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Question #41248

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ . Let $C_{ij}$ and $M_{ij}$ denote the cofactor and minor of the element $a_{ij}$ of matrix $A$ , respectively. Define vectors $\vec{u}_i = \sum_{j=1}^3 \left( C_{ij} + \text{sgn}(x^2 - 4x + 3) M_{ij} \right) \hat{e}_j$ for $i = 1, 2, 3$ , where $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the standard orthogonal unit vectors. If $V(x)$ is the volume of the parallelepiped formed by the vectors $\vec{u}_1, \vec{u}_2, \vec{u}_3$ , find the value of $\int_{0}^{3} V(x) dx$ .

Options

A.
128
B.
192
C.
256
D.
384

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

Minors and Cofactors Sign Convention|Scalar Triple Product Volume Logic|Signum Function|Modulus Function

Why This Question Is Tricky

Common Mistakes

This question tests the precise relationship between minors and cofactors ( $M_{ij} = (-1)^{i+j} C_{ij}$ ) within the context of a scalar triple product (volume).

A common trap is treating $M_{ij}$ and $C_{ij}$ as identical or differing by a global negative sign.

If a student incorrectly assumes $M_{ij} = C_{ij}$ , they will factor out $(1+s)$ and compute $\det(U) = (1+s)^3 \det(C) = 8 \Delta^2 = 128$ , leading to the incorrect option 128.

Furthermore, the Signum function forces students to evaluate the determinant under two different structural conditions (checkerboard vs. anti-checkerboard), rewarding those who correctly apply Jacobi's theorem for sub-determinants of the cofactor matrix.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Minors and Cofactors Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free