A point on curve xy2 = 1 which is at minimum distance from the or

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 Multiple Choice QuestionsMultiple Choice Questions

61.

The function f (x) = x2 e- 2x, x > 0. Then the maximum value of f (x) is :

  • 1e

  • 12e

  • 1e2

  • 4e4


62.

The angle between the tangents at those points on the curve x = t2 + 1 and y = t2 - t - 6 where it meets x-axis is :

  • ±tan-1429

  • ±tan-1549

  • ±tan-11049

  • ±tan-1829


63.

If θ is semi vertical angle of a cone of maximum volume and given slant height, then tan(θ) is equal to

  • 2

  • 1

  • 2

  • 3


64.

A man of 2 m height walks at a uniform speed of 6 km/h away from a lamp post of 6 m height. The rate at which the length of his shadow increase, is

  • 2 km/h

  • 1 km/ h

  • 3 km/h

  • 6 km/h


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65.

If y = 4x-5 is a tangent to the curve y2 = px3 + q at (2, 3), then

  • p = 2, q = - 7

  • p = - 2, q = 7

  • p = - 2, q = - 7

  • p = 2, q = 7


66.

A missile is fired from the fround level rises x metre vertically upwards in t second, where x = 100t - 252t2. The maximum height recahed is

  • 200 m

  • 125 m

  • 160 m

  • 190 m


67.

If the curves x2 = 9A(9 - y) and x2 = A(y + 1) intersect orthogonally, then the value of A is

  • 3

  • 4

  • 5

  • 7


68.

If f (x) = 3x4 + 4x3 - 12x2 + 12, then f(x) is

  • increasing in (- , - 2) and in (0, 1)

  • increasing in (- 2, 0) and in (1, )

  • decreasing in (- 2, 0) and in (0, 1)

  • decreasing in (- , - 2) and in (1, )


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69.

Gas is being pumped into a spherical balloon at the rate of 30 ft3/min. Then, the rate at which the radius increases when it reaches the value 15 ft is

  • 115π ft/min

  • 130π ft/min

  • 120 ft/min

  • 125 ft/min


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70.

A point on curve xy2 = 1 which is at minimum distance from the origin is

  • (1, 1)

  • (1/4, 2)

  • (21/6, 2- 1/3)

  • (2- 1/3, 21/6)


D.

(2- 1/3, 21/6)

Let (t2, 1/t) be a point on the curve. If its distance from origin is λ, then

λ2 = t4 + 1t2

     ddtλ2 = 4t3 - 2t3    d2dt2λ2 = 12t2 + 6t4 > 0Now, ddtλ2 = 0  t = 2- 16Hence, required point is 2- 1/3, 21/6


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