CBSE Class 12 Mathematics Linear Programming Notes

Download CBSE Class 12 Mathematics Linear Programming Notes in PDF format. All Revision notes for Class 12 Mathematics have been designed as per the latest syllabus and updated chapters given in your textbook for Mathematics in Class 12. Our teachers have designed these concept notes for the benefit of Class 12 students. You should use these chapter wise notes for revision on daily basis. These study notes can also be used for learning each chapter and its important and difficult topics or revision just before your exams to help you get better scores in upcoming examinations, You can also use Printable notes for Class 12 Mathematics for faster revision of difficult topics and get higher rank. After reading these notes also refer to MCQ questions for Class 12 Mathematics given on studiestoday

Revision Notes for Class 12 Mathematics Chapter 12 Linear Programming

Class 12 Mathematics students should refer to the following concepts and notes for Chapter 12 Linear Programming in Class 12. These exam notes for Class 12 Mathematics will be very useful for upcoming class tests and examinations and help you to score good marks

Chapter 12 Linear Programming Notes Class 12 Mathematics

 

TOPIC 11

LINEAR PROGRAMMING

KEY CONCEPTS

Linear Programming Problem : Linear programming problem is one that is concerned with finding the optimal value ( maximum or minimum value ) of a linear function of several variables called objective function

Feasible Region:Feasible region is the region which is common to all the linear constraints (linear inequalities )

Important LPP are

1. Diet Problems

2. Manufacturing Problems

3. Transportation Problems

Steps for solving a LPP

Solving linear programming problem using Corner Point Method. The method comprises of the following steps:

1. Convert the word problem into mathematical formulation by using given constraints.

2. Solve the linear inequations formed in step 1 and plot the graph.

3. Find the feasible region of the linear programming problem and determine its corner points either by inspection or by solving the two

equations of the lines intersecting at that point.

4. Evaluate the objective function Z = ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points.

5. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.

(ii) In case, the feasible region is unbounded, we have:

6. (a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value.

 (b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value.

Problems of LPP

1. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs.5 and Rs.4 per unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost, but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically. Explain the importance of balanced diet.

Hint : Let x units of food A, y units of food B are mixed Z = 5x + 4y 200x+100y ≥ 4000, 40x + 40y ≥ 1400, x ≥ 0, y ≥ 0 2m Correct graph 2m

Points (50,0), (20,15), (5,30), (0, 40) and minimum cost is at (5, 30) 2m

2. 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.Which computer would you prefer to buy?

Solution:-Let the merchant stock x desktop models and y portable models. Therefore, x ≥ 0 and y ≥ 0

The cost of a desktop model is Rs 25000 and of a portable model is Rs 4000. However, the merchant can invest a maximum of Rs 70 lakhs.

The monthly demand of computers will not exceed 250 units.

CBSE Class 12 Mathematics Linear Programming

The profit on a desktop model is Rs 4500 and the profit on a portable model is Rs 5000.

Total profit, Z = 4500x + 5000y

Thus, the mathematical formulation of the given problem is

CBSE Class 12 Mathematics Linear Programming

subject to the constraints,

CBSE Class 12 Mathematics Linear Programming

The feasible region determined by the system of constraints is as follows

CBSE Class 12 Mathematics Linear Programming

The corner points are A (250, 0), B (200, 50), and C (0, 175).

The values of Z at these corner points are as follows.

CBSE Class 12 Mathematics Linear Programming

CBSE Class 12 Mathematics Linear Programming

The maximum value of Z is 1150000 at (200, 50).

Thus, the merchant should stock 200 desktop models and 50 portable models to get the maximum profit of Rs 1150000.

3. A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1costs Rs. 4 per unit and F2costsRs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets

the minimal nutritional requirements.

Solution:-

. Minimize Z = 4x + 6y

CBSE Class 12 Mathematics Linear Programming

Cost will be minimum when 24 units of F1& 4/3 units of F2 will be mixed and minimum cost will be Rs 104

CBSE Class 12 Mathematics Linear Programming

4. One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming  hat there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solvegraphically.

Solution:-Let x and y be the no. of cakes of type I & II respectively. Then according to question 

CBSE Class 12 Mathematics Linear Programming

CBSE Class 12 Mathematics Linear Programming
20 cakes of first kind and 10 cakes of 2nd kind should be made to get max. numbers of cake. 

5: A firm has to transport 1200 packages using large vans,which can carry 200 packages each& small vans which can carry 80 packages each.The cost of engaging each large van is Rs 400 7 each small van is Rs 200. Not more than 3000 is to be spent on the job& the no. of large vans 
cannot exceed the no. of small vans. Formulate the problem as CPP, given that the objective is to minimize the cost.

CBSE Class 12 Mathematics Linear Programming
Let x large vans and y small vans are engaged

Then LPP is

To minimize Z = 400 x + 200y

Subject to constraints

x≥o, y≥ 0

400x+200y≤3000 =>2x+y ≤15

200x+80y≥1200 =>5x+2y ≥30

and x ≤ y

Please click the link below to download CBSE Class 12 Mathematics Linear Programming.

More Study Material

CBSE Class 12 Mathematics Chapter 12 Linear Programming Notes

We hope you liked the above notes for topic Chapter 12 Linear Programming which has been designed as per the latest syllabus for Class 12 Mathematics released by CBSE. Students of Class 12 should download and practice the above notes for Class 12 Mathematics regularly. All revision notes have been designed for Mathematics by referring to the most important topics which the students should learn to get better marks in examinations. Studiestoday is the best website for Class 12 students to download all latest study material.

Notes for Mathematics CBSE Class 12 Chapter 12 Linear Programming

Our team of expert teachers have referred to the NCERT book for Class 12 Mathematics to design the Mathematics Class 12 notes. If you read the concepts and revision notes for one chapter daily, students will get higher marks in Class 12 exams this year. Daily revision of Mathematics course notes and related study material will help you to have a better understanding of all concepts and also clear all your doubts. You can download all Revision notes for Class 12 Mathematics also from www.studiestoday.com absolutely free of cost in Pdf format. After reading the notes which have been developed as per the latest books also refer to the NCERT solutions for Class 12 Mathematics provided by our teachers

Chapter 12 Linear Programming Notes for Mathematics CBSE Class 12

All revision class notes given above for Class 12 Mathematics have been developed as per the latest curriculum and books issued for the current academic year. The students of Class 12 can rest assured that the best teachers have designed the notes of Mathematics so that you are able to revise the entire syllabus if you download and read them carefully. We have also provided a lot of MCQ questions for Class 12 Mathematics in the notes so that you can learn the concepts and also solve questions relating to the topics. All study material for Class 12 Mathematics students have been given on studiestoday.

Chapter 12 Linear Programming CBSE Class 12 Mathematics Notes

Regular notes reading helps to build a more comprehensive understanding of Chapter 12 Linear Programming concepts. notes play a crucial role in understanding Chapter 12 Linear Programming in CBSE Class 12. Students can download all the notes, worksheets, assignments, and practice papers of the same chapter in Class 12 Mathematics in Pdf format. You can print them or read them online on your computer or mobile.

Notes for CBSE Mathematics Class 12 Chapter 12 Linear Programming

CBSE Class 12 Mathematics latest books have been used for writing the above notes. If you have exams then you should revise all concepts relating to Chapter 12 Linear Programming by taking out a print and keeping them with you. We have also provided a lot of Worksheets for Class 12 Mathematics which you can use to further make yourself stronger in Mathematics

Where can I download latest CBSE Class 12 Mathematics Chapter 12 Linear Programming notes

You can download notes for Class 12 Mathematics Chapter 12 Linear Programming for latest academic session from StudiesToday.com

Can I download the Notes for Chapter 12 Linear Programming Class 12 Mathematics in Pdf format

Yes, you can click on the link above and download notes PDFs for Class 12 Mathematics Chapter 12 Linear Programming which you can use for daily revision

Are the revision notes available for Chapter 12 Linear Programming Class 12 Mathematics for the latest CBSE academic session

Yes, the notes issued for Class 12 Mathematics Chapter 12 Linear Programming have been made available here for latest CBSE session

How can I download the Chapter 12 Linear Programming Class 12 Mathematics Notes pdf

You can easily access the link above and download the Class 12 Notes for Mathematics Chapter 12 Linear Programming for each topic in Pdf

Is there any charge for the Class 12 Mathematics Chapter 12 Linear Programming notes

There is no charge for the notes for CBSE Class 12 Mathematics Chapter 12 Linear Programming, you can download everything free of charge

Which is the best online platform to find notes for Chapter 12 Linear Programming Class 12 Mathematics

www.studiestoday.com is the best website from which you can download latest notes for Chapter 12 Linear Programming Mathematics Class 12

Where can I find topic-wise notes for Class 12 Mathematics Chapter 12 Linear Programming

Come to StudiesToday.com to get best quality topic wise notes for Class 12 Mathematics Chapter 12 Linear Programming

Can I get latest Chapter 12 Linear Programming Class 12 Mathematics revision notes as per CBSE syllabus

We have provided all notes for each topic of Class 12 Mathematics Chapter 12 Linear Programming as per latest CBSE syllabus