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Question #66687

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} x & 1 & 1 \\ 1 & x-2 & 1 \\ 1 & 1 & x-2 \end{pmatrix}$ and $B = \begin{pmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 3 & 1 & x-2 \end{pmatrix}$ . Let $f(x) = \frac{|A|}{|B|}$ . A sequence of matrices $M_n$ is constructed such that $|M_n| - |M_{n-1}| = (f(x))^n$ for $n \ge 2$ . If the sequence of determinants $|M_n|$ converges to a finite limit as $n \to \infty$ , find the sum of all possible integers $x \in [-5, 5]$ .

Options

A.
8
B.
10
C.
11
D.
14

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

Singular vs Non-Singular Matrices|Method of Differences|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

The question tests the Method of Differences applied to determinants, leading to a geometric progression whose convergence requires $|f(x)| < 1$ .

The core trap lies in solving the rational inequality.

Students frequently cross-multiply variable denominators without considering their signs.

If they cross-multiply the simplified fraction, they incorrectly obtain $x \in (\sqrt{2}, \infty)$ , missing the negative interval and the integer $-1$ .

If they cross-multiply the original unsimplified fraction, they assume $|B| > 0$ and miss $x=2$ .

Furthermore, the removable singularity at $x=3$ acts as a secondary trap; failing to exclude it adds an invalid integer to the sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Singular vs Non-Singular Matrices
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free