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Question #41041

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B,$ and $C$ be three events in a sample space such that $P(A) = \frac{1}{2}$ , $P(B) = \frac{1}{3}$ , $P(C) = \frac{1}{4}$ , $P(A \cap B) = \frac{1}{4}$ , $P(B \cap C) = \frac{1}{6}$ , $P(C \cap A) = \frac{1}{8}$ , and $P(A \cap B \cap C) = \frac{1}{12}$ . Let $M$ be a $3 \times 3$ matrix such that its determinant is given by $|M| = P(\overline{A} \cap \overline{B} \cap \overline{C})$ . If $N = \text{adj}(2M)$ , find the value of $584 \sqrt{|N|}$ .

Options

A.
1752
B.
438
C.
876
D.
2920

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

De Morgan’s Laws in Probability|Determinant Scaling Property|Adjoint-Determinant Relation

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of probability theory (Inclusion-Exclusion Principle and De Morgan's Laws) with linear algebra (determinant properties).

The 'Lethal Collision' occurs when students must apply the scalar scaling property of determinants ( $|kM| = k^n|M|$ ) in conjunction with the adjoint determinant property ( $|\text{adj}(A)| = |A|^{n-1}$ ).

A common trap is incorrectly scaling the matrix by assuming $|2M| = 2|M|$ or $|2M| = 2^2|M|$ , leading to the incorrect options 438 and 876.

Another trap is failing to apply De Morgan's Law correctly, using the union probability instead of the complement, leading to 2920.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free