University of Liverpool, Bengaluru Campus
Study at a world-renowned UK university in India | Admissions open for UG & PG programs.
Search Colleges, Exams, Schools & more
Login
NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming
Linear Programming-Students understand a series of Mathematical methods applied to optimize various situations of real life. This article elaborates how students can maximize or minimize any expression subject to certain constraints. Concepts like linear inequalities, feasible region, objective function, optimal solution, graph based approach to solve various LPPs will cover and also will familiarize the students with real applications of mathematics in various fields of commerce, finance and management, etc. Compiled with due care by subject matter experts at Careers360 based on the updated syllabus of CBSE,
This Story also Contains
- NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming: Download Free PDF
- NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming: Exercise Questions
- Linear Programming Class 12 NCERT Solutions: Exercise-wise
- Class 12 Maths NCERT Chapter 12: Extra Question
- Linear Programming Class 12 Chapter 12: Topics
- NCERT Class 12 Maths Chapter 12: Important Formulae
- Approach to Solve Questions of Linear Programming Class 12
- Why are Class 12 Maths Chapter 12 Linear Programming Question Answers Important?
- Chapter Summary of NCERT Solutions for Class 12 Maths Chapter 12 - Linear Programming
- Expert Review of NCERT Solutions for Class 12 Maths Chapter 12 - Linear Programming
- NCERT Solutions for Class 12 Maths: Chapter Wise
these NCERT Solutions for class 12 Mathematics chapter provide error-free solutions in step by step form for every question mentioned in the textbook. The NCERT Solutions for class 12 help students not only to conceptualise graphically but also helps to solve word problems that often require a thorough understanding of Mathematical modeling. This chapter is crucial from exam perspective like JEE Main, JEE Advanced and consistent practice ensures good grasp of conceptual as well as problem-solving skills.
NCERT Solutions for Class 12 Maths Chapter 12
Linear Programming: Download Free PDF
Students who wish to access the Class 12 Maths Chapter 12 NCERT Solutions can click on the link below to download the complete solution in PDF.
NCERT Solutions for Class 12 Maths Chapter 12
Linear Programming: Exercise Questions
Here are the NCERT Class 12 Maths Chapter 12 Linear Programming question answers with clear and detailed solutions.
| Linear Programming Class 12 Question Answers Exercise: 12.1 Page number: 403-404 Total questions: 10 |
Answer:
The region determined by constraints, $x+y\leq 4,x\geq 0,y\geq 0.$ is as follows,
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
|
The maximum value of Z is 16 at $D(0,4)$
Answer:
The region determined by constraints, $x+2y\leq 8,3x+2y\leq 12,x\geq 0,y\geq 0.$ is as follows,
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 $B(4,0)$
Answer:
The region determined by constraints, $3x + 5y \leq 15$ , $5x+2y\leq 10$ , $x\geq 0,y\geq 0$ is as follows :
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 $D(\frac{20}{19},\frac{45}{19})$
Answer:
The region determined by constraints $x+3y\geq 3,x+y\geq 2,x,y\geq 0.$ is as follows,
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 is:
|
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 the resulting half plane has a point in common with the feasible region or not.
We can see that a feasible region has no common point with. $Z = 3x + 5y$
Hence, Z has a minimum value of 7 at $B(\frac{3}{2},\frac{1}{2})$
Answer:
The region determined by constraints, $x+2y\leq 10,3x+y\leq 15,x,y\geq 0$ is as follows,
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 $B(4,3)$
Answer:
The region determined by constraints $2x+y\geq 3,x+2y\geq 6,x,y\geq 0.$ is as follows,
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
|
The 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 $Z = x + 2y$.
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,
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 the maximum value is 600 at all points joining line segment $B(60,30)$ and $D(120,0)$
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,
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 $D(0,200)$.
Answer:
The region determined by constraints $x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0.$ is as follows,
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 the resulting half-plane has a point in common with a feasible region or not.
We can see that the resulting feasible region has a common point with the feasible region.
Hence, Z =1 is not the maximum value; Z has no maximum value.
Answer:
The region determined by constraints $x-y\leq -1,-x+ y\leq 0,x,y,\geq 0.$ is as follows,
There is no feasible region, and thus, Z has no maximum value.
Linear Programming Class 12 NCERT Solutions: Exercise-wise
Check the NCERT Solutions of Linear Programming Class 12 Maths Chapter 12 exercise from the following link
Class 12 Maths NCERT Chapter 12: Extra Question
Question: In an LPP, if the objective function $z=a x+b y$ has the same maximum value on two corner points of the feasible region, then the number of points at which $z_{\max }$ occurs is:
Solution:
In an LPP, if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points gives the same maximum value. Hence, the number of points at which Zmax occurs is infinite.
Hence, the correct answer is "infinite".
CBSE Class 12th Syllabus: Subjects & Chapters
Select your preferred subject to view the chapters
Linear Programming Class 12 Chapter 12: Topics
Here is the list of important topics that are covered in Class 12 Chapter 12 Linear Programming.
- 12.1 Introduction
- 12.2 Linear Programming Problem and its Mathematical Formulation
NCERT Class 12 Maths Chapter 12: Important Formulae
Feasible Region: The feasible region, or solution region, of a linear programming problem is the common area determined by all the constraints, including the non-negativity constraints (x ≥ 0, y ≥ 0).
Infeasible Solution: Any point within or on the boundary of the feasible region represents a feasible solution to the constraints. Points outside the feasible region are considered infeasible solutions.
Optimal Solution: An optimal solution is any point within the feasible region that provides the optimal value (maximum or minimum) of the objective function.
Fundamental Theorems in Linear Programming
Optimality at Corner Points: For a linear programming problem with a feasible region represented as a convex polygon, if the objective function Z = ax + by has an optimal value, this optimal value must occur at one of the corner points (vertices) of the feasible region.
Existence of Maxima and Minima: If the feasible region R is bounded, then the objective function Z has both a maximum and a minimum value on R, and each of these values occurs at a corner point (vertex) of R. If R is unbounded, a maximum or minimum may not exist. However, if it does exist, it must occur at a corner point of R.
Corner Point Method: The corner point method is used to solve a linear programming problem and consists of the following steps:
Find the feasible region of the linear programming problem and determine its corner points (vertices).
Evaluate the objective function Z = ax + by at each corner point. Let M and m represent the largest and smallest values obtained at these points.
If the feasible region is bounded, M and m respectively represent the maximum and minimum values of the objective function.
If the feasible region is unbounded, then:
-
M is the maximum value of the objective function if the open half-plane determined by ax + by > M has no points in common with the feasible region.
-
m is the minimum value of the objective function if the open half-plane determined by ax + by < M has no points in common with the feasible region.
JEE Main Highest Scoring Chapters & Topics
Just Study 40% Syllabus and Score upto 100%
Download EBookTheorem 1: Let R be the feasible region (convex polygon) for a linear programming problem and let $\mathrm{Z}=a x+b y$ be the objective function. When Z has an optimal value (maximum or minimum), where the variables $x$ and $y$ are subject to constraints described by linear inequalities, this optimal value must occur at a corner point* (vertex) of the feasible region.
Theorem 2: Let R be the feasible region for a linear programming problem, and let $\mathrm{Z}=a x+b y$ be the objective function. If R is bounded ${ }^{* *}$, then the objective function Z has both a maximum and a minimum value on R, and each of these occurs at a corner point (vertex) of $R$.
Approach to Solve Questions of Linear Programming Class 12
Using these approaches, students can tackle the Linear Programming Class 12 Chapter 12 Question Answers with greater confidence.
- Determine whether the problem requires maximisation or minimisation.
- Check if variables are negative or non-negative. If they are non-negative, then they will satisfy a set of linear constraints.
- Plot all the constraints carefully on the graph paper based on their inequalities. Find and shade the feasible region, bounded or unbounded, that satisfies all constraints simultaneously.
- Shortcut tricks: Label all the axes and lines clearly in the graph paper to understand the representation. Be aware of the mistakes made during the plotting or solving of intersections. If the region is unbounded, the optimal value may not exist.
Why are Class 12 Maths Chapter 12 Linear Programming Question Answers Important?
This chapter helps you understand how to make the best possible decisions using maths. It shows how real-life problems can be solved by finding maximum or minimum values. These Class 12 Maths chapter 12 Linear Programming question answers make these ideas easier to learn and apply through step-by-step examples. Here are some more points on why these question answers are important.
- These solutions teach us how to form equations from word problems and solve them using graphs and logical thinking.
- Students learn to apply linear programming to real-world situations such as business, economics, and resource management.
- Dealing with Class 12 Maths chapter 12 Linear Programming question answers builds our problem-solving skills for higher studies and entrance exams.
- It also connects our maths learning to practical fields like data analysis, operations research, and optimisation.
Chapter Summary of NCERT Solutions for Class 12 Maths Chapter 12 - Linear Programming
This chapter includes an elaboration of various concepts of Linear Programming by using graphical techniques. NCERT Solutions explain the problems in an easy manner with detailed descriptions.The students practice various problems based on application related to 10 questions included in 1 textbook exercises that aid in improving concept and numericals skills of the students and building their confidence and accuracy. Practice of the questions in this chapter improves interpretation skills, drawing of graphs, logic application ability and graphic thinking ability that helps in their exams. Through this chapter the student is able to solve various problems on the optimisation in practical form.
Expert Review of NCERT Solutions for Class 12 Maths Chapter 12 - Linear Programming
This is the most applicable chapter in Class 12 Maths, since the concept helps in solving real-time optimization problems by means of modeling. With a clear understanding of how graphical methods and feasible regions are solved, you can solve the questions of linear programming even faster and more accurately. Preparing graphical representation in an adept manner and learning solutions for every question available in the NCERT textbook would enable you to achieve accuracy, efficiency and get prepared for board examinations as well as competitive exams such as JEE Main and Advanced. These were the suggestions from our seasoned Mathematics faculty members of Careers360.
NCERT Solutions for Class 12 Maths: Chapter Wise
Given below is the chapter-wise list of the NCERT Class 12 Maths solutions with their respective links:
Also, read,
- NCERT Exemplar Class 12 Maths Solutions Chapter 12 Linear Programming
- NCERT Notes Class 12 Maths Chapter 12 Linear Programming
NCERT Solutions for Class 12 Subject-wise
Here, you can find the NCERT Solutions for other subjects as well.
- NCERT solutions class 12 chemistry
- NCERT solutions for class 12 physics
- NCERT solutions for class 12 biology
Class-wise NCERT Solutions
Here, you can find the NCERT Solutions for classes 9 to 11.
NCERT Books and NCERT Syllabus
Here, you can find the NCERT books and syllabus for class 12.
- NCERT Books Class 12 Maths
- NCERT Syllabus Class 12 Maths
- NCERT Books Class 12
- NCERT Syllabus Class 12
Also, check,
Confused between CGPA and Percentage?
Get your results instantly with our calculator!
💡 Conversion Formula used is: Percentage = CGPA × 9.5
Frequently Asked Questions (FAQs)
Q: What is Linear Programming?
A:
Linear Programming is a technique used to find the optimal (maximum or minimum) value of an objective function under the condition of some constraints.
Q: Why is Linear Programming important for Class 12 students?
A:
Linear programming is important because students get to know the real world applications of optimization.
Q: Which topics are covered in this chapter?
A:
Topics covered in the linear programming chapter includes linear inequalities, feasible regions, objective functions, optimization problems, and graphically solving problems.
Q: How do NCERT Solutions help in Linear Programming?
A:
NCERT solutions of chapter 12 can ease the work of students in solving graphical and application based problems due to step-by-step approach.
Q: Is Linear Programming needed for JEE Main, and other similar tests?
A:
Absolutely, the essential basics of linear programming can be useful for both the general and board examinations and various similar competition examinations
Q: Which topic requires the most practice in this chapter?
A:
Students should put most practice into finding the feasible region and plotting the optimal solution on a graph.
Q: How can students score well in Linear Programming?
A:
By practicing NCERT questions and accurately plotting the graph students can definitely secure a higher score.
Q: What are the common mistakes students make in this chapter?
A:
Mistakes typically made by the students while dealing with linear programming includes graphing, identifying feasible regions and choosing the correct vertex as the optimal solution.
Q: Are NCERT Solutions enough for CBSE Board exam preparation?
A:
Yes, NCERT solutions provide thorough explanations and solutions of every textbook question.
Articles
Related Stories
|
Upcoming School Exams
Certifications By Top Providers
Explore Top Universities Across Globe
Questions related to CBSE Class 12th
On Question asked by student community
Have a question related to CBSE Class 12th ?
Hello Ananya,
Please specify the class for which you need the question papers. I am providing Class 10 and 12 papers.
Here are the links to the CBSE Half-yearly Question Papers (2025-2026).
Hello Ananya,
Please specify the class for which you need the question papers. I am providing Class 10 and 12 papers.
Here are the links to the CBSE Half-yearly Question Papers (2025-2026).
Hello Pawan,
CBSE Class 10 Mathematics 2026 and previous year question paper:
https://school.careers360.com/boards/cbse/cbse-class-10-question-paper-2026
CBSE Class 12 Mathematics 2026 and previous year question paper:
https://school.careers360.com/boards/cbse/cbse-previous-year-question-papers-class-12-maths
Hello Dharani,
Check the link below to download NCERT Class 12 previous year question papers in PDF format for all subjects.
https://school.careers360.com/boards/cbse/cbse-previous-year-question-papers-class-12
Hello Vipin,
Check the link below to download CBSE Class 12 question papers in PDF format for all subjects, including Mathematics.
https://school.careers360.com/boards/cbse/cbse-previous-year-question-papers-class-12
Popular CBSE Class 12th Questions
A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is
| Option 1)
|
Option 2)
|
| Option 3)
|
Option 4)
|
An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range
| Option 1)
|
Option 2)
|
| Option 3)
|
Option 4)
|
A particle is projected at 600 to the horizontal with a kinetic energy . The kinetic energy at the highest point
| Option 1)
|
Option 2)
|
| Option 3)
|
Option 4)
|
In the reaction,
| Option 1)
|
Option 2)
|
| Option 3)
|
Option 4)
|
How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?
| Option 1)
0.02 |
Option 2)
3.125 × 10-2 |
| Option 3)
1.25 × 10-2 |
Option 4)
2.5 × 10-2 |
Colleges After 12th
Applications for Admissions are open.
University of Liverpool, Bengaluru Campus
ApplyStudy at a world-renowned UK university in India | Admissions open for UG & PG programs.
Victoria University, Delhi NCR
ApplyApply for UG & PG programmes from Victoria University, Delhi NCR Campus
Illinois Tech Mumbai
ApplyAdmissions open for UG & PG programs at Illinois Tech Mumbai
University of Aberdeen Mumbai
ApplyApply for UG & PG courses at University of Aberdeen, Mumbai Campus
University of York, Mumbai
ApplyUG & PG Admissions open for CS/AI/Business/Economics & other programmes.
University of Bristol, Mumbai Enterprise Campus
ApplyBristol's expertise meets Mumbai's innovation. Admissions open for UG & PG programmes
CBSE Class 12th Notifications
Never miss CBSE Class 12th update
Get timely CBSE Class 12th updates directly to your inbox. Stay informed!