Download AKTU B-Tech 1st Sem 2016-2017 RME 101 Elements Of Mechanical Engineering Question Paper

VASAVI COLLEGE OF ENGINEERING (Autonomous), HYDERABAD

B.E. II Year I-Semester Examinations, December-2016

Subject: Mathematics – III (Common to all Branches)

Time: 3 hours Max. Marks: 70

Note: Answer ALL questions from Part-A and any FIVE from Part-B

Part-A (10 x 2 = 20 Marks)

1. Form the partial differential equation by eliminating the arbitrary constants a and b from z = (x + a)(y + b).

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2. Solve (D - 2D' - 3)z = 0.

3. State Dirichlet’s conditions for a function f(x) in the interval (c, c + 2p) for Fourier series expansion.

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4. If f(x) = x in -p < x < p, find the Fourier coefficient a_n.

5. Write the various possible solutions of one-dimensional heat equation.

6. A tightly stretched string with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points a velocity ?x(l - x), find u(x, 0).

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7. State Cauchy’s integral formula.

8. Find the residue of f(z) = z² / (z - 1)²(z + 2) at z = -2.

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9. Define a bilinear transformation.

10. Find the fixed points of the transformation w = (z - i) / (z + i).

Part-B (5 x 10 = 50 Marks)

1. a) Solve (x² - yz)p + (y² - zx)q = z² - xy.

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   b) Solve (D² + D - 6)z = y cos x.

2. Expand f(x) = x² as a Fourier series in the interval (-p, p).

3. A homogeneous rod of length l has its ends A and B kept at 0°C and 100°C respectively, until steady state conditions prevail. The temperature at B is suddenly reduced to 0°C and kept so. Find the temperature distribution in the rod at time t.

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4. Find the analytic function f(z) = u + iv, if u - v = e^x(cos y - sin y).

5. Evaluate ?_c (z² - z + 2) / (z - 1) dz, where c is the circle |z| = 1/2 using Cauchy’s Integral formula.

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6. Evaluate ?₀^2p d? / (13 + 5 sin ?) using contour integration.

7. a) Find the bilinear transformation which maps the points z = 1, i, -1 into w = 2, i, -2.

   b) Discuss the transformation w = z².

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