Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 17 Scheme 17MAT31 Engineering Mathematics III Question Paper

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Third Semester B.E. Degree Examination, June/July 2019

Engineering Mathematics - III

Time: 3 hrs. Max. Marks: 100

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

Module-1

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1. a. Obtain the Fourier series of the function f(x) = x² in -p < x < p and hence deduce
   1/1² + 1/2² + 1/3² + ... = p²/6 (08 Marks)
2. b. Obtain the Half Range Fourier cosine series for the f(x) = sin x in [0, p]. (06 Marks)
3. c. Obtain the constant term and the coefficients of first sine and cosine terms in the Fourier expansion of y given
   X: 0 1 2 3 4 5
   

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   y: 9 18 24 28 26 20 (06 Marks)

OR

1. a. Obtain the Fourier series of
   f(x) =
   { -k, -p < x < 0
   

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   { k, 0 < x < p
   [0, 2p] and hence deduce that 1 - 1/3 + 1/5 - 1/7 + ... = p/4 (08 Marks)
2. b. Find the Fourier half range cosine series of the function f(x) = 2x - x² in [0, 3]. (06 Marks)
3. c. Express f(x) = { 1, |x| < a ; { 0, |x| > a as a Fourier Integral. (06 Marks)

Module-2

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1. a. Find the Fourier transform of f(x) = { 1, |x| < a ; { 0, |x| > a and hence deduce ?(sin x)/x dx = p/2 (08 Marks)
2. b. Find the Fourier sine transform of e^-ax/x ; a > 0 and hence evaluate ?tan^-1(x)/x dx (06 Marks)
3. c. Obtain the z-transform of cos n? and sin n?. (06 Marks)

OR

1. a. Find the Fourier transform of f(x) = e^-a|x|, a > 0 (08 Marks)

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2. b. Find the Fourier cosine transform of f(x) where
   f(x) = { x ; 0 < x < 1
   { 2-x; 1 < x < 2 (06 Marks)
3. c. Solve u_n+2 + 6u_n+1 + 9u_n = 2ⁿ with u₀ = u₁ = 0 using z-transform. (06 Marks)

Module-3

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1. a. Fit a straight line y = ax + b for the following data by the method of least squares.
   x: 0 1 2 3 4
   y: 1 1.8 3.3 4.5 6.3 (08 Marks)
2. b. Calculate the coefficient of correlation for the data:
   x: 1 2 3 4 5 6 7 8 9
   

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   y: 9 8 10 12 11 13 14 16 15 (06 Marks)
3. c. Compute the real root of x log₁₀ x - 1.2 = 0 by the method of false position. Carry out 3 iterations in (2, 3). (06 Marks)

OR

1. a. Fit a second degree parabola to the following data y = a + bx + cx².
   x: 0 1 2 3 4 5
   

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   y: 6 4 3 6 11 18 (06 Marks)
2. b. If ? is the angle between two regression lines, show that
   tan ? = (s_x s_y / (s_x² + s_y²)) * (1 - r²) / r (08 Marks)
3. c. Using Newton Raphson method, find the real root of the equation 3x = cos x + 1 near x = 0.5. Carry out 3 iterations. (06 Marks)

Module-4

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1. a. From the following table, estimate the number of students who obtained marks between 40 and 45.
   Marks: 30-40 40-50 50-60 60-70 70-80
   No. of students: 31 42 51 35 31 (08 Marks)
2. b. Use Newton's divided difference formula to find f(9) for the data:
   x: 5 7 11 13 17
   

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   f(x): 150 392 1452 2366 5202 (06 Marks)
3. c. Find the approximate value of ?₀^p/2 vcos ? d? by Simpson's 1/3^rd rule by dividing [0, p/2] into 6 equal parts. (06 Marks)

OR

1. a. The area A of a circle of diameter d is given for the following values:
   d: 80 85 90 95 100
   

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   A: 5026 5674 6362 7088 7854
   Calculate A when d = 105 using an appropriate interpolation formula. (08 Marks)
2. b. Using Lagrange's interpolation formula to find the polynomial which passes through the points (0, -12), (1, 0), (3, 6), (4, 12). (06 Marks)
3. c. Evaluate ?₄^5.2 log_e x dx taking 6 equal parts by applying Weddle's rule. (06 Marks)

Module-5

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1. a. If F = 3xy i - y² j, evaluate ?_C F.dr where 'C' is arc of parabola y = 2x² from (0, 0) to (1, 2) (08 Marks)
2. b. Evaluate by Stokes theorem ?_C (sin z dx + cos x dy + sin y dz), where C is the boundary of the rectangle 0 < x < p, 0 < y < 1, z = 1. (06 Marks)
3. c. Prove that the necessary condition for the I = ?f(x,y,y')dx to be extremum is ?f/?y - d/dx (?f/?y') = 0 (06 Marks)

OR

1. a. Using Green's theorem evaluate ?_C (3x² - 8y²)dx + (4y - 6xy)dy, where C is the boundary of the region bounded by the lines x = 0, y = 0, x + y = 1. (08 Marks)

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2. b. Find the external value of ?₀^p/2 (y'² - y² + 4y cos x) dx: Given that y(0) = 0, y(p/2) = 0. (06 Marks)
3. c. Prove that the shortest distance between two points in a plane is along a straight line joining them. (06 Marks)