For all real values of a0, a1, a2, a3 satisfying a0 +&n

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 Multiple Choice QuestionsMultiple Choice Questions

31.

If f(x) is an odd differentiable function defined on - ,  such that f'(3) = 2, then f'(- 3) is equal to

  • 0

  • 1

  • 2

  • 4


32.

If f(x) = tan-1logex2logex2 + tan-13 + 2logx1 - 6logx

  • x2

  • x

  • 1

  • 0


33.

The number of points at which the function f(x) = maxa - x, a + x, b, -  < x < , 0 < a < b cannot be differentiable, is

  • 0

  • 1

  • 2

  • 3


34.

If f(x) is a function such that f'(x) = (x - 1)2(4 - x), then

  • f(0) = 0

  • f(x) is increasing in (0, 3)

  • x = 4 is a critical point of f(x)

  • f(x) is decreasing in (3, 5)


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35.

Let f: R  R be a continuous function which satisfies f(x) = 0xf(t)dt. Then, the value of f(loge5) is

  • 0

  • 2

  • 5

  • 3


36.

Let f: [2, 2]  R  be a continuous function such that f(x) assumes only irrational values. If f(2) = 2, then

  • f(0) = 0

  • f(2 - 1) = 2 - 1

  • f(2 - 1) = 2 + 1

  • f(2 - 1) = 2


37.

Let [x] denotes the greatest integer less than or equal to x. Then, the value of α f · which the function

f(x) = sin- x2- x2, x  0α,              x = 0 is continuous at x = 0, is

  • α = 0

  • α = sin- 1

  • α = sin1

  • α = 1


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38.

For all real values of a0, a1, a2, a3 satisfying a0 + a12 + a23 + a34 = 0, the equation a0 + a1x + a2x + a3x3 = 0 has a real root in the interval

  • [0, 1]

  • [- 1, 0]

  • [1, 2]

  • [- 2, - 1]


A.

[0, 1]

Let f(x) = a3x44 + a2x33 + a1x22 + a0x f(0) = 0, f(1) = a34 + a23 + a12 + a0 = 0 f(0) = f(1) f'(x) = 0  has atleast one real root in [0, 1]   according to Rolle's theorem f'(x) = a3x3 + a2x2 + a1x + a0Hence, a3x3 + a2x2 + a1x + a0 must has a real root in the interval [0, 1].


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39.

Let f:R  R  be defined asf(x) = 0,              x is irrationalsinx,      x is rationalThen, which of the following is true?

  • f is discontinuous for all x

  • f is continuous for all x

  • f is discontinuous at x = kπ, where k is an integer

  • f is continuous at x = kπ, where k is an integer


40.

If limx0axex - blog1 + xx2 = 3, then the value of a and b are, respectively

  • 2, 2

  • 1, 2

  • 2, 1

  • 2, 0


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