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Question #41207

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $k$ for which the quadratic equations $x^2 - 2x + 1 - k = 0$ and $x^2 - 2\sqrt{k}x + k - 1 = 0$ have at least one common real root. Let $A$ be a $2 \times 2$ matrix with real entries such that its trace is $k^2 - 4k + 5$ and its determinant is $k-2$ . If $k \in S$ is chosen such that the trace of $A$ is minimized, which of the following is equal to $A^{2024}$ ?

Options

A.
I
B.
A
C.
2024A
D.
A - I

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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

Idempotent and Involutory Properties|Condition for Common Roots|Maximum/Minimum of Quadratic Expressions

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of polynomial root analysis, functional optimization, and matrix algebra.

A common trap is attempting to use the complex cross-multiplication condition for common roots, which becomes algebraically intractable with the $\sqrt{k}$ term.

Students must instead solve the quadratics directly by completing the square.

Furthermore, students might overlook the implicit domain constraint $k \ge 0$ required for real coefficients.

The core architectural trap lies in shifting the problem from matrix operations to minimizing a quadratic trace function, which then elegantly collapses the matrix into an idempotent form via the Cayley-Hamilton theorem, bypassing the need for explicit matrix elements.

Quick Facts About This Question

Question Information

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