Before You Begin

Question #41084

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be the solution to the differential equation $x \frac{dy}{dx} = 1$ on the domain $x \in \mathbb{R} \setminus \{0\}$ , satisfying $y(e) = 1$ . A real number $x$ is chosen uniformly at random from the set $S = [-e^2, e^3] \setminus \{0\}$ . Let $E_1$ be the event that $y(x) \ge 1$ and $E_2$ be the event that $y(x) \le 2$ . Find the conditional probability $P(\overline{E_1} \cup \overline{E_2} \mid x < e^2 + e)$ .

Options

A.
$\frac{3}{2e + 1}$
B.
$\frac{2}{e + 1}$
C.
$\frac{2e - 2}{2e + 1}$
D.
$\frac{e - 1}{e + 1}$

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

De Morgan’s Laws in Probability|Conditional Probability Foundations|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of differential equations, geometric probability, and De Morgan's Laws.

The primary trap lies in the integration of $1/x$ .

Students frequently forget the modulus, incorrectly yielding $y(x) = \ln x$ instead of $\ln|x|$ .

This restricts their perceived domain to $x > 0$ , prematurely truncating the sample space and the favorable event space.

When they subsequently apply De Morgan's Law ( $\overline{E_1} \cup \overline{E_2} = \overline{E_1 \cap E_2}$ ) to find the conditional probability, the missing negative interval $[-e^2, -e]$ skews the ratio of lengths, leading directly to the distractor option $\frac{2}{e+1}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free