CAT Quantitative Aptitude Questions | CAT Algebra - Functions questions

The question is about Greatest Integer Function. A question which combines the function with the probability. Greatest integer function with square in it.. Functions is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept of Functions questions in the CAT Exam. Make sure you practice a lot of questions in CAT Functions Algebra to push your CAT preparation in the right direction. Functions as an idea is a fabulous idea if one can wrap their head around it the right way, and it has been one of the staple ideas tested by the CAT exam historically.

Question 4: [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

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Functions: 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.

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, √10)

[x²] = 16
x² ∈ [16, 17)
=> x ∈ [4, √17)
In the given range [x²] = 25 only when x = 5
So [x²] = [x]² when x ∈ [3, √10] or [4, √10) or 5.
Probability = \\frac{10 - 3 + √17 - 4}{2}\\)
\\frac {3.16 - 3 + 4.12 - 4}{2}\\) = \\frac{0.28}{2}\\) = 0.14

The question is "If x lies between 3 and 5, 5 inclusive, what is the probability that [x²] = [x]²?"

Hence the answer is "Roughly 0.14"

Choice C is the correct answer.