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Question #41583

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, \dots, z_6$ be the distinct roots of the equation $z^6 - 7iz^3 - 12 = 0$ . Consider the differential equation $\left( \sum_{k=1}^{6} |z_k|^2 \right) \det(M) + e^{\frac{d^2y}{dx^2}} + x \frac{dy}{dx} = 0$ where $M$ is the matrix $M = \begin{pmatrix} 0 & i \left(\frac{d^4y}{dx^4}\right)^3 & \left(\frac{dy}{dx}\right)^2 \\ -i \left(\frac{d^4y}{dx^4}\right)^3 & 0 & x^3 \\ -\left(\frac{dy}{dx}\right)^2 & -x^3 & 0 \end{pmatrix}$ Which of the following represents the order and degree of the given differential equation?

Options

A.
Order: $4$ , Degree: $3$
B.
Order: $4$ , Degree: Not defined
C.
Order: $2$ , Degree: $1$
D.
Order: $2$ , Degree: Not defined

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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

Order and Degree Definition|Skew-Symmetric Determinant|Algebraic Manipulation of i|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify extraneous complex information and apply fundamental matrix properties before evaluating differential equation parameters.

The primary trap is the intimidating $3 \times 3$ matrix containing high-order derivatives and complex terms, paired with a sum of moduli of complex roots.

Students who fail to recognize that $M$ is an odd-order skew-symmetric matrix (whose determinant is identically zero) will waste time calculating the roots and incorrectly expanding the determinant, leading them to falsely identify the order as 4.

Furthermore, the secondary trap lies in the definition of degree: even if they simplify the equation to $e^{y''} + xy' = 0$ , they might incorrectly state the degree as 1 instead of recognizing that the transcendental function of the derivative makes the degree undefined.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Order and Degree Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free