Question

[x] = greatest integer less than equal to x. If x lies between 3 and 5. What is the probability than [x²] = [x]²?

1. Roughly 0.64
2. Roughly 0.5
3. Roughly 0.14
4. Roughly 0.36

Explanatory Answer

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 ∈
[x²] = 16 => x² ∈ [16, 17)
=> x ∈
In the given range [x²] = 25 only when x = 5
So [x²] = [x]² when x ∈ or or 5.

Probability = = = 0.14

Answer Choice (C)