CAT Practice : Functions

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The greatest integer function is one of the most interesting around. Input can be anything, output is discrete - this function is a logician's delight. You can create gazillion questions; none of which can be solved by plugging in formulae.

Greatest Integer Function

    Q.5: [x] = greatest integer less than or equal to x. If x lies between 3 and 5,5 inclusive, what is the probability that [x2] = [x]2?
    1. Roughly 0.64
    2. Roughly 0.5
    3. Roughly 0.14
    4. Roughly 0.36

 

  • Correct Answer
    Choice C. Roughly 0.14

Explanatory Answer

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Detailed Solution

Let us take a few examples.
[32] = [3]2
[3.52] = 12 [3.5]2 = 9
[42] = 16 [4]2 = 16
For x ∈ (3, 5). [x]2 can only take value 9, 16 and 25.

Let us see when [x2] will be 9, 16 or 25.
If [x2] = 9,
x2 ∈ [9, 10)
=> x ∈ [3, )

[x2] = 16
x2 ∈ [16, 17)
=> x ∈ [4, )

In the given range [x2] = 25 only when x = 5
So [x2] = [x]2 when x ∈ [3, ] or [4, ) or 5.
Probability =
= 0.14
Answer choice (c)


Correct Answer: Roughly 0.14



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