Download JNTUA B.Tech 1-1 R15 2019 June 15A54101 Mathematics I Question Paper

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Code: 15A54101

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B.Tech I Year I Semester (R15) Supplementary Examinations June/July 2019

MATHEMATICS - I

(Common to all branches)

Time: 3 hours

Max. Marks: 70

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PART - A

(Compulsory Question)

Answer the following: (10 X 02 = 20 Marks)

1. Solve (4xy + 3y² – x)dx + x(x + 2y)dy = 0.
2. Solve y" – 6y' + 9y = 5e-2x.

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3. Solve x²y'' + xy' + 9y = 0.
4. Solve (1 + x)²y'' + (1 + x)y' + y = 0.
5. Expand ex cos y in a Taylor's series about the point (1,p/4).
6. Find the radius of curvature for the curve x²y = a(x² + y²) at the point (-2a, 2a).
7. Evaluate ?^vx₁ ?^vx_x² (x² + y²)dydx.

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8. Evaluate ?^x₀ ?^y₀ xy dydx by changing the order of integration.
9. Given A = x²yz i + y²zx j + z²xy k find div? & curl ?.
10. If F = x²i + xy j evaluate ? F. dr from (0,0) to (1,1) along: (i) The line y=x. (ii) The parabola y = vx.

PART - B

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

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UNIT - I

2. Solve dy/dx + x sin 2y = x³cos²y.

OR

3. Solve y'" + 4y' – 12y = e2x – 3 sin 2x.

UNIT - II

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4. Solve by the method of variation of parameters y" + a²y = sec ax.

OR

5. Solve (3x + 2)²y" + 3(3x + 2)y' - 36y = 8x² + 4x + 1.

UNIT - III

6. If x = a cos hu cos v and y = a sin hu sin v prove that ?(x,y)/?(u,v) = a²(cos h 2u - cos 2v).

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OR

7. Find the extreme values of the function f (x, y) = x³ + 3xy² – 3x² – 3y² + 4.

UNIT - IV

8. Evaluate ?¹₀ ?^v(1-x²)₀ ?^v(1-x²-y²)₀ dz dy dx / v(1-x²-y²-z²)

OR

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9. Change the integral ?^a₀ ?^v(a²-x²)₀ x² / v(x² + y²)dy dx into polars and hence evaluate the same.

UNIT - V

10. Find div F and curl F where F = V(x³ + y³ + z³ – 3xyz).

OR

11. Verify Stoke's theorem for the vector F = (x² + y²)i – 2xy j taken round the rectangle bounded by x=0, x=a, y=0, y=b.

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