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Question #41799

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{-2, -1, 0, 1, 2\}$ . An ordered triplet $(x, y, z)$ of distinct numbers is chosen randomly from $S$ to form the matrix $A = \begin{pmatrix} x & |x| & 1 \\ y & |y| & 1 \\ z & |z| & 1 \end{pmatrix}$ Let $E$ be the event that $\det(\text{adj} A) \ge 16$ and $F$ be the event that the matrix $A$ is invertible. The value of the conditional probability $P(E \mid F)$ is

Options

A.
$\frac{5}{8}$
B.
$\frac{5}{9}$
C.
$\frac{1}{2}$
D.
$\frac{2}{3}$

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

Conditional Probability Constraints|Adjoint-Determinant Relationship|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

The student must calculate the probability of a determinant condition being met, where the matrix elements are defined by modulus functions.

The 'Lethal Collision' occurs when the student fails to account for the sign-flip of the modulus inside the determinant expansion, leading to incorrect adjoint properties and a collapse of the conditional sample space.

Specifically, students often miss the case where two variables are negative and one is zero, which makes the determinant zero despite the variables having mixed signs.

This acts as a hidden constraint in the conditional probability sample space $n(F)$ , trapping those who blindly assume mixed signs guarantee invertibility.

Quick Facts About This Question

Question Information

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