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Question #41051

Difficulty: 8/10 | 🟢 Free

The Question

A real number $x$ is chosen uniformly at random from the interval $[0, 3]$ . Let $E_1$ be the event that $[x^2] - 3[x] + 2 \le 0$ , and $E_2$ be the event that $x^2 - 3x + 2 \le 0$ , where $[\cdot]$ denotes the greatest integer function. The probability of the event $E_1 \cup E_2$ is:

Options

A.
$\frac{\sqrt{5} - 1}{3}$
B.
$\frac{2}{3}$
C.
$\frac{\sqrt{5} + \sqrt{2} - 2}{3}$
D.
$\frac{1}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Algebra of Events|Sample Space Partitioning|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly partition a continuous sample space when events are defined by step functions with different arguments (here, $[x]$ and $[x^2]$ ).

The 'Lethal Collision' occurs when students incorrectly assume $[x^2] = [x]^2$ , leading to a false continuous interval $E_1 = [1, 3)$ and an incorrect probability of $2/3$ .

Alternatively, students might treat the discrete step function as continuous, ignoring the GIF entirely, yielding $1/3$ .

The rigorous solution demands partitioning the sample space at integer boundaries of $x$ , evaluating the piecewise conditions for $[x^2]$ , and correctly applying the algebra of events to find the measure of the union without double-counting the intersection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Algebra of Events
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free