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Question #41006

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = (\sin 2x)^{\sin 2x}$ be defined on its maximal domain $D \subset (0, 2\pi)$ where the base is strictly positive. Let $A = \{x \in D : f'(x) = 0\}$ and $B = \{x \in (0, 2\pi) \setminus D : x \text{ is a limit point of } D \text{ and } \lim_{t \to x, t \in D} f(t) \text{ exists finitely}\}$ . A sample space is defined as the set of discrete points $S = A \cup B$ . If three distinct elements are chosen at random from $S$ , what is the probability that exactly one of them belongs to $A$ ?

Options

A.
$\frac{3}{14}$
B.
$\frac{8}{55}$
C.
$\frac{3}{28}$
D.
$\frac{1}{15}$

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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

Sample Space Construction|Multiplication Theorem|Logarithmic Differentiation Domain|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of logarithmic differentiation, domain constraints, and probability sample space construction.

The primary trap lies in blindly applying the chain rule to $f(x) = u^v$ and solving $f'(x) = 0$ without verifying if the roots fall within the strict domain $u > 0$ .

Students commonly include extraneous roots ( $3\pi/4, 7\pi/4$ ) where the original function is undefined, incorrectly inflating $n(A)$ to 8.

Furthermore, it tests the rigorous definition of limits at boundary points of a domain.

Failing either the domain check or the boundary limit evaluation leads to an incorrect sample space cardinality, acting as a topological filter before the probabilistic Multiplication Theorem can be applied.

Quick Facts About This Question

Question Information

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