Q.The corner points of the feasible region determined by the system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. The objective function is $Z = 4x + 3y$. Compare the quantity in Column A and Column B, where Column A is the Maximum of $Z$ and Column B is $325$.
(A) The quantity in column A is greater
(B) The quantity in column B is greater
(C) The two quantities are equal
(D) The relationship cannot be determined on the basis of the information supplied

Concept: Linear Programming Graphical Method — the maximum of $Z$ occurs at a corner point of the feasible region.

Step 1: Evaluate $Z = 4x + 3y$ at each given corner point.

• $(0, 0)$: $Z = 4(0) + 3(0) = 0$
• $(0, 40)$: $Z = 4(0) + 3(40) = 120$
• $(20, 40)$: $Z = 4(20) + 3(40) = 80 + 120 = 200$
• $(60, 20)$: $Z = 4(60) + 3(20) = 240 + 60 = 300$
• $(60, 0)$: $Z = 4(60) + 3(0) = 240$ …

The graphical method for linear programming rests on a beautiful geometric fact: when you have a linear objective function and a convex feasible region (a polygon), the optimum — maximum or minimum — will always occur at one of the vertices (corner points). Why? Because the objective function represents a family of parallel lines; as you slide them in the direction of increase, the last point of contact with the polygon is always a corner. So you never need to check interior points — just test the vertices.

Here we are given five corner points and the objective function $Z = 4x + 3y$. Column A is the maximum value of $Z$ over this region. Column B is the fixed number $325$. The question is simply: which is larger?

Let’s evaluate $Z$ at each corner.

1. At $(0, 0)$:

   $Z = 4(0) + 3(0) = 0$

2. At $(0, 40)$:

   $Z = 4(0) + 3(40) = 120$

3. At $(20, 40)$:

   $Z = 4(20) + 3(40) = 80 + 120 = 200$

4. At $(60, 20)$:

   $Z = 4(60) + 3(20) = 240 + 60 = 300$

5. At $(60, 0)$:

   $Z = 4(60) + 3(0) = 240$

The largest value among these is $300$, which occurs at $(60, 20)$. …

Method: Optimising over given corner points (corner-point evaluation)

When a question already hands you the corner points of a bounded feasible region, you never sketch anything — you just test the objective at each vertex. This is the corner-point method that underlies every CBSE Class 12 Linear Programming solution.

Steps

Step 1: Trust the Corner Point Theorem.

For a linear objective over a bounded convex polygon, the maximum and minimum each occur at a vertex. No interior or edge point can beat the best corner, so the finite list of corners is all you need.

Step 2: Substitute every corner into $Z=ax+by$.

$Z=ax+by$ …

Common Mistakes

Mistake 1: Guessing the maximum is at the point "farthest from the origin."

Why it's wrong: the optimum depends on the objective's coefficients, not on distance. Correct approach: substitute each corner into $Z=4x+3y$ and compare the actual values.

Mistake 2: Reporting the maximum of $Z$ instead of answering the comparison asked. …