Algebra

Linear Programming

Medium

core

Consider the following L.L.P.

Minimize z = 30x - 30y + 1800;

subject to x + y ≤ 30, x ≤ 15, y ≤ 20, x + y ≥ 15 and x, y ≥ 0.

Then it attains its optimal value at the point

(0, 20)

(20, 40)

(10, 20)

(0, 15)

✅ Correct Option: 1

The objective function to minimize is:

$z = 30x - 30y + 1800$

Subject to constraints:

$x + y \leq 30$
$x \leq 15$
$y \leq 20$
$x + y \geq 15$
$x, y \geq 0$

The corner points of the feasible region are found by intersecting the constraint lines:

At $x = 0$ and $x + y = 15$ :

$y = 15$

Point: $(0, 15)$

At $x = 0$ and $y = 20$ :

Point: $(0, 20)$

At $y = 20$ and $x + y = 30$ :

$x + 20 = 30$

$x = 10$

Point: $(10, 20)$

At $x = 15$ and $x + y = 30$ :

$15 + y = 30$

$y = 15$

Point: $(15, 15)$

At $x = 15$ and $x + y = 15$ :

$15 + y = 15$

$y = 0$

Point: $(15, 0)$

Evaluating $z = 30x - 30y + 1800$ at each corner point:

At $(0, 15)$ :

$z = 30(0) - 30(15) + 1800$

$z = 0 - 450 + 1800$

$z = 1350$

At $(0, 20)$ :

$z = 30(0) - 30(20) + 1800$

$z = 0 - 600 + 1800$

$z = 1200$

At $(10, 20)$ :

$z = 30(10) - 30(20) + 1800$

$z = 300 - 600 + 1800$

$z = 1500$

At $(15, 15)$ :

$z = 30(15) - 30(15) + 1800$

$z = 450 - 450 + 1800$

$z = 1800$

At $(15, 0)$ :

$z = 30(15) - 30(0) + 1800$

$z = 450 - 0 + 1800$

$z = 2250$

The minimum value is $z = 1200$ at point $(0, 20)$ .

Therefore, the optimal value is attained at $(0, 20)$ .

2026: 30 May Shift 1

For a LPP, the objective function is $z=4x+3y$ and the feasible region determined by a set of linear constraints is shown in the graph as a shaded portion. Which one of the following statement is true ?

2025: 14 May Shift 2

The region represented by the constraints $x \geq 0, y \geq 0$ of an LPP is

2025: 16 May Shift 1

The corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 3/2), (1, 2) and (4, 0). If the objective function is Z = ax + by, where 'a' and 'b' are positive, then the condition on 'a' and 'b' so that the maximum of Z occurs at (1, 2) and (4, 0) is:

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:

2025: 30 May Shift 2

In an LPP, the feasible region represented by the set off constraints $2x + 3y \leq 18$ , $x + y \leq 10$ , $x \geq 0$ , $y \geq 0$ is

2025: 3 June Shift 2

Which of the following terms are associated with a linear programming problem?

(A) Constraints

(B) Independent events

(D) Objective function

Choose the correct answer from the options given below:

Consider the following L.L.P. Minimize z = 30x - 30y + 1800; subject to x + y ≤ 30, x ≤ 15, y ≤ 20, x + y ≥ 15 and x, y ≥ 0. Then it attains its optimal value at the point

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Consider the following L.L.P.

Minimize z = 30x - 30y + 1800;

subject to x + y ≤ 30, x ≤ 15, y ≤ 20, x + y ≥ 15 and x, y ≥ 0.

Then it attains its optimal value at the point