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Question #41853

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, \infty) \to \mathbb{R}$ be defined as $f(x) = \begin{vmatrix} \int_1^x e^{t^2} dt & x+\ln(x) & |x-1| \\ x^2 & x & x^2+1 \\ 1 & 1 & 2 \end{vmatrix}$ If $f(x)$ is differentiable at $x=1$ , let $L = f'(1)$ . If $f(x)$ is non-differentiable at $x=1$ , let $L = \text{ND}$ . Find the value of $L$ .

Options

A.
-2
B.
2
C.
0
D.
\text{ND}

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

Algebra of Functions|Differentiation of Determinants|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

The question tests the differentiation of determinants containing non-differentiable entries.

A common trap is blindly applying the determinant differentiation rule (sum of determinants with differentiated rows).

Since the entry $|x-1|$ is non-differentiable at $x=1$ , students incorrectly conclude that the entire determinant function $f(x)$ is non-differentiable (Option ND).

However, the cofactor of $|x-1|$ is $x^2-x$ , which evaluates to 0 at $x=1$ .

Expanding the determinant first reveals that the non-differentiable term is multiplied by a function that smooths it out at $x=1$ , making $f(x)$ differentiable.

This forces students to analyze the algebraic structure of the function rather than relying on mechanical differentiation rules.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Algebra of Functions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free