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Question #41910

Difficulty: 8/10 | 🟢 Free

The Question

Let $U = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $V = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ . Define the matrix-valued function $F(x) = e^{xU} e^{xV}$ for $x \in \mathbb{R}$ . Let $M(x) = F''(x) + \begin{pmatrix} 1+\sin x & 2 \\ 3 & 2-\sin x \end{pmatrix}$ , where $F''(x)$ is the second element-wise derivative of $F(x)$ . Evaluate the limit: $\lim_{x \to 0} \frac{\det(M(x))}{x}$

Options

A.
-1
B.
-4
C.
0
D.
1

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

The L'Hôpital's Rule Misuse|Differentiation of Determinants|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of matrix exponentials, differentiation of matrix functions, and L'Hôpital's rule.

The primary trap lies in the non-commutativity of matrices $U$ and $V$ .

A common mistake is to assume $e^{xU}e^{xV} = e^{x(U+V)}$ , which incorrectly yields $F''(0) = (U+V)^2 = I$ .

If a student falls for this, they will construct an incorrect matrix $M(x)$ , leading to a determinant of $3\cosh x - 5\sinh x + \sin x - \sin^2 x - 3$ .

Applying L'Hôpital's rule to this erroneous determinant yields a trap answer of $-4$ .

This perfectly filters students who blindly apply scalar exponential properties to matrices.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	The L'Hôpital's Rule Misuse
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free