Let f : R → R be a function defined by f(x) = Min {x + 1, |x|

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 Multiple Choice QuestionsMultiple Choice Questions

151.

Let f : N → Y be a function defined as f (x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}.Show that f is invertible and its inverse is 

  • straight g space left parenthesis straight y right parenthesis space equals space fraction numerator 3 straight y space plus space 4 over denominator 3 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals space 4 plus fraction numerator straight y space plus space 3 over denominator 4 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals fraction numerator straight y space plus space 3 over denominator 4 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals fraction numerator straight y space plus space 3 over denominator 4 end fraction
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152.

Let R be the real line. Consider the following subsets of the plane R × R.
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x − y is an integer}. Which one of the following is true?

  • neither S nor T is an equivalence relation on R

  • both S and T are equivalence relations on R

  • S is an equivalence relation on R but T is not 

  • S is an equivalence relation on R but T is not 

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153.

Let f(x) = open curly brackets table attributes columnalign left end attributes row cell left parenthesis straight x minus 1 right parenthesis space sin space open parentheses fraction numerator 1 over denominator straight x minus 1 end fraction close parentheses end cell row cell 0 comma space space space space space space space space space space space space space space space space space space space space space space space space space space space if space straight x space equals 1 space space space space space space space space space space end cell end table close comma space if space straight x space not equal to space 1Then which one of the following is true?

  • f is neither differentiable at x = 0 nor at x = 1

  • f is differentiable at x = 0 and at x = 1

  • f is differentiable at x = 0 but not at x = 1 

  • f is differentiable at x = 0 but not at x = 1 

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154.

The largest interval lying in  open parentheses negative straight pi over 2 comma straight pi over 2 close parentheses spacefor which the function open square brackets straight f left parenthesis straight x right parenthesis space equals 4 to the power of negative straight x squared end exponent space plus space cos to the power of negative 1 end exponent space open parentheses straight x over 2 minus 1 close parentheses plus space log space left parenthesis cos space straight x right parenthesis close square brackets is defined, is

  • [0, π]

  • open parentheses negative straight pi over 2 comma straight pi over 2 close parentheses
  • [-π/4, π/2)

  • [-π/4, π/2)

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155.

Let f : R → R be a function defined by f(x) = Min {x + 1, |x| + 1}. Then which of the following is true ?

  • f(x) ≥ 1 for all x ∈ R

  • f(x) is not differentiable at x = 1

  • f(x) is differentiable everywhere

  • f(x) is differentiable everywhere


C.

f(x) is differentiable everywhere

f(x) = min{x + 1, |x| + 1}
f(x) = x + 1 ∀ x ∈ R.

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156.

The function f: R ~ {0} → R given by
straight f left parenthesis straight x right parenthesis space equals space 1 over straight x minus fraction numerator 2 over denominator straight e to the power of 2 straight x end exponent minus 1 end fraction
can be made continuous at x = 0 by defining f(0) as

  • 2

  • -1

  • 1

  • 1

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157.

The number of values of x in the interval [0, 3π] satisfying the equation 2sin2 x + 5sinx − 3 = 0 is

  • 4

  • 6

  • 1

  • 1

111 Views

158.

The set of points where x f(x) = x /1+|x| is differentiable is

  • (−∞, 0) ∪ (0, ∞)

  • (−∞, −1) ∪ (−1, ∞)

  • (−∞, ∞)

  • (−∞, ∞)

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159.

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} be a relation on the set A = {3, 6, 9, 12}. The relation is

  • reflexive and transitive only

  • reflexive only

  • an equivalence relation

  • an equivalence relation

172 Views

160.

Let f : (-1, 1) → B, be a function defined by straight f left parenthesis straight x right parenthesis space equals space tan to the power of negative 1 end exponent space fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction comma space then f is both one-one and onto when B is the interval

  • open parentheses 0 comma space straight pi over 2 close parentheses
  • [0, π/2)

  • open parentheses negative straight pi over 2 comma straight pi over 2 close parentheses
  • open parentheses negative straight pi over 2 comma straight pi over 2 close parentheses
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