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Question #41609

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} -3 & -4 & 4 \\ -3 & -1 & 3 \\ -9 & -8 & 10 \end{pmatrix}$ and $X(t)$ be the solution to the system of differential equations $\frac{dX}{dt} = A X$ with initial condition $X(0) = \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$ . A vector $Y(t)$ is defined implicitly by the relation $M(t) Y(t) = X(t)$ , where $M(t) = A^2 - t A + (t^2 - 7)I$ . Let $S$ be the set of all distinct times $t > 0$ for which the explicit conversion to $Y(t)$ fails (i.e., $M(t)$ is singular). If $\alpha = \sum_{t \in S} t$ , find the value of $\frac{1}{\alpha^2} \sqrt{\text{Tr}(X(\ln \alpha) X^T(\ln \alpha))}$ .

Options

A.
42
B.
21
C.
252
D.
126

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Implicit to Explicit Conversion|Characteristic Equation|Inversion and Determinant Product

Why This Question Is Tricky

Common Mistakes

This problem tests the connection between systems of differential equations, matrix exponentials, and the spectral mapping theorem.

The primary trap lies in the implicit-to-explicit conversion $Y(t) = M(t)^{-1}X(t)$ .

Students often attempt to compute the determinant of the $3 \times 3$ matrix $M(t)$ algebraically, leading to a massive degree-6 polynomial in $t$ that is highly prone to calculation errors.

The elegant approach uses the property that the eigenvalues of $M(t)$ are $P(\lambda_i)$ , where $P(x) = x^2 - tx + t^2 - 7$ .

Setting these to zero identifies the singularities.

Furthermore, recognizing $X(0)$ as an eigenvector bypasses the need to compute the full matrix exponential $e^{At}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Implicit to Explicit Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free