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Question #40459

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $x$ satisfying the inequality $\frac{x^2 - 2x - 3}{x^2 - 4} \le 0$ Let $k = \sum_{x \in S} x$ . If $A$ is a $3 \times 3$ matrix whose $(m, n)^{\text{th}}$ entry is given by $a_{mn} = (k + i^{m+n})^2$ for $1 \le m, n \le 3$ , where $i = \sqrt{-1}$ , then the value of $\det(A)$ is:

Options

A.
-256
B.
-4000
C.
256
D.
0

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

Algebraic Manipulation of i|Factor Theorem in Determinants|Cross Multiplication Trap|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

The question tests two distinct concepts: solving rational inequalities and evaluating determinants via matrix multiplication.

The primary trap is the 'Cross Multiplication Trap' in the inequality.

Students who incorrectly cross-multiply by $x^2 - 4$ without considering its sign will obtain $(x-3)(x+1) \le 0 \implies x \in [-1, 3]$ , leading to an incorrect sum $k=5$ and the trap answer $-4000$ .

The determinant evaluation requires recognizing that $a_{mn}$ can be decoupled into a product of two matrices, avoiding a massive algebraic expansion and elegantly utilizing the properties of $i$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Algebraic Manipulation of i
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free