A cone of maximum volume is being cut from sphere, then ratio bet

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

Domain of y = cot-1x is

  • - , 0

  • 0, 

  • - , 

  • None of these


2.

If limx0log3 + x - log3 - xx = k, then the value of k will be

  • 0

  • - 13

  • 23

  • - 23


3.

For the function 3x4 - 8x3 + 12x2 - 48x + 25 in the interval [1, 3]. The value of maxima and minima are

  • 16, - 39

  • - 16, 39

  • 6, - 9

  • None of these


4.

f(x) = x3 - 27x + 5 is an increasing function when

  • x < - 3

  • x >3

  • x  - 3

  • x <3


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5.

In the interval 0, π2 function log(sin(x)) is

  • increasing

  • decreasing

  • neither increasing nor decreasing

  • None of the above


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6.

A cone of maximum volume is being cut from sphere, then ratio between height of cone and diameter of sphere is

  • 23

  • 13

  • 34

  • 14


A.

23

Let r be the radius ofsphere and OC = x

 Height of cone = CD                            = r +xand radius of base of cone = BC                            = OB2 - OC2                           = r2 - x2Volume of cone,V = 13πr2 - x2r + x  = 13πr3 + r2x - x2r - x3On differentiating two times w.r.t. x, we get        dVdx = 13πr2 - 2xr - 3x2and d2Vdx2 = 13π- 2r - 6xFor maxima or minima, put dVdx = 0              r2 - 2xr - 3x2 = 0 r2 - 3xr + xr - 3x2 = 0           r - 3xr + x = 0                               x = r3  r  - x

At x = r3, f''r3 < 0 f(x) is maximum at x = r3Hence, height of the cone, h = r + x    h = r + r3   h = 4r3 h2r = 23


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7.

The nth derivative of e2x + 5 is

  • 22nn! e2x + 5

  • 2n e2x + 5

  • 22n e2x + 5

  • 2nn e2x + 5


8.

If 1 - x2 + 1 - y2 = ax - y, then dydx is equal to

  • 1 - x21 - y2

  • 1 - y21 - x2

  • x2 - 11 - y2

  • y2 - 11 - x2


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9.

Function f(x) = x - 1,   x < 22x - 3, x  2 is continuous function for

  • all real values of x

  • only x = 2

  • only real values of x, when x  2

  • the integer values of x


10.

If g(x) = x2 + x - 2 and gof(x) = 2x2 - 5x+ 2, then f(x) is equal to

  • 2x - 3

  • 2x + 3

  • 2x2 + 3x + 1

  • 2x2 - 3x - 1


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