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Question #41706

Difficulty: 8/10 | 🟢 Free

The Question

Let $G(x)$ be a generating function whose first derivative is given by $G'(x) = x^4 - 12x^3 + 46x^2 - 60x$ . A random variable $X$ follows a binomial distribution $B(n, p)$ with $n=9$ and $p=\frac{1}{2}$ . Let $A$ be the event that $G(x)$ is strictly decreasing at $x = X$ , and let $B$ be the event that $G(x)$ is strictly concave at $x = X$ . Find the conditional probability $P(B \mid A)$ .

Options

A.
$\frac{42}{127}$
B.
$\frac{21}{128}$
C.
$\frac{63}{256}$
D.
$\frac{42}{125}$

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

Sample Space Partitioning|Increasing/Decreasing Strictness|Concavity Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to partition a discrete sample space using higher-order derivative analysis (concavity and monotonicity).

The primary trap lies in the factorization of $G'(x)$ and $G''(x)$ ; students often incorrectly assume all roots of the cubic/quartic are real or fail to check the discriminant of the quadratic factor $x^2 - 6x + 10$ , leading to an incorrect range for Event A.

Furthermore, the strict inequalities for decreasing ( $G'(x) < 0$ ) and concave ( $G''(x) < 0$ ) behavior force the exclusion of boundary points (like $x=1, 3, 5$ ), which is a classic lethal collision between continuous calculus bounds and discrete probability spaces.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sample Space Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free