Question 171 markMCQSection A

The feasible region of a linear programming problem with objective function Z=5x+7yZ = 5x + 7y is shown below. The maximum value of Z - minimum value of Z is

Feasible region of a linear programming problem with vertices (0,2), (3,4) and (7,0)
Linear Programming
Linear Programming - Feasible Region

Options

(A)8
(B)29
(C)35
(D)43

Official answer

The correct option is (D) 43.


Reasoning: Evaluating Z=5x+7yZ = 5x + 7y at each corner point of the feasible region, the maximum occurs at (3,4)(3, 4) giving Z = 43, and the minimum occurs at the origin (0,0)(0, 0) giving Z = 0, so Max Z−Min Z=43−0=43\text{Max } Z - \text{Min } Z = 43 - 0 = 43.

Answer keywords:linear programmingfeasible regionobjective functioncorner point theoremoptimal valueZ = 5x + 7ymaximum and minimum value

Marking scheme

  1. 1.1 mark: correct option (D) 43 — no partial marking for MCQ.

Hint

Evaluate Z at every corner (vertex) of the shaded feasible region and subtract the smallest value from the largest.

Quick oral answer

By the Corner Point Theorem, the maximum and minimum of a linear objective function on a bounded feasible region occur at its vertices, so I evaluate Z at each corner and subtract the smallest from the largest.

Common mistakes

  1. 1.Evaluating Z at only one or two corner points instead of all vertices of the feasible region.
  2. 2.Confusing the maximum value alone (43) with the required difference (Max Z−Min Z\text{Max } Z - \text{Min } Z).
  3. 3.Misreading the coordinates of a vertex from the graph, leading to an incorrect Z value.

Analysis and explanation

This is a corner-point-theorem application on a bounded feasible region.


Why (D) is correct: By the Corner Point Theorem, the optimal (maximum/minimum) value of a linear objective function over a bounded feasible region occurs at one of its corner points. For the region shown, evaluating Z=5x+7yZ = 5x + 7y at each vertex — (0,0),(7,0),(3,4),(0,2)(0,0), (7,0), (3,4), (0,2) — gives Z = 0, 35, 43, and 14 respectively. The maximum is 43 (at (3,4)) and the minimum is 0 (at the origin), so Max Z−Min Z=43\text{Max } Z - \text{Min } Z = 43.


Why the distractors are wrong:

  • (A) 8 and (B) 29 do not correspond to any valid pair of corner-point Z-values for this region; they likely result from picking the wrong pair of vertices or an arithmetic slip.
  • (C) 35 is the value of Z at (7,0) alone, not the required difference — a common error is stopping after computing Z at just one boundary vertex instead of comparing all corners.

Interesting facts

The Corner Point Theorem (also called the Fundamental Theorem of Linear Programming) guarantees that for a bounded feasible region, the optimal value of a linear objective function always occurs at a vertex, never strictly inside the region — a fact that follows from the convexity of the feasible region.

Linear programming was developed by George Dantzig in 1947 as the Simplex Method to solve resource-allocation problems for the US Air Force; CBSE's graphical corner-point method is a simplified two-variable version of this same idea.

Question details

Q17 of 44 questions

Board
CBSE
Class
Class 12
Subject
Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Linear Programming
Topic
Linear Programming - Feasible Region
Bloom's level
apply
Syllabus topic
Unit 4: Linear Programming — Graphical Method, Corner Point Theorem
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What is the Corner Point Theorem used to solve this MCQ?

It states that for a bounded feasible region, the maximum and minimum values of a linear objective function occur at one of the corner points (vertices) of the region, so you only need to evaluate Z at each vertex rather than every point in the region.

Why is the minimum value taken as 0 in this feasible region?

Because the origin (0,0)(0,0) is a vertex of this particular bounded feasible region, and Z=5x+7yZ = 5x + 7y evaluates to 0 there, which is the smallest of all the corner values.