Before You Begin

Question #41276

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be a non-zero $3 \times 3$ real skew-symmetric matrix such that $M^3 + 9M = O$ , where $O$ is the null matrix. For $x > 0$ , let $A(x) = \frac{2}{x} I - M$ , where $I$ is the $3 \times 3$ identity matrix. If $f(x) = \text{Tr}(\text{adj}(A(x)))$ and $g(x) = \det(A(x))$ , then the value of $\lim_{x \to \infty} \frac{x(f(x) - 9)}{g(x)}$ is

Options

A.
$\frac{1}{3}$
B.
$\frac{2}{3}$
C.
$\frac{4}{9}$
D.
$\frac{3}{2}$

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

Adjoint-Determinant Relationship|Skew-Symmetric Determinant|Infinite Limit Behavior

Why This Question Is Tricky

Common Mistakes

This question tests the deep relationship between the eigenvalues of a matrix, its determinant, and the trace of its adjoint.

The trap lies in the limit evaluation as $x \to \infty$ : the matrix $A(x)$ approaches $-M$ , causing $\det(A(x)) \to 0$ and $\text{Tr}(\text{adj}(A(x))) \to 9$ .

This creates a complex $0/0$ indeterminate form.

Students who attempt to explicitly construct the adjoint matrix will be overwhelmed by algebraic complexity.

The elegant 'Lethal Collision' bypass requires utilizing the eigenvalues of the skew-symmetric matrix ( $0, \pm 3i$ ) to directly compute the polynomials for the determinant and the trace of the adjoint, transforming a daunting matrix limit into a straightforward rational function limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Adjoint-Determinant Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free