Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 15 Scheme 15MAT31 Engineering Mathematics III Question Paper

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RV 15MAT31

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

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Engineering Mathematics - III

Time: 3 hrs.

Max. Marks: 80

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 for the function :
   f(x) =
   -p in -p < x < 0
   x in 0 < x < p
   Hence deduce that p² / 8 = S ( 2n-1 )² (08 Marks)

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2. b. Express y as a Fourier series up to the second harmonics, given :
   x 0 p/4 p/2 3p/4 p 5p/4 3p/2 7p/4 2p
   y 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98 (08 Marks)

OR

1. a. Obtain the Fourier series for the function f(x) = 2x - x² in 0 < x < 2p. (08 Marks)

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2. b. Obtain the constant term and the first two coefficients in the only Fourier cosine series for given data :
   x 0 1 2 3 4 5
   y 4 8 15 7 6 2 (08 Marks)

Module-2

1. a. Find the Fourier transform of e^-ax. (06 Marks)

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2. b. Find the Fourier sine transform of 1/x, a > 0 . (05 Marks)
3. c. Obtain the z - transform of sin n? and cos n? (05 Marks)

OR

1. a. Find the inverse cosine transform of F(a) = 1 - a, 0 < a <1
   0, a >1
   

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   Hence evaluate ?₀⁸ (sin² t / t² )dt . (06 Marks)
2. b. Find inverse Z - transform of (3z² + 2z) / ((5z -1)(5z + 2)). (05 Marks)
3. c. Solve the difference equation y_n+2 - 6y_n+1 + 9y_n = 2ⁿ with y₀ = 0, y₁ = 0 , using z - transforms. (05 Marks)

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Module-3

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1. a. Find the lines of regression and the coefficient of correlation for the data :
   x 1 2 3 4 5 6 7
   y 9 8 10 12 11 13 14 (06 Marks)
2. b. Fit a second degree polynomial to the data :
   x 0 1 2 3 4
   

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   y 1 1.8 1.3 2.5 6.3 (05 Marks)
3. c. Find the real root of the equation x sin x + cos x = 0 near x = p, by using Newton — Raphson method upto four decimal places. (05 Marks)

OR

1. a. In a partially destroyed laboratory record, only the lines of regression of y on x and x on y are available as 4x — 5y + 33 = 0 and 20x — 9y = 107 respectively. Calculate x¯, y¯ and the coefficient of correlation between x and y. (06 Marks)
2. b. Fit a curve of the type y = ae^bx to the data :
   

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   x 5 15 20 30 35 40
   y 10 14 25 40 50 62 (05 Marks)
3. c. Solve cos x = 3x — 1 by using Regula — Falsi method correct upto three decimal places, (Carryout two approximations). (05 Marks)

Module-4

1. a. Given f(40) = 184, f(50) = 204, f(60) = 226, f(70) = 250, f(80) = 276, f(90) = 304. Find f(38) using Newton's forward interpolation formula. (06 Marks)

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2. b. Find the interpolating polynomial for the data :
   x 0 1 2 5
   Y 2 3 12 147
   By using Lagrange's interpolating formula. (05 Marks)
3. c. Use Simpson's 1/3 rule to evaluate ?₀^0.3 (1 - 8x³)^1/2 dx considering 3 equal intervals. (05 Marks)

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OR

1. a. The area of a circle (A) corresponding to diameter (D) is given below :
   D 80 85 90 95 100
   A 5026 5674 6362 7088 7854
   Find the area corresponding to diameter 105, using an appropriate interpolation formula. (06 Marks)

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2. b. Given the values :
   x 5 7 11 13 17
   f(x) 150 392 1452 2366 5202
   Evaluate f(9) using Newton's divided difference formula. (05 Marks)
3. c. Evaluate ?₀¹ (1/(1+x)) dx by Weddle's rule taking seven ordinates. (05 Marks)

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Module-5

1. a. Using Green's theorem, evaluate ? (2x² - y²)dx + (x² + y²)dy where C is the triangle formed by the lines x = 0, y = 0 and x + y = 1. (06 Marks)
2. b. Verify Stoke's theorem for f = (2x — y)i — yz² j — y² zk for the upper half of the sphere x² + y² + z² = 1. (05 Marks)
3. c. Find the extremal of the functional ?_x1^x2 { y² + (y')² + 2ye^x }dx (05 Marks)

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OR

1. a. Using Gauss divergence theorem, evaluate ? f.ds , where f = 4xzi — y² j + yzk and s is the surface of the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1. (05 Marks)
2. b. A heavy cable hangs freely under the gravity between two fixed points. Show that the shape of the cable is a Catenary. (06 Marks)
3. c. Find the extremal of the functional I = ?₀^p/2 (y'² - y² + 4y cos x) dx , given that y(0) = 0 = y(p/2). (05 Marks)

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