Download JNTUA B.Tech 1-1 R13 2019 Dec 13A54101 Mathematics I Question Paper

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Code: 13A54101

B.Tech I Year (R13) Supplementary Examinations December 2019

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MATHEMATICS – I

(Common to all branches)

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. Solve (x² - y²)dx = 2xydy .
2. Solve the differential equation : (D⁴ – 2D³ + 2D² – 2D+1)y = 0

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3. Solve the differential equation: (x²D² -3xD+1)y = 0 .
4. If U = log(x³+ y³+ z³-3xyz) then find (?U/?x) + (?U/?y) + (?U/?z)
5. If x = r cos?, y = rsin ? then find ?(x, y)/?(r,?)?
6. If f(x, y) = x² +3xy² -3x² -3y² +4 then find critical points.
7. Evaluate ?₀¹?_x^vx dy dx.

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8. Evaluate ?₀^a?₀^x?₀^x+y e^x+y+z dxdydz .
9. Find Curl F for F = zi +xj + yk .
10. State Gauss divergence theorem.

PART - B

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

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UNIT - I

2. (a) Solve (D⁴ + 2D² + 1)y = e^x cos x.
   (b) Find the orthogonal trajectories of the family of confocal conics x²/a²+? + y²/b²+? = 1, where ? is a parameter.

   OR

3. (a) Solve (D² - 2D + 1)y = x e^x sin x.
   

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   (b) Using method of variation of parameter Solve (D² + 4)y = tan 2x.

UNIT - II

4. (a) Verify Rolle's theorem for the function f(x) = sin x / e^x in [0,p].
   (b) Find the coordinates of the centre of curvature at (at², 2at) on the parabola y² = 4ax.

   OR

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5. (a) Find the maximum and minimum values of x³ + 3xy² – 15x² – 15y² + 72x.
   (b) Find C of Cauchy's Mean Value theorem for f (x) = sinx, g(x) = cos x in a, b.

UNIT - III

6. (a) Trace the curve y² (a - x) = x² (a + x).
   (b) Find the surface area of the solid of revolution of one loop of the curve r² = a² cos 2? about the initial line.

   OR

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7. (a) Evaluate ?₀^a ?₀^va2-x2 dx dy / (1+x²+y²).
   (b) By changing the order of integration, evaluate ?₀^4a ?_x²/4a^2vax dy dx.

UNIT - IV

8. (a) Find the Laplace transform of: (i) {sin³t. cost}. (ii) {t² sin 2t}.
   

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   (b) Apply Convolution theorem to find L^-1 {1/(s+1)(s²+1)}.

   OR

9. (a) Find the Laplace transform of e^-3t (2 cos 5t – 3 sin 5t).
   (b) Solve the D.E y" – 2y '- 8y = 0, y(0) = 3, y '(0) = 6. Using Laplace transform, bounded by x = 0, x = 1; y = 0, y = 1; z = 0, z = 1.

UNIT - V

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10. (a) Find the directional derivative of xyz² + xz at (1, 1, 1) in the direction of i+2j+3k.
    (b) Verify Green's theorem for ?_C [(xy + y²)dx + x² dy] where C is bounded by y = x and y = x².

    OR

11. Verify Gauss divergence theorem for F = 4xyi + 4zk where S is the surface of the cube bounded by x = 0, x = 1; y = 0, y = 1; z = 0, z = 1.

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