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

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#### Solution

**y ≤ x**

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#### RELATED QUESTIONS

Which of the following statements is correct?

**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 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 requires 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.

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.

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.

**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:**

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

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

The value of objective function is maximum under linear constraints

Which of the following is correct?

Objective function of LPP is ______.

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, 0), (3, 0), (2, 1), `(0, 7/3)` the maximum value of z = 4x + 5y is ______.

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 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

**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 =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 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 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?

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 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 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.

**Choose the correct alternative :**

Which of the following is correct?

Objective function of LPP is ______.

**Choose the correct alternative :**

Feasible region; the set of points which satify.

**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

**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

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**Choose the correct alternative :**

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

**Choose the correct alternative :**

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

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

The feasible region is the set of point which satisfy.

Maximize z = 5x + 2y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

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

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

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

LPP is related to efficient use of limited resources

The variables involved in LPP are called ______

Constraints are always in the form of ______ or ______.

A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are ₹ 5 and ₹ 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.

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 = 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

Given an L.P.P maximize Z = 2x_{1} + 3x_{2} subject to the constrains x_{1} + x_{2} ≤ 1, 5x_{1} + 5x_{2} ≥ 0 and x_{1} ≥ 0, x_{2} ≥ 0 using graphical method, we observe

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

Maximise Z = 4x_{1} + x_{2} subject to the constraints x_{1} + x_{2} ≤ 50; 3x_{1} + x_{2} ≤ 90 and x_{1} ≥ 0, x_{2} ≥ 0.

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

Maximize Z = 60x_{1} + 15x_{2} subject to the constraints: x_{1} + x_{2} ≤ 50; 3x_{1} + x_{2} ≤ 90 and x_{1}, 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.

The set of feasible solutions of LPP is a ______.

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:

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.**

2y - 5x ≥ 0