A feasible region of a system of linear inequalities is said to be _____ if it can be enclosed within a circle

NCERT Exemplar · Q30

Q. A feasible region of a system of linear inequalities is said to be _________ if it can be enclosed within a circle.

Sikkim CbseShort· 1mImportance★★★★★

67% · 45/67 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Unbounded Feasible Region

Unbounded Feasible Region

Stand in an open field and you can walk as far as you like in some direction without ever hitting a wall. Stand in a closed hall and walls stop you on every side. A feasible region in linear programming is the set of all points satisfying every constraint; calling it unbounded means that in at least one direction you can keep moving forever and still stay inside it.

Note

Unbounded does not mean "huge area" — it means the region has no boundary in some direction, so it stretches off to infinity there.

How a region becomes unbounded

Each constraint is a linear inequality such as $x \geq 0$ , $y \geq 0$ , or $x + y \leq 10$ , and each cuts the plane into a half-plane. The feasible region is the overlap of these half-planes. If the constraints fail to "close off" the region in some direction, it runs on without end.

Example. With only $x \geq 0$ and $y \geq 0$ , the region is the entire first quadrant — you can travel as far as you wish along either axis, so it is unbounded. Add $x + y \leq 10$ and the region shrinks to the triangle with vertices $(0,0)$ , $(10,0)$ , $(0,10)$ — now bounded.

Why it matters for the objective

In an LP problem you maximise or minimise $Z = ax + by$ over the feasible region.

If the region is bounded, both the maximum and the minimum of $Z$ exist and each occurs at a corner point.
If the region is unbounded, an optimum may fail to exist — you might push $Z$ larger and larger by moving out along the open direction.

Watch out

An unbounded region does not automatically destroy the answer. For $Z = -x + y$ on the first quadrant, as $x \to \infty$ the term $-x$ drives $Z \to -\infty$ , so a maximum can still sit at a corner. Always compare the objective's direction with the open direction of the region.

Exam technique

Graph the constraints and evaluate $Z$ at every corner point. …

Start your 14-day free trial to unlock the full solution →

The key idea is that a feasible region which can be enclosed within a circle is bounded — it does not extend infinitely in any direction. The answer is bounded.

Why This Matters: The Shape of Feasibility

When you solve a system of linear inequalities, the set of all points that satisfy every inequality is called the feasible region. This region can take two fundamental shapes:

Bounded: The region is "closed in" — you can draw a circle (or any finite shape) around it, and the entire region lies inside. Think of a polygon or a triangle.
Unbounded: The region stretches out to infinity in at least one direction. No matter how large a circle you draw, part of the region will always be outside it.

The question is asking for the word that describes a feasible region that can be enclosed within a circle. That word is bounded.

Step-by-Step Reasoning

Understand the definition of "bounded" in geometry.

A set of points in the plane is called bounded if there exists some finite number $R > 0$ such that the entire set lies inside a circle of radius $R$ centered at some point. In other words, you can "fence it in" with a finite boundary.
Apply this to a feasible region.

A feasible region is the intersection of half-planes (from linear inequalities). If this intersection is a closed polygon (like a triangle, rectangle, or any shape with finite area), then you can always find a circle large enough to cover it. Such a region is bounded.
Contrast with the opposite case. …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ29 — In a LPP if the objective function Z = ax + by has the same maximum va…→Next questionQ31 — A corner point of a feasible region is a point in the region which is …

Q.A feasible region of a system of linear inequalities is said to be _________ if it can be enclosed within a circle.

Concept understanding — Unbounded Feasible Region

Unbounded Feasible Region

How a region becomes unbounded

Why it matters for the objective

Exam technique

Why This Matters: The Shape of Feasibility

Step-by-Step Reasoning

Method: Applying the definition of a bounded region

The reasoning pattern

Common Mistakes

Q.A feasible region of a system of linear inequalities is said to be _________ if it can be enclosed within a circle.

Why This Matters: The Shape of Feasibility

Step-by-Step Reasoning

🎓Unlock everything free for 14 days

Unlock everything free for 14 days