If the volume of spherical ball is increasing at the rate of 4&pi

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 Multiple Choice QuestionsMultiple Choice Questions

181.

The maximum value of fx = logxxx  0, x  1 is

  • e

  • 1e

  • e2

  • 1e2


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182.

If the volume of spherical ball is increasing at the rate of 4π cm3/s, then the rate of change of its surface area when the volume is 288 π cm3, is

  • 43π cm2/s

  • 23π cm2/s

  • 4π cm2/s

  • 2π cm2/s


A.

43π cm2/s

Let V and r be the volume and radius of spherical ball, respectively.

Volume of spherical ball = 43πr3

       V = 43πr3     ...i 288π = 43πr3   given, V = 288 cm3    288 = 43r3       r3 = 72 × 3 = 8 × 27       r3 = 23 × 33             taking cube roots both sides        r = 6

On differentiating Eq. (i) w.r.t. 't', we get

     dVdt = 4πr2drdt  4π = 4πr2drdt       given dVdt = 4π cubic cm/s    1 = 62drdt      r = 6 drdt = 136Now, surface area of spherical ball, (s) = 4πr2    s = 4πr2

On differentiating both sides, w.r.t. 't', we get

    dsdt = 4 × 2πrdrdt           = 8 × π × 6 × 136           r = 6 and drdt = 136 dsdt = 43π cm2/s


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183.

The equation of displacement of a particle is x(t) = 5t2 - 7t + 3. The acceleration at the moment when its velocity becomes 5 m/sec is

  • 3 m/sec2

  • 7 m/sec2

  • 10 m/sec2

  • 8 m/sec2


184.

The mean value of the function fx = 2ex + 1 on the interval [0, 2] is

  • 2 - loge2e2 + 1

  • 2 + loge2e2 + 1

  • 2 + loge2e2 - 1

  • - 2 + loge2e2 - 1


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185.

The function y = 2x - x2

  • increases in (0, 1) but decreases in (1, 2)

  • decreases in (0, 2)

  • increases m (1, 2) but decreases in (0, 1)

  • increases in (0, 2)


186.

The interval in which the function y = x - 2sinx0  x  2π increases throughout is

  • 5π3, 2π

  • 0, π3

  • π3, 5π3

  • 0, π4


187.

The points of the curve y = x3 + x - 2 at which its tangent are parallel to the straight line y = 4x - 1 are

  • (2, 7), (- 2, - 11)

  • (0, 2), (21/3, 21/3)

  • (- 21/3, - 21/3), (0, - 4)

  • (1, 0), (- 1, - 4)


188.

The equation of the normal to the curve y = - x + 2 at the point of its intersection with the bisector of the first quadrant is

  • 4x - y + 16 = 0

  • 4x - y = 16

  • 2x - y - 1 = 0

  • 2x - y + 1 = 0


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189.

The angle at which the curve y = x2 and the curve x = 53cost, y = 54sint intersect is

  • tan-1241

  • tan-1412

  • - tan-1241

  • 2tan-1412


190.

The maximum value of the function y = 2tanx - tan2x over 0, π2 is

  • 1

  • 3

  • 2


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