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Question #41283

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{r=1}^n r^{104}$ . Consider the real number $L$ defined by $L = \lim_{n \to \infty} \left( \frac{105 S_n}{n^{105}} \right)^{\frac{n}{105}}$ Let $S$ be the set of all $4 \times 4$ real matrices $A$ satisfying $A^T = -A$ and $A A^T = \left( \ln L - \frac{1}{2} \right) I$ , where $I$ is the $4 \times 4$ identity matrix. Find the number of elements in $S$ .

Options

A.
0
B.
1
C.
4
D.
$\infty$

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

Skew-Symmetric Determinant|Orthogonal Matrix Conditions|The 1^∞ Form Shortcut|Sum of Powers of Natural Numbers

Why This Question Is Tricky

Common Mistakes

This problem tests the precise evaluation of $1^\infty$ limits involving sums of powers, and the properties of real skew-symmetric matrices.

A common trap is approximating $S_n \approx n^{105}/105$ and ignoring the $n^{104}/2$ term, leading to $L=1$ and the impossible condition $AA^T = -1/2 I$ (yielding 0 solutions).

Another trap is correctly finding $AA^T = 0$ but incorrectly assuming infinitely many non-zero matrices satisfy this (confusing real matrices with complex ones or zero divisors in rings).

For real matrices, $AA^T = 0$ strictly implies $A=0$ , yielding exactly 1 solution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Skew-Symmetric Determinant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free