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Question #41724

Difficulty: 8/10 | 🟢 Free

The Question

Let $V(x)$ be the volume of a right circular cone of height $x$ inscribed in a sphere of radius $3$ . Let $x_0$ be the height that maximizes this volume, and let $\lambda = \frac{9}{32\pi} V(x_0)$ . For each integer $n \ge 3$ , let $X_n$ be a random variable following the binomial distribution $B(n, p_n)$ , where $p_n = \frac{\lambda}{n}$ . Evaluate the limit: $\lim_{n \to \infty} n \left( e^\lambda P(X_n \le 1) - 4 \right)$

Options

A.
-9
B.
-6
C.
0
D.
9

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

Binomial Distribution Tail-Ends|Geometry Constraint Modeling|Critical Point vs Stationary Point|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous transition from a discrete Binomial distribution to a continuous Poisson approximation, specifically targeting the 'L'Hopital's Pitfall' in limit evaluations.

Students often incorrectly assume that since $\lim_{n \to \infty} P(X_n \le 1) = 4e^{-3}$ , the limit of the difference multiplied by $n$ is simply $0 \times \infty = 0$ .

Alternatively, if they attempt to use L'Hopital's rule by setting $x = 1/n$ , they frequently miscalculate the derivative of the exponential-logarithmic composite function $(1 - \lambda x)^{1/x}$ by ignoring the second-order Taylor expansion term ( $-\lambda^2 x^2 / 2$ ) of the logarithm, which is critical for extracting the correct tail-end remainder.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Binomial Distribution Tail-Ends
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free