Download JNTUA B.Tech 1-1 R13 2014 June Regular 13A54101 Mathematics I Question Paper

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Code: 13A54101

Time: 3 hours

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B.Tech I Year (R13) Regular Examinations June/July 2014

MATHEMATICS - I

(Common to all branches)

Part - A

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(Compulsory Question)

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

1. Solve d²y/dx² + 1.5 dy/dx + 0.5y = 0.
2. Solve (e^x + 1) cos xdx + e^x sin xdy = 0.
3. Find Taylor's series expansion for tan^-1 x about (1, 1).

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4. Find the radius of the curvature at the origin for the curve 2x + 3y² + 4x²y + xy – y² + 2x = 0.
5. Find the asymptote of y = (x² + 2x - 1) / x
6. Evaluate ?₀¹ e^{x + y} dydx.
7. Find L{Cos²t}.
8. Find L^-1{e^-3s/s}.

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9. Show that ?. (rⁿr) = (n + 3)rⁿ.
10. State Stokes theorem.

Part - B

Answer all five units (5 X 10 = 50 M)

UNIT - I

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A mass m suspended from one end of a spring is subjected to a periodic force f = f₀ sinat in the direction of its length. The force f is measured positive vertically downwards and at time t = 0, m is at rest. If the spring constant is K, prove that the displacement of m at time t is given by

x = f₀ / m(p²-a²) (sinat-sinpt) where p² = k/m. Neglect the damping effects.

OR

Solve (x²D² + xD + 1)y = logxsin(logx).

UNIT - II

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Discuss the maxima and minima of sinx siny sin(x + y).

OR

Prove that the evolute of the cycloid x = a(t - sint), y = a(1 - cost) is another cycloid.

UNIT - III

Find the length of the arc of the parabola y² = 4ax cut off by the straight line y = x.

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OR

Evaluate ?₀¹ ?₀¹?₀¹ logz dz dx dy.

UNIT - IV

Using convolution theorem solve the IVP: y"(t) + 3y'(t) + 2y(t) = e^-t, y(0) = 0, y'(0) = -1

OR

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Find L^-1{s/(s²+a²)²-a²b²}.

UNIT - V

For a solenoidal vector f, prove that ?x(?x(?x(?xf))) = ?⁴f.

OR

Evaluate ?_c [(2xy³ – y²cosx)dx + (1 – 2ysinx + 3x²y²)dy] where C is the arc of the parabola 2x = py² from (0, 0) to (p, 1).

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