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Question #41330

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x)$ be a $3 \times 3$ matrix with differentiable real-valued functions as entries. For all $x \in \mathbb{R}$ , the traces of the powers of $A(x)$ are given by: $\text{tr}(A(x)) = x$ $\text{tr}(A^2(x)) = 5x^2 - 22$ $\text{tr}(A^3(x)) = 10x^3 - 18x^2 - 18$ Let $f(x) = \text{tr}(\text{adj}(A(x)) \cdot A'(x))$ . If $\alpha$ and $\beta$ are the roots of the equation $f(x) = 0$ , find the value of $3|\alpha - \beta|^2$ .

Options

A.
4
B.
12
C.
22
D.
66

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

Rolle's Theorem Root Existence|System of Linear Equations (Cramer/Inverse)|Adjoint-Determinant Relation

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Matrix Algebra (Newton's sums for invariants) and Calculus (Jacobi's formula).

The primary trap lies in the Adjoint-Determinant differential relation.

Students often fail to recognize Jacobi's formula, $\frac{d}{dx}\det(A) = \text{tr}(\text{adj}(A) A')$ , which elegantly bypasses the need to know the individual entries of $A(x)$ or $A'(x)$ .

A common fatal mistake is incorrectly assuming that the trace of a product equals the product of traces, leading to $f(x) = \text{tr}(\text{adj}(A)) \text{tr}(A') = e_2(x) \cdot 1 = -2x^2 + 11 = 0$ .

This yields an incorrect squared difference of $22$ and a trap answer of $66$ .

The problem rigorously filters for students who understand deep structural matrix calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Rolle's Theorem Root Existence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free