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## Chapter 6: Linear Programming

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 6 Linear Programming Exercise 6.1 [Pages 97 - 99]

A manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry and then sent to 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 are as follows:

Gadgets |
Foundry |
Machine Shop |

A | 10 | 5 |

B | 6 | 4 |

Time available (hours) | 60 | 35 |

Profit on the sale of A is ₹ 30 and B is ₹ 20 per unit. Formulate the L.P.P. to have maximum profit.

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

Nutrient\Fodder |
Fodder 1 |
Fodder2 |

Nutrient A | 2 | 1 |

Nutrient B | 2 | 3 |

Nutrient C | 1 | 1 |

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

A company manufactures two types of chemicals A and 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.

Raw Material \Chemical |
A |
B |
Availability |

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. Formulate the problem as L.P.P. to maximize the profit.

A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 on 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, 60 hours per week respectively. Formulate the Linear programming problem to maximize the profit.

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 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. If maximum availability of Machine M_{1} is 10 hours and that of Machine M_{2} is 12 hours, then formulate the L.P.P. to maximize the profit.

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:

Raw Material\Fertilizers |
F_{1} |
F_{2} |
Availability |

A | 2 | 3 | 40 |

B | 1 | 4 | 70 |

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

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 6 Linear Programming Exercise 6.2 [Page 101]

**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 L.P.P. by graphical method :**

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

**Solve the following L.P.P. 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. by graphical method :**

Maximize : Z = 10x + 25y subject to 0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5 also find maximum value of z.

**Solve the following L.P.P. by graphical method :**

Maximize: Z = 3x + 5y subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0 also find maximum value of Z.

**Solve the following L.P.P. by graphical method :**

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

**Solve the following L.P.P. by graphical method :**

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

**Solve the following L.P.P. by graphical method :**

Minimize: Z = 6x + 2y subject to x + 2y ≥ 3, x + 4y ≥ 4, 3x + y ≥ 3, x ≥ 0, y ≥ 0.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 6 Linear Programming Miscellaneous Exercise 6 [Pages 102 - 105]

**Choose the correct alternative :**

The value of objective function is maximize under linear constraints.

at the centre of feasible region

at (0, 0)

at any vertex of feasible region.

The vertex which is at maximum distance from (0, 0).

**Choose the correct alternative :**

Which of the following is correct?

Every LPP has on optional solution

Every LPP has unique optional solution.

If LPP has two optional solution then it has infinitely many solutions.

The set of all feasible solutions of LPP may not be a convex set.

**Choose the correct alternative:**

Objective function of LPP is

A constraint

A function to be maximized or minimized

A relation between the decision variables

A feasible region

Equation of straight line

**Choose the correct alternative :**

The maximum value of z = 5x + 3y. subject to the constraints

235

`(235)/(9)`

`(235)/(19)`

`(235)/(3)`

**Choose the correct alternative :**

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

56

`65

55

66

**Choose the correct alternative :**

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

(36, 25)

(20, 35)

(35, 20)

(40, 15)

**Choose the correct alternative :**

Of all the points of the feasible region the optimal value of z is obtained at a point

inside the feasible region.

at the boundary of the feasible region.

at vertex of feasible region.

on x - axis.

**Choose the correct alternative :**

Feasible region; the set of points which satify.

The objective function.

All of the given constraints.

Some of the given constraints

Only non-negative constrains

**Choose the correct alternative :**

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

x = 0, y = `(1)/(2)`

x = `(1)/(2)`, y = 0

x = 1, y = – 2

x = y = `(1)/(2)`

**Choose the correct alternative :**

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

(0, 0), (4, 0), (3, 1), (0, 4).

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

(0, 0), `(7/2, 0), (3, 1)`, (5, 7).

(6, 0), (4, 0), (3, 1), (0, 7).

**Choose the correct alternative :**

The corner points of the feasible region are (0, 0), (2, 0), `(12/7, 3/7)` and (0,1) then the point of maximum z = 7x + y

(0, 0)

(2, 0)

`(12/7, 3/7)`

(0, 1)

**Choose the correct alternative :**

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

12

13

`(35)/(2)`

0

**Choose the correct alternative :**

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

(2, 2)

(0, 10)

(4, 0)

(2, 4)

**Choose the correct alternative :**

The half plane represented by 3x + 2y ≤ 0 constraints the point.

`(1, 5/2)`

(2, 1)

(0, 0)

(5, 1)

**Choose the correct alternative :**

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

(0, 0)

(2, 2)

(3, 4)

(1, 1)

**Fill in the blank :**

Graphical solution set of the in equations x ≥ 0, y ≥ 0 is in _______ quadrant

**Fill in the blank :**

The region represented by the in equations x ≤ 0, y ≤ 0 lines in _______ quadrants

**Fill in the blank :**

The optimal value of the objective function is attained at the _______ points of feasible region.

**Fill in the blank :**

The region represented by the inequality y ≤ 0 lies in _______ quadrants.

**Fill in the blank :**

The constraint that a factory has to employ more women (y) than men (x) is given by_______

**Fill in the blank :**

“A gorage employs eight men to work in its shownroom and repair shop. The constraints that there must be at least 3 men in showroom and at least 2 men in repair shop are ______ and _______ respectively.

**Fill in the blank :**

A train carries at least twice as many first class passengers (y) as second class passengers (x) The constraint is given by_______

**Fill in the blank :**

A dish washing machine holds up to 40 pieces of large crockery (x) This constraint is given by_______.

**State whether the following is True or False :**

The region represented by the inequalities x ≥ 0, y ≥ 0 lies in first quadrant.

True

False

**State whether the following is True or False :**

The region represented by the inqualities x ≤ 0, y ≤ 0 lies in first quadrant.

True

False

**State whether the following is True or False :**

The optimum value of the objective function of LPP occurs at the center of the feasible region.

True

False

**State whether the following is True or False :**

Graphical solution set of x ≤ 0, y ≥ 0 in xy system lies in second quadrant.

True

False

**State whether the following is True or False :**

Saina wants to invest at most ₹ 24000 in bonds and fixed deposits. Mathematically this constraints is written as x + y ≤ 24000 where x is investment in bond and y is in fixed deposits.

True

False

**State whether the following is True or False :**

The point (1, 2) is not a vertex of the feasible region bounded by 2x + 3y ≤ 6, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.

True

False

**State whether the following is True or False :**

The feasible solution of LPP belongs to only quadrant I.

True

False

**Solve the following problem :**

Maximize Z = 5x_{1} + 6x_{2} Subject to 2x_{1} + 3x_{2} ≤ 18, 2x_{1} + x_{2} ≤ 12, x ≥ 0, x_{2} ≥ 0

**Solve the following problem :**

Minimize Z = 4x + 2y Subject to 3x + y ≥ 27, x + y ≥ 21, x ≥ 0, y ≥ 0

**Solve the following problem :**

Maximize Z = 6x + 10y Subject to 3x + 5y ≤ 10, 5x + 3y ≤ 15, x ≥ 0, y≥0

**Solve the following problem :**

Minimize Z = 2x + 3y Subject to x – y ≤ 1, x + y ≥ 3, x ≥ 0, y ≥ 0

**Solve the following problem :**

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 problem :**

Maximize Z = 60x + 50y Subject to x + 2y ≤ 40, 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

**Solve the following problem :**

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 following table.

Product/Machines | Chair (x) |
Table (y) |
Available time (hours) |

Assembling | 3 | 3 | 36 |

Finishing | 5 | 2 | 50 |

Polishing | 2 | 6 | 60 |

Formulate and solve the following Linear programming problems using graphical method.

**Solve the following problem :**

A company manufactures bicyles and tricycles, each of which must be processed through two machines A and B Maximum availability of machine A and B is respectively 120 and 180 hours. 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 ₹ 180 for a bicycle and ₹ 220 on a tricycle, determine the number of bicycles and tricycles that should be manufacturing in order to maximize the profit.

**Solve the following problem :**

A factory produced two types of chemicals A and B The following table gives the units of ingredients P & Q (per kg) of Chemicals A and B as well as minimum requirements of P and Q and also cost per kg. of chemicals A and B.

Ingredients per kg. /Chemical Units | A (x) |
B (y) |
Minimum requirements in |

P | 1 | 2 | 80 |

Q | 3 | 1 | 75 |

Cost (in ₹) | 4 | 6 |

Find the number of units of chemicals A and B should be produced so as to minimize the cost.

**Solve the following problem :**

A Company produces mixers and processors Profit on selling one mixer and one food processor is ₹ 2000 and ₹ 3000 respectively. Both the products are processed through three machines A, B, C The time required in hours by each product and total time available in hours per week on each machine are as follows:

Machine/Product | Mixer per unit | Food processor per unit | Available time |

A | 3 | 3 | 36 |

B | 5 | 2 | 50 |

C | 2 | 6 | 60 |

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

**Solve the following problem :**

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

**Solve the following problem :**

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

**Solve the following problem :**

A firm manufactures two products A and B on which profit earned per unit is ₹ 3 and ₹ 4 respectively. The product A requirs one minute of processing time on M_{1} and 2 minutes on M_{2}. B requires one minutes on M_{1} and one minute 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.

**Solve the following problem :**

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 unit 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 manufacture per month to maximize profit? How much is the maximum profit?

## Chapter 6: Linear Programming

## Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 6 - Linear Programming

Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 6 (Linear Programming) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 6 Linear Programming are Introduction of Linear Programming, Linear Programming Problem (L.P.P.), Mathematical Formulation of Linear Programming Problem.

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