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Question #41902

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} 1 & 1 & 1 \\ \cos(\pi x) & 1 & 1 \\ \cos(\pi x) & |x-1| & 1 \end{vmatrix}$ . Let $S$ be the number of points $c \in (0, 2)$ such that $f'(c) = 0$ . Find the value of $S$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Differentiation of Determinants|Mean Value Theorem Applicability|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of Rolle's Theorem on a determinant function containing a non-differentiable modulus term.

The trap is two-fold: First, students observe $f(0) = f(2) = 0$ and reflexively apply Rolle's Theorem, assuming at least one root exists for $f'(x) = 0$ .

Second, if they differentiate the determinant row-wise, they encounter the derivative of $|x-1|$ .

A common fatal mistake is assuming the derivative of $|x-1|$ at $x=1$ is 0 (or ignoring it because it's multiplied by a term that evaluates to 0), failing to realize the derivative becomes undefined, not zero.

By rigorously expanding and checking left/right derivatives, one finds $f'(1)$ is undefined and $f'(x)$ is strictly non-zero elsewhere, yielding 0 solutions.

Quick Facts About This Question

Question Information

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