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

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NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1 (Ex 12.1) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 12 Linear Programming Exercise 12.1 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all exercise solutions in your emails.


Class:

NCERT Solutions for Class 12

Subject:

Class 12 Maths

Chapter Name:

Chapter 12 - Linear Programming

Exercise:

Exercise - 12.1

Content-Type:

Text, Videos, Images and PDF Format

Academic Year:

2023-24

Medium:

English and Hindi

Available Materials:

  • Chapter Wise

  • Exercise Wise

Other Materials

  • Important Questions

  • Revision Notes

Competitive Exams after 12th Science

Access NCERT Solutions for Class-12 Maths Chapter 12 Linear Programming

Exercise 12.1

1: Maximise $Z=3x+4y$

Subject to the constraints: $x+y\ge 4$, $x\ge 0$, $y\ge 0$  

Ans: The constraints: $x+y\ge 4$, $x\ge 0$, and $y\ge 0$, determine the feasible region as shown below: 


(Image will be uploaded soon)


The values of $Z$at the corner points $O(0,0),A(4,0),B(0,4)$ of the feasible region are: 

Corner point

$Z=3x+4y$


$O(0,0)$

$0$


$A(4,0)$

$12$


$B(0,4)$

$16$

Maximum 

Hence, the maximum value of $Z$ is  $16$ at the point $B(0,4)$. 


2: Minimise $Z=3x+4y$.

Subject to constraints $x+2y\le 8$, $3x+3y\le 12$,$x\ge 0$ and $y\ge 0$.

Ans: The constraints $x+2y\le 8$, $3x+3y\le 12$,$x\ge 0$ and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


The values of $Z$at the corner points $O(0,0),A(4,0),B(2,3)$ and $C(0,4)$ of the feasible region are: 

Corner point

$Z=3x+4y$


$O(0,0)$

$0$


$A(4,0)$

$-12$

Minimum 

$B(2,3)$

$6$


$C(0,4)$

$16$


Hence, the minimum value of $Z$ is $-12$ at the point $A(4,0)$. 


3: Maximise $Z=5x+3y$

Subject to the constraints: $3x+5y\le 15$, $5x+2y\le 10$, $x\ge 0$, $y\ge 0$  

Ans: The constraints: $3x+5y\le 15$, $5x+2y\le 10$,$x\ge 0$, and $y\ge 0$ determine the feasible region as shown below:


(Image will be uploaded soon)


The values of $Z$at the corner points $O(0,0),A(2,0),B(0,3)$ and $C(\frac{20}{19},\frac{45}{19})$ of the feasible region are: 

Corner point

$Z=5x+3y$


$O(0,0)$

$0$


$A(2,0)$

$10$


$B(0,3)$

$9$


$C(\frac{20}{19},\frac{45}{19})$

$\frac{235}{19}$

Maximum 

Hence, the maximum value of $Z$ is $\frac{235}{19}$ at the point$C(\frac{20}{19},\frac{45}{19})$. 


4: Minimise $Z=5x+3y$

Subject to the constraints: $x+3y\ge 3$, $x+y\ge 2$, $x\ge 0$, $y\ge 0$  

Ans: The constraints $x+3y\ge 3$, $x+y\ge 2$,$x\ge 0$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


Therefore, the feasible region is unbounded.

The values of $Z$at the corner points $A(3,0),B(\frac{3}{2},\frac{1}{2})$ and $C(0,2)$ of the feasible region are: 

Corner point

$Z=5x+3y$


$A(3,0)$

$9$


$B(\frac{3}{2},\frac{1}{2})$

$7$

Smallest

$C(0,2)$

$10$


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

So, we draw the graph of the inequality, $3x+5y<7$, and check if the resulting half-plane has common points with the feasible region.

As the feasible region has no common point with $3x+5y<7$, the minimum value of $Z$ is $7$ at the point $B(\frac{3}{2},\frac{1}{2})$. 


5: Maximise $Z=3x+2y$

Subject to the constraints: $x+2y\le 10$, $3x+5y\le 15$, $x\ge 0$, $y\ge 0$  

Ans: The constraints, $x+2y\le 10$, $3x+5y\le 15$,$x\ge 0$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


The values of $Z$at the corner points $A(5,0),B(4,3)$ and $C(0,5)$ of the feasible region are: 

Corner point

$Z=3x+2y$


$A(5,0)$

$15$


$B(4,3)$

$18$

Maximum

$C(0,5)$

$10$


Hence, the maximum value of $Z$ is $18$ at the point $B(4,3)$. 


6: Minimise $Z=x+2y$

Subject to the constraints: $2x+y\ge 8$, $x+2y\ge 6$, $x\ge 0$, $y\ge 0$  

Ans: The constraints $2x+y\ge 8$, $x+2y\ge 6$,$x\ge 0$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


The values of $Z$at the corner points $A(6,0)$ and $B(0,3)$ of the feasible region are: 

Corner point

$Z=x+2y$

$A(6,0)$

$6$

$B(0,3)$

$6$

As the value of $Z$ at points $A(6,0)$ and $B(0,3)$ is same, we need to take any other point, for example, $(2,2)$ on line $x+2y=6$

Now, $Z=6$ .

Thus, the minimum value of $Z$ occurs at more than $2$ points. Hence, we can say that the value of $Z$ is minimum at every point on the line $x+2y=6$.


7: Minimise and Maximise $Z=5x+10y$

Subject to the constraints: $x+2y\le 120$, $x+y\ge 60$, $x-2y\ge 60$,$x\ge 0$, $y\ge 0$  

Ans: The constraints $x+2y\le 120$, $x+y\ge 60$,$x-2y\ge 60$, $x\ge 0$, and $y\ge 0$ determine the feasible region as shown below:  


(Image will be uploaded soon)


The values of $Z$at the corner points $A(60,0),B(120,0),C(60,30)$ and $D(40,20)$ of the feasible region are:

Corner point

$Z=5x+10y$


$A(60,0)$

$300$

Minimum

$B(120,0)$

$600$

Maximum

$C(60,30)$

$600$

Maximum 

$D(40,20)$

$400$


Hence, the minimum value of $Z$ is $300$ at the point $A(60,0)$and the maximum value of $Z$ is $600$ at all the points lying on the line segment joining the points $B(120,0)$ and $C(60,30)$. 


8: Minimise and Maximise $Z=x+2y$

Subject to the constraints: $x+2y\ge 100$, $2x-y\le 0$, $2x+y\le 200$,$x\ge 0$, $y\ge 0$  

Ans: The constraints $x+2y\ge 100$, $2x-y\le 0$,$2x+y\le 200$, $x\ge 0$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


The values of $Z$at the corner points $A(0,50),B(20,40),C(50,100)$ and $D(0,200)$ of the feasible region are: 

Corner point

$Z=x+2y$


$A(0,50)$

$100$

Minimum

$B(20,40)$

$100$

Minimum

$C(50,100)$

$250$

 

$D(0,200)$

$400$

Maximum

Hence, the maximum value of $Z$ is $400$ at the point $D(0,200)$and the minimum value of $Z$is $100$ at all the points lying on the line segment joining the points $A(0,50)$ and $B(20,40)$.


9: Maximise $Z=-x+2y$

Subject to the constraints: $x\ge 3$, $x+y\ge 5$, $x+2y\ge 6$ , $y\ge 0$  

Ans: The constraints $x\ge 3$, $x+y\ge 5$,$x+2y\ge 6$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


It is visible that the feasible region is unbounded. 

The values of $Z$ at corner points $A(6,0),B(4,1)$ and $C(3,2)$are given as:

Corner point

$Z=-x+2y$

$A(6,0)$

$-6$

$B(4,1)$

$-2$

$C(3,2)$

$1$

Since, the feasible region is unbounded, $Z=1$ may or may not be the maximum value. 

So, we will draw the graph of the inequality, $-x+2y>1$, and check if the resulting half-plane has common points with the feasible region.

As the resulting feasible region has common points with the feasible region. Therefore, $Z=1$ is not the maximum value. Also, $Z$ has no maximum value. 


10: Maximise $Z=x+y$

Subject to the constraints: $x-y\le -1$, $-x+y\le 0$, $x\ge 0$ , and $y\ge 0$  

Ans: The constraints, $x-y\le -1$, $-x+y\le 0$,$x\ge 0$, and $y\ge 0$ determine the feasible region as shown below: 


(Image will be uploaded soon)


It can be seen that there is no feasible region. Thus, $Z$ does not have any maximum value.

NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming (Ex 12.1) Exercise 12.1

NCERT solutions for class 12 maths chapter 12 exercise 12.1 linear programming teaches kids the method of solving linear programming questions, the usage of theorems and the corner point method. According to the first theorem, let R be the feasible place (convex polygon) for a linear programming problem and the objective characteristic is given by Z = ax + by. 


The top-of-the-line value (maximum or minimal) of Z must occur at a nook point or vertex of the feasible place, in which the variables x and y are problems to constraints defined by linear inequalities. The subsequent theorem states that allow Z = ax + through being the goal function and let R be the feasible region for a linear programming problem. Then will Z have both a maximum and a minimum value on R, only if R is bounded and each one of these values occurs at a nook factor (vertex) of R. The elegance 12 maths NCERT answers chapter 12 exercise 12.1 has 10 problems that see the use of these theorems in detail.


While trying the question in the NCERT answers class 12 maths chapter 12 exercise 12.1 linear programming students want to maintain in mind whether the problem is asking them to minimize or maximize the value of Z. Additionally, they ought to shade the correct place in order to get the right answer set. The hyperlink given can be used to access the scrollable PDF version of the solutions.

NCERT Solutions for Class 12 Maths PDF Download

 

NCERT Solution Class 12 Maths of Chapter 12 All Exercises

Chapter 12 - Linear Programming Exercises in PDF Format

Exercise 12.1

10 Questions & Solutions (10 Long Answers)

Exercise 12.2

11 Questions & Solutions (11 Long Answers)

 

NCERT Solutions for Class 12 Math Chapter 12 Linear Programming Exercise 12.1

Opting for the NCERT solutions for Ex 12.1 Class 12 Math is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 12.1 Class 12 Math NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

 

Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 12 students who are thorough with all the concepts from the Math textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 12 Math Chapter 12 Exercise 12.1 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.

 

Besides these NCERT solutions for Class 12 Math Chapter 12 Exercise 12.1, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it. 

 

Do not delay any more. Download the NCERT solutions for Class 12 Math Chapter 12 Exercise 12.1 from Vedantu website now for better exam preparation. If you have the Vedantu app in your phone, you can download the same through the app as well. The best part of these solutions is that they can be accessed both online and offline as well.

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

1. What are the steps to solve linear programming?

Here are the steps to solve linear programming:


  • Determine the factors that influence decisions.

  • Compose the objective function.

  • Determine the constraints

  • Select the approach to the linear programming issue.

  • Construct the graph

  • Identify the feasible region

  • Find the optimum point

2. What do you mean by Linear programming?

When a linear function is exposed to various constraints, it is maximised or reduced using the mathematical modelling technique known as "linear programming." In commercial planning and industrial engineering, this approach has worked well for guiding quantitative decisions in the physical and social sciences, though to a lesser extent.

3. How many questions are there in Class 12 Maths Chapter 12 Linear Programming (Ex 12.1) Exercise 12.1?

Class 12 Maths Chapter 12 Linear Programming (Ex 12.1) Exercise 12.1 consists of a total of ten conceptual questions.

4. Why should I practice class 12 maths NCERT chapter 12  Linear Programming (Ex 12.1) Exercise 12.1?

The NCERT book and solutions have been prepared by the best and most highly skilled educators and scholars and have created the content in such a way that all the maths concepts could be understood by each student. Practising exercise 12.1 will help you in clearing your basics about the concept of Linear programming and will help as you go further through the chapter. The NCERT math book has explained the concept of linear programming, on which the first exercise is based, with the help of solved examples that are written in the easiest way.

5. Where can I find the NCERT solutions for Class 12 Maths chapter 12  Linear Programming (Ex 12.1) Exercise 12.1?

NCERT solutions for class 12 maths chapter 12 linear programming (Ex 12.1) exercise 12.1 are free to download in pdf format from Vedantu. Teachers at Vedantu are highly skilled and experts in their subjects, and they have curated these NCERT solutions according to the latest CBSE pattern and guidelines.