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Question #41202

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} \text{sgn}(x^2 - x - 2) + 1 & \cos^{-1}(x) \\ \sin(2\pi x) & \text{sgn}(x + 1) \end{pmatrix}$ , where $\text{sgn}(\cdot)$ denotes the signum function. If $S$ is the set of all real values of $x$ for which $A$ is an idempotent matrix, then the value of $4 \sum_{x \in S} x^2$ is equal to

Options

A.
8
B.
10
C.
12
D.
14

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

Singular vs Non-Singular Matrices|Idempotent and Involutory Properties|Signum Function Discontinuity|Domain Constraint Intersections

Why This Question Is Tricky

Common Mistakes

This question tests the properties of idempotent matrices combined with domain constraints and piecewise discontinuities.

Students often forget that for a $2 \times 2$ matrix, $A^2 = A$ implies either $A=O, I$ or $\text{Tr}(A)=1$ and $\det(A)=0$ .

The trap lies in the signum function: evaluating it at the boundary $x=-1$ yields a different matrix structure than $x \in (-1, 1]$ , but both cases elegantly satisfy $\text{Tr}(A)=1$ .

Failing to check the boundary point $x=-1$ separately due to the discontinuity of the signum function will lead to missing a solution, resulting in an incorrect sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Singular vs Non-Singular Matrices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free