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Question #67295

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} e^{x^2 \sin(1/x)} & \pi \\ 0 & e^{-x^2 \sin(1/x)} \end{pmatrix}$ for $x \neq 0$ . Let $n \in \mathbb{N}$ be the solution to the equation: $\lim_{x \to 0} \frac{\sum_{k=1}^n \text{Tr}(A(x)^k) - 2n}{x^4 \sin^2(1/x)} = 385$ If $B = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ , find the value of $\text{Tr}(B^n)$ .

Options

A.
123
B.
144
C.
89
D.
199

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

Matrix Equality|Matrix Trace Properties|Sum of Powers of Natural Numbers|L'Hopital's Pitfall

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

This question tests the student's ability to bypass the lethal L'Hopital's pitfall by recognizing the structure of the trace operator and employing the Method of Dominant Terms (Taylor expansion).

If a student blindly applies L'Hopital's rule to the limit involving $x^2 \sin(1/x)$ , the chain rule generates $1/x$ terms that oscillate and falsely appear to diverge, acting as a high-level filter.

Furthermore, it connects matrix trace properties to recurrence relations (Lucas numbers), demanding recursive logic where the limit's evaluation is a strict prerequisite for the matrix power trace computation.

Quick Facts About This Question

Question Information

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