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Question #41903

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x & \sqrt{x-1} \\ 0 & x \end{pmatrix}$ and $B(x) = \begin{pmatrix} \sqrt{3-x} & 0 \\ x & \sqrt{3-x} \end{pmatrix}$ . Let $S$ be the largest subset of $\mathbb{R}$ such that for all $x \in S$ , the matrices $A(x)$ and $B(x)$ have real entries. Let $M(x) = A(x)B(x) - B(x)A(x)$ and $f(x) = \text{Tr}(M(x)^2)$ . If $\alpha = \min(S)$ and $\beta = \max(S)$ , find the value of $f'_+(\alpha) + f'_-(\beta)$ , where $f'_+$ and $f'_-$ denote the right-hand and left-hand derivatives with respect to $x$ .

Options

A.
44
B.
76
C.
0
D.
\text{Does not exist}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Differentiation of Determinants|Matrix Multiplication Non-Commutativity|System of Inequalities

Why This Question Is Tricky

Common Mistakes

This problem tests the non-commutativity of matrix multiplication and the differentiability of matrix functions under domain constraints.

The primary trap lies in the domain restriction $S = [1, 3]$ imposed by the square roots in the matrix entries.

Students often attempt to use the matrix product rule $(AB)' = A'B + AB'$ directly.

At the boundaries $\alpha=1$ and $\beta=3$ , the derivatives of the individual matrices $A'(1)$ and $B'(3)$ are undefined due to division by zero.

This leads students to incorrectly conclude that the derivative of the trace does not exist.

However, by first computing the commutator $M(x) = [A, B]$ , the undefined terms either cancel out or are neutralized when squared, revealing that the scalar function $f(x)$ is a simple polynomial $2x^3 - 2x^2$ , which is perfectly differentiable on the closed interval.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Differentiation of Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free