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Question #41297

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} 1 & x & x^2 \\ 1 & [x] & [x]^2 \\ 1 & x+[x] & (x+[x])^2 \end{pmatrix}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $f(x) = \det(\text{adj}(A(x)))$ . If $S$ is the set of all points in the interval $(-4, 5)$ where $f(x)$ is not differentiable, find the value of $\sum_{k \in S} (k+2)$ .

Options

A.
15
B.
20
C.
18
D.
17

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

Factor Theorem in Determinants|Adjoint-Determinant Relation|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Factor Theorem on determinants and the Adjoint-Determinant relation $\det(\text{adj}(A)) = |A|^{n-1}$ .

The primary trap lies in the differentiability analysis of the resulting step-function dependent expression.

Students often mistakenly assume that the presence of $[x]$ and $\{x\}$ implies non-differentiability at all integers.

Furthermore, if a student incorrectly uses $|A|$ instead of $|A|^2$ for the adjoint determinant, the function remains non-differentiable at $x=0$ and $x=1$ , leading to the trap answer 20.

The squaring effect from the adjoint relation smooths out the 'corners' at $x=0$ and $x=1$ , making the function differentiable specifically at those two integer points.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Factor Theorem in Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free