Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2017 Jan 2016 Dec 15MAT21 Engineering Mathematics II Question Paper

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15MAT21

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Second Semester B.E. Degree Examination, Dec.2016/Jan.2017

Engineering Mathematics - II

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Time: 3 hrs.

Max. Marks: 80

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

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Module-1

1. a. Solve (D-2)² y = 8(e^2x + x²) by inverse differential operator method. (06 Marks)
2. b. Solve (D² – 4D + 3) y = e^x cos 2x by inverse differential operator method. (05 Marks)
3. c. Solve by the method of variation of parameters y" – 6y' + 9y = e^3x/x. (05 Marks)



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OR

1. a. Solve (D²_ 1)y = x sin 3x by inverse differential operator method. (06 Marks)
2. b. Solve (D³ – 6D² + 11D – 6)y = e^-2x by inverse differential operator method. (05 Marks)
3. c. Solve (D² + 2D + 4) y = 2x² + 3 e^-2x by the method of undetermined coefficient. (05 Marks)



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Module-2

1. a. Solve x³ y'"' + 3x² y" + xy' + 8y = 65 cos(log x). (06 Marks)
2. b. Solve xy p² + p(3x² - 2y²) - 6xy = 0. (05 Marks)
3. c. Solve the equation y²(y – x) = x⁴ p² by reducing into Clairaut's form, taking the substitution y^-1 = v and x^-1 = u. (05 Marks)



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OR

1. a. Solve (2x + 3)² y" - (2x + 3) y' - 12y = 6x. (06 Marks)
2. b. Solve p² + 4x³p - 12x²y = -12x⁴. (05 Marks)
3. c. Solve p³ – 4xy p + 8y² = 0. (05 Marks)



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Module-3

1. a. Obtain the partial differential equation by eliminating the arbitrary function. z = f(x+at) + g(x-at) (06 Marks)
2. b. Solve ?²z/?x?y = sin x sin y, for which ?z/?y = -2 sin y, when x = 0 and z = 0, when y is an odd multiple of p/2. (05 Marks)
3. c. Find the solution of the wave equation ?²u/?t² = a² ?²u/?x² by the method of separation of variables. (05 Marks)



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OR

1. a. Obtain the partial differential equation by eliminating the arbitrary function F(x + my + nz, x² + y² + z²) = 0. (06 Marks)
2. b. Solve ?²z/?y² - ?z/?y = z, given that, when y = 0, z = e^x and ?z/?y = e^x. (05 Marks)
3. c. Derive one dimensional heat equation ?u/?t = C² ?²u/?x². (05 Marks)



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Module-4

1. a. Evaluate ?(x + y + z) dy dx dz, limits z=x to 2x, y=0 to x, x=0 to 1. (06 Marks)
2. b. Evaluate ? xy dy dx, limits x=0 to 4a, y=x²/4a to v(ax), by changing the order of integration. (05 Marks)
3. c. Evaluate ?₀⁴ x^1/2 (4-x)^-1/2 dx by using Beta and Gamma function. (05 Marks)



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OR

1. a. Evaluate ?₀⁸?₀⁸ e^-x²-y² dx dy by changing to polar co-ordinates. Hence show that ?₀⁸ e^-x² dx = v(p)/2. (06 Marks)
2. b. Find by double integration, the area lying inside the cardioid r = a(1-cos ?). (05 Marks)
3. c. Obtain the relation between beta and gamma function in the form B(m, n) = G(m)G(n) / G(m+n). (05 Marks)



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Module-5

1. a. Find i) L {e^3t (2 cos 5 t - 3 sin 5 t)}. (06 Marks)
2. b. If a periodic function of period 2a is defined as f(t) = t, if 0 < t < a; 2a - t, if a < t < 2a. Then show that L{f(t)}= (1/s²) tan h(as/2). (05 Marks)
3. c. Solve the equation by Laplace transform method. y'' + 2y' - y'-2y = 0. Given y(0) = y' (0) = 0. (05 Marks)



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OR

1. a. Find L^-1 {s/(s²+4)(s²+9)}. (06 Marks)
2. b. Find L^-1 {1/(s²+a²)²} by using Convolution theorem. (05 Marks)
3. c. Express f(t) = sin t, 0 < t < p; sin 2t, p < t < 2p; sin 3t, t > 2p in terms of unit step function and hence find its Laplace transforms. (05 Marks)



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