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Question #41819

Difficulty: 8/10 | 🟢 Free

The Question

Let $X, Y,$ and $Z$ be independent random variables, each chosen uniformly from the set $\{1, 2, 3, \dots, 10\}$ . Consider the system of linear equations $A \mathbf{v} = \mathbf{b}$ , where $A = \begin{pmatrix} 0 & X & Y \\ -X & 0 & Z \\ -Y & -Z & 0 \end{pmatrix} \quad \text{and} \quad \mathbf{b} = \begin{pmatrix} Z \\ Y \\ X \end{pmatrix}$ Let $R$ be the rank of the augmented matrix $[A | \mathbf{b}]$ . The expected value of $R$ , denoted by $E[R]$ , is:

Options

A.
$\frac{749}{250}$
B.
$\frac{747}{250}$
C.
$\frac{499}{250}$
D.
$3$

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

Random Variable Expectation|System of Linear Equations (Cramer/Inverse)|Skew-Symmetric Determinant

Why This Question Is Tricky

Common Mistakes

This question fuses linear algebra with probability, specifically targeting the properties of skew-symmetric matrices.

The primary trap is the reflexive assumption that a $3 \times 4$ augmented matrix with non-zero random entries will almost surely have full row rank (rank 3), leading students to incorrectly guess $E[R] = 3$ .

Furthermore, students who attempt Cramer's rule will find $\det(A)=0$ and might erroneously conclude the system is always inconsistent.

The elegant mathematical pivot requires recognizing that for a skew-symmetric matrix, the left null space equals the right null space.

This allows a quick dot product to find the consistency condition $X^2+Z^2=Y^2$ , seamlessly transitioning the problem into counting Pythagorean triples within the sample space.

Quick Facts About This Question

Question Information

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