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Linear programming is not just about finding the best solution—it’s about understanding the limits of possibility. Linear programming is a mathematical technique that is used for maximising or minimising a linear objective function, subject to a set of linear constraints. In the NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming, students will learn about the application of the systems of linear inequalities/equations to solve some real-life problems of various types. In real-life situations, a linear programming problem is like getting a maximum profit using limited resources and fulfilling customers' demand without going over budget.
In CBSE 12th results 2025, Shiv Nadar School recorded 100%. Praneel Munshi topped the CBSE Board exams 2025 with 98.80% in science stream. 21.10% students scored 95% and above.
Number of students with 100% marks in CBSE Class 12 results 2025:
Linear programming teaches us that decisions are not just numbers—they are balanced strategies. The objective of these class 12 NCERT solutions is to provide students with quality study material along with clear explanations. These solutions of NCERT are prepared by experienced careers360 experts following the latest CBSE syllabus.
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.
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.
Theorem 1: Let R be the feasible region (convex polygon) for a linear programming problem and let
Theorem 2: Let R be the feasible region for a linear programming problem, and let
Class 12 Maths chapter 12 solutions Exercise: 12.1 Page number: 403-404 Total questions: 10 |
Question 1: Solve the following Linear Programming Problems graphically: Maximise
Answer:
The region determined by constraints,
The region A0B represents the feasible region
The corner points of the feasible region are
Maximize
The value of these points at these corner points are :
Corner points | ||
12 | ||
0 | ||
16 | maximum |
The maximum value of Z is 16 at
Question 2: Solve the following Linear Programming Problems graphically: Minimise
Answer:
The region determined by constraints,
The corner points of feasible region are
The value of these points at these corner points are :
Corner points | ||
6 | ||
-12 | Minimum | |
0 | ||
16 |
The minimum value of Z is -12 at
Question 3: Solve the following Linear Programming Problems graphically: Maximise
Answer:
The region determined by constraints,
The corner points of feasible region are
The value of these points at these corner points are :
Corner points | ||
9 | ||
0 | ||
10 | ||
Maximum |
The maximum value of Z is
Question 4: Solve the following Linear Programming Problems graphically: Minimise
Answer:
The region determined by constraints
The feasible region is unbounded as shown.
The corner points of the feasible region are
The value of these points at these corner points is:
Corner points | ||
9 | ||
7 | Minimum | |
10 | ||
The feasible region is unbounded, therefore, 7 may or may not be the minimum value of Z.
For this, we draw
We can see that a feasible region has no common point with.
Hence, Z has a minimum value of 7 at
Question 5: Solve the following Linear Programming Problems graphically: Maximise
Answer:
The region determined by constraints,
The corner points of feasible region are
The value of these points at these corner points are :
Corner points | ||
15 | ||
18 | Maximum | |
10 | ||
The maximum value of Z is 18 at
Question 6: Solve the following Linear Programming Problems graphically: Minimise
Answer:
The region determined by constraints
The corner points of the feasible region are
The value of these points at these corner points are :
Corner points | |
6 | |
6 |
The value of Z is the same at both points.
If we take any other point like
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
Answer:
The region determined by constraints,
The corner points of feasible region are
The value of these points at these corner points are:
Corner points | ||
400 | ||
600 | Maximum | |
300 | Minimum | |
600 | maximum |
The minimum value of Z is 300 at
Answer:
The region determined by constraints
The corner points of the feasible region are
The value of these points at these corner points are :
Corner points | ||
100 | Minimum | |
100 | Minimum | |
250 | ||
400 | Maximum |
The minimum value of Z is 100 at all points on the line segment joining points
The maximum value of Z is 400 at
Answer:
The region determined by constraints
The corner points of the feasible region are
The value of these points at these corner points are :
Corner points | ||
- 6 | minimum | |
-2 | ||
1 | maximum | |
The feasible region is unbounded, therefore, 1 may or may not be the maximum value of Z.
For this, we draw
We can see 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
There is no feasible region, and thus, Z has no maximum value.
Also, read,
Question: In an LPP, if the objective function
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".
- 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.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
Also, read,
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In Class 12 Maths, Linear Programming finds real-life applications in optimizing resource allocation, such as maximizing profits or minimizing costs in scenarios like production planning, diet planning, and transportation problems.
In Linear Programming Problems (LPP), a feasible solution satisfies all constraints, while an optimal solution is a feasible solution that either maximizes or minimizes the objective function.
Linear equations and linear inequalities are the types of linear programming problems in NCERT Class 12. The problems in LPP basically consist of the problems that include the calculation of the minimum or maximum value.
There are 10 questions in NCERT Class 12 Maths Chapter 12.
Graphical Method: Owing to the importance of linear programming models in various industries, many types of algorithms have been developed over the years to solve them. Some famous mentions include the Simplex method, the Hungarian approach, and others
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Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
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