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Question #40961

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{bmatrix} x^3+1 & x^2+x & x+1 \\ 9 & 6 & 3 \\ 28 & 12 & 4 \end{bmatrix}$ . Let $f(x) = \sqrt{\det(A(x) A^T(x))}$ . If $L$ and $R$ denote the left-hand derivative and right-hand derivative of $f(x)$ at $x=2$ respectively, find the value of $R - L$ .

Options

A.
72
B.
-72
C.
36
D.
0

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

Right-Hand vs Left-Hand Derivatives|Factor Theorem in Determinants|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize determinant properties, the Factor Theorem, and non-differentiability at roots of absolute value functions.

The primary trap lies in the simplification of $\sqrt{\det(A(x)A^T(x))}$ .

Students often incorrectly simplify this to $\det A(x)$ instead of $|\det A(x)|$ , completely missing the cusp at $x=2$ and erroneously concluding that the function is differentiable (yielding $R-L=0$ ).

Furthermore, expanding the determinant brutally consumes time and invites algebraic errors; the intended 'Lethal Collision' rewards recognizing row dependencies at $x=-1, 2, 3$ to instantly construct the polynomial via the Factor Theorem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Right-Hand vs Left-Hand Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free