Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 17 Scheme 17MAT411 Engineering Mathematics IV Question Paper

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CBCS SCHEME

17MAT41

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Fourth Semester B.E. Degree Examination, June/Jul 2019

Engineering Mathematics - IV

Time: 3 hrs.

Max. Marks: 100

Note: Answer any FIVE full questions, choosing ONE full question from each module.

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Module-1

1. a. If y' + y + 2x = 0, y(0) = -1 then find y(0.1) by using Taylor's series method. Consider upto third order derivative term. (06 Marks)
2. b. Find y(0. 2) by using modified Euler's method, given that y' = x + y, y(0) = 1. Take h = 0.1 and carry out two modifications at each step. (07 Marks)
3. c. If y' = (x² + y²)^-1, y(0) = 2, y(0.2) = 2.0933, y(0.4) = 2.1755, y(0.6) = 2.2493 then find y(0.8) by Milne's method. (07 Marks)

OR

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1. a. Use Taylor's series method to find y(0.1) from y' = 3x + y², y(0) = 1. Consider upto fourth derivative term. (06 Marks)
2. b. Use Runge - Kutta method to find y(0.1) from y' = x+y, y(0) = -1. (07 Marks)
3. c. Use Adam - Bashforth method to find y(0.4) from y' = 1+xy, y(0) = 1, y(0.1) = 1.0025, y(0.2) = 1.0101, y(0.3) = 1.0228. (07 Marks)

Module-2

1. a. Express x³ - 5x² + 6x + 1 in terms of Legendre polynomials. (06 Marks)

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2. b. Find y(0.1), by using Runge-Kutta method, given that y" + xy' + y = 0, y(0) = 1 y'(0) = 0. (07 Marks)
3. c. Solve Bessel's operation leading to J_n(x). (07 Marks)

OR

1. a. Prove that J_-1/2(x) = v(2/(px)) cos x. (06 Marks)
2. a. Prove that J_1/2(x) = v(2/(px)) sin x. (redundant)

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3. b. Find y(0.4) by using Milne's method, given that y(0) = 1, y'(0) = 1, y(0.1) = 1.0998, y'(0.1) = 0.9946, y(0.2) = 1.1987, y'(0.2) = 0.9773, y(0.3) = 1.2955, y'(0.3) = 0.946. (07 Marks)
4. c. State and prove Rodrigue's formula. (07 Marks)

Module-3

1. a. Derive Cauchy - Riemann equations in Cartesian coordinates. (06 Marks)
2. b. Find an analytic function f(z) = u + iv in terms of z, given that u = e^2x(x cos 2y - y sin 2y). (07 Marks)

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3. c. Evaluate ? (sin pz² + cos pz²) / ((z-1)(z-2)) dz, c is |z|= 3 by residue theorem. (07 Marks)

OR

1. a. Prove that (?/?x)² + (?/?y)² |f(z)|² = 4|f'(z)|². (06 Marks)
2. b. Discuss the transformation W = z². (07 Marks)
3. c. Find a bilinear transformation that maps the points i, 0, -i in Z-plane into -1, i, 1 in W plane respectively. (07 Marks)

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Module-4

1. a. In a sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2, out of 1000 such samples, how many would be expected to contain atleast 3 defective parts? (06 Marks)
2. b. If X is a normal variate with mean 30 and standard deviation 5, find the probabilities that i) 26 < X = 40 ii) X > 45 iii) |X-30| > 5. Given that F(0.8) = 0.288, F(2.0) = 0.4772, F(3) = 0.4987, F(1) = 0.3413. (07 Marks)
3. c. The joint density function of two continuous random variables X and Y is given by f(x, y) = { K xy, 0 = x = 4, 1 = y = 5 ; 0, otherwise } Find i) K ii) E(x) iii) E(2x + 3y). (07 Marks)

OR

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1. a. Derive mean and standard deviation of the Poisson distribution. (06 Marks)
2. b. The joint probability distribution for two random variables X and Y as follows :
   

   X\Y	-2	-1	4	5
   0	0.1	0.2	0	0.3
   2	0.2	0.1	0.3	0

   Find i) Expectations of X, Y, XY ii) SD of X and Y iii) Covariance of X, Y iv) Correlation of X and Y. (07 Marks)
3. c. In a certain town the duration of shower has mean 5 minutes. What is the probability that a shower will last for i) 10 minutes or more ii) Less than 10 minutes iii) Between 10 and 12 minutes. (07 Marks)

Module-5

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1. a. A group of boys and girls were given in Intelligence test. The mean score, SD score and numbers in each group are as follows :
   

   	Mean	SD	N
   Boys	74	8	12
   Girls	70	10	10

   Is the difference between the means of the two groups significant at 5% level of significance? Given that t_0.05 = 2.086 for 20 d.f. (06 Marks)
2. b. The following table gives the number of accidents that take place in an industry during various days of the week. Test if accidents are uniformly distributed over the week.
   

   Day	Mon	Tue	Wed	Thu	Fri	Sat
   No. of accidents	14	18	12	11	15	14

   Given that ?² = 11.07 at 5 d.f. (07 Marks)
3. c. Find the unique fixed probability vector for the regular stochastic matrix.
   

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   A=

   0	1	0
   1/6	1/2	1/3
   0	2/3	1/3

   (07 Marks)

OR

1. a. Define the following terms : i) Type I error and type II error. ii) Transient state. iii) Absorbing state. (06 Marks)
2. b. A certain stimulus administered to each of the 12 patients resulted in the following increases of blood pressure : 5, 2, 8, -1, 3, 0, -2, 1, 5, 0, 4, 6. Can it be concluded that the stimulus will be general be accompanied by an increase in blood pressure. Given that t 0.05 = 2.2 for 11 d.f. (07 Marks)
3. c. If P =

   0	2/3	1/3
   1/2	1/2	0
   1/2	0	1/2

   Find the corresponding stationary probability vector. (07 Marks)

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