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Question #70541

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & b & c \end{pmatrix}$ be an orthogonal matrix with $\det(A) = 1$ . Let $B = 3A - 3I$ , where $I$ is the $3 \times 3$ identity matrix. A differentiable vector function $\mathbf{x}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{pmatrix}$ satisfies the differential equation $\frac{d\mathbf{x}}{dt} = B \mathbf{x}(t) + \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ with the initial condition $\mathbf{x}(0) = \begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$ . Find the value of $\lim_{t \to \infty} \left( 3x_1(t) + 6x_2(t) + 12x_3(t) \right)$ .

Options

A.
8
B.
10
C.
12
D.
Does not exist

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

General vs Particular Solutions|Orthogonal Matrix Conditions|Singular vs Non-Singular Matrices|Linear Dependency of Rows

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to handle systems of differential equations where the coefficient matrix is singular.

A common trap is assuming that the steady-state particular solution can be found using $\mathbf{x}_p = -B^{-1}\mathbf{v}$ .

Since $\det(B) = 0$ (a direct consequence of $A$ being a rotation matrix, meaning it has an eigenvalue of 1), $B$ is non-invertible.

Students who blindly apply the inverse formula will incorrectly conclude that no steady state exists.

Furthermore, failing to use the $\det(A)=1$ condition leads to the wrong matrix $B$ , which actually results in a system with no steady state, punishing incomplete analysis of the orthogonal matrix properties.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	General vs Particular Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free