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Question #41270

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x)$ be a $3 \times 3$ matrix defined as: $A(x) = \begin{pmatrix} \frac{x}{x-5} & \frac{x^2}{x-5} & \frac{x^3}{x-5} \\ \frac{4}{x-4} & \frac{16}{x-4} & \frac{64}{x-4} \\ \frac{6}{x-8} & \frac{36}{x-8} & \frac{216}{x-8} \end{pmatrix}$ Find the sum of all integers $x$ that satisfy the inequality $\det(A(x)) \le 0$ .

Options

A.
19
B.
23
C.
28
D.
36

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

Determinant Product Rule|Elementary Row Operation Invariance|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix operations and algebraic inequalities, specifically targeting the 'Equivalent Inequality' principle.

Students often apply elementary row operations or factor out terms to simplify a determinant, inadvertently canceling out factors like $(x-4)$ from the numerator and denominator.

The lethal trap lies in applying the Wavy Curve (Method of Intervals) on the simplified expression without accounting for the domain constraints of the original matrix entries.

Because $x=4$ makes the original matrix undefined, it must be excluded from the final solution set.

Failing to check the initial domain leads to the inclusion of the extraneous integer $4$ , resulting in the classic trap answer of $23$ .

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