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Question #66607

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & 1 & 1 \\\\ x & [x] & \{x\} \\\\ x^2 & [x]^2 & \{x\}^2 \end{pmatrix}$ , where $[x]$ and $\{x\}$ denote the greatest integer and fractional part functions respectively. If $B = \text{adj}(A)$ , which of the following is equal to $\det(\text{adj}(AB))$ ?

Options

A.
$\{x\}^6 [x]^6 (x - 2\{x\})^6$
B.
$\{x\}^4 [x]^4 (x - 2\{x\})^4$
C.
$\{x\}^2 [x]^2 (x - 2\{x\})^2$
D.
$-\{x\}^3 [x]^3 (x - 2\{x\})^3$

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

Determinant Product Rule|Adjoint-Determinant Relation|Linear Independence vs Dependence|Integral Part and Fractional Part

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the synthesis of determinant properties with piecewise functions.

The primary trap lies in the sequential application of the adjoint-determinant relation.

Students often incorrectly assume $\det(AB) = \det(A)$ when $B = \text{adj}(A)$ , or miscalculate the scalar multiplication of the identity matrix, leading to lower powers of $\det(A)$ like 2 or 4.

Furthermore, the problem implicitly tests the linear dependence condition: at integer values of $x$ , $\{x\} = 0$ , making columns 1 and 2 identical (since $x = [x]$ ).

The matrix becomes singular ( $\det(A)=0$ ), causing the adjoint to collapse to a zero matrix.

The correct algebraic power (6) naturally preserves this singularity, ensuring the determinant of the adjoint remains zero at these critical integer boundaries.

Quick Facts About This Question

Question Information

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