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Question #40409

Difficulty: 8/10 | 🟢 Free

The Question

Let $B$ be a $3 \times 3$ matrix with real entries and $I$ be the $3 \times 3$ identity matrix. Define the matrix-valued function $A(x) = B \cos x + I |\sin x|$ and let $f(x) = \det(A(x))$ . Which of the following conditions on $B$ is both necessary and sufficient to guarantee that $f(x)$ is differentiable at $x = \pi$ ?

Options

A.
$\text{tr}(\text{adj}(B)) = 0$
B.
$\text{tr}(B) = 0$
C.
$\det(B) = 0$
D.
$\text{tr}(B^2) = 0$

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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

Square Root and Modulus Logic|Differentiation of Determinants|Matrix Trace Properties|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question creates a lethal collision between the non-differentiability of the modulus function and the coefficients of the characteristic polynomial of a matrix.

Students typically attempt to differentiate the determinant directly using Jacobi's formula or get overwhelmed by the modulus.

The trap lies in recognizing that the determinant of $B \cos x + I |\sin x|$ is a disguised characteristic polynomial.

The non-differentiable $|\sin x|$ term appears linearly with the coefficient $\text{tr}(\text{adj}(B))$ , forcing students to deduce that this specific trace must vanish to smooth out the jump discontinuity at $x = \pi$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Square Root and Modulus Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free