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Question #41042

Difficulty: 8/10 | 🟢 Free

The Question

A real number $x$ is chosen uniformly at random from the interval $[-1, 5]$ . Let $f(x) = |x^2 - 4x + 3|$ . Let $E_1$ be the event that $f(x) \ge 3$ and $E_2$ be the event that $f(x) \le \frac{24}{25}$ . Find the probability of the event $E_1' \cap E_2'$ .

Options

A.
$\frac{4}{15}$
B.
$\frac{1}{5}$
C.
$\frac{23}{30}$
D.
$\frac{11}{15}$

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

De Morgan’s Laws in Probability|Bijectivity and Inverse|Function Definition (The 'One-to-One' Constraint)

Why This Question Is Tricky

Common Mistakes

This question tests the application of De Morgan's Laws in continuous probability spaces, combined with the critical trap of non-injective function preimages.

Students often incorrectly assume bijectivity or forget the absolute value's negative branch when finding the preimage of a range.

If a student ignores the absolute value, they find $E_2$ as $x \in [0.6, 3.4]$ (length 2.8), leading to the distractor $1/5$ .

If they only take the positive square root branch of the inverse, they get length 1.4, leading to $23/30$ .

The correct approach requires carefully partitioning the domain to account for the 4-to-1 mapping in the inner region.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free