State whether the following statement is True or False: The half plane represented by 3x + 4y ≥ 12 includes the point (4, 3) - Mathematics and Statistics

True or False

State whether the following statement is True or False:

The half-plane represented by 3x + 4y ≥ 12 includes the point (4, 3)


  • True

  • False




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Chapter 2.6: Linear Programming - Q.2 (B)


A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit of product  A  requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LPP.

A rubber company is engaged in producing three types of tyres AB and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in number of tyres per day, are as follows:

Plant A B C
I 50 100 100
II 60 60 200

The monthly demand for tyre AB and C is 2500, 3000 and 7000 respectively. If plant I costs Rs 2500 per day, and plant II costs Rs 3500 per day to operate, how many days should each be run per month to minimize cost while meeting the demand? Formulate the problem as LPP.

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

  A B
Grinding 1 2
Turning 3 1
Assembling 6 3
Testing 5 4

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

Vitamins A and B are found in two different foods F1 and F2. One unit of food F1contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F1 and F2 cost Rs 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

  Product A Product B Weekly capacity
Department 1 3 2 130
Department 2 4 6 260
Selling price per unit Rs 25 Rs 30  
Labour cost per unit Rs 16 Rs 20  
Raw material cost per unit Rs 4 Rs 4  

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is ₹400 and each small van is ₹200. Not more than ₹3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost

The solution set of the inequation 2x + y > 5 is

Which of the following sets are convex?

Let X1 and X2 are optimal solutions of a LPP, then

The maximum value of Z = 4x + 2y subjected to the constraints 2x + 3y ≤ 18, x + y ≥ 10 ; xy ≥ 0 is

The optimal value of the objective function is attained at the points

The maximum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80; xy ≥ 0 is

Consider a LPP given by
Minimum Z = 6x + 10y
Subjected to x ≥ 6; y ≥ 2; 2x + y ≥ 10; xy ≥ 0
Redundant constraints in this LPP are 

The objective function Z = 4x + 3y can be maximised subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6; xy ≥ 0

If the constraints in a linear programming problem are changed

Which of the following is not a convex set?

A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is ₹ 50 each on a toy of type A and ₹ 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution. 

Feasible region is the set of points which satisfy ______.

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

Choose the correct alternative:

The constraint that in a college there are more scholarship holders in FYJC class (X) than in SYJC class (Y) is given by

Choose the correct alternative:

How does a constraint, “A washing machine can hold up to 8 kilograms of cloths (X)” can be given?

Tyco Cycles Ltd manufactures bicycles (x) and tricycles (y). The profit earned from the sales of each bicycle and a tricycle are ₹ 400 and ₹ 200 respectively, then the total profit earned by the manufacturer will be given as ______

By spending almost ₹ 250, Rakhi bought some kg grapes (x) and some dozens of bananas (y), then as a constraint this information can be expressed by ______

A doctor prescribed 2 types of vitamin tablets, T1 and T2 for Mr. Dhawan. The tablet T1 contains 400 units of vitamin and T2 contains 250 units of vitamin. If his requirement of vitamin is at least 4000 units, then the inequation for his requirement will be ______

Heramb requires at most 400 calories from his breakfast. Every morning he likes to take oats and milk. If each bowl of oats and a glass of milk provides him 80 calories and 50 calories respectively, then as a constraint this information can be expressed as ______

Ganesh owns a godown used to store electronic gadgets like refrigerator (x) and microwave (y). If the godown can accommodate at most 75 gadgets, then this can be expressed as a constraint by ______

Ms. Mohana want to invest at least ₹ 55000 in Mutual funds and fixed deposits. Mathematically this information can be written as ______

Determine the minimum value of Z = 3x + 2y (if any), if the feasible region for an LPP is shown in Figue.

Solve the following LPP graphically:
Maximise Z = 2x + 3y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0

Minimise Z = 3x + 5y subject to the constraints:
x + 2y ≥ 10
x + y ≥ 6
3x + y ≥ 8
x, y ≥ 0

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is ______.

Feasible region (shaded) for a LPP is shown in the Figure Minimum of Z = 4x + 3y occurs at the point ______.

The common region determined by all the linear constraints of a LPP is called the ______ region.

In maximization problem, optimal solution occurring at corner point yields the ____________.

A type of problems which seek to maximise (or, minimise) profit (or cost) form a general class of problems called.

Conditions under which the object function is to be maximum or minimum are called ______.


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