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Question #40749

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in the $xy$ -plane defined by the system of inequalities: $y \le 3x$ $2y \ge x$ $2x + y \ge 8$ If the set of all possible values of $\alpha$ for which the point $(\alpha, \alpha^2)$ lies in the region $R$ is the interval $[m, n]$ , then the value of $m+n$ is:

Options

A.
$5$
B.
$\frac{5}{2}$
C.
$\frac{24}{5}$
D.
$\frac{7}{2}$

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

Point-Line Relative Position|System of Inequalities|Graphical Feasible Region

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the relative position of a point (constrained to a parabola) within a system of linear inequalities forming an unbounded feasible region.

The lethal trap occurs when students incorrectly assume the region must be a closed bounded polygon (the triangle formed by the lines) and reverse the third inequality to $2x + y \le 8$ , leading to the interval $[1/2, 2]$ and the trap answer $5/2$ .

Another common error is bypassing the inequality intersection entirely and merely summing the x-coordinates of the region's vertices ( $8/5$ and $16/5$ ), leading to $24/5$ .

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