Before You Begin

Question #41587

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be a $3 \times 3$ orthogonal matrix and $N$ be a $3 \times 3$ nilpotent matrix of index 3. Consider the differential equation given by $\det \left( P^T \left[ \left( \frac{d^3y}{dx^3} \right)^{\frac{2}{3}} I + \left( \frac{d^2y}{dx^2} \right)^{\frac{3}{4}} N + \left( \frac{dy}{dx} \right)^{\frac{4}{5}} N^2 \right] P \right) = x^2 + 1$ where $I$ is the $3 \times 3$ identity matrix. If $m$ and $n$ are the order and degree of this differential equation respectively, find the value of $m + n$ .

Options

A.
5
B.
11
C.
23
D.
43

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Order and Degree Definition|Orthogonal Matrix Conditions|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize matrix algebra with the definitions of order and degree in differential equations.

The primary trap lies in the fractional powers of the lower derivatives.

A student who incorrectly expands the determinant or fails to recognize that the nilpotent terms vanish will attempt to rationalize the fractional powers (e.g., $3/4$ and $4/5$ by raising the equation to the power of 20), leading to an artificially inflated degree (such as 40, yielding an answer of 43).

By applying the spectral properties of nilpotent matrices, the complex operator collapses into a simple second-degree, third-order equation, rewarding conceptual rigor over blind algebraic manipulation.

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