Algebra

Linear Programming

Medium

common

Consider the following L.P.P.

Minimize z = 400x + 300y subject to 100x + 200y ≥ 12000, 300x + 400y ≥ 20000, 200x + 100y ≥ 15000 and x, y ≥ 0. Then

Its feasible region is bounded and optimal point is (60, 30)

Its feasible region is unbounded and optimal point is (60, 30)

Its feasible region is bounded and optimal point is (0, 110)

Its feasible region is unbounded and optimal point is (30, 70)

✅ Correct Option: 2

The given constraints are:

$100x + 200y \geq 12000$

$300x + 400y \geq 20000$

$200x + 100y \geq 15000$

$x, y \geq 0$

Simplifying the constraints by dividing:

$x + 2y \geq 120$

$3x + 4y \geq 200$

$2x + y \geq 150$

$x, y \geq 0$

All constraints are of " $\geq$ " type, meaning the feasible region consists of points above or to the right of all constraint lines. The region extends infinitely upward and rightward.

Therefore, the feasible region is unbounded.

For minimum value in an unbounded region, the corner points at the intersections of constraint lines need to be examined.

Finding the intersection of $x + 2y = 120$ and $2x + y = 150$ :

From $2x + y = 150$ :

$y = 150 - 2x$

Substituting into $x + 2y = 120$ :

$x + 2(150 - 2x) = 120$

$x + 300 - 4x = 120$

$-3x = -180$

$x = 60$

Therefore:

$y = 150 - 2(60)$

$y = 150 - 120$

$y = 30$

Corner point: $(60, 30)$

Checking at $(60, 30)$ :

$x + 2y = 60 + 60 = 120 \geq 120$ ✓

$3x + 4y = 180 + 120 = 300 \geq 200$ ✓

$2x + y = 120 + 30 = 150 \geq 150$ ✓

All constraints are satisfied.

The objective function at $(60, 30)$ :

$z = 400x + 300y$

$z = 400(60) + 300(30)$

$z = 24000 + 9000$

$z = 33000$

This gives the minimum value as $(60, 30)$ is at the corner of the unbounded feasible region.

The feasible region is unbounded and the optimal point is $(60, 30)$ .

Correct Option: 2

2026: 30 May Shift 1

In reference to a linear programming problem (LPP), which of the following statements are true?

A. The optimal value of the objective function is attained at the points given by intersections of inequation with the axes only.

B. The objective function of a LPP is a linear function to be optimized.

C. Every LPP admits an optimal solution.

D. The region represented by the inequation system, $x, y\geq0$ , $x+2y\leq4, 3x+y\geq3, 4x+3y\geq6$ is bounded in the first quadrant.

Choose the correct answer from the options given below:

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The corner points of the bounded feasible region of a linear programming problem are A(15,0), B(40,0), C(4, 18), D(6, 12) and objective function is $z = 20x+10y$ , then which of the following are TRUE?

(A) Value of z at point A(15, 0) = 300.

(B) Value of z at point B(40, 0) = 800.

(D) Maximum value of z = 800.

Choose the correct answer from the options given below:

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The sum of the x-coordinates of the corner points of the feasible region for the LPP: Minimize $z = 3x + 2y$ subject to constraints $x + y \leq 14$ , $x \geq 4$ , $x \leq 8, y \geq 0$ is

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The number of corner points of the feasible region of an LPP determined by the constraints $x - y \geq 0$ , $2y \leq x + 2, x \geq 0, y \geq 0$ is

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If the minimum value of the objective function $Z = ax + by$ of an LPP occurs at two points $(3, 5)$ and $(5, 3)$ , then

Consider the following L.P.P. Minimize z = 400x + 300y subject to 100x + 200y ≥ 12000, 300x + 400y ≥ 20000, 200x + 100y ≥ 15000 and x, y ≥ 0. Then

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Consider the following L.P.P.

Minimize z = 400x + 300y subject to 100x + 200y ≥ 12000, 300x + 400y ≥ 20000, 200x + 100y ≥ 15000 and x, y ≥ 0. Then