Relations and Functions

1

Let f(x) = min{2x2, 52 − 5x}, where x is any positive real number. Then the maximum possible value of f(x) is

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2

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

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3

The domain of sin-1[log3(x/3)] is

  1. [-9, -1]
  2. [-9, 1]
  3. [-1,9]
  4. [1, 9]
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4

Let W denote the words in the English dictionary. Define the relation R by:

R = {(x, y) ε W × W | the words x and y have at least one letter in common}. Then R is

  1. reflexive, symmetric and not transitive
  2. reflexive, symmetric and transitive
  3. not reflexive, symmetric and transitive
  4. reflexive, not symmetric and transitive
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5

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

  1. (-∞,1/3] ; 3x2 - 2x + 1
  2. [2,∞) ; 2x3 - 3x2 -12x + 6
  3. (-∞,∞) ; x3 - 3x2 + 3x + 3
  4. (-∞,-4] ; x3 + 6x2 + 6
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6

The domain of the function f(x) = 1/√(|x| - x) is

  1. (0,∞)
  2. (-∞,0)
  3. (-∞,∞)
  4. (-∞,∞) - {0}
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7

The function f(x) = log (x + √(x2 +1)) is

  1. an even function
  2. an odd function
  3. neither an even nor an odd function
  4. a periodic function
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8

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

  1. not symmetric
  2. transitive
  3. reflexive
  4. a function
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9

If a ϵ R and the equation -3(x - [x])2 + 2(x - [x]) + a2 = 0 (where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval

  1. (-1, 0) U (0, 1)
  2. (-2, -1)
  3. (1, 2)
  4. (-∞, -2) U (2, ∞)
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10

For real x, let f(x) = x3 + 5x + 1, then

  1. f is one-one and onto R
  2. f is onto R but not one-one
  3. f is neither one-one nor onto R
  4. f is one-one but not onto R
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11

A function f from the set of natural numbers to integers defined by

f(n) = (n-1)/2, when n is odd

f(n) = -n/2, when n is even

  1. one-­one and onto both
  2. one-­one and but not onto
  3. neither one-­one nor onto
  4. onto but not one-­one
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