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Question #40967

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x) = \begin{pmatrix} x & |x|^3 \\ 0 & -x \end{pmatrix}$ . Let $Y(x)$ be a $2 \times 2$ matrix-valued function satisfying the differential equation $\frac{dY}{dx} = M(x) Y(x)$ with the initial condition $Y(0) = I$ , where $I$ is the $2 \times 2$ identity matrix. Let $f(x)$ be the entry in the first row and second column of $Y(x)$ . If $k$ is the maximum integer such that $f(x)$ is $k$ -times continuously differentiable on $\mathbb{R}$ , find the ordered pair $\left(k, f(\sqrt{2})\right)$ .

Options

A.
$\left(3, \frac{e^2 - 3}{2e}\right)$
B.
$\left(3, \frac{e^2 - 1}{2e}\right)$
C.
$\left(2, \frac{e^2 - 3}{2e}\right)$
D.
$\left(4, \frac{e^2 - 1}{2e}\right)$

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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

Polynomial Continuity/Differentiability Link, General vs Particular Solutions, Locus of a Moving Point, Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of matrix differential equations and the continuity/differentiability of piecewise polynomial-exponential functions.

A common trap is assuming the solution to $\frac{dY}{dx} = M(x)Y$ is $Y(x) = e^{\int M(x) dx}$ .

This is only true if $[M(x), \int M(x) dx] = 0$ , which fails here.

Students falling for this trap will incorrectly obtain $f(\sqrt{2}) = \frac{e^2-1}{2e}$ .

Furthermore, analyzing the differentiability of $f(x)$ requires recognizing the behavior of $|x|^3$ under integration, leading to a function that is exactly 3 times continuously differentiable at the origin.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Polynomial Continuity/Differentiability Link
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free