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Question #41813

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_n = [a_{ij}]$ be an $n \times n$ matrix where $a_{ij} = i + j$ for $1 \le i, j \le n$ . Let $p = \frac{|adj(A_n)| + 1}{|A_n|^{n-1} + 2}$ be the probability of success in a single trial of a Bernoulli experiment. Let $X$ be the number of successes in $n$ independent trials of this experiment. If $P(X=2) = P(X=3) > 0$ , then the variance of $X$ is equal to

Options

A.
$\frac{5}{4}$
B.
$\frac{5}{2}$
C.
$\frac{3}{4}$
D.
1

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Bernoulli Trials Logic|Adjoint-Determinant Relationship|Inversion and Determinant Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle cyclic dependencies between matrix dimensions and probability distributions.

The trap lies in the expression for $p$ : students often blindly apply the property $|adj(A)| = |A|^{n-1}$ without evaluating $|A|$ .

If they fail to recognize that $A_n$ is a singular matrix for $n \ge 3$ (due to linearly dependent rows), they cannot determine $p$ .

Furthermore, the condition $P(X=2) = P(X=3) > 0$ acts as a reverse-constraint, forcing $n \ge 3$ and validating the singularity of $A_n$ , creating a beautiful logical loop that collapses to a single valid integer $n=5$ .

Quick Facts About This Question

Question Information

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