Class 12 RD Sharma Solutions – Chapter 30 Linear Programming – Exercise 30.2 | Set 2

Last Updated : 20 May, 2021

Question 11. Minimize Z = 5x + 3y

Subject to

2x + y â‰¥ 10

x + 3y â‰¥ 15

x â‰¤ 10

y â‰¤ 8

x, y â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

2x + y = 10, x + 3y = 15, x = 10, y = 8

Area shown by 2x + y â‰¥ 10:

The line 2x + y = 10 meets the coordinate axes at A(5, 0) and B(0, 10) respectively.

After connecting these points we will get the line 2x + y = 10.

Thus,

(0,0) does not assure the in equation 2x + y â‰¥ 10.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation 2x + y â‰¥ 10.

The area represented by x + 3y â‰¥ 15:

The line x + 3y = 15 connects the coordinate axes at C(15, 0) and D(0, 5) respectively.

After connecting these points we will get the line x + 3y = 15.

Thus,

(0,0) assure the in equation x + 3y â‰¥ 15.

The area in xy plane which does not have the origin represents the solution set of the in equation x + 3y â‰¥ 15.

The line x = 10 is the line that passes through the point (10, 0) and is parallel to Y axis.x â‰¤ 10 is the area to the left of the line

x = 10.

The line y = 8 is the line that passes through the point (0, 8) and is parallel to X axis.y â‰¤ 8 is the area below the line y = 8.

The area shows by x â‰¥ 0 and y â‰¥ 0:

Thus,

All point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shows by the in equations x â‰¥ 0 and y â‰¥ 0.

The suitable area determined by the system of constraints, 2x + y â‰¥ 10, x + 3y â‰¥ 15, x â‰¤ 10, y â‰¤ 8, x â‰¥ 0 and y â‰¥ 0 are as

follows.

The corner points of the suitable region are

E(3, 4),

H

F(10, 8) and

G(1, 8).

The values of Z at these corner points are as follows.

Hence,

The minimum value of Z is 27 at the point F(3, 4).

Therefore,

x = 3 and y =4 is the best solution of the given LPP.

Hence, the best value of Z is 27.

Question 12.  Minimize Z = 30x + 20y

Subject to

x + y â‰¤ 8

x + 4y â‰¥ 12

5x + 8y = 20

x, y â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

x + y = 8, x + 4y = 12, x = 0 and y = 0

5x + 8y = 20 is already an equation.

The area shows by x + y â‰¤ 8:

The line x + y = 8 connects the coordinate axes at A(8, 0) and B(0, 8) respectively.

After connecting these points we will get the line x + y = 8.

Thus,

(0,0) assures the in equation x + y â‰¤ 8.

Thus,

The area in xy plane which have the origin represents the solution set of the in equation x + y â‰¤ 8.

The area shows by x + 4y â‰¥ 12:

The line x + 4y = 12 connects the coordinate axes at C(12, 0) and D(0, 3) respectively.

After connecting these points we will get the line x + 4y = 12.

Thus,

(0,0) assure the in equation x + 4y â‰¥ 12.

Thus,

The area in xy plane which does not have the origin shows the solution set of the in equation x + 4y â‰¥ 12.

The line 5x + 8y = 20 is the line that passes through E(4, 0) and F .

The area shows by x â‰¥ 0 and y â‰¥ 0:

Thus,

All point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shows by the in equations x â‰¥ 0 and y â‰¥ 0.

The suitable area determined by the system of constraints, x + y â‰¤ 8, x + 4y â‰¥ 12, 5x + 8y = 20, x â‰¥ 0 and y â‰¥ 0 are as follows.

The corner points of the suitable area are B(0,8), D(0,3), G .
The values of Z at these corner points are as follows.

Hence,

The minimum value of Z is 60 at the point D(0,3).

Therefore,

x = 0 and y =3 is the best solution of the given LPP.

Therefore,

The best value of Z is 60.

Question 13.  Maximize Z = 4x + 3y

Subject to

3x + 4y â‰¤ 24

8x + 6y â‰¤ 48

x â‰¤ 6000

x â‰¤ y

x, y â‰¥ 0

Solution:

Here, we need to maximize Z = 4x + 3y

Convert the given in equations into equations, we will get the following equations:

3x + 4y = 24, 8x + 6y = 48, x = 5 , y = 6, x = 0 and y = 0.

The line 3x + 4y = 24 connects the coordinate axis at A(8, 0) and B(0,6).

Connects these points to get the line 3x + 4y = 24.

Thus,

(0, 0) assure the in equation 3x + 4y â‰¤ 24.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

The line 8x + 6y = 48 connects the coordinate axis at C(6, 0) and D(0,8).

Connect these points to get the line 8x + 6y = 48.

Thus,

(0, 0) assure the in equation 8x + 6y  â‰¤ 48.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

x = 5 is the line passing through x = 5 parallel to the Y axis.

y = 6 is the line passing through y = 6 parallel to the X axis.

The area shown by x â‰¥ 0 and y â‰¥ 0:

Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a suitable scale.

The corner points of the feasible region are O(0, 0), G(5, 0), F , E  and B(0, 6).

The values of Z at these corner points are as follows.

Here we can see that the maximum value of the objective function Z is 24 which is at F  and E .

Therefore,

The best value of Z is 24.

Question 14. Minimize Z = x – 5y + 20

Subject to

x – y â‰¥ 0

-x + 2y â‰¥ 2

x â‰¥ 3

y â‰¤  4

x, y â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

x âˆ’ y = 0, âˆ’ x + 2y = 2, x = 3, y = 4, x = 0 and y = 0.

The area shown by x âˆ’ y â‰¥ 0 or x â‰¥ y:

The line x âˆ’ y = 0 or x = y passes through the origin.The area to the right of the line x = y will assure the given in equation.

Now we will check by taking an example like if we take a point (4, 3) to the right of the line x = y .

Here, x â‰¥ y.

Thus,

It assure the given in equation.

Take a point (4, 5) to the left of the line x = y. Here, x â‰¤ y. That means it does not assure the given in equation.

The area shown by âˆ’ x + 2y â‰¥ 2:

The line âˆ’ x + 2y = 2 connects the coordinate axes at A(âˆ’2, 0) and B(0, 1) respectively.

After connecting these points we will get the line âˆ’ x + 2y = 2.

Thus,

(0,0) does not assure the in equation âˆ’ x + 2y â‰¥ 2.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation âˆ’ x + 2y â‰¥ 2 .

The line x = 3 is the line that passes through the point (3, 0) and is parallel to Y axis. x â‰¥ 3 is the area to the right of the

line x = 3.

The line y = 4 is the line that passes through the point (0, 4) and is parallel to X axis. y â‰¤ 4 is the area below the line y = 4.

The area shown by x â‰¥ 0 and y â‰¥ 0:

Hence,

All point in the first quadrant assure these in equations.

Thus, the first quadrant is the area shown by the in equations x â‰¥ 0 and y â‰¥ 0.

The suitable area determined by the system of constraints x âˆ’ y â‰¥ 0,âˆ’ x + 2y â‰¥ 2, x â‰¥ 3, y â‰¤ 4, x â‰¥ 0 and y â‰¥ 0 are as follows.

The corner points of the suitable area are C , D(3, 3), E(4, 4) and F(6, 4).

The values of Z at these corner points are as follows.

Hence,

The minimum value of Z is 4 at the point E(4, 4).

Therefore,

x = 4 and y = 4 is the best solution of the given LPP.

Hence,

The best value of Z is 4.

Question 15.  Maximize Z = 3x + 5y

Subject to

x + 2y â‰¤ 20

x + y â‰¤ 15

y â‰¤ 15

x, y â‰¥ 0

Solution:

Here we need to maximize Z = 3x + 5y

Convert the given in equations into equations, we will get the following equations:

x + 2y = 20, x + y = 15, y = 5, x = 0 and y = 0.

The line x + 2y = 20 connects the coordinate axis at A(20, 0) and B(0,10).

Connect these points to get the line x + 2y = 20.

Thus,

(0, 0) assume the in equation x + 2y â‰¤ 20.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

The line x + y = 15 connect the coordinate axis at C(15, 0) and D(0,15).

Connect these points to get the line x + y = 15.

Thus,

(0, 0) assume the in equation x + y â‰¤ 15.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

y = 5 is the line passing through (0, 5) and parallel to the X axis. The area below the line y = 5 will assure the given in equation.

The area shown by x â‰¥ 0 and y â‰¥ 0:

Hence,

All the point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a suitable scale.

The corner points of the suitable area are O(0, 0), C(15, 0), E(10, 5) and F(0, 5)

The values of Z at these corner points are as follows.

Here we can see that the maximum value of the objective function Z is 55 which is at E(10, 5).

Therefore,

The best value of Z is 55.

Question 16. Minimize Z = 3x1 + 5x2

Subject to

x1 + x2 â‰¥ 3

x1 + x2 â‰¥ 2

x1, x2 â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

x1 + 3x2 = 3, x1 + x2 = 2, x1 = 0 and x2 = 0

The area shown by x1 + 3x2 â‰¥ 3 :

The line x1 + 3x2 = 3 connects the coordinate axes at A(3, 0) and B(0, 1) respectively.

After connecting these points we will get the line x1 + 3x2 = 3.

Thus,

(0,0) does not assure the in equation x1 + 3x2 â‰¥ 3.

Thus,

The area in the plane which does not have the origin represents the solution set of the in equation x1 + 3x2 â‰¥ 3.

The area shown by x1 + x2 â‰¥ 2:

The line x1 + x2 = 2 connects the coordinate axes at C(2, 0) and D(0, 2) respectively.

After connecting these points we will get the line x1 + x2 = 2.

Thus,

(0,0) does not assure the in equation x1 + x2 â‰¥ 2.

Thus,

The area having the origin represents the solution set of the in equation x1 + x2 â‰¥ 2.

The area shown by x1 â‰¥ 0 and x2 â‰¥ 0:

Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x1 â‰¥ 0 and x2 â‰¥ 0.

The suitable area determined by the system of constraints, x1 + 3x2 â‰¥ 3 , x1 + x2 â‰¥ 2, x1 â‰¥ 0, and x2 â‰¥ 0, are as follows.

The corner points of the suitable area are O(0, 0), B(0, 1), E  and C(2, 0).

The values of Z at these corner points are as follows.

Hence,

The minimum value of Z is 0 at the point O(0, 0).

Hence,

x1 = 0 and x2 = 0 is the best solution of the given LPP.

Therefore,

The best value of Z is 0.

Question 17.  Maximize Z = 2x + 3y

Subject to

x + y â‰¥ 1

10x + y â‰¥ 5

x + 10y â‰¥ 1

x, y â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

x + y = 1, 10x +y = 5, x + 10y = 1, x = 0 and y = 0

The area shown by x + y â‰¥ 1:

The line x + y = 1 connects the coordinate axes at A(1, 0) and B(0,1) respectively.

After connecting these points we will get the line x + y = 1.

Thus,

(0,0) does not assure the in equation x + y â‰¥ 1.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation x + y â‰¥ 1.

The area shown by 10x +y â‰¥ 5:

The line 10x +y = 5 connect the coordinate axes at C  and D(0, 5) respectively.

After connecting these points we will get the line 10x +y = 5.

Thus,

(0,0) does not assure the in equation 10x +y â‰¥ 5.

Thus, the area which does not have the origin represents the solution set of the in equation 10x +y â‰¥ 5.

The area shown by x + 10y â‰¥ 1:

The line x + 10y = 1 connect the coordinate axes at A(1, 0) and F  respectively.

After connecting these points we will have the line x + 10y = 1.

Thus,

(0,0) does not assure the in equation x + 10y â‰¥ 1.

Thus, the area which does not have the origin represents the solution set of the in equation x + 10y â‰¥ 1.

The area shown by x â‰¥ 0 and y â‰¥ 0:

Here,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x â‰¥ 0, and y â‰¥ 0.

The suitable area determined by the system of constraints x + y â‰¥ 1, 10x +y â‰¥ 5, x + 10y â‰¥ 1, x â‰¥ 0, and y â‰¥ 0, are as follows.

The suitable area is unbounded.

Hence,

The maximum value is infinity i.e. the solution is unbounded.

Question 18. Maximize Z = -x1 + 2x2

Subject to

-x1 + 3x2 â‰¤ 10

x1 + x2 â‰¤ 6

x1 – x2 â‰¤ 2

x1, x2 â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

âˆ’x1 + 3x2 = 10, x1 + x2 = 6, x1 + x2 = 2, x1 = 0 and x2 = 0

The area shown by âˆ’x1 + 3x2 â‰¤ 10:

The line âˆ’x1 + 3x2 = 10 coincide the coordinate axes at A(âˆ’10, 0) and B  respectively.

After connecting these points we will get the line âˆ’x1 + 3x2 = 10.

Thus,

(0,0) satisfies the in equation âˆ’x1 + 3x2 â‰¤ 10 .

Thus,

The area region in the plane which have the origin shows the solution set of the in equation

âˆ’x1 + 3x2 â‰¤ 10.

The area shown by x1 + x2 â‰¤ 6:

The line x1 + x2 = 6 connects the coordinate axes at C(6, 0) and D(0, 6) respectively.

After connecting these points we will get the line x1 + x2 = 6.

Thus,

(0,0) assure the in equation x1 + x2 â‰¤ 6.

Thus,

The area having the origin represents the solution set of the in equation x1 + x2 â‰¤ 6.

The area shown by x1âˆ’ x2 â‰¤ 2:

The line x1 âˆ’ x2 = 2 coincide the coordinate axes at E(2, 0) and F(0, âˆ’2) respectively.

After connecting these points we will get the line x1 âˆ’ x2 = 2.

Thus,

(0,0) assure the in equation x1âˆ’ x2 â‰¤ 2.

Thus,

The area having the origin represents the solution set of the in equation x1âˆ’ x2 â‰¤ 2.

The area shown by x1 â‰¥ 0 and x2 â‰¥ 0:

Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the region shown by the in equations x1 â‰¥ 0 and x2 â‰¥ 0.

The suitable area determined by the system of constraints, âˆ’x1 + 3x2 â‰¤ 10, x1 + x2 â‰¤ 6, x1âˆ’ x2 â‰¤ 2, x1 â‰¥ 0, and x2 â‰¥ 0, are as follows.

The corner points of the assure area are O(0, 0), E(2, 0), H(4, 2), G(2, 4) and B .

The values of Z at these corner points are as follows.

Here we can see that the maximum value of the objective function Z is  which is at B .

Question 19. Maximize Z = -x + y

Subject to

-2x + y â‰¤ 1

x  â‰¤ 2

x + y â‰¤ 3

x, x â‰¥ 0

Solution:

Here we need to maximize Z = x + y

Convert the given in equations into equations, we will get the following equations:

âˆ’2x + y = 1, x = 2, x + y = 3, x = 0 and y = 0.

The line âˆ’2x + y = 1 coincide the coordinate axis at A  and B(0, 1).

Now connect these points to get the line âˆ’2x + y = 1 .

Thus,

(0, 0) assure the in equation âˆ’2x + y â‰¤ 1.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

x = 2 is the line passing through (2, 0) and parallel to the Y axis.

The area below the line x = 2 will assure the given in equation.

The line x + y = 3 coincide the coordinate axis at C(3, 0) and D(0, 3).

Connect these points to get the line x + y = 3.

Thus,

(0, 0) assure the in equation x + y â‰¤ 3.

Thus,

The area in xy-plane that have the origin shows the solution set of the given equation.

The area shown by x â‰¥ 0 and y â‰¥ 0:

Hence,

All the points in the first quadrant assures these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a satisfactory scale.

The corner points of the suitable area are O(0, 0),G(2, 0), E(2, 1) and F

The values of Z at these corner points are as follows.

Here we can see that the maximum value of the objective function Z is 3 which is at E(2, 1) and F .

Therefore,

The best value of Z is 3.

Question 20. Maximize Z = -3x1 + 4x2

Subject to

x1 – x2 â‰¤ -1

x1 + x2  â‰¤ 0

x1, x2 â‰¥ 0

Solution:

Convert the given in equations into equations, we will get the following equations:

x1 âˆ’ x2 = âˆ’1, âˆ’x1 + x2 = 0, x1 = 0 and x2 = 0

The area shown by x1 âˆ’ x2 â‰¤ âˆ’1:

The line x1 âˆ’ x2 = âˆ’1 coincide the coordinate axes at A(âˆ’1, 0) and B(0, 1) respectively.

After connecting these points we will get the line x1 âˆ’ x2 = âˆ’1.

Thus,

(0,0) does not assures the in equation x1 âˆ’ x2 â‰¤ âˆ’1 .

Thus,

The area in the plane which does not have the origin shows the solution set of the in equation x1 âˆ’ x2 â‰¤ âˆ’1.

The area shown by âˆ’x1 + x2 â‰¤ 0 or x1 â‰¥ x2:

The line âˆ’x1 + x2 = 0 or x1 = x2 is the line passing through (0, 0).

The area to the right of the line x1 = x2 will assure the given in equation âˆ’x1 + x2 â‰¤ 0.

If we mark a point (1, 3) to the left of the line x1 = x2

Here, 1â‰¤3 which is not assuring the in equation x1 â‰¥ x2

Hence,

The area to the right of the line x1 = x2 will assure the given in equation âˆ’x1 + x2 â‰¤ 0.

The area shown by x1 â‰¥ 0 and x2 â‰¥ 0:

Thus,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x1 â‰¥ 0 and x2 â‰¥ 0.

The suitable area determined by the system of constraints, x1 âˆ’ x2 â‰¤ âˆ’1, âˆ’x1 + x2 â‰¤ 0, x1 â‰¥ 0, and x2 â‰¥ 0, are as follows.

Here we can see that the suitable area of the given LPP does not exist.

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