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Question #41348

Difficulty: 8/10 | 🟢 Free

The Question

Let the curve $C$ be defined by $y = f(x)$ , where $f(x) = \begin{vmatrix} \sin(\pi x) & \cos(\pi x) & 2e^{x-1} \\ -\cos(\pi x) & \sin(\pi x) & \ln(x) \\ -2e^{x-1} & -\ln(x) & \tan(\pi x) \end{vmatrix}$ Let $N(x)$ be the function representing the normal line to the curve $C$ at $x = 1$ . Find the value of $N(1 + 3\pi)$ .

Options

A.
1
B.
-1
C.
$9\pi^2$
D.
$-9\pi^2$

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

Tangent/Normal Slope Interplay|Differentiation of Determinants|Skew-Symmetric Determinant

Why This Question Is Tricky

Common Mistakes

This problem tests the differentiation of determinants and the interplay between tangent and normal slopes, while setting a trap using skew-symmetric matrices.

At $x=1$ , the matrix is skew-symmetric, so its determinant is zero.

Students often incorrectly assume that because the matrix evaluates to a skew-symmetric form at a point, the function itself is identically zero or has a zero derivative, leading them to conclude the tangent is horizontal and the normal is undefined.

By requiring the evaluation of the normal line at a specific point, the question forces students to correctly differentiate the determinant (either by expansion or Jacobi's formula) to find the non-trivial slope, rewarding conceptual depth over rote memorization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Tangent/Normal Slope Interplay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free