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Question #41919

Difficulty: 8/10 | 🟢 Free

The Question

Let $X(t)$ be a $3 \times 3$ matrix of differentiable functions satisfying the differential equation $X'(t) = A(t)X(t) + X(t)B(t)$ for $t \in (0, \pi)$ , with the initial condition $X\left(\frac{\pi}{2}\right) = I_3$ . Given that: $A(t) = \begin{pmatrix} t & \sin t & \cos t \\ -\sin t & -t & e^t \\ -\cos t & -e^t & \frac{2t}{t^2+1} \end{pmatrix}, \quad B(t) = \begin{pmatrix} \cot t & t^2 & \ln t \\ -t^2 & \cot t & \sqrt{t} \\ -\ln t & -\sqrt{t} & \cot t \end{pmatrix}$ If $f(t) = \det(X(t))$ , find the value of $\lim_{t \to 0^+} \frac{f(t)}{t^3}$ .

Options

A.
$\frac{4}{\pi^2+4}$
B.
$\frac{2}{\pi^2+4}$
C.
$\frac{8}{\pi^2+4}$
D.
0

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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

Logarithmic Integration Modulus|Differentiation of Determinants|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to differentiate a determinant of a matrix product/sum without explicitly solving the matrix differential equation.

The primary trap lies in the non-commutativity of matrices: students often attempt to solve $X'(t) = A(t)X(t) + X(t)B(t)$ by assuming scalar-like exponential integration $X(t) = e^{\int A} e^{\int B}$ , which is mathematically invalid since $A(t)$ and $B(t)$ do not commute with their own integrals or with each other.

This leads to a non-existent or incorrect solution set.

The correct approach bypasses finding $X(t)$ entirely by applying Jacobi's formula and the cyclic property of the trace operator, reducing a complex non-commutative matrix problem into a straightforward scalar logarithmic integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Logarithmic Integration Modulus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free