Let f be differentiable for all x. If f(1) = - 2 and f′(x) ≥

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

41.

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

  • Interval Function
    (-∞, ∞) x3 – 3x2 + 3x + 3
  • Interval Function
    [2, ∞) 2x3 – 3x2 – 12x + 6
  • Interval Function
    (-∞, 1/3] 3x2 – 2x + 1
  • Interval Function
    (-∞, 1/3] 3x2 – 2x + 1
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42.

Suppose f(x) is differentiable x = 1 and limit as straight h rightwards arrow 0 of 1 over straight h space straight f left parenthesis 1 plus straight h right parenthesis space equals space 5 space comma then space straight f apostrophe left parenthesis 1 right parenthesis space equals

  • 3

  • 4

  • 5

  • 5

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43.

Let f be differentiable for all x. If f(1) = - 2 and f′(x) ≥ 2 for x ∈ [1, 6] , then

  • f(6) ≥ 8

  • f(6) < 8

  • f(6) < 5

  • f(6) < 5


A.

f(6) ≥ 8

As f(1) = - 2 & f′(x) ≥ 2 ∀ x ∈ [1, 6]
Applying Lagrange’s mean value theorem

fraction numerator straight f left parenthesis 6 right parenthesis space minus straight f left parenthesis 1 right parenthesis over denominator 5 end fraction space equals space straight f apostrophe space left parenthesis straight c right parenthesis space greater or equal than 2
rightwards double arrow space straight f left parenthesis 6 right parenthesis space greater or equal than space 10 plus space straight f left parenthesis 1 right parenthesis
rightwards double arrow space straight f left parenthesis 6 right parenthesis space greater or equal than space 10 minus 2
rightwards double arrow space straight f left parenthesis 6 right parenthesis space greater or equal than space 8

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44.

If f is a real-valued differentiable function satisfying |f(x) – f(y)| ≤ (x – y)2 , x, y ∈ R and f(0) = 0, then f(1) equals

  • -1

  • 0

  • 2

  • 2

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45.

If cos-1 x - cos-1 y/2 = α, then 4x2 − 4xy cos α + y2 is equal to

  • 2 sin 2α

  • 4

  • 4 sin2 α

  • 4 sin2 α

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46.

If straight x dy over dx space equals space straight y space left parenthesis log space straight y space minus space log space straight x space plus 1 right parenthesis commathen the solution of the equation is

  • straight y space log space open parentheses straight x over straight y close parentheses space equals cx
  • space straight x space log space open parentheses straight y over straight x close parentheses space equals space cy
  • log space open parentheses straight y over straight x close parentheses space equals space cx
  • log space open parentheses straight y over straight x close parentheses space equals space cx
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47.

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of 50 cm3 /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is

  • fraction numerator 1 over denominator 36 space straight pi end fraction space cm divided by min
  • fraction numerator 1 over denominator 18 space straight pi end fraction space cm divided by min
  • fraction numerator 1 over denominator 54 space straight pi end fraction space cm divided by min
  • fraction numerator 1 over denominator 54 space straight pi end fraction space cm divided by min
161 Views

48.

A particle is projected from a point O with velocity u at an angle of 60o with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity then is given by

  • u/3

  • u/2

  • 2u/3

  • 2u/3

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49.

If a1, a2, a3,…, an,… are in G.P., then the determinant

space increment space equals space open vertical bar table row cell log space straight a subscript straight n end cell cell log space straight a subscript straight n plus 1 end subscript end cell cell log space straight a subscript straight n plus 2 end subscript end cell row cell log space straight a subscript straight n plus 3 end subscript end cell cell log space straight a subscript straight n plus 4 end subscript end cell cell log space straight a subscript straight n plus 5 end subscript end cell row cell log space straight a subscript straight n plus 6 end subscript end cell cell log space straight a subscript straight n plus 7 end subscript end cell cell log space straight a subscript straight n plus 8 end subscript end cell end table close vertical bar space is space equal space to

  • 1

  • 0

  • 4

  • 4

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50.

A real valued function f(x) satisfies the functional equation f(x – y) = f(x) f(y) – f(a – x) f(a + y) where a is a given constant and f(0) = 1, f(2a – x) is equal to

  • –f(x)

  • f(x)

  • f(a) + f(a – x)

  • f(a) + f(a – x)

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