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Question #41200

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a twice differentiable, strictly decreasing function on $\mathbb{R}$ . For all $x \in \mathbb{R}$ , the rows of the matrix $M = \begin{pmatrix} y'' & e^x & 1 \\ |y'| & 1 & 0 \\ y & x & 1 \end{pmatrix}$ are linearly dependent. If $m$ and $n$ are the order and degree of the differential equation satisfied by $y = f(x)$ respectively, then find the value of $2m + 3n$ .

Options

A.
7
B.
10
C.
8
D.
5

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|Linear Dependency of Rows|Order and Degree Definition|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of matrix singularity with the strict definition of the degree of a differential equation.

The primary trap lies in the presence of the modulus function $|y'|$ within the determinant expansion.

Students commonly assume that to find the degree, they must globally rationalize the equation by isolating the modulus and squaring both sides, which incorrectly yields a degree of 2.

However, by utilizing the given condition that the function is strictly decreasing, the modulus resolves linearly ( $|y'| = -y'$ ).

This reveals that the differential equation is actually linear, possessing a degree of 1.

Failing to apply the domain-specific behavior of the modulus before determining the degree is a classic JEE Advanced pitfall.

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