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Question #41127

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ skew-symmetric matrix with real entries. Let $f(x) = \det(A - xI)$ . If the function $g: \mathbb{R} \to \mathbb{R}$ defined by $g(x) = f(x) + 245|x - \det(A)|$ is bijective, then the minimum possible value of $\text{Tr}(A^T A)$ is

Options

A.
$490$
B.
$245$
C.
$980$
D.
$0$

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

Bijectivity and Inverse|Skew-Symmetric Determinant|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This problem tests the properties of skew-symmetric matrices and the calculus of non-differentiable functions.

Students must first identify that the determinant of an odd-order skew-symmetric matrix is zero, simplifying the modulus term.

The core trap lies in analyzing the bijectivity of the resulting piecewise polynomial.

Students often incorrectly assume that a polynomial with a modulus is injective if its underlying polynomial is monotonic, failing to check the derivative's sign change across the non-differentiable cusp at $x=0$ .

By forcing $g'(x) \le 0$ for $x > 0$ , the minimum trace is correctly identified, penalizing those who ignore the piecewise nature or the trace formula $\text{Tr}(A^T A) = 2\sum a_{ij}^2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Bijectivity and Inverse
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free