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Question #40456

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x)$ be a $3 \times 3$ matrix whose $(i, j)^{\text{th}}$ entry is given by $m_{ij} = \sec^2 x + \cos((i-j)x)$ for $1 \le i, j \le 3$ and $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ . Let $C(x)$ be the cofactor matrix of $M(x)$ . If $I = \int_{-\pi/4}^{\pi/4} \frac{\det(C(x))}{\sin^4 x (1-\cos x)^4} \, dx$ then the value of $I$ is: \ (Assume the integrand is continuously extended at $x=0$ )

Options

A.
$\frac{128}{3}$
B.
$\frac{64}{3}$
C.
$\frac{256}{3}$
D.
32

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

Lagrange Identity Equivalents|Triangular/Diagonal Matrix Shortcut|Inversion and Determinant Product

Why This Question Is Tricky

Common Mistakes

This question tests the 'Determinant Product' and 'Cofactor Matrix' properties, acting as a severe trap for students who attempt to expand the $3 \times 3$ trigonometric determinant directly (which leads to an intractable algebraic mess).

The logical pivot requires recognizing that the entries $m_{ij}$ can be decoupled into a dot product, allowing $M(x)$ to be factored as $A B^T$ .

Furthermore, it tests the inversion-related property $\det(C) = (\det M)^{n-1}$ , linking matrix algebra to determinant evaluation before resolving into a clean definite integral with a removable singularity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Lagrange Identity Equivalents
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free