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Question #41197

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_n = [a_{ij}]$ be a $4 \times 4$ real skew-symmetric matrix for each $n \in \mathbb{N}$ , such that its elements satisfy: $a_{12}a_{34} - a_{13}a_{24} + a_{14}a_{23} = e^{1/n^2}$ Let $B$ be a $4 \times 4$ real matrix satisfying $B^2 = I$ and $\text{Tr}(B) = 0$ . Define the matrix sequence $M_n = (A_n^T B A_n)^{-1}$ . Evaluate the infinite sum: $S = \sum_{n=1}^{\infty} \tan^{-1} \left( -\frac{1}{2} \ln(\det(M_n)) \right)$

Options

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

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

Infinite Series involving Inverse Tangent|Skew-Symmetric Determinant|Inversion and Determinant Product|Idempotent and Involutory Properties

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix determinant properties and telescoping inverse trigonometric series.

The primary trap lies in the 'Inversion and Determinant Product' property; students often incorrectly evaluate $\det((A_n^T B A_n)^{-1})$ by missing the inverse, leading to a sign error in the logarithm and a divergent or negative series.

Additionally, the problem uses an involutory matrix $B$ with zero trace to test eigenvalue logic—misidentifying $B$ as idempotent would lead to $\det(B)=0$ , breaking the logarithmic sequence.

Finally, it requires recognizing the Pfaffian of a $4 \times 4$ skew-symmetric matrix to correctly find $\det(A_n)$ , separating rote learners from those with deep conceptual clarity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Infinite Series involving Inverse Tangent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free