PYQ (JEE Main & Advanced)

High-Yield JEE Main Concepts for Linear Programming

1. Bounded vs. Unbounded Regions

In JEE Main, questions often test whether the feasible region is bounded or unbounded. A bounded feasible region is enclosed and finite, so a linear objective function always attains both a maximum and a minimum at some corner point of the region. An unbounded feasible region extends indefinitely in at least one direction, but this does not automatically imply that a maximum or minimum does not exist. The existence of an optimum depends on the direction in which the objective function increases or decreases. For example, in an unbounded region, a minimization problem may still have a finite minimum if the objective function cannot decrease indefinitely inside the region.

2. Infinite Optimal Solutions

A very important JEE concept is the case of multiple optimal solutions. If the objective function $Z = ax + by$ has the same optimal value at two adjacent corner points of the feasible region, then every point on the line segment joining those two corner points also gives the same optimal value. This happens because the objective-function line becomes parallel to that side of the feasible region.

3. Infeasible Regions

Sometimes the constraints contradict each other, producing no common region. In such cases, the feasible region is empty, and no feasible solution exists. For instance, if one inequality demands the solution to lie below one line and another demands it to lie above a parallel or conflicting line in such a way that no overlap is possible, then neither a maximum nor a minimum exists because there is no admissible point.

4. Parameter Relations (Finding $p$ and $q$)

A common PYQ pattern gives a feasible region and states that the maximum or minimum of $Z = px + qy$ occurs at two specified corner points. In that case, the objective function must take equal values at those two points. So if the optimum occurs at $(x_1, y_1)$ and $(x_2, y_2)$, then:

$$px_1 + qy_1 = px_2 + qy_2$$

This equation gives the required relation between $p$ and $q$. This idea is based on the property of infinitely many optimal solutions.