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Question #41069

Difficulty: 8/10 | 🟢 Free

The Question

Let $k = \int_{1/e}^e \frac{|\ln x| - \ln x}{x} dx$ . Let $A$ be the set consisting of the distinct elements of $\{k-2, 0, k\}$ . Let $S = \{ (x, y, z) \mid x, y, z \in A \}$ be a set of vectors in $\mathbb{R}^3$ . If three distinct vectors are chosen randomly from $S$ , what is the probability that they are linearly independent?

Options

A.
$\frac{656}{975}$
B.
$\frac{29}{56}$
C.
$\frac{8}{9}$
D.
$\frac{176}{225}$

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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

Complementary Counting|Combinatorial Probability|Linear Independence vs Dependence|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of definite integration, combinatorial geometry, and probability.

The primary trap lies in the evaluation of the logarithmic integral with a modulus.

Students who ignore the absolute value property or the sign change of $\ln x$ at $x=1$ will incorrectly evaluate $k=0$ .

This leads to the degenerate set $A = \{-2, 0\}$ , which drastically alters the vector space to $S = \{-2, 0\}^3$ .

Calculating the probability of linear independence for this incorrect sample space yields the trap answer $29/56$ .

Furthermore, the problem demands rigorous complementary counting to find linearly dependent triplets in $\mathbb{R}^3$ , requiring students to systematically classify coplanar vectors into cases involving the zero vector, collinear pairs, and distinct coplanar lines, punishing incomplete combinatorial casework.

Quick Facts About This Question

Question Information

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