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Question #41241

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_k = \begin{pmatrix} \frac{1}{k(k+1)} & 2k-1 \\ -(2k-1) & 0 \end{pmatrix}$ for $k \in \mathbb{N}$ . Let $M_n = \sum_{k=1}^n A_k$ . If $S$ is the set of all natural numbers $n$ such that the matrix $M_n$ has real eigenvalues, find the number of elements in $S$ .

Options

A.
0
B.
1
C.
2
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Matrix Trace Properties|Method of Differences|The Empty Set Trap

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix trace properties, telescoping series, and quadratic discriminants.

The 'Empty Set Trap' is deployed here: candidates successfully apply the method of differences to construct the matrix $M_n$ and its characteristic equation, but often assume a valid integer solution must exist.

By failing to rigorously validate the domain constraint ( $n \in \mathbb{N}$ ) against the discriminant inequality $2n(n+1) \le 1$ , students guess a small integer or make algebraic slips.

The realization that the discriminant is strictly negative for all natural numbers, yielding an empty set (0 elements), is a hallmark of JEE Advanced's rigorous domain validation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Matrix Trace Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free