The sum of two positive numbers is given. If the sum of their cub

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 Multiple Choice QuestionsMultiple Choice Questions

251.

P is the point of contact of the tangent form the origin to the curve y = loge(x). The length of the perpendicular drawn form the origin to the normal at P is

  • 12e

  • 1e

  • 2e2 + 1

  • e2 +1


252.

For the curve 4x5 = 5y4, the ratio of the cube of the subtangent at a point on the curve the square of the subnormal at the same point is

  • 445

  • 544

  • 4454

  • 544


253.

The set of real values of x for which f(x) = xlogx is increasing is

  • x :x  e

  • empty

  • {x : x < e}

  • {1}


254.

A sphere increases its volume at the rate of π cc/s. The rate at which its surface area increases when the radius is 1 cm is

  • 2π sq cm/s

  • π sq cm/s

  • 3π2 sq cm/s

  • π2 sq cm/s


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255.

If a ball is thrown vertically upwards and the height 's' reached in time 't' is given by s = 22t - 11t2, then the total distance travelled by the ball is

  • 44 units

  • 33 units

  • 11 units

  • 22 units


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256.

The sum of two positive numbers is given. If the sum of their cubes is minimum, then

  • they are equal

  • one is twice the other

  • they are unequal

  • one is thrice the other


A.

they are equal

Let x and y be two positive numbers According to questionx + y = a   y = a - xLet  z = x3 + y3       z = x3 + a - x3       ...iOn dtlferentiatmg Eq. (i) w.r.t. 'x', we getdzdx = 3x2 +3a - x2- 1      = 3x2 - a - x2      = 3x +a - xx - a + x      = 3a2x - a             ...iiFor maxunum or minimum, dzdx = 0 3a2x - a = 0       3a  0       2x - a = 0                x = a2On differentiating Eq. (ii) w.r.t 'x', we get    d2zdx2 = 6a d2zdx2 > 0 z has a minimum value at x = a2Therefore, the numbers are a2 and a - a2 = a2.


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257.

The perimeter of a sectoris a constant. Ifits area is to be maximum, then the sectorial angle is

  • πc6

  • πc4

  • 4c

  • 2c


258.

The tangent to the curve xy = 25 at any point on it cuts the coordinate axes at A and B, then the area of the OAB is

  • 50 sq units

  • 25 sq units

  • 75 sq units

  • 100 sq units


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259.

The length of the subtangent, ordinate and the subnormal are in

  • AP

  • HP

  • GP

  • Arithmetico-geometric progression


260.

The maximum value of xe-x is

  • e

  • 1e

  • - e

  • 1e


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