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Question #40740

Difficulty: 8/10 | 🟢 Free

The Question

Let $D$ be the set of all real numbers $m$ satisfying the inequality $\log_{0.5}(m-1) \ge -1$ . Let $R$ be the region in the first quadrant consisting of all points $(x, y)$ such that $x \ge 1$ and $0 \le y \le \sup_{m \in D} \left( \frac{2m}{x^2} - \frac{m^2}{x^3} \right)$ If $A$ is the area of the region $R$ , find the value of $A$ .

Options

A.
$\ln 3 + \frac{3}{2}$
B.
2
C.
$\frac{3}{2}$
D.
\text{Unbounded}

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

General Equation Transformations|Bounded vs Unbounded Regions|Sign Flipping Rule|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question engineers a 'Lethal Collision' between the sign-flipping rule in logarithmic inequalities and compound function boundaries.

Students must first determine the parameter domain $D$ .

Failing to flip the inequality sign for the base $0.5 < 1$ yields the incorrect domain $m \in [3, \infty)$ .

When finding the envelope $h(x) = \sup f(x,m)$ , this error places the vertex $m=x$ inside the domain for all $x \ge 3$ , resulting in $h(x) = 1/x$ and an unbounded area (Option D).

Furthermore, students who incorrectly swap the order of integration and supremum ( $\sup \int f(x,m) dx$ ) will obtain a finite area of 2 (Option B), punishing a lack of rigorous understanding of uniform convergence and order of operations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	General Equation Transformations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free