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

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Code: 15A54101

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B.Tech I Year I Semester (R15) Supplementary Examinations November/December 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. (a) Solve (x + y + 1) dy/dx = 1.
2. (b) Solve dy/dx + y cos x + sin y + y = 0.

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3. (c) Find the envelope of the family of the lines y = mx + v(1 + m)², m being the parameter.
4. (d) Find the evolute of the parabola y² = 4ax .
5. (e) Evaluate ? y dy dx (limits: x from 0 to 2, y from 0 to x).
6. (f) Using cylindrical coordinates, find the volume of the cylinder with base radius a, and height h.
7. (g) Find the extrema of f(x, y) = a² – x² – y².

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8. (h) Verify Euler's theorem for u(x, y, z) = xy + yz + zx.
9. (i) If R = xi +yj +zk, show that ? · R = 3 and ? × R = 0.
10. (j) If F = 3xyi – y²j, then evaluate ? F · dR, where C is the curve y = 2x² from (0, 0) to (1, 2).

PART - B

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

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UNIT - I

1. Solve (D² + 4D + 3)y = e^x cos2x – cos3x – 3x².

   OR

2. Find the equation of the tangent at any point (x, y) to the curve x³ + y³ = a³ show that the portion of the tangent intercepted between the axes is of constant length.

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UNIT - II

1. (a) Using the method of variation of parameters, solve d²y/dx² + 4y = tan 2x .

   (b) A rectangle sheet of metal of length 6 meters is given. Four equal squares are removed from the corners. The sides of this sheet are now turned up to form an open rectangular box. Find approximately, the height of the box, such that the volume of the box is maximum.

   OR

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2. The deflection of a strut of length / with one end built-in and the other end subjected to the end thrust p, satisfies d²y/dx² + a²y = a²(l – x). Find the deflection y of the strut at a distance x from the built-in end.

UNIT - III

1. (a) A rectangular box open at the top is to have a volume of 32 cft. Find the dimensions of the box requiring least material for its construction.

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   (b) Find the points on the surface z² = xy + 1 nearest to the origin.

   OR

2. (a) Discuss the maxima and minima of f(x, y) = sin x sin y sin (x + y).

   (b) Find the Taylor's series expansion of tan^-1(y/x) about (1, 1).

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UNIT - IV

1. Evaluate ? xyz dx dy dz over the positive octant of the sphere x² + y² + z² = a² .

   OR

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2. Find the volume bounded by the cylinders x² + y² = 4, y + z = 4 and z = 0.

UNIT - V

1. (a) Verify Stoke's theorem for A = y²i + xyj – xzk where S is the hemisphere x² + y² + z² = a², z = 0.

   (b) Evaluate ?_s F · ds where F = 4xi - 2y²j+z²k and S is the surface bounding the region x² + y² = 4, z = 0 and z = 3.

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   OR

2. Verify Green's theorem for ?_C [(xy + y²)dx + x²dy], where C is bounded by y = x and y = x².

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