The altitude of the right circular cone of maximum volume that ca

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 Multiple Choice QuestionsMultiple Choice Questions

191.

The values of a and b for which the function y = aloge(x ) + bx2 + x, has extremum at the points x1 = 1 and x2 = 2 are

  • a = 23, b = - 16

  • a = - 23, b = - 16

  • a = - 23, b = 16

  • a = - 13, b = - 16


192.

The value of maxima of 1xx is

  • 1ee

  • ee

  • e

  • e1/e


193.

A point particle moves along a straight line such that x = t, where t is time. Then, ratio of acceleration to cube of the velocity is

  • - 1

  • - 0.5

  • - 3

  • - 2


194.

The tangents to curve y = x3 - 2x2 + x - 2 which are parallel to straight line y = x, are

  • x + y = 2 and x - y = 8627

  • x - y = 2 and x - y = 8627

  • x - y = 2 and x + y = 8627

  • x + y = 2 and x + y = 8627


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195.

If two sides of a triangle are given, then the area of the triangle will be maximum, if. the angle between the given sides is

  • π3

  • π4

  • π6

  • π2


196.

If f(x) = 803x4 + 8x3 - 18x2 + 60, then the points of local maxima for the function f(x) are

  • 1, 3

  • - 3, 1

  • - 1, 3

  • - 1, - 3


197.

The adjacent sides of a rectangle with given parameter as 200 cm and enclosing minimum area are

  • 20 cm and 80 cm

  •  50 cm and 50 cm40 cm and 60 cm

  • 50 cm and 50 cm

  • 30 cm and 70 cm


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198.

The altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is

  • r2

  • r3

  • 3r4

  • 4r3


D.

4r3

Let R be the radius and h be the height of cone. OA = h - rIn OAB,       r2 = R2 + h - r2  r2 = R2 + h2 + r2 - 2rh R2 = 2rh - h2The volume V of the cone is given by       V = 13πR2h          = 13πh2rh - h2 = 13π2rh2 - h3On differentiating w.r.t. h, we get   dVdh = 13π4rh - 3h2For maximum and minimum, put dVdh = 0 4rh = 3h2   4r = 3h

          h = 4r3Now, d2Vdh2 = 13π4r - 6hAt           h = 4r3d2Vdh2h = 4r3 = 13π4r - 6 × 4r3                      = π34r - 8r                      = - 43 < 0 V is maximum when h = 4r3.

Hence, volume of the cone is maximum when h = 4r3, which is the attitude of cone.


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199.

Let f(x) = x(x - 1)2, the point at which f(x) assumes maximum and minimum are respectively

  • 13, 1

  • 1, 13

  • 3, 1

  • None of these


200.

Rectangles are inscribed ina circle of radius r. The dimensions of the rectangle which has the maximum area, are

  • r, r

  • 2r, 2r

  • 2r, 2r

  • None of the above


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