Download JNTUA B.Tech 1-1 R15 2016 Dec Regular 15A54101 Mathematics I Question Paper

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R15

Code: 15A54101

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B.Tech I Year I Semester (R15) Regular & Supplementary Examinations December 2016

MATHEMATICS - I

(Common to CE, EEE, CSE, ECE, ME, EIE and IT)

Time: 3 hours Max. Marks: 70

PART - A

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(Compulsory Question)

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1. Answer the following: (10 X 02 = 20 Marks)
   1. Find the orthogonal trajectories of the family of parabolas through the origin and foci on the y axis.
   2. Find the complementary function (D³ + 2D)y = e^2x + cos(3x + 7).
   3. x² d²y + 3x dy = 0 has the general solution dx² dx

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   4. Find P. I(?² – 4? + 1)^-1 sin z.
   5. If u = e^x+y, v = e^x+y, then find J.
   6. Find the radius of curvature at any point of the cardioids s = 4 asin ?
   7. ? (x² + y²)dxdy = D: y = x, y² = x.
   8. Evaluate ?₀¹ dx?₁² dy ?₀³ xyzdz.

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   9. x (? x ) is
   10. Evaluate ?_C y²dx - 2x²dy along the parabola y = x² from (0, 0)to (2, 4).

PART - B

(Answer all five units, 5 X 10 = 50 Marks)

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UNIT - I

2. Solve: x(x - 1) dy - y = x² (x - 1)³. dx
3. Solve: (D³ + 2D² – 3D)y = xe^3x.

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UNIT - II

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4. Solve: (D² + a²)y = tan ax by the method of variation of parameters.

   OR

5. The deflection y of a strut of length l with one end built-in and other end subjected to the end thrust P, satisfies d²y + a²y = a²R (1-x/l). Find the deflection y of the strut at a distance x from the built-in end.

UNIT - III

6. (a) If u = sin^-1 x+y then show that xux + yuy = 3 tan u. (b) If u = x + y + z, uv = y + z, uvw = z, then prove ?(x,y,z) = u²v. ?(u,v,w)

   OR

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7. (a) Find the points on the surface z² = xy + 1 nearest to the origin. (b) Find the radius of curvature at (3,3) on the curve x³ + xy² — 6y² = 0.

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UNIT - IV

8. Evaluate ?₀¹?₀^v(1-x²) xy²dxdy by changing the order of integration.

   OR

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9. Evaluate ? ? ? xy²zdxdydz taken through the positive octant of the sphere: x² + y² + z² = a².

UNIT - V

10. (a) Find the directional derivative of f = xy + yz + zx in the direction of vector i + 2j + 2k at the point (1, 2, 0). (b) Find curl f where f = grad (x³ + y³ + z³ – 3xyz).

    OR

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11. Evaluate by Green's theorem ?(y - sin x)dx + cosx dy where C is triangle enclosed the lines y = 0, x = p/2, py = 2x.

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