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CBSE Board Examination 2026 · Section A · 1 mark

Q17: Maximum minus minimum of Z=5x+7yZ = 5x + 7y on a feasible region

Board 2026Ch 12 · Linear Programming
The question

The feasible region of a linear programming problem with objective function Z=5x+7yZ = 5x + 7y is shown below :

0 1 2 3 4 5 6 7 1 2 (3, 4)

The maximum value of Z − minimum value of Z is

(A) 88

(B) 2929

(C) 3535

(D) 4343

Exam answer

The corner points of the feasible region are (0,0)(0, 0)⁠, (0,2)(0, 2)⁠, (3,4)(3, 4) and (7,0)(7, 0)⁠.

Why?

Corner point method: when the feasible region is bounded, the maximum and the minimum of ZZ both occur at corner points.

Read the corners from the figure: the origin, (0,2)(0, 2) on the yy-axis, the marked point (3,4)(3, 4)⁠, and (7,0)(7, 0) on the xx-axis.

Z(0,0)=0Z(0,2)=14Z(3,4)=15+28=43Z(7,0)=35\begin{aligned} Z(0, 0) &= 0 \\ Z(0, 2) &= 14 \\ Z(3, 4) &= 15 + 28 = 43 \\ Z(7, 0) &= 35 \end{aligned}
Why?

Substitute each corner into Z=5x+7yZ = 5x + 7y⁠:

Z(0,0)=5(0)+7(0)=0+0=0Z(0,2)=5(0)+7(2)=0+14=14Z(3,4)=5(3)+7(4)=15+28=43Z(7,0)=5(7)+7(0)=35+0=35\begin{aligned} Z(0, 0) &= 5(0) + 7(0) \\ &= 0 + 0 = 0 \\[4pt] Z(0, 2) &= 5(0) + 7(2) \\ &= 0 + 14 = 14 \\[4pt] Z(3, 4) &= 5(3) + 7(4) \\ &= 15 + 28 = 43 \\[4pt] Z(7, 0) &= 5(7) + 7(0) \\ &= 35 + 0 = 35 \end{aligned}

The largest of 0, 14, 43, 35 is 43. The smallest is 0.

Maximum Z=43Z = 43⁠, minimum Z=0Z = 0⁠, so the difference is 43−0=4343 - 0 = 43⁠.

Why?

Watch out: the origin is a corner of the shaded region, so the minimum is 0. Missing it gives 43−14=2943 - 14 = 29⁠, option (B).

The question asks for maximum minus minimum, not for the maximum alone.

Answer: (D) 4343

Where this comes from

Chapter 12, Linear Programming: Corner point method.

Question type: Both maximum and minimum, and a tie along an edge

How to spot it: click to see

Type 3. Both maximum and minimum, and a tie along an edge

Spot it: "Minimise and maximise ZZ". Often one rule, like x−2y≥0x - 2y \ge 0⁠, is a line through the origin.

Here: Read the corners off the graph: (0,0)(0, 0)⁠, (0,2)(0, 2)⁠, (3,4)(3, 4)⁠, (7,0)(7, 0)⁠. One table of Z=5x+7yZ = 5x + 7y gives both the maximum, 43, and the minimum, 0, so the difference is 43.

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