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Question #41572

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x,y)$ be a $3 \times 3$ matrix defined as: $M(x,y) = \begin{pmatrix} 3x^2 & -\sqrt{6}xy & y^2 \\ \sqrt{6}xy & 3x^2-y^2 & -\sqrt{6}xy \\ y^2 & \sqrt{6}xy & 3x^2 \end{pmatrix}$ Let $R$ be the region in the $xy$ -plane satisfying $\det(M(x,y)) \le 1$ . Evaluate the integral: $I = \iint_R \mathrm{tr}\left( \mathrm{adj}(M(x,y)) \left( x \frac{\partial M}{\partial x} + y \frac{\partial M}{\partial y} \right) \right) dx dy$

Options

A.
$\frac{\pi\sqrt{3}}{2}$
B.
$\frac{\pi\sqrt{3}}{4}$
C.
$\pi\sqrt{3}$
D.
$\frac{3\pi\sqrt{3}}{8}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Region Ambiguity|Homogeneous Function Homogeneity|Differentiation of Determinants

Why This Question Is Tricky

Common Mistakes

This question masterfully combines linear algebra and multivariable calculus to trap students.

The primary trap is the 'Differentiation of Determinants' property; students who fail to recognize Jacobi's formula ( $d(\det M) = \mathrm{tr}(\mathrm{adj}(M) dM)$ ) will attempt to manually differentiate the matrix and multiply by its adjugate, leading to an insurmountable algebraic mess.

Once reduced to a gradient dot product, the 'Homogeneous Function' trap requires applying Euler's Theorem to bypass direct differentiation of the determinant.

Finally, evaluating the integral over the region demands recognizing the elliptical symmetry and applying a Jacobian transformation, punishing those who attempt tedious Cartesian integration with ambiguous limits.

Quick Facts About This Question

Question Information

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