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

LPP is related to efficient use of limited resources

#### Options

True

False

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#### Solution

**True**

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#### RELATED QUESTIONS

Which of the following statements is correct?

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

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

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 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 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.

**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:**

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.

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 ______.

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

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

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 ______.

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 ______.

**Solve the following LPP:**

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

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

4x - 18 ≥ 0

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

5y - 12 ≥ 0

**Sketch the graph of the following inequation in XOY co-ordinate system:**

|x + 5| ≤ y

**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 = 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:**

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 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?

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 per unit. 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.

**Choose the correct alternative :**

Which of the following is correct?

**Choose the correct alternative :**

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

**Choose the correct alternative :**

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

**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

**Choose the correct alternative :**

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

**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.

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

The feasible solution of LPP belongs to only quadrant I.

Maximize z = 10x + 25y subject to x + y ≤ 5, 0 ≤ x ≤ 3, 0 ≤ y ≤ 3

Minimize z = 7x + y subjected to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0

Minimize z = 2x + 4y is subjected to 2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0 show that the minimum value of z occurs at more than two points

Maximize z = −x + 2y subjected to constraints x + y ≥ 5, x ≥ 3, x + 2y ≥ 6, y ≥ 0 is this LPP solvable? Justify your answer.

x − y ≤ 1, x − y ≥ 0, x ≥ 0, y ≥ 0 are the constant for the objective function z = x + y. It is solvable for finding optimum value of z? Justify?

**Choose the correct alternative:**

The feasible region is

**Choose the correct alternative:**

Z = 9x + 13y subjected to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, 0 ≤ x, y was found to be maximum at the point

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

Objective function of LPP is a relation between the decision variables

The variables involved in LPP are called ______

Constraints are always in the form of ______ or ______.

The constraint that in a particular XII class, number of boys (y) are less than number of girls (x) is given by ______

A company produces two types of products say type A and B. Profits on the two types of product are ₹ 30/- and ₹ 40/- per kg respectively. The data on resources required and availability of resources are given below.

Requirements |
Capacity available per month |
||

Product A |
Product B |
||

Raw material (kgs) | 60 | 120 | 12000 |

Machining hours/piece | 8 | 5 | 600 |

Assembling (man hours) | 3 | 4 | 500 |

Formulate this problem as a linear programming problem to maximize the profit.

A company manufactures two models of voltage stabilizers viz., ordinary and auto-cut. All components of the stabilizers are purchased from outside sources, assembly and testing is carried out at the company’s own works. The assembly and testing time required for the two models are 0.8 hours each for ordinary and 1.20 hours each for auto-cut. Manufacturing capacity 720 hours at present is available per week. The market for the two models has been surveyed which suggests a maximum weekly sale of 600 units of ordinary and 400 units of auto-cut. Profit per unit for ordinary and auto-cut models has been estimated at ₹ 100 and ₹ 150 respectively. Formulate the linear programming problem.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 6x_{1} + 8x_{2} subject to constraints 30x_{1} + 20x_{2 }≤ 300; 5x_{1} + 10x_{2} ≤ 110; and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 22x_{1} + 18x_{2} subject to constraints 960x_{1} + 640x_{2} ≤ 15360; x_{1} + x_{2} ≤ 20 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 40x_{1} + 50x_{2} subject to constraints 3x_{1} + x_{2} ≤ 9; x_{1} + 2x_{2} ≤ 8 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Minimize Z = 20x_{1} + 40x_{2} subject to the constraints 36x_{1} + 6x_{2} ≥ 108; 3x_{1} + 12x_{2} ≥ 36; 20x_{1} + 10x_{2} ≥ 100 and x_{1}, x_{2} ≥ 0.

A solution which maximizes or minimizes the given LPP is called

In the given graph the coordinates of M_{1} are

The maximum value of the objective function Z = 3x + 5y subject to the constraints x ≥ 0, y ≥ 0 and 2x + 5y ≤ 10 is

**Solve the following linear programming problem graphically.**

Minimize Z = 200x_{1} + 500x_{2} subject to the constraints: x_{1} + 2x_{2} ≥ 10; 3x_{1} + 4x_{2} ≤ 24 and x_{1} ≥ 0, x_{2} ≥ 0.

The maximum value of Z = 3x + 5y, subject to 3x + 2y ≤ 18, x ≤ a, y ≤ 6, x, y ≥ 0 is ______.

The LPP to maximize Z = x + y, subject to x + y ≤ 1, 2x + 2y ≥ 6, x ≥ 0, y ≥ 0 has ________.

The values of θ satisfying sin7θ = sin4θ - sinθ and 0 < θ < `pi/2` are ______

Which of the following can be considered as the objective function of a linear programming problem?

The minimum value of z = 5x + 13y subject to constraints 2x + 3y ≤ 18, x + y ≥ 10, x ≥ 0, y ≥ 2 is ______

The point which provides the solution of the linear programming problem, Max.(45x + 55y) subject to constraints x, y ≥ 0, 6x + 4y ≤ 120, 3x + 10y ≤ 180, is ______

Solve the following LP.P.

Maximize z = 13x + 9y,

Subject to 3x + 2y ≤ 12,

x + y ≥ 4,

x ≥ 0,

y ≥ 0.

Shamli wants to invest ₹ 50, 000 in saving certificates and PPF. She wants to invest atleast ₹ 15,000 in saving certificates and at least ₹ 20,000 in PPF. The rate of interest on saving certificates is 8% p.a. and that on PPF is 9% p.a. Formulation of the above problem as LPP to determine maximum yearly income, is ______.

The maximum value of Z = 9x + 13y subject to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0 is ______.

For the following shaded region, the linear constraint are:

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

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

Two kinds of foods A and B are being considered to form a weekly diet. The minimum weekly requirements of fats, Carbohydrates and proteins are 12, 16 and 15 units respectively. One kg of food A has 2, 8 and 5 units respectively of these ingredients and one kg of food B has 6, 2 and 3 units respectively. The price of food A is Rs. 4 per kg and that of food B is Rs. 3 per kg. Formulate the L.P.P. and find the minimum cost.

Food F_{1} contains 2, 6, 1 units and food F_{2} contains 1, 1, 3 units of proteins, carbohydrates, fats respectively per kg. 8, 12 and 9 units of proteins, carbohydrates and fats is the weekly minimum requirement for a person. The cost of food F_{1} is Rs. 85 and food F_{2} is Rs. 40 per kg. Formulate the L.P.P. to minimize the cost.

**Sketch the graph of the following inequation in XOY co-ordinate system.**

x + y ≤ 0

**Sketch the graph of the following inequation in XOY co-ordinate system.**

2y - 5x ≥ 0