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Question #41721

Difficulty: 8/10 | 🟢 Free

The Question

A point $z$ is chosen uniformly at random from the region $S = \{z \in \mathbb{C} : 4 \le |z| \le 5\}$ . Let $w = u + iv$ be a complex number defined by the transformation $w = z + \frac{16}{z}$ . A Bernoulli trial is considered a success if $w$ satisfies both of the following conditions: (i) $\det \begin{pmatrix} u & v & 1 \\ 8 & 0 & 1 \\ -8 & 0 & 1 \end{pmatrix} \ge 0$ (ii) $|w - 8| + |w + 8| \le 12$ Let $X$ be the number of successes in $n = 2024$ independent trials. Find the value of $E(X^2 + X + 1)$ , where $E(\cdot)$ denotes the expected value.

Options

A.
0
B.
1
C.
2024
D.
2025

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Bernoulli Trials Logic|Determinant as Geometric Area/Volume|Locus Problems via Transformation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate complex transformations, determinant-based geometric constraints, and probability distributions, while avoiding a critical feasibility trap.

The Joukowsky transformation $w = z + 16/z$ acts as a complex distractor, baiting students into mapping the annulus to the $w$ -plane to find the area of intersection with an assumed ellipse.

Students often blindly apply the area formula for an ellipse upon seeing the sum of distances to foci, calculating $2a=12 \implies a=6$ and $2ae=16 \implies ae=8$ , leading to a negative $b^2 = 36 - 64 = -28$ .

If they ignore the negative sign or fail to check the domain, they will calculate a non-zero probability.

The trap is that the triangle inequality $|w - 8| + |w + 8| \ge 16$ makes the condition $|w - 8| + |w + 8| \le 12$ an empty set.

Consequently, the probability $p$ collapses to 0, nullifying the binomial distribution parameters and trivially reducing the expected value expression to 1.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bernoulli Trials Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free