1. ⇒ (MHT CET 2023 12th May Evening Shift )

The maximum value of $z = 7 x + 8 y$ subject to the constraints $x + y \leq 20, y \geq 5, x \leq 10, x \geq 0, y \geq 0$ is

A. 150

B. 160

C. 110

D. 180

Correct answer option is (B)

MHT CET 2023 12th May Evening Shift Mathematics - Linear Programming Question 2 English Explanation

Feasible region lies on the origin side of lines $x + y = 20, x = 10$ and on non-origin side of $y = 5$ .

$∴$ Corner points of the feasible region ant $A (0, 5), B (10, 5), C (10, 10)$ and $D (0, 20)$

$z$ at $A (0, 5) = 40$

$z$ at $B (10, 5) = 110$

$z$ at $C (10, 10) = 150$

$z$ at $D (0, 20) = 160$

$∴$ Maximum value of z is 160.

2. ⇒ (MHT CET 2023 12th May Morning Shift )

For a feasible region OCDBO given below, the maximum value of the objective function $z = 3 x + 4 y$ is

MHT CET 2023 12th May Morning Shift Mathematics - Linear Programming Question 1 English

A. 70

B. 100

C. 110

D. 130

Correct answer option is (C)

Corner points of the given feasible region are $O (0, 0), C (10, 10), D (10, 20), B (0, 25)$

$∴ z$ at $C (10, 10) = 70$ ,

$z$ at $D (10, 20) = 110$ ,

$z$ at $B (0, 25) = 100$

$∴$ The maximum value of $z$ is 110.

3. ⇒ (MHT CET 2023 11th May Evening Shift )

The maximum value of $z = 3 x + 5 y$ subject to the constraints $3 x + 2 y \leq 18, x \leq 4, y \leq 6, x, y \geq 0$ , is

A. 27

B. 36

C. 42

D. 30

Correct answer option is (B)

MHT CET 2023 11th May Evening Shift Mathematics - Linear Programming Question 3 English Explanation

Objective function $z = 3 x + 5 y$

The corner points of the feasible region are $O (0, 0), A (4, 0), B (4, 3), C (2, 6)$ and $D (0, 6)$

$∴ Z$ at $A (4, 0) = 12$

$Z$ at $B (4, 3) = 27$

$Z$ at $C (2, 6) = 36$

$Z$ at $D (0, 6) = 30$

$∴$ Maximum value of $Z$ is 36.

4. ⇒ (MHT CET 2023 11th May Morning Shift )

For the following shaded area, the linear constraints except $x, y \geq 0$ are

MHT CET 2023 11th May Morning Shift Mathematics - Linear Programming Question 5 English

A. $2 x + y \leq 2, x - y \leq 1, x + 2 y \leq 8$

B. $2 x + y \geq 2, x - y \leq 1, x + 2 y \leq 8$

C. $2 x + y \geq 2, x - y \geq 1, x + 2 y \leq 8$

D. $2 x + y \geq 2, x - y \geq 1, x + 2 y \geq 8$

Correct answer option is (B)

Shaded region lies on non-origin side of $2 x + y = 2$ and on origin side of the lines $x - y = 1$ and $x + 2 y = 8$

$∴ 2 x + y \geq 2, x - y \leq 1, x + 2 y \leq 8$

5. ⇒ (MHT CET 2023 10th May Evening Shift )

The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function $3 x + 5 y$ , subject to the linear constraints given by this system of inequations is

MHT CET 2023 10th May Evening Shift Mathematics - Linear Programming Question 6 English

A. 19.5

B. 21

C. 15

D. 19.8

Correct answer option is (A)

MHT CET 2023 10th May Evening Shift Mathematics - Linear Programming Question 6 English Explanation

Let the corner points of the feasible region be A, B, C, D.

Solving equations $y = 3$ and $2 x + 3 y = 12$ , we get

$A = (1.5, 3)$

Similarly,

$\begin{aligned} B = (4, 3) \\ C = (4, 7) \\ D = (\frac{3}{5}, \frac{18}{5}) \end{aligned}$

Let $Z = 3 x + 5 y$

$∴$ Value of $Z$ at point $A = 19.5$

Value of $Z$ at point $B = 27$

Value of $Z$ at point $C = 47$

Value of $Z$ at point $D = \frac{99}{5}$

$∴$ The minimum value of $Z$ is 19.5.

⇒Linear programming