x+20y—z=-18, 2x-3y+20z=25 [04]
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(c) Construct an Interpolating polynomial which takes the following values :
x 0 1 2 3 4 5 6 7
y 1 2 4 7 11 16 22 29 [07]
OR
(c) Obtain Cubic spline for subinterval 0 < x <1 & 1< x <2 from the following data:
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x 0 1 2 3S(x) 1 2 33 244 [07]
Q3
(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. [03]
| x | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|
| y | 1.02 | 0.94 | 0.89 | 0.79 | 0.71 | 0.62 | 0.55 |
(b) Using Newton’s forward formula, find the value of f(1.6) [04]
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| x | 1 | 1.4 | 1.8 | 2.2 |
|---|---|---|---|---|
| S(x) | 3.49 | 4.82 | 5.96 | 6.5 |
(c) Use Euler’s method to find y(2) from the differential equation dy/dx = x+2y, y(1) =1 with h=0.1 [03]
OR
(c) Using Simpson’s 1/3 rule, evaluate ? 1/(1+x2) dx by taking 4 sub intervals. [04]
(c) Evaluate ?01 exp(-x2) dx using the Gaussian Integration formula with n = 3. [04]
Q.4
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(a) Solve dy/dx = x2 + y2, y(0) = 1.1 to find y(0.2) and y(0.4) by Milne’s Predictor — Corrector method. [03]
(b) There are two boxes A and B containing 4 white, 3 red and 3 white, 7 red balls respectively. A box is chosen at random and a ball is drawn from it, If the ball is white, find the probability that it is from box A. [04]
(c) An unbiased coin is tossed 6 times. Find the probability of getting (1) exactly 4 heads (2) at least 4 heads. [07]
(d) Eleven school boys were given a test in a Subject. They were given a month’s further coaching and a second test of equal category was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching?
| Boys | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks 1st test | 23 | 20 | 19 | 21 | 18 | 20 | 18 | 17 | 23 | 16 | 19 |
| Marks 2nd test | 24 | 19 | 22 | 18 | 20 | 22 | 20 | 20 | 23 | 20 | 17 |
(At 5% level of significance for n=10 d. f. t0.05 =2.228) [07]
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OR
(a) Two people are selected at random from a group of seven men and five women. Find the Probability that both are men or both are women. [03]
(b) 100 Electric bulbs are found to be defective in a lot of 5000 bulbs. Find the probability that at the most 3 bulbs are defective in a box of 100 bulbs. [04]
(c) A dice is thrown 150 times and the following results are obtained. [07]
| Number turned up | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 19 | 23 | 28 | 17 | 32 | 31 |
Test the Hypothesis that the dice is unbiased at 5% level of significance.
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(At 5% level of significance for n=5 d.f. ?20.05 =11.07)
Q.5
(a) Find the standard deviation of a group of data points: 101.8, 103.2, 104.0, 102.5, 103.5 [03]
(b) Define Chi-square Test. State (a) conditions to apply test (b) application of test [04]
(c) Represent the following information in form of a network. Find average duration time or expected time of each activity and obtain the critical path. [07]
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| Activity | 1-2 | 1-3 | 2-4 | 3-4 | 3-5 | 4-5 | 4-6 | 5-7 | 5-8 | 9-6 | 9-10 | 6-10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Optimistic time | 4 | 1 | 8 | 3 | 2 | 3 | 3 | 4 | 4 | 2 | 4 | 4 |
| Most Likely time | 9 | 5 | 10 | 6 | 4 | 7 | 7 | 8 | 7 | 7 | 8 | 7 |
| Pessimistic time | 14 | 8 | 17 | 8 | 5 | 10 | 10 | 9 | 14 | 10 | 18 | 9 |
OR
(a) Compute the Median from the data: [03]
| Class | 0-30 | 30-60 | 60-90 | 90-120 | 120-150 | 150-180 |
|---|---|---|---|---|---|---|
| Frequency | 8 | 13 | 22 | 27 | 18 | 7 |
(b) A bag Contains 5 white and 7 black balls. Find the expectation of a man who is allowed to draw two balls from the bag and who is to receive one rupee for each black ball and two rupees for each white ball. [04]
(c) Draw PERT — diagram after finding out expected time & find critical path. [07]
| Activity | Sequence | Optimistic Time | Most Likely Time | Pessimistic Time |
|---|---|---|---|---|
| A | 1-2 | 7 | 12 | 13 |
| B | 1-3 | 7 | 10 | 12 |
| C | 2-5 | 8 | 13 | 15 |
| D | 3-5 | 10 | 12 | 22 |
| E | 5-6 | 10 | 14 | 18 |
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