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Question #40308

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $x$ satisfying the inequality $\sqrt{x^2 - 4x + 3} \le 2 - |x - 2|$ If $S$ is expressed as the union of disjoint intervals, let $L$ be the sum of the lengths of these intervals. Find the value of $L$ .

Options

A.
$\frac{1}{2}$
B.
$\frac{5}{2}$
C.
$\frac{1}{4}$
D.
2

Check Your Answer

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

Set Complement Boundaries|Modulus Inequality Logic|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question deploys a 'Lethal Collision' trap by combining a modulus inequality with a strict domain restriction from a square root.

Students frequently focus entirely on the algebraic manipulation of the modulus (case-splitting or squaring) and forget to intersect their final solution with the domain of the square root ( $x^2 - 4x + 3 \ge 0$ ).

If the domain constraint is ignored, the algebraic solution yields the continuous interval $[3/4, 13/4]$ with a length of $5/2$ , which is prominently placed as a distractor.

The correct path requires recognizing the set complement boundary (the 'hole' in the domain between 1 and 3) which fractures the solution into two disjoint intervals, drastically changing the total length.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Complement Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free