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

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

The values of x, y and z for the system of equations x + 2y + 3z = 6, 3x - 2y + z = 2 and 4x + 2y + z = 7 are respectively

  • 1, 1, 1

  • 1, 2, 3

  • 1, 3, 2

  • 2, 3, 1


2.

If the determinant  = 3- 2sin3θ- 78cos2θ- 11142 = 0, then the value of sinθ is

  • 13or 1

  • 12 or 32

  • 0 or 12

  • None of these


3.

The relation R in R defined by R = {(a, b): a  b3), is

  • reflexive

  • symmetric

  • transitive

  • None of these


4.

The value of 2tan-1csctan-1x - tancot-1x is

  • tan-1x

  • tan(x)

  • cot(x)

  • csc-1x


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5.

Let f (x + y) = f(x) + f(y) for all x and y. If the function f(x) is continuous at x = 0, then f(x) is continuous

  • only at x = 0

  • at x  R - 0

  • for all x

  • None of these


6.

Let fx = x2sin1x, x  00,             x = 0. Then, f(x) is continuous but not differentiable at x = 0, if

  • n  0, 1

  • n  [1, )

  • n  - , 0

  • n = 0


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7.

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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8.

If in a ABCsin3A + sin3B + sin3C = 3sinAsinBsinC, then the value of determinant abcbcacab is equal to

  • 0

  • 1

  • 2

  • 3


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9.

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


10.

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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