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Question #41312

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \sin(\pi x) \cos(\pi \{x\}) & (x-2)^2 \\ \sin(\pi x) & x-2 \end{vmatrix}$ , where $\{x\}$ denotes the fractional part of $x$ . Let $S$ be the set of all points in the interval $(0, 5)$ where $f(x)$ is not differentiable. Find the sum of all elements in $S$ .

Options

A.
8
B.
10
C.
0
D.
7

Check Your Answer

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

Chain Rule Inner Function Differentiability|Differentiation of Determinants|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The question tests the differentiability of a determinant whose entries involve the fractional part function {x} inside a trigonometric function.

The trap lies in blindly applying the chain rule to differentiate the determinant, which would incorrectly suggest the function is differentiable everywhere or non-differentiable at all integers.

By forcing students to evaluate the left-hand and right-hand derivatives at integer points, the 'Lethal Collision' occurs: the jump discontinuity of {x} at integers flips the sign of the cosine term, making the derivative undefined except at the specific root x=2 where the algebraic multiplier (x-2) annihilates the discontinuity.

Students who assume non-differentiability at all integers will incorrectly sum to 10.

Quick Facts About This Question

Question Information

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