Download JNTUA B.Tech 1-1 R15 2017 Dec Supple 15A54101 Mathematics I Question Paper

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Code: 15A54101

R15

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B.Tech I Year I Semester (R15) Regular & Supplementary Examinations December 2017

MATHEMATICS - I

(Common to all branches)

Time: 3 hours

Max. Marks: 70

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PART - A

(Compulsory Question)

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

1. Define an ordinary differential equation with example.

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2. Find the general solution of (4D² + 4D + 1)y = 0
3. To solve the D.E (D² + a²)y = tan ax by the method of variation of parameters find 'B' when P.I = Ax + Bx.
4. Transform the Cauchy's homogeneous differential equation x² d²y/dx² + x dy/dx into a linear differential equation with constant coefficients.
5. If u = x² + y² + z², x = e^t, y = e^tsint, z = e^tcost then find du/dt
6. If x = r cos ?, y = r sin ?, then find J(x,y)/J(r,?)

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7. Evaluate ? ?(x + y)dydx.
8. Evaluate ? e^x+y+zdx dy dz.
9. If f = xy²i + 2x²yzj – 3yz²k then find divf at (1,-1,1).
10. State the Gauss divergence theorem.

PART - B

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

UNIT - I

1. (a) Solve (x² – ay)dx + (y² – ax)dy = 0.
   (b) Solve (xy sinxy + cosxy)ydx + (xysinxy + xcosxy)xdy = 0
   OR

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2. (a) A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6 a.m to 9 a.m. How many grams will be present at noon?
   (b) If a voltage of 20 Cos 5t is applied to a series circuit consisting of 10 ohm resistor and 2 Henry inductor, determine the current at any time t.

UNIT - II

4. Solve (x²D² + 3xD + 1)y = 1/(1-x)²
   OR

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5. A horizontal tie-rod is freely pinned at each end. It carries a uniform load w/b per unit length and has a horizontal pull P. Find the central deflection and the maximum bending moment taking the origin at one of its ends.

UNIT - III

6. (a) Verify Taylor's theorem for f(x) = (1 - x)^5/2 with Lagrange's form of remainder 2 terms in the interval [0,1].
   (b) If u = yz/x, v = zx/y, w = xy/z show that ?(u,v,w)/?(x,y,z) = 4.
   OR

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7. (a) Examine for minimum and maximum values of sin x + sin y + sin(x + y).
   (b) Find the radius of curvature of the curve x²y = a(x² + y²) at (-2a, 2a).

UNIT - IV

8. (a) Evaluate ? x²y²(x + y)dy dx.
   (b) Evaluate ?₀^log2 ?₀^x ?₀^x+logy e^x+y+z dz dx dy
   

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   OR
9. (a) Find the whole area of the lemniscates r² = a²cos2?.
   (b) Find the volume bounded by the xy plane, the cylinder x²+y²=1 and the plane x+y+z=3.

UNIT - V

10. (a) Find the angle between the surfaces x² + y² + z² = 9 and z = x² + y² - 3 at the point (2,-1,2).
    

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    (b) Use divergence theorem to evaluate ? F.ds where F = 4xi – 2y²j + z²k and S is the surface bounded by the region x² + y² = 4, z = 0 and z = 3.
    OR
11. Verify stokes theorem for F = (x² + y²)i – 2xyj taken round the rectangle bounded by the lines x = ± a, y=0, y=b.

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