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Question #41818

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $n$ satisfying the inequality $|n^2 - 6n + 5| \le 5 - n$ . A number $k$ is chosen uniformly at random from $S$ . Let the random variable $X$ be defined as the maximum real root of the quadratic equation $x^2 - (2^k + k^2)x + k^2 2^k = 0$ Find the expected value $E(X)$ .

Options

A.
$\frac{39}{4}$
B.
$\frac{32}{3}$
C.
$\frac{63}{5}$
D.
$\frac{69}{4}$

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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

Random Variable Expectation|Quadratic Roots in Sequences|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

This question tests the integration of probability distributions with algebraic inequalities.

The primary trap lies in solving the modulus inequality $|f(n)| \le g(n)$ .

Students often only solve $f(n) \le g(n)$ and forget the lower bound $-g(n) \le f(n)$ , leading to an incorrect sample space $S = \{0, 1, 2, 3, 4, 5\}$ .

This boundary condition trap causes them to incorrectly include $k=3$ and $k=4$ , yielding the trap expectation $32/3$ .

Additionally, it tests the careful evaluation of the maximum of two functions, $2^k$ and $k^2$ , which swap dominance.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Random Variable Expectation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free