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Question #41409

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x)$ be a $3 \times 3$ matrix defined as: $M(x) = \begin{pmatrix} 2e^x + e^{-x} + 1 & 2e^{2x} + e^{-2x} + 1 & 2e^{3x} + e^{-3x} + 1 \\ e^x + 2e^{-x} + 1 & e^{2x} + 2e^{-2x} + 1 & e^{3x} + 2e^{-3x} + 1 \\ e^x + e^{-x} + 2 & e^{2x} + e^{-2x} + 2 & e^{3x} + e^{-3x} + 2 \end{pmatrix}$ Evaluate the integral: $I = \int_{-\ln 2}^{\ln 2} \frac{e^{2x} |\det(M(x))|}{4(e^x + 1)} \, dx$

Options

A.
$\frac{1}{6}$
B.
$\frac{3}{2} - 2\ln 2$
C.
$\frac{5}{6} - \ln 2$
D.
$\frac{11}{6}$

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

Substitution (Change of Variable), Determinant Product Rule, Coplanarity Condition, Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the Determinant Product Rule and the critical analysis of integrands involving absolute values of determinants.

The matrix $M(x)$ is deliberately constructed as the product of a constant matrix and a variable matrix $B(x)^T$ .

Students who attempt direct expansion will face a massive algebraic wall.

Using $\det(M) = \det(A)\det(B^T)$ simplifies the expression.

The 'Lethal Collision' occurs because the rows of $B(x)^T$ represent vectors that become coplanar at $x=0$ , causing the determinant to vanish and change sign.

Failing to recognize this coplanarity condition and blindly integrating without splitting the domain at the singularity $x=0$ leads to the incorrect distractor $\frac{3}{2} - 2\ln 2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Substitution (Change of Variable)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free