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Question #41395

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x^2-1 & x-1 \\ 1-x & x-1 \end{pmatrix}$ for $x \in \mathbb{R}$ . Let $M(x)$ be defined as $M(x) = |\det(A(x))| A(x)^{-1}$ when $A(x)$ is non-singular, and $M(x) = \mathbf{0}$ when $A(x)$ is singular. A function $f:\mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \int_{0}^{x} \text{Tr}(M(t)) \, dt$ Let $S$ be the set of all $x$ -coordinates of the inflection points of the curve $y = f(x)$ . Find the sum of the elements of $S$ .

Options

A.
$-\frac{5}{2}$
B.
$-\frac{1}{2}$
C.
$-2$
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

Concavity Analysis|Matrix Trace Properties|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

The trap tests the assumption that matrix polynomials and their traces are always infinitely differentiable, ignoring the domain restrictions and absolute value signatures introduced by the determinant in the inverse.

Students often blindly use $A^{-1} = \text{adj}(A)/\det(A)$ and cancel the determinant with the absolute value, incorrectly concluding $f'(x) = \text{Tr}(\text{adj}(A(x))) = x^2+x-2$ .

This leads to a single false inflection point at $x = -1/2$ .

The lethal collision occurs at the singular point $x = -2$ , where the matrix transition forces a sign flip in the trace, creating a non-differentiable point in $f'(x)$ that serves as a hidden inflection point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Concavity Analysis
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free