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Question #41225

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the particular solution to the exact differential equation $(2x y e^{x^2})dx + (e^{x^2} + 2y)dy = 0$ satisfying $f(0) = 1$ . Let the function $g(x)$ be defined as $g(x) = \lim_{n \to \infty} \frac{246 x^{2n} + \frac{1}{246}}{x^{2n} + 1}$ Define the matrix $A(x)$ as $A(x) = \begin{pmatrix} f(x) & f(g(x)) \\ e^{x^2} & e^{(g(x))^2} \end{pmatrix}$ The product of all real values of $x$ for which the matrix $A(x)$ is singular is:

Options

A.
$1$
B.
$-1$
C.
$246^2$
D.
$246^{-2}$

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

Singular vs Non-Singular Matrices|Exact Differential Recognition|Composition Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of exact differential equations, functional analysis, and limit-defined piecewise functions.

The 'Lethal Collision' occurs when students assume the composition $f(g(x))$ inherits the global properties of $f(x)$ without analyzing the jump discontinuities of $g(x)$ .

The exact DE yields a function $f(x)$ that makes the matrix determinant equation reduce to $|x| = |g(x)|$ .

Students often fall into the trap of assuming $g(x)$ is continuous or ignoring the absolute value, missing the negative roots or the roots in the $|x| < 1$ domain.

The '246' constants act as a distractor, but carefully evaluating the piecewise limit reveals exactly four symmetric roots whose product elegantly collapses to 1.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Singular vs Non-Singular Matrices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free