#### Chapters

## Chapter 7: Linear Programming

#### Exercise 7.1 [Pages 232 - 233]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Exercise 7.1 [Pages 232 - 233]

Solve graphically : x ≥ 0

Solve graphically : x ≤ 0

Solve graphically : y ≥ 0

Solve graphically : y ≤ 0

Solve graphically : x ≥ 0 and y ≥ 0

Solve graphically : x ≤ 0 and y ≥ 0

Solve graphically : x ≤ 0 and y ≤ 0

Solve graphically : x ≥ 0 and y ≤ 0.

Solve graphically : 2x – 3 ≥ 0

Solve graphically : 2y – 5 ≥ 0

Solve graphically : 3x + 4 ≤ 0

Solve graphically : 5y + 3 ≤ 0

Solve graphically : x +2y ≤ 6

Solve graphically : 2x – 5y ≥10

Solve graphically : 3x + 2y ≥ 0

Solve graphically : 5x – 3y ≤ 0

Solve graphically : 2x + y ≥ 2 and x – y ≤ 1

Solve graphically : x – y ≤ 2 and x + 2y ≤ 8

Solve graphically : x + y ≥ 6 and x + 2y ≤ 10

Solve graphically : 2x + 3y≤ 6 and x + 4y ≥ 4

Solve graphically : 2x + y ≥ 5 and x – y ≤ 1

#### Exercise 7.2 [Page 234]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Exercise 7.2 [Page 234]

**Find the feasible solution of the following inequation:**

3x + 2y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥** **0

**Find the feasible solution of the following inequation:**

2x + 3y ≤ 6, x + y ≥ 2, x ≥ 0, y ≥** **0

**Find the feasible solution of the following inequation:**

3x + 4y ≥ 12, 4x + 7y ≤ 28, y ≥ 1, x ≥ 0.

**Find the feasible solution of the following inequation:**

x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0.

**Find the feasible solution for the following system of linear inequations:**

0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4

**Find the feasible solution of the following inequations:**

x - 2y ≤ 2, x + y ≥ 3, - 2x + y ≤ 4, x ≥ 0, y ≥ 0

A company produces two types of articles A and B which require silver and gold. Each unit of A requires 3 gm of silver and 1 gm of gold, while each unit of B requires 2 gm of silver and 2 gm of gold. The company has 6 gm of silver and 4 gm of gold. Construct the inequations and find feasible solution graphically

A furniture dealer deals in tables and chairs. He has ₹ 15,000 to invest and a space to store at most 60 pieces. A table costs him ₹ 150 and a chair ₹ 750. Construct the inequations and find the feasible solution.

#### Exercise 7.3 [Pages 237 - 378]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Exercise 7.3 [Pages 237 - 378]

A manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry and then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for production of A and B and the number of man-hours available for the firm is as follows :

Gadgets |
Foundry |
Machine shop |

A | 10 | 5 |

B | 6 | 4 |

Time available (hour) | 60 | 35 |

Profit on the sale of A is ₹ 30 and B is ₹ 20 per units. Formulate the LPP to have maximum profit.

In a cattle breading firm, it is prescribed that the food ration for one animal must contain 14. 22 and 1 units of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit of these two contains the following amounts of these three nutrients:

Fodder → |
Fodder 1 | Fodder 2 |

Nutrient ↓ |
||

Nutrients A | 2 | 1 |

Nutrients B | 2 | 3 |

Nutrients C | 1 | 1 |

The cost of fodder 1 is ₹ 3 per unit and that of fodder 2 ₹ 2. Formulate the LPP to minimize the cost.

A company manufactures two types of chemicals Aand B. Each chemical requires two types of raw material P and Q. The table below shows number of units of P and Q required to manufacture one unit of A and one unit of B and the total availability of P and Q.

Chemical→ | A |
B |
Availability |

Raw Material ↓ | |||

P | 3 | 2 | 120 |

Q | 2 | 5 | 160 |

The company gets profits of ₹ 350 and ₹ 400 by selling one unit of A and one unit of B respectively. (Assume that the entire production of A and B can be sold). How many units of the chemicals A and B should be manufactured so that the company gets a maximum profit? Formulate the problem as LPP to maximize profit.

A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 in magazines A and B per copy. These are processed on three Machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II, and 2 hours on machine III. Magazine B requires 3 hours on machine I, 2 hours on machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, and 60 hours per week respectively. Formulate the LPP to determine weekly production of magazines A and B, so that the total profit is maximum.

A manufacturer produces bulbs and tubes. Each of these must be processed through two machines M_{1} and M_{2}. A package of bulbs requires 1 hour of work on Machine M_{1} and 3 hours of work on Machine M_{2}. A package of tubes requires 2 hours on Machine M_{1} and 4 hours on Machine M_{2}. He earns a profit of ₹ 13.5 per package of bulbs and ₹ 55 per package of tubes. Formulate the LPP to maximize the profit, if he operates the machine M_{1}, for almost 10 hours a day and machine M_{2} for almost 12 hours a day.

A company manufactures two types of fertilizers F_{1} and F_{2}. Each type of fertilizer requires two raw materials A and B. The number of units of A and B required to manufacture one unit of fertilizer F_{1} and F_{2} and availability of the raw materials A and B per day are given in the table below:

Fertilizers→ | F_{1} |
F_{2} |
Availability |

Raw Material ↓ | |||

A | 2 | 3 | 40 |

B | 1 | 4 | 70 |

By selling one unit of F_{1} and one unit of F_{2}, the company gets a profit of ₹ 500 and ₹ 750 respectively. Formulate the problem as LPP to maximize the profit.

A doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carbohydrates and proteins are 18, 28, 14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is ₹ 4.5 per unit and that of food B is ₹ 3.5 per unit. Form the LPP, so that the sick person’s diet meets the requirements at a minimum cost.

If John drives a car at a speed of 60 km/hour, he has to spend ₹ 5 per km on petrol. If he drives at a faster speed of 90 km/hour, the cost of petrol increases ₹ 8 per km. He has ₹ 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as L.P.P.

The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs ₹ 20 per kg and sand costs of ₹ 6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the L.P.P. for the cost to be minimum.

#### Exercise 7.4 [Page 241]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Exercise 7.4 [Page 241]

**Solve the following LPP by graphical method:**

Maximize z = 11x + 8y, subject to x ≤ 4, y ≤ 6, x + y ≤ 6, x ≥ 0, y ≥ 0,

**Solve the following LPP by graphical method:**

Maximize z = 4x + 6y, subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.

**Solve the following LPP by graphical method:**

Maximize z = 7x + 11y, subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0.

**Solve the following L.P.P graphically:**

Maximize: Z = 10x + 25y

Subject to: x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0

**Solve the following LPP by graphical method:**

Maximize: z = 3x + 5y

Subject to: x + 4y ≤ 24

3x + y ≤ 21

x + y ≤ 9

x ≥ 0, y ≥ 0

**Solve the following LPP by graphical method:**

Minimize Z = 7x + y subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0

**Solve the following LPP by graphical method:**

Minimize z = 8x + 10y, subject to 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

**Solve the following LPP by graphical method:**

Minimize z = 6x + 21y, subject to x + 2y ≥ 3, x + 4y ≥ 4, 3x + y ≥ 3, x ≥ 0, y ≥ 0.

#### Miscellaneous exercise 7 [Pages 242 - 243]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Miscellaneous exercise 7 [Pages 242 - 243]

**Select the appropriate alternatives for each of the following question:**

The value of objective function is maximum under linear constraints

at the centre of feasible region

at (0, 0)

at a vertex of feasible region

the vertex which is of maximum distance from (0, 0).

**Select the appropriate alternatives for each of the following question:**

Which of the following is correct?

Every LPP has an optimal solution

A LPP has unique optimal solution

If LPP has two optimal solutions, then it has infinite number of optimal solutions

The set of all feasible solution of LPP may not be convex set

**Select the appropriate alternatives for each of the following question:**

Objective function of LPP is

a constraint

a function to be maximized or minimized

a relation between the decision variables

equation of a straight line

**Select the appropriate alternatives for each of the following question:**

The maximum value of z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is

235

`235/9`

`235/19`

`235/3`

**Select the appropriate alternatives for each of the following question:**

The maximum value of z = 10x + 6y subject to the constraints 3x + y ≤ 12, 2x + 5y ≤ 34, x, ≥ 0, y ≥ 0 is

56

65

55

66

**Select the appropriate alternatives for each of the following question:**

The point of which the maximum value of x + y subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x, ≥ 0, y ≥ 0 is is obtained at

(30, 25)

(20, 35)

(35, 20)

(40, 15)

**Select the appropriate alternatives for each of the following question:**

Of all the points of the feasible region, the optimal value of z obtained at the point lies

inside the feasible region

at the boundary of the feasible region

at vertex of feasible region

outside the feasible region

**Select the appropriate alternatives for each of the following question:**

Feasible region is the set of points which satisfy

the objective function

all the given constraints

some of the given constraints

only one constraint

**Select the appropriate alternatives for each of the following question:**

Solution of LPP to minimize z = 2x + 3y, such that x ≥ 0, y ≥ 0, 1 ≤ x + 2y ≤ 10 is

x = 0, y = `1/2`

x = `1/2`, y = 0

x = 1, y = 2

x = `1/2`, y = `1/2`

**Select the appropriate alternatives for each of the following question:**

The corner points of the feasible solution given by the inequation x + y ≤ 4, 2x + y ≤ 7, x ≥ 0, y ≥ 0 are

(0, 0), (4, 0), (7, 1), (0, 4)

(0, 0), `(7/2, 0)`, (3, 1), (0, 4)

(0, 0), `(7/2, 0)`, (3, 1), (0, 7)

(0, 0), (4, 0), (3, 1), (0, 7)

**Select the appropriate alternatives for each of the following question:**

The corner points of the feasible solution are (0, 0), (2, 0), `(12/7, 3/7)`, (0, 1). Then z = 7x + y is maximum at

(0, 0)

(2, 0)

`(12/7, 3/7)`

(0, 1)

**Select the appropriate alternatives for each of the following question:**

If the corner points of the feasible solution are (0, 0), (3, 0), (2, 1), `(0, 7/3)` the maximum value of z = 4x + 5y is

12

13

`35/3`

0

**Select the appropriate alternatives for each of the following question:**

If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0), then the point of minimum z = 3x + 2y is

(2, 2)

(0, 10)

(4, 0)

(3, 4)

**Select the appropriate alternatives for each of the following question:**

The half-plane represented by 3x + 2y < 8 contains the point

`(1, 5/2)`

(2, 1)

(0, 0)

(5, 1)

**Select the appropriate alternatives for each of the following question:**

The half-plane represented by 4x + 3y >14 contains the point

(0, 0)

(2, 2)

(3, 4)

(1, 1)

#### Miscellaneous exercise 7 [Pages 243 - 245]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Linear Programming Miscellaneous exercise 7 [Pages 243 - 245]

**Solve each of the following inequations graphically using XY-plane:**

4x - 18 ≥ 0

**Solve each of the following inequations graphically using XY-plane:**

- 11x - 55 ≤ 0

**Solve each of the following inequations graphically using XY-plane:**

5y - 12 ≥ 0

**Solve each of the following inequations graphically using XY-plane:**

y ≤ - 3.5

**Sketch the graph of the following inequations in XOY-coordinate system:**

|x + 5| ≤ y

Solve graphically : 2x + y ≥ 2 and x – y ≤ 1

**Find graphical solution for the following system of linear in equation:**

3x + 4y ≤ 12, x - 2y ≥ 2, y ≥ - 1

**Solve the following LPP:**

Maximize z = 5x_{1} + 6x_{2} subject to 2x_{1} + 3x_{2} ≤ 18, 2x_{1} + x_{2} ≤ 12, x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following LPP:**

Maximize z = 4x + 2y subject to 3x + y ≤ 27, x + y ≤ 21, x ≥ 0, y ≥ 0.

**Solve the following LPP:**

Maximize z = 6x + 10y subject to 3x + 5y ≤ 10, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.

**Solve the following LPP:**

Maximize z = 2x + 3y subject to x - y ≥ 3, x ≥ 0, y ≥ 0.

**Solve the following LPP:**

Maximize z = 4x_{1} + 3x_{2} subject to

3x_{1} + x_{2} ≤ 15, 3x_{1} + 4x_{2} ≤ 24, x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following LPP:**

Maximize z =60x + 50y_{ } subject to

x + 2y ≤ 40, 3x + 2y ≤ 60, x ≥ 0, y ≥ 0.

**Solve the following LPP:**

Minimize z = 4x + 2y subject to

3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0.

A carpenter makes chairs and tables. Profits are ₹ 140 per chair and ₹ 210 per table. Both products are processed on three machines: Assembling, Finishing and Polishing. The time required for each product in hours and availability of each machine is given by the following table:

Product → | Chair (x) |
Table (y) |
Available time (hours) |

Machine ↓ | |||

Assembling | 3 | 3 | 36 |

Finishing | 5 | 2 | 50 |

Polishing | 2 | 6 | 60 |

Formulate the above problem as LPP. Solve it graphically

A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.

If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

A chemical company produces two compounds, A and B. The following table gives the units of ingredients, C and D per kg of compounds A and B as well as minimum requirements of C and D and costs per kg of A and B. Find the quantities of A and B which would give a supply of C and D at a minimum cost.

Compound |
Minimum requirement |
||

A | B | ||

Ingredient C Ingredient D |
1 3 |
2 1 |
80 75 |

Cost (in Rs) per kg | 4 | 6 | - |

A company produces mixers and food processors. Profit on selling one mixer and one food processor is Rs 2,000 and Rs 3,000 respectively. Both the products are processed through three machines A, B, C. The time required in hours for each product and total time available in hours per week on each machine arc as follows:

Machine |
Mixer |
Food Processor |
Available time |

A | 3 | 3 | 36 |

B | 5 | 2 | 50 |

C | 2 | 6 | 60 |

How many mixers and food processors should be produced in order to maximize the profit?

A chemical company produces a chemical containing three basic elements A, B, C, so that it has at least 16 litres of A, 24 litres of B and 18 litres of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 litres of A, 12 litres of B and 2 litres of C. Each unit of compound II has 2 litres of A, 2 litres of B and 6 litres of C. The cost per unit of compound I is ‘ 800 and that of compound II is ‘ 640. Formulate the problems as LPP and solve it to minimize the cost.

A person makes two types of gift items A and B requiring the services of a cutter and a finisher. Gift item A requires 4 hours of the cutter's time and 2 hours of finisher's time. Fifth item B requires 2 hours of the cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available time respectively every month. The profit on one gift item of type A is ₹ 75 and on one gift item of type, B is ₹ 125. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?

A firm manufactures two products A and B on which profit earned per unit ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M_{1} and M_{2}. The product A requires one minute of processing time on M_{1} and two minutes of processing time on M_{2}, B requires one minute of processing time on M_{1} and one minute of processing time on M_{2}. Machine M_{1} is available for use for 450 minutes while M_{2} is available for 600 minutes during any working day. Find the number of units of product A and B to be manufactured to get the maximum profit.

A firm manufacturing two types of electrical items A and B, can make a profit of ₹ 20 per unit of A and ₹ 30 per unit of B. Both A and B make use of two essential components a motor and a transformer. Each unit of A requires 3 motors and 2 transformers and each units of B requires 2 motors and 4 transformers. The total supply of components per month is restricted to 210 motors and 300 transformers. How many units of A and B should be manufactured per month to maximize profit? How much is the maximum profit?

## Chapter 7: Linear Programming

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 7 - Linear Programming

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 7 Linear Programming are Linear Inequations in Two Variables, Linear Programming Problem (L.P.P.), Lines of Regression of X on Y and Y on X Or Equation of Line of Regression, Graphical Method of Solving Linear Programming Problems, Linear Programming Problem in Management Mathematics.

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