### Probability Question

**Question:**

[x]
= greatest integer less than equal to x. If x lies between 3 and 5. What is the
probability than [x

^{2}] = [x]^{2}?
A. Roughly
0.64

B. Roughly
0.5

C. Roughly
0.14

D. Roughly
0.36

**Correct Answer:**

**(C)**

**Explanation:**

Let
us take a few examples.

[3

^{2}] = [3]^{2}
[3.5

^{2}] =12 [3.5]^{2}= 9
[4

^{2}] = 16 [4]^{2}= 16
For
x Î (3, 5). [x]

^{2}can only take value 9, 16 and 25.
Let
us see when [x

^{2}] will be 9, 16 or 25
If
[x

^{2}] = 9, x^{2}[9, 10)
=> x[3, )

[x

^{2}] = 16 => x^{2}[16, 17)
=> x [4, )

In
the given range [x

^{2}] = 25 only when x = 5
So
[x

^{2}] = [x]^{2}when x [3, ] or [4, ) or 5.
Probability =

= = 0.14

**Difficulty Level 2**

Labels: 2iim Chennai, CAT 2013 questions, CAT 2013 Solutions, CAT Probability, CAT questions

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