CBSE Sample Paper 2023-24 · Section A · 1 mark

Q11: Which inequality is not a constraint of the shaded feasible region

Sample 2023-24Ch 12 · Linear Programming

The question

The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below.

The given graph, redrawn.

Which of the following is not a constraint to the given Linear Programming Problem?

(a) $x + y \ge 2$

(b) $x + 2y \le 10$

(d) $x - y \le 1$

Exam answer

The shaded region lies on the same side of $x - y = 1$ as the origin $O(0, 0)$ ⁠.

Why?

A line $ax + by = c$ splits the plane into two half-planes. On one side $ax + by < c$ ⁠, and on the other $ax + by > c$ ⁠.

To find which inequality a region satisfies, test one point that is not on the line. The origin is the easiest.

At $O$ ⁠, $x - y = 0 < 1$ ⁠, so the region satisfies $x - y \le 1$ ⁠, not $x - y \ge 1$ ⁠.

Why?

Check with a corner of the region, $(0, 2)$ ⁠: $x - y = 0 - 2 = -2 \le 1$ ⁠, so $x - y \ge 1$ fails there.

Watch out: points on the line itself satisfy both $\le$ and $\ge$ ⁠, so always test a point off the line.

For $x + y = 2$ ⁠: at $O$ ⁠, $x + y = 0 < 2$ ⁠. The region is on the other side, so $x + y \ge 2$ ⁠.

For $x + 2y = 10$ ⁠: at $O$ ⁠, $x + 2y = 0 < 10$ ⁠. The region is on the same side, so $x + 2y \le 10$ ⁠.

So (a), (b) and (d) are constraints, and (c) is not.

Why?

Check with the corner $(0, 5)$ ⁠: $0 + 5 = 5 \ge 2$ and $0 + 2(5) = 10 \le 10$ ⁠.

The question asks for the inequality that is not a constraint, so the answer is (c).

Answer: (c) $x - y \ge 1$

Where this comes from

Chapter 12, Linear Programming: Reading constraints from a feasible region.

Question type: Read the rules from a graph

How to spot it: click to see

Type 7. Read the rules from a graph

Spot it: the graph of the region is given, and the question asks for its constraints, or for the rule that is missing.

Here: The region lies on the origin's side of $x - y = 1$ ⁠, where $x - y = 0 \lt 1$ ⁠, so its rule is $x - y \le 1$ ⁠. Checked the same way, (a), (b) and (d) are rules, so (c) is not.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Linear Programming in past papers: question 22 of 27 · see them all