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Question #41275

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = [x]\hat{i} + 2\hat{j} + \hat{k}$ , $\vec{b} = 2\hat{i} + [x]\hat{j} + \hat{k}$ , and $\vec{c} = \hat{i} + 2\hat{j} + [x]\hat{k}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $N$ be the $3 \times 3$ matrix whose rows are the vectors $\vec{a} \times \vec{b}$ , $\vec{b} \times \vec{c}$ , and $\vec{c} \times \vec{a}$ . If $S$ is the set of all $x \in [-5, 5]$ such that $\det(\text{adj}(\text{adj}(N))) = 0$ , then the value of $\int_S 2x \, dx$ is:

Options

A.
3
B.
0
C.
1
D.
8

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Adjoint-Determinant Relationship|Vector Triple Product Expansion|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of linear algebra (adjoint properties and vector triple product) with calculus (integration over piecewise constant functions).

The primary trap lies in the Greatest Integer Function (GIF).

Students often incorrectly assume that finding the roots of the determinant equation yields discrete points (e.g., $x = 1, 2, -3$ ), leading them to evaluate the integral as 0.

Furthermore, the expansion of the scalar triple product of cross products $[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}] = [\vec{a}, \vec{b}, \vec{c}]^2$ must be recognized to simplify the adjoint determinant condition $\det(\text{adj}(\text{adj}(N))) = \det(N)^4$ without brute-force computation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Adjoint-Determinant Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free