Chapter 30 : Linear programming - Rd Sharma Solutions for Class 12-science Maths CBSE

Chapter 30 - Linear programming Excercise Ex. 30.1

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Question 2

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Chapter 30 - Linear programming Excercise Ex. 30.2

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Question 26

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:

2x + 4y £ 8

3x + y £ 6

X + y £ 4

X ³ 0, y ³ 0

Solution 26

 

Converting the inequations into equations, we obtain the lines

2x + 4y = 8, 3x + y = 6, x + y = 4, x = 0, y = 0.

These lines are drawn on a suitable scale and the feasible region of the LPP is shaded in the graph.

  

 

From the graph we can see the corner points as (0, 2) and (2, 0).

 

  

Chapter 30 - Linear programming Excercise Ex. 30.3

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Question 10

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. food p costs Rs. 60 kg and Food Q costs Rs. 80 kg. Food P contains 3 units / kg of Vitamin A and 5 units/ kg of Vitamin B while food Q contains 4 unit / kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

Solution 10

  

 

  

 

  

Question 11

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. 

Solution 11

 

  

  

 

 

  

Question 12

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240units of calcium, atleasy 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

Solution 12

  

 

  

 

  

Question 13

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 unit of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture pre bag?

Solution 13

  

 

  

 

  

 

Note: Answer given in the book is incorrect.

Question 14

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food

Vitamin A

Vitamin B

Vitamin C

X

1

2

3

Y

2

2

1

 

One kg of food X costs Rs. 16 and one kg of food costs Rs. 20. Find the least cost of the mixture which will produce the required diet?

Solution 14

  

  

 

 

  

Question 15

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, acid potash and chlorine in a bag of each brand are given in the Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

Kg per bag

 

Brand P

Brand Q

Nitrogen

3

3.5

Phosphoric acid

1

2

Potash

3

1.5

Chlorine

1.5

2

 

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

Solution 15

  

 

  

 

  

Chapter 30 - Linear programming Excercise Ex. 30.4

Question 1

If a young man drives his vehicle at 25km/hr, he has to spend Rs 2per km on petrol. if he drives it as a fast of 40 km/hr, the petrol cost increase to Rs 5 per km. He has Rs 100 to speed on petrol and travel a maximum distance in one hour time with less polution . Express this problem as an LPP and solve it graphically. What value do you find hear?

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Question 15

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Question 32

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

Solution 32

Question 33

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Question 34

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Question 35

A manufacturer makes two products, A and B. Product A sells at Rs. 200 each and takes ½ hour to make. Product B sells at Rs.300 each and takes 1 hour to make. There is a permanent order for 14 units of product A and 16 units of product B. A working week consist of 40 hours of production and the weekly turn over must not be less than Rs. 1000. If the profit on each of product A is Rs. 20 and an product B is Rs. 30, then how many of each should be produced so that the profit is maximum? Also find the maximum profit.

Solution 35

  

 

 

  

 

  

 

 

Question 36

If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2 per km on petrol. If he drives is at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5/ per km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an LPP solve the same.

Solution 36

  

 

 

  

 

  

 

Question 37

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Question 38

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Question 39

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Question 40

A factory makes tennis rackets and cricket bats A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hour of machine time and 24 hours of craftman’s time. If the profit on racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.    

Solution 40

Question 41

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and his profit on the desktop model is Rs 4500 and on the portable model is RS 5000. Make an LPP and solve it graphically.    

Solution 41

Question 42

A cooperative society of formers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so at to maximise the total profit of the society?

Solution 42

  

 

  

 

  

Question 43

A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 1200 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Solution 43

  

  

 

 

  

 

Question 44

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftrnan's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time.

  1. What number of rackets and bats must be made if the factory is to Work at full capacity?
  2. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.
Solution 44

  

  

 

 

  

Question 45

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.

Solution 45

 

  

  

 

 

 

  

Question 46

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

Solution 46

  

  

 

 

  

Question 47

There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs. 6/kg and F2 costs Rs. 5 /kg, determine how much of each type of fertiliser should be used so that nutrient requirement are met at a minimum cost. What is the minimum cost?

Solution 47

  

 

  

 

 

  

Question 48

A manufacturer has three machines I, II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.

The number of hours required for producing 1 unit of each of M and N on the three machines are given in the following table:

 

Items

Number of hours required on machines

 

I

II

III

M

1

2

1

N

2

1

1.25

 

She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximize her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

Solution 48

  

 

  

 

 

  

Question 49

There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:

To/from

Cost (in Rs.)

 

A

B

C

P

160

100

150

Q

100

120

100

 

How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost?

Solution 49

Let x and y units of commodity be transported from factory P to the depots at A and B respectively.

Then (8 - x - y) units will be transported to depot at C.

The flow is shown below.

 

  

 

  

 

  

Question 50

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

 

Types of Toys

Machines

 

I

II

III

A

12

18

6

B

6

0

9

 

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs. 7.50 and that on each toy of type B is Rs.5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Solution 50

   

 

  

Question 51

An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

Solution 51

  

  

 

  

Question 52

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Solution 52

  

  

 

  

Chapter 30 - Linear programming Excercise Ex. 30.5

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Chapter 30 - Linear programming Excercise Ex. 30RE

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,5000 and Rs 9,000 respectivley. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more thatn 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainages from this land. How much land should be allocated to each crop so as to miximise the total profit of the society?

Solution 7

Question 8

A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of  Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Solution 12

Corner points formed for:

Corner Points

Profit

Remarks

(0, 0)

Z = 100 × 0 + 120 × 0 = 0

 

(0, 10)

Z = 100 × 0 + 120 × 10 = 1200

 

(3, 8)

Z = 100 × 3 + 120 × 8 = 1260

Maximum

(17/3, 0)

Z = 100 × 17/3 + 120 × 0 = 1700/3

 

 

Revenue is maximum when x = 3, y = 8.

Maximum Profit = Rs. 1260

Question 13

Solution 13


Question 14

Solution 14




Question 15

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium. 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

Solution 15

Question 16

Solution 16

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Solution 17

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Solution 18



Question 19

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Question 22

Solution 22


Question 23

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

Solution 23

Let us plot the constraints and find the feasible region through graph.

 

Thus the profit is maximum when the dealer buys 8 electronic sewing machines and 12 manual operated sewing machines.

 

Question 24

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs. 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Solution 24

Thus profit is maximum and is equal to Rs.1680

The company should manufacture 12 Type A machines and 6 Type B machines to maximize their profit.

 

Chapter 30 - Linear programming Excercise MCQ

Question 1

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

  1. half plane that contains the origin
  2. open half plane not contains the origin
  3. whole xy-plane except the points lying on the line

2x + y =5

  1. none of these

 

Solution 1

Correct option: (b)

  

Question 2

Objective function of a LPP is

  1. a constraint
  2. a function to be optimized
  3. a relation between the variable
  4. none of these

 

Solution 2

Correct option: (b)

Objective function of a LPP is always maximized or minimized. Hence, it is optimized.

 

Question 3

Which of these following sets are convex?

  1. {(x, y):x2 + y21}
  2. {(x, y):y2 x}
  3. {(x, y):3x2 + 4y2 5}
  4. {(x, y): y 2, y 4 }

 

Solution 3

Correct option: (d)

Set of points between two parallel lines. Hence, set is connected. Set is convex.

Question 4

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

  1. X = λ X1 + (1- λ) X2 , λ R is also an optimal solution
  2. X = λ X1 + (1- λ) X2 , 0 ≤ λ ≤ 1 gives an optimal solution
  3. X = λ X1 + (1+ λ) X2 , 0≤ λ ≤ 1 gives an optimal solution
  4. X = λ X1 + (1+ λ) X2 , λ R given an optional solution
Solution 4

Correct option: (b)

  

Question 5

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

  1. 36
  2. 40
  3. 20
  4. None of these

 

Solution 5

Correct option: (d)

  

From the graph we conclude that no feasible region exist.

 

Question 6

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

  1. given by intersection of inequations with the axes only
  2. given by intersection of inequations with x-axis only
  3. given by corner points of the feasible region
  4. none of these

 

Solution 6

Correct option: (c)

The maximum value of the objective function is attained at the points given by corner points of the feasible region.

Question 7

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

  1. 320
  2. 300
  3. 230
  4. none of these
Solution 7

Correct option: (d)

If we put x=0 and y=0 in all the equations then we get contradiction. Hence, region is on open half plane not containing origin. The region is unbounded we can not find the maximum value of the feasible region.

Question 8

Consider a LPP given by

Minimum Z = 6x + 10y

Subjected to x 6; y 2; 2x + y 10; x, y 0

Redundant constraints in this LPP are

  1. x ≥ 0, y ≥ 0
  2. x ≥ 6, 2x + y ≥ 10
  3. 2x + y ≥ 10
  4. none of these

 

Solution 8

Correct option: (c)

  

 

Minimum Z will be at 2x + y 10. 

 

Question 9

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

  1. At only one point
  2. At two points only
  3. At an infinite number of points
  4. None of these

 

Solution 9

Correct option: (c)

  

  

 

Question 10

If the constraints in a linear programming problem are changed

  1. the problem is to be re-evaluated
  2. solution is not defined
  3. the objective function has to be modified
  4. the change in constraints is ignored

 

Solution 10

Correct option: (a)

Optimization of objective function is depend on constraints. Hence, if the constraints in a linear programming problem are changed the problem is to be re-evaluated.

 

Question 11

Which of the following statement is correct?

  1. Every LPP admits an optimal solution
  2. A LPP admits unique optimal solution
  3. If a LPP admits two optimal solution it has an infinite number of optimal solution
  4. The set of all feasible solutions of a LPP is not a converse set

 

Solution 11

Correct option: (c)

Optimal solution of LPP has three types.

  1. Unique
  2. Infinite
  3. Does not exist.

Hence, it has infinite solution if it admits two optimal solution.

Question 12

Which of the following is not a convex set?

  1. {(x, y): 2x + 5y <7}
  2. {(x, y):x2 + y2 ≤ 4}
  3. {x:|x|= 5}
  4. {(x, y): 3x2+2y2 ≤ 6}

 

Solution 12

Correct option: (c)

As |x|=5 will only on x-axis. Hence, set is not connected to any two points between the set.

Hence, it is not convex.

Question 13

By graphical method, the solution of linear programming problem

Maximize Z= 3x1 + 5x2

Subject to 3x1 + 2x2 18

 x1 4

x2 6

x1 0, x2  0, is

  1. x1 = 2, x2 = 0, Z = 6
  2. x1 = 4, x2 = 6, Z = 36
  3. x1 = 4, x2 = 3, Z = 27
  4. x1 = 4, x2 = 6, Z = 42

 

 

Solution 13

Correct option: (b)

  

  

 

Question 14

The region represented by the inequation system x, y 0, y 6, x + y 3 is

  1. unbounded in first quadrant
  2. unbounded in first and second quadrants
  3. bounded in first quadrant
  4. none of these

 

Solution 14

Correct option: (c)

As region is on origin side it is always bounded. Also, given that x,y 0 it is bounded in the first quadrant.

 

NOTE: Answer not matching with back answer.

Question 15

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

  1. (30, 25)
  2. (20, 35)
  3. (35, 20)
  4. (40, 15)

 

Solution 15

Correct option: (d)

  

  

Question 16

The value of objective function is maximum under linear constraints

  1. at the centre of feasible region
  2. at (0, 0)
  3. at any vertex of feasible region
  4. the vertex which is maximum distance from (0, 0)

 

Solution 16

Correct option: (c)

To find maximum or minimum value of the region we use the coordinates of the vertices of feasible region. Hence, the value of objective function is maximum under linear constraints at any vertex of the feasible region.

 

Note: Answer not matching with back answer.

Question 17

The corner points of the feasible region determined by the following system of linear inequalities:

2x + y 10, x + 3y 15, x, y 0 are (0, 0), (5, 0),

 (3, 4) and (0,5). Let Z = p x + q y, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is

  1. p = q
  2. p = 2q
  3. p = 3q
  4. q = 3p

 

Solution 17

Correct option: (d)

Given that Z=px + qy

Maximum value at (3, 4) = maximum value at (0, 5)

3p+4q=5q

q=3p