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Question #40958

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} 0 & x & 1 & 241 \\ -x & 0 & 1 & x \\ -1 & -1 & 0 & x^2 \\ -241 & -x & -x^2 & 0 \end{vmatrix}$ .

If $g(x) = (f(x))^{1 + \sin^2(\pi x)}$ , find the value of $g'(1)$ .

Options

A.
$964$
B.
$482$
C.
$2$
D.
$0$

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

Logarithmic Differentiation|Skew-Symmetric Determinant|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the student's meta-knowledge of determinant properties, specifically that the determinant of an even-ordered skew-symmetric matrix is the square of its Pfaffian.

If a student attempts to differentiate the determinant directly using Jacobi's formula or row-wise addition, the algebraic complexity becomes an insurmountable nightmare, especially when compounded by the exponential function $1 + \sin^2(\pi x)$ .

The trap lies in the exponent: students who fail to use logarithmic differentiation or miss the perfect square property will either get bogged down in massive polynomial expansions or incorrectly assume the determinant evaluates to just the Pfaffian (yielding the distractor 2).

The '241' acts as a numerical bait to make manual expansion look even more intimidating.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Logarithmic Differentiation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free