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NCERT Solutions for Exercise 12.1 Class 12 Maths Chapter 12 - Linear Programming

NCERT Solutions for Exercise 12.1 Class 12 Maths Chapter 12 - Linear Programming

Edited By Ramraj Saini | Updated on Dec 04, 2023 11:02 AM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 12 Exercise 12.1

NCERT Solutions for Exercise 12.1 Class 12 Maths Chapter 12 Linear Programming are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for exercise 12.1 Class 12 Maths chapter 12 gives practice questions to understand linear programming problems. There are 10 questions explained in exercise 12.1 Class 12 Maths. There are many methods to solve linear programming problems. Here in this Class 12 NCERT Mathematics chapter, only graphical methods for solving linear programming problems are discussed. In the NCERT Solutions for class 12 maths chapter 12 exercise 12.1 all the 10 questions are solved using graphs. Class 12 Maths chapter 12 exercise 12.1 solves problems related to maximising or minimising linear functions subjected to certain constraints. These constraints are a set of linear inequalities.

This Story also Contains
  1. NCERT Solutions For Class 12 Maths Chapter 12 Exercise 12.1
  2. Access NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1
  3. Linear Programming Class 12 Chapter 12-Exercise: 12.1
  4. More About NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1
  5. Benefits of NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1
  6. Key Features Of NCERT Solutions for Exercise 12.1 Class 12 Maths Chapter 12
  7. Also see-
  8. NCERT Solutions Subject Wise
  9. Subject Wise NCERT Exemplar Solutions

12th class Maths exercise 12.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise enumerated in NCERT Book together using the link provided below.

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Access NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1

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Linear Programming Class 12 Chapter 12-Exercise: 12.1

Question:1 Solve the following Linear Programming Problems graphically: Maximise Z = 3x + 4y Subject to the constraints x+y\leq 4,x\geq 0,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+y\leq 4,x\geq 0,y\geq 0. is as follows,

1627031435613

The region A0B represents the feasible region

The corner points of the feasible region are B(4,0),C(0,0),D(0,4)

Maximize Z = 3x + 4y

The value of these points at these corner points are :

Corner points
Z = 3x + 4y

B(4,0)
12

C(0,0)
0

D(0,4)
16
maximum
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The maximum value of Z is 16 at D(0,4)

Question:2 Solve the following Linear Programming Problems graphically: Minimise z=-3x+4y Subject to . x+2y\leq 8,3x+2y\leq 12,x\geq 0,y\geq 0. Show that the minimum of Z occurs at more than two points

Answer:

The region determined by constraints, x+2y\leq 8,3x+2y\leq 12,x\geq 0,y\geq 0. is as follows,

1627031511366

The corner points of feasible region are A(2,3),B(4,0),C(0,0),D(0,4)

The value of these points at these corner points are :

Corner points
z=-3x+4y

A(2,3)
6

B(4,0)
-12
Minimum
C(0,0)
0

D(0,4)
16

The minimum value of Z is -12 at B(4,0)

Question:3 Solve the following Linear Programming Problems graphically: Maximise Z = 5x + 3y Subject to 3x + 5y \leq 15 , 5x+2y\leq 10 , x\geq 0,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, 3x + 5y \leq 15 , 5x+2y\leq 10 , x\geq 0,y\geq 0 is as follows :

1627031555890

The corner points of feasible region are A(0,3),B(0,0),C(2,0),D(\frac{20}{19},\frac{45}{19})

The value of these points at these corner points are :

Corner points
Z = 5x + 3y

A(0,3)
9

B(0,0)
0

C(2,0)
10

D(\frac{20}{19},\frac{45}{19})
\frac{235}{19}
Maximum

The maximum value of Z is \frac{235}{19} at D(\frac{20}{19},\frac{45}{19})

Question:4 Solve the following Linear Programming Problems graphically: Minimise Z = 3x + 5y Such that x+3y\geq 3,x+y\geq 2,x,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x+3y\geq 3,x+y\geq 2,x,y\geq 0. is as follows,

1627031646530

The feasible region is unbounded as shown.

The corner points of the feasible region are A(3,0),B(\frac{3}{2},\frac{1}{2}),C(0,2)

The value of these points at these corner points are :

Corner points
Z = 3x + 5y

A(3,0)
9

B(\frac{3}{2},\frac{1}{2})
7
Minimum
C(0,2)
10


The feasible region is unbounded, therefore 7 may or may not be the minimum value of Z .

For this, we draw 3x + 5y< 7 and check whether resulting half plane has a point in common with the feasible region or not.

We can see a feasible region has no common point with. Z = 3x + 5y

Hence, Z has a minimum value of 7 at B(\frac{3}{2},\frac{1}{2})

Question:5 Solve the following Linear Programming Problems graphically: Maximise Z = 3x + 2y Subject to x+2y\leq 10,3x+y\leq 15,x,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+2y\leq 10,3x+y\leq 15,x,y\geq 0 is as follows,

1627031733350

The corner points of feasible region are A(5,0),B(4,3),C(0,5)

The value of these points at these corner points are :

Corner points
Z = 3x + 2y

A(5,0)
15

B(4,3)
18
Maximum
C(0,5)
10


The maximum value of Z is 18 at B(4,3)

Question:6 Solve the following Linear Programming Problems graphically: Minimise Z = x + 2y Subject to 2x+y\geq 3,x+2y\geq 6,x,y\geq 0.

Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints 2x+y\geq 3,x+2y\geq 6,x,y\geq 0. is as follows,

1627031776022

The corner points of the feasible region are A(6,0),B(0,3)

The value of these points at these corner points are :

Corner points
Z = x + 2y
A(6,0)
6
B(0,3)
6

Value of Z is the same at both points. A(6,0),B(0,3)

If we take any other point like (2,2) on line Z = x + 2y , then Z=6.

Thus the minimum value of Z occurs at more than 2 points .

Therefore, the value of Z is minimum at every point on the line Z = x + 2y .

Question:7 Solve the following Linear Programming Problems graphically: Minimise and Maximise z=5x+10y Subject to x+2y\leq 120,x+y\geq 60,x-2y\geq 0,x,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+2y\leq 120,x+y\geq 60,x-2y\geq 0,x,y\geq 0 is as follows,

1627031823610

The corner points of feasible region are A(40,20),B(60,30),C(60,0),D(120,0)

The value of these points at these corner points are :

Corner points
z=5x+10y

A(40,20)
400

B(60,30)
600
Maximum
C(60,0)
300
Minimum
D(120,0)
600
maximum

The minimum value of Z is 300 at C(60,0) and maximum value is 600 at all points joing line segment B(60,30) and D(120,0)

Question:8 Solve the following Linear Programming Problems graphically: Minimise and Maximise z=x+2y Subject to x+2y\geq 100,2x-y\leq 0,2x+y\leq 200,x,y,\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x+2y\geq 100,2x-y\leq 0,2x+y\leq 200,x,y,\geq 0 is as follows,

1627031869634

The corner points of the feasible region are A(0,50),B(20,40),C(50,100),D(0,200)

The value of these points at these corner points are :

Corner points
z=x+2y

A(0,50)
100
Minimum
B(20,40)
100
Minimum
C(50,100)
250

D(0,200)
400
Maximum

The minimum value of Z is 100 at all points on the line segment joining points A(0,50) and B(20,40) .

The maximum value of Z is 400 at D(0,200) .

Question:9 Solve the following Linear Programming Problems graphically: Maximise Z = -x+2y Subject to the constraints: x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0. is as follows,

1627032034587

The corner points of the feasible region are A(6,0),B(4,1),C(3,2)

The value of these points at these corner points are :

Corner points
Z = -x+2y

A(6,0)
- 6
minimum
B(4,1)
-2

C(3,2)
1
maximum

The feasible region is unbounded, therefore 1 may or may not be the maximum value of Z.

For this, we draw -x+2y> 1 and check whether resulting half plane has a point in common with a feasible region or not.

We can see the resulting feasible region has a common point with a feasible region.

Hence , Z =1 is not maximum value , Z has no maximum value.

Question:10 Solve the following Linear Programming Problems graphically: Maximise Z = x + y, Subject to x-y\leq -1,-x+ y\leq 0,x,y,\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x-y\leq -1,-x+ y\leq 0,x,y,\geq 0. is as follows,

1627032109317

There is no feasible region and thus, Z has no maximum value.

More About NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1

In order to understand the concepts well it is important to solve the NCERT syllabus exercise questions. The NCERT solutions for Class 12 Maths chapter 12 exercise 12.1 helps in solving the first exercise of the chapter linear programming. Students can make use of Class 12 Maths chapter 12 exercise 12.1 solutions for preparation of board exams as well as engineering entrance exam like JEE Mains.

Also Read| Linear Programming Class 12th Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1

  • Class 12th Maths chapter 12 exercise 12.1 are prepared by the best faculties of Mathematics
  • All main topics are covered and exercise 12.1 Class 12 Maths gives answers to all the questions and are in detail
  • Students can use Class 12 Maths chapter 12 exercise 12.1 to prepare for CBSE exams.

Key Features Of NCERT Solutions for Exercise 12.1 Class 12 Maths Chapter 12

  • Comprehensive Coverage: The solutions encompass all the topics covered in ex 12.1 class 12, ensuring a thorough understanding of the concepts.
  • Step-by-Step Solutions: In this class 12 maths ex 12.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
  • Accuracy and Clarity: Solutions for class 12 ex 12.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
  • Conceptual Clarity: In this 12th class maths exercise 12.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
  • Inclusive Approach: Solutions for ex 12.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
  • Relevance to Curriculum: The solutions for class 12 maths ex 12.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT Solutions Subject Wise

Subject Wise NCERT Exemplar Solutions

Frequently Asked Question (FAQs)

1. What is the number of exercises in the NCERT chapter linear programming?

There are three exercises including miscellaneous.

2. How many questions are there in exercise 12.1 Class 12 Maths?

Ten questions are explained in the NCERT Class 12 chapter exercise 1

3. What number of solved examples are given before the NCERT Class 12 chapter linear programming exercise 12.1?

There are 5 solved examples before exercise 12.1

4. Why to solve NCERT exercises?

Solving NCERT exercise give more conceptual understanding and students will be able to clear their doubts and can understand the are where they have to improve.

5. How to use NCERT Solutions for Class 12 Maths chapter 12 exercise 12.1?

First understand the concepts and practice solved example. Then move on to the exercise and try to solve it yourself. If you have any doubts look in to the Class 12 Maths chapter 12 exercise 12.1 solutions.

6. What is the importance of the Class 12 NCERT Maths chapter 12 linear programming for CBSE Class 12 board exams?

For CBSE Class 12 Maths exam one question of 5 marks is expected from the chapter linear programming. 

7. Give the pattern of linear programming questions of Class 12 NCERT exercise 12.1 .

The linear programming questions will have an objective function. Either maximise or minimize the it according to the given constrains.

8. What method is adopted in the class 12 NCERT chapter 12 problems?

Graphical method is used to solve the problems in Class 12 chapter 12

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i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

You can use them people also used problem

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Divyan 06 August,2024
i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

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Since you already have a 12th-grade qualification with 84%, you meet the eligibility criteria and can apply for the Medhavi Scholarship exam. Preparing well for the exam can increase your chances of receiving a higher scholarship.

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Yuvan 01 September,2024
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024
Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

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kumardurgesh1802 10 July,2024
I have completed my 12th with arts cbse. I want to repeat my 12th with pcmb. How to become a private candidate and what are deadline to be able to appear in 2025 exam?

Hello student,

If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:

  • No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
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  • Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
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  • Schools might not offer classes for private students, so focus on self-study or coaching.

Remember , these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

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Good luck with your studies!

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kumardurgesh1802 10 July,2024
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