Which of the following sets is not convex?

Correct option is A

Solution:

$$\textbf{Option A: } X = \{ (x, y) \mid x^2 + y^2 \geq 5 \} \\\text{This represents the exterior (including boundary) of a circle centered at the origin.} \\\text{It is not convex because a line segment joining two points in this region may pass through the interior of the circle,} \\\text{which is not included in the set. So, this set is} \, \textbf{not convex}. \\\ \\\textbf{Option B: } X = \{ (x, y) \mid |x| \leq 3, \, |y| \leq 2 \} \\\text{This represents a closed rectangle. The line segment joining any two points in this set lies entirely within the set.} \\\Rightarrow \textbf{Convex} \\\ \\\textbf{Option C: } X = \{ (x, y) \mid 5x - y = 4 \} \\\text{This is a straight line, and all lines are convex sets.} \\\Rightarrow \textbf{Convex} \\\ \\\textbf{Option D: } X = \{ (x, y) \mid 2x - y \leq 3 \} \\\text{This is a half-plane (including the boundary line), and half-planes are convex.} \\\Rightarrow \textbf{Convex} \\\\\boxed{\text{Correct Answer: Option A (not convex)}}$$

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