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Question #41552

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x) = \begin{pmatrix} x^3 - \lambda x & e^{|x|} \sin x & \cos x \\ -e^{|x|} \sin x & x^3 - \lambda x & \sqrt{1+x^2} \\ -\cos x & -\sqrt{1+x^2} & x^3 - \lambda x \end{pmatrix}$ , where $\lambda$ is a real parameter. Let $f(x) = \det(M(x))$ . If the rows of $M(x)$ are linearly dependent at $x = 2$ , then the value of the integral $\int_{-2}^{2} \left( \frac{f(x)}{2 + \cos x} + \frac{3\lambda}{2} x^2 \right) dx$ is equal to:

Options

A.
16
B.
32
C.
64
D.
0

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

Symmetry Utilization|Skew-Symmetric Determinant|Linear Dependency of Rows

Why This Question Is Tricky

Common Mistakes

This question tests the ability to identify the characteristic polynomial of a skew-symmetric matrix disguised as a complex determinant.

The 'Linear Dependency' trap forces students to evaluate the determinant at $x=2$ .

Brute-force expansion leads to an unmanageable equation, but recognizing the $M(x) = yI + S$ structure immediately reveals that $y(2)=0$ .

Furthermore, students must utilize the skew-symmetric property to deduce that the determinant is an odd function, turning an analytically impossible integration into a trivial polynomial evaluation.

Failing to check for rank-deficiency or symmetry traps the student in a calculus nightmare.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Symmetry Utilization
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free