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Question #41922

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} x^2 + |x-1| & x^3 + |x-2| & x^4 + |x-3| \\ x^2 - |x-1| & x^3 - |x-2| & x^4 - |x-3| \\ x^2 & x^3 & x^4 \end{vmatrix}$ . Let $D$ be the set of all real numbers $x$ at which $f(x)$ is not differentiable. If $D$ is a non-empty set, let $S = \sum_{x \in D} x$ . If $D$ is the empty set, let $S = \sum_{k=1}^{5} f'(k)$ . Find the value of $S$ .

Options

A.
0
B.
6
C.
10
D.
15

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

Linear Independence vs Dependence|Differentiation of Determinants|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's mathematical maturity in recognizing linear dependence within a matrix before blindly applying differentiation rules.

The trap lies in the modulus functions $|x-1|, |x-2|$ , and $|x-3|$ , which are non-differentiable at $x=1, 2, 3$ .

A student who mechanically applies the determinant differentiation formula $\frac{d}{dx}|A| = |A_1'| + |A_2'| + |A_3'|$ will encounter undefined derivatives at these critical points and incorrectly conclude that $f(x)$ is non-differentiable, leading to the trap answer of $S = 1+2+3=6$ .

However, because the rows are linearly dependent, the vector space geometry collapses, making the determinant identically zero and thus differentiable everywhere.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Linear Independence vs Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free