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Question #40746

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the region in the $xy$ -plane defined by the inequalities $|x - 2y| \le 4$ and $|2x + y| \le 5$ . If a point $P(x, y)$ is chosen at random from the region $R$ , the probability that $P$ satisfies the inequality $|x + y| \ge 2$ is

Options

A.
$\frac{1}{6}$
B.
$\frac{1}{3}$
C.
$\frac{2}{3}$
D.
$\frac{5}{6}$

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

Point-Line Relative Position|Modulus Function Breakdown|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' in 2D coordinate geometry and probability.

Students frequently drop the absolute value prematurely, solving only for $x+y \ge 2$ and ignoring the symmetric region $x+y \le -2$ .

This leads to calculating only half the valid area (yielding the trap answer 1/6).

Furthermore, the problem requires a solid grasp of linear transformations (Jacobian) or advanced locus geometry to evaluate areas of regions bounded by non-standard parallel lines, discriminating rote learners from those with deep geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Point-Line Relative Position
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free