Download JNTUA B.Tech 1-1 R13 2014 Dec Supply 13A54101 Mathematics I Question Paper

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Code: 13A54101

R13

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B.Tech I Year (R13) Supplementary Examinations December/January 2014/2015

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 (D³ + 1)y = 0.
2. (b) Solve dy/dx = (x + y + 2)² = 0.

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3. (c) Expand e^x+y in a neighborhood of (1, 1).
4. (d) Find the envelope of the family of curves y = mx + m² for different values of 'm'.
5. (e) Find the asymptotes of y³ – x²y + 2y² + 4y + x.
6. (f) Find the quadrature of the rectangular hyperbola y = k²/x from x = a to x = b.
7. (g) L{e^at cosh bt} =

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8. (h) L^-1{s/(s²+a²)} =
9. (i) Prove that a . (?r) = a.r/r³, a is a constant vector.
10. (j) State Green's theorem.

PART - B

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

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UNIT - I

2. The deflection of a strut of length ℓ with one end built - in and the other end subjected to the end thrust P, satisfies d²y/dx² + a²y = P/a²R (ℓ-x). Find the deflection y of the strut at a distance x from the built - in end.

OR

3. Verify Maclaurin's theorem for f(x) = (1-x)^5/2 with Lagrange form of remainder up to 3 terms with x = 1.

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UNIT - II

4. Solve (D² - 4D)y = e^x + sin3x cos2x.

OR

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5. Find the radius of curvature at any point P(at², 2at) on the parabola y² = 4ax. Show that it is (S)^3/2/va Where S is the focus of the parabola?

UNIT - III

6. Find the volume of the solid generated by revolution of the loop of the curve y² (a-x) = x² (a + x) about the axis.

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OR

7. Evaluate the integral ?_y=0¹ ?_x=y^vy xdxdy/(x²+y²).

UNIT - IV

8. Find the Laplace transform for f(t) = (vt - 1/vt)³.

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OR

9. The triangular wave function defined by f(t) = {t, 0<t<a; (2a - t, a< t <2a} and f(t + 2a) = f(t). Find Laplace transform of f(t).

UNIT - V

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10. Find the directional derivative of Ø(x, y, z) = xy + yz + zx in the direction of -2i + j + 2 k at the point (1,2,0).

OR

11. If F = 2xzi - xj + y²k evaluate ? F dv where V is the region bounded by the surface x = 0, y = 0, x = 2, y = 6, z = x², z = 4.

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