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Question #41036

Difficulty: 8/10 | 🟢 Free

The Question

A real number $x$ is chosen uniformly at random from the interval $[-4, 4]$ . Let $A$ be the event that $x$ satisfies the inequality $\frac{x^3 - 2x^2 - 4x + 8}{x+1} \ge 0$ . Let $B$ be the event that $\lim_{n \to \infty} \left( \frac{n^2 + 3xn + 1}{n^2 + xn + 1} \right)^n \ge e^2$ . Find the conditional probability $P(B \mid A)$ .

Options

A.
$\frac{3}{7}$
B.
$\frac{2}{3}$
C.
$\frac{3}{8}$
D.
$\frac{1}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Conditional Probability Foundations|The 1^∞ Form Shortcut|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of $1^\infty$ limit forms with continuous probability spaces defined by rational inequalities.

The primary trap lies in the Wavy Curve method (Method of Intervals).

By presenting the numerator as a cubic polynomial $x^3 - 2x^2 - 4x + 8$ , students must factor it to reveal a repeated root $(x-2)^2(x+2)$ .

A common mistake is failing to recognize the even multiplicity, leading students to incorrectly alternate the sign across all critical points (yielding an incorrect interval of $[-2, -1) \cup [2, 4]$ ).

This yields an erroneous sample space length of 3 for Event A, directly corrupting the denominator of the conditional probability $P(B|A)$ and leading to the trap answer 2/3.

Quick Facts About This Question

Question Information

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