Download GTU B.Tech 2020 Winter 4th Sem 2141703 Numerical Techniques And Statistical Methods Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER-IV (NEW) EXAMINATION - WINTER 2020
Subject Code:2141703 Date:09/02/2021
Subject Name:Numerical Techniques & Statistical Methods
Time:02:30 PM TO 04:30 PM Total Marks:47

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Instructions:

1. Attempt any THREE questions from Q.1 to Q.6 .
2. Q.7 is compulsory.
3. Make suitable assumptions wherever necessary.
4. Figures to the right indicate full marks.

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Q.1

1. (a) An approximate value of p is 3.1428571 and its true value is 3.1415925. Compute the absolute and relative errors. [03]
2. (b) An incomplete frequency distribution is given as follows:
   

   Variable	10-20	20-30	30-40	40-50	50-60	60-70	70-80
   Frequency	12	30	f1	65	f2	25	18

   Given that the median value is 46, determine the value of f1 and f2 . [04]

Q.2

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1. (a) Solve the following system by Gauss-Seidal iteration method:
   27x + 6y—z =85
   6x +15y +2z =172
   X+ y+54z=110 [07]
2. (b) Find the value of f(5) by using Lagrange’s'interpolation method for the given data
   

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   x	1	2	3	4	7
   f(x)	2	4	8	16	128

   [03]

Q.3

1. (a) Solve the equation x² —3x-5=0 by using Newton Raphson method, correct up to four decimal places. [04]
2. (b) Use Simpson’s 1/3 and Simpson’s 3/8 rule to evaluate the following integral
   ∫ 1/(1+x) dx from 0 to 1 [07]

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Q.4

1. (a) Use Euler’s method with h =0.1 to find the solution of the equation dy/dx = x² + y² , given that y(0) = 0 in the range 0 < x < 5. [03]
2. (b) Find first derivatives at x = 1.1 from the following table
   

   X	1	1.2	1.4	1.6	1.8	2.0
   f(x)	0	0.1280	0.5440	1.2960	2.4320	4.0000

   [04]
3. (c) Using Milne’s Predictor Corrector method to obtain y(0.4) by solving dy/dx =2e^x — y given that y(0) =2, y(0.1) =2.01, y(0.2) =2.04, y(0.3) =2.09 . [07]

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Q.5

1. (a) A can hit a target 4 times in 5 shots, B, 3 times in 4 shots and C, 2 times in 3 shots. They fire a volley (it means one shot each). Find the probability that the target will be hit. [03]
2. (b) Out of 800 families with 4 children each how many families would be expected to have (i) atleast one boy (ii) 2 boys and 2 girls Assume equal probabilities for boys and girls. [04]
3. (c) Five coins are tossed 3200 times and the following results are obtained:
   

   Number of heads	0	1	2	3	4	5
   Frequency	80	570	1100	900	500	50

   If χ² for 5 d. f at 5% level of significance be 11.07, test the hypothesis that the coins are unbiased. [07]

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Q.6

1. (a) A factory produces razor blades. The probability of its being defective is 1/500. In 10,000 packets of 10 blades each, calculate the approximate numbers of packets have 2 defective blades. [03]
2. (b) If Skulls are classified as A, B, C according as the length and breadth index as under 75, between 75 and 80, or over 80; find the approximately mean and the standard deviation of the classes in which A are 58% , B are 38% and C are 4% given f(0.20)=0.08 and f (1.75)=0.46 [04]
3. (c) In a bolt factory, machines A, B, and C manufacture 25%, 35% and 40% of the total of their output 5%, 4% and 2% are defective bolts. A bolt is drawn at random from the product and is found to be defective. What are the chances that it was manufactured by machines A, B, or C? [07]

Q.7

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In an experiment on immunization of cattle from tuberculoses the following result were obtained:

Examine the effect of vaccine in controlling the susceptibility to tuberculoses. (Given that for 5% level of significance χ² for 1 d. f. is 3.841, for 2 d. f. is 5.99 and for 3 d. f. is 7.815). [05]

OR

The heights of 9 males of a given locality are found to be 45, 47, 50, 52, 48, 47, 49, 53, 51 inches. Is it reasonable to believe that the average height is differ significantly from assumed mean 47.5 inches? (Given that for 5% level of significance t for 7 d. f. is 2.365, for 8 d. f. is 2.306 and for 9 d. f. is 2.262) [05]

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