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Question #41209

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x & [x] & \{x\} \\ [x] & \{x\} & x \\ \{x\} & x & [x] \end{pmatrix}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . Let $P$ be the $3 \times 3$ matrix obtained by interchanging the first and second rows of the identity matrix $I_3$ . If $B(x) = \text{adj}(P) A(x)^T P$ and $f(x)$ is the scalar triple product of the vectors formed by the first, second, and third rows of $B(x)$ respectively, find the value of $16 \int_{-1}^{2} f(x) dx$ .

Options

A.
$24$
B.
$-24$
C.
$12$
D.
$48$

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

Transpose of Product|Scalar Triple Product Volume Logic|Row/Column Exchange Parity|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of matrix operations, determinant properties, and piecewise integration.

The primary trap lies in the 'Row/Column Exchange Parity' combined with the adjoint matrix determinant.

Students often incorrectly assume $\det(\text{adj}(P)) = \det(P)$ or forget that $\det(P) = -1$ for a row swap, leading to a sign error (Option B).

A secondary trap is the algebraic simplification of the anti-circulant determinant involving $x, [x], \{x\}$ ; failing to use the identity $x = [x] + \{x\}$ makes the integration seemingly impossible and tests the student's algebraic resilience.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Transpose of Product
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free