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Question #41337

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} 2\{x\} & [x]^2 - 4[x] + 3 \\ [x]^2 - 8[x] + 15 & -2\{x\} \end{pmatrix}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . Let $S$ be the set of all $x \in [0, 10]$ such that $A(x)$ is a nilpotent matrix. Find the sum of all elements in $S$ .

Options

A.
$15 + \sqrt{3}$
B.
$15 + 2\sqrt{3}$
C.
$9$
D.
$15$

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

Step Function Discontinuities|Nilpotent Matrix Logic|Idempotent and Involutory Properties

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix nilpotency and piecewise function analysis.

A common trap is assuming that $\det(A) = 0$ requires the polynomial in $[x]$ to be exactly zero, leading students to only find the integer roots $x \in \{1, 3, 5\}$ .

By introducing the $2\{x\}$ terms, the determinant equation becomes a bounded Diophantine-style equation $4\{x\}^2 = -P([x])$ .

Students must recognize that the fractional part can compensate for specific non-zero integer values of the polynomial, revealing hidden non-integer solutions.

This effectively filters out candidates who lack rigorous bounding techniques.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Step Function Discontinuities
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free