List of top Mathematics Questions on Linear Programming

Maximize $ Z = 2x+3y $, subject to the constraints:
$ x+y \le 2 $
$ 2x+y \le 3 $
$ x,y \ge 0 $

For the given linear programming problem,
Minimum Z = 6x + 10y
subject to the constraints
$ x \ge 6 $; $ y \ge 2 $; $ 2x+y \ge 10 $; $ x,y \ge 0 $,
the redundant constraints are:

For any Linear Programming Problem (LPP), choose the correct statement:
A. There exists only finite number of basic feasible solutions to LPP
B. Any convex combination of k - different optimum solution to a LPP is again an optimum solution to the problem
C. If a LPP has feasible solution, then it also has a basic feasible solution
D. A basic solution to AX = b is degenerate if one or more of the basic variables vanish

Which one of the following set of constraints does the given shaded region represent?

The feasible region is bounded by the inequalities: \[ 3x + y \geq 90, \quad x + 4y \geq 100, \quad 2x + y \leq 180, \quad x, y \geq 0 \] If the objective function is $ Z = px + qy $ and $ Z $ is maximized at points $ (6, 18) $ and $ (0, 30) $, then the relationship between $ p $ and $ q $ is:

Solve the following LPP graphically: Maximize: \[ Z = 2x + 3y \] Subject to: \[ \begin{aligned} x + 4y &\leq 8 \quad \text{(1)} \\ 2x + 3y &\leq 12 \quad \text{(2)} \\ 3x + y &\leq 9 \quad \text{(3)} \\ x &\geq 0,\quad y \geq 0 \quad \text{(non-negativity constraints)} \end{aligned} \]

The maximum value of $Z = 3x + 4y$ subject to the constraints $x + y \leq 1$, $x \geq 0$, $y \geq 0$ is:

In an LPP, corner points of the feasible region determined by the system of linear constraints are $(1,1), (3,0), (0,3)$. If $Z = ax + by$, where $a>0, b>0$ is to be minimized, the condition on $a$ and $b$ so that the minimum of $Z$ occurs at $(3, 0)$ and $(1, 1)$ will be:

A person wants to invest 75,000 in options A and B, which yield returns of 8% and 9% respectively. He plans to invest at least 15,000 in Plan A, 25,000 in Plan B, and keep Plan A ≤ Plan B. Formulate the LPP to maximize the return.

If the system of linear equations
x + y + z = 2, 2x + y − z = 3, 3x + 2y + kz = 4
has a unique solution, then:

The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15), and (0, 30). If the objective function is Z = αx + βy, α, β > 0, the condition on α and β so that maximum of Z occurs at corner points (5, 5) and (0, 20) is:

Minimize $Z = -50x + 20y$ subject to the constraints: \[ 2x - y \geq -5, \quad 3x + y \geq 3, \quad 2x - 3y \leq 12, \quad x \geq 0, \quad y \geq 0. \] Then which of the following is/are true:
(A) Feasible region is unbounded.
(B) $Z$ has no minimum value.
(C) The minimum value of $Z$ is 100.
(D) The minimum value of $Z$ is -300.

Optimize $Z = 3x + 9y$ subject to the constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]

Find the minimum value of ( z = x + 3y ) under the following constraints:

• x + y ≤ 8
• 3x + 5y ≥ 15
• x ≥ 0, y ≥ 0

Let the two variables $ x $ and $ y $ satisfy the following conditions: \[ x + y \leq 50, \quad x + 2y \leq 80, \quad 2x + y \geq 20, \quad x, y \geq 0. \] Then the maximum value of $ Z = 4x + 3y $ is:

Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]

Minimize Z = 5x + 3y \text{ subject to the constraints} \[ 4x + y \geq 80, \quad x + 5y \geq 115, \quad 3x + 2y \leq 150, \quad x \geq 0, \quad y \geq 0. \]

Maximize $ z = 8000x + 12000y $ subject to constraints: \[ 3x + 4y \leq 60, \quad x + y \leq 30, \quad x \geq 0, \quad y \geq 0. \]

Find the minimum value of \[ Z = 50x + 70y \] $\text{under the following constraints by graphical method:}$ \[ 2x + y \geq 8, \] \[ x + 2y \geq 10, x \geq 0, y \geq 0. \]

Find the maximum value of $ Z = 3x + 4y $ under the constraints \[ x + y \leq 4, x \geq 0, y \geq 0. \]

Find the maximization of $ z = x + y $, under the following constraints: \[ x - y \leq -1, -x + y \leq 0, x \geq 0, y \geq 0. \]

By graphical method solve the LPP under the following constraints: \[ x + 2y \geq 10, \] \[ 3x + 4y \leq 24, \] \[ x \geq 0, \, y \geq 0, \] then find the minimum value of $ z = 200x + 500y $.

Minimize $Z = 6x + 3y$ under the constraints: \[ 4x + y \geq 80, x + 5y \geq 115, 3x + 2y \leq 150, x \geq 0, \; y \geq 0. \]

Maximize $Z = 8000x + 12000y$ under the constraints: \[ 9x + 12y \leq 180, 3x + 4y \leq 60, x + 3y \leq 30, x \geq 0, \; y \geq 0 \]

Solve the inequality $3x + 4y \leq 12$, $4x + 3y \leq 12$, $x \geq 0$, $y \geq 0$ by graphical method.