Download Free PDF of Linear Programming Exercise 12.1 NCERT Solutions for Class 12 Maths
Linear Programming in Class 12 Maths unlocks the ability to solve real-world resource problems using mathematical logic. Here, detailed NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1 help you master the stepwise formulation of constraints and the graphical method, essential for board exam success. This chapter is important, typically carrying a 5-mark weightage in the CBSE Board, making your understanding not just useful, but scoring, too.
Table of Content
This chapter teaches you how to identify feasible regions, analyze linear inequalities, and maximize or minimize objective functions—all skills required for queries like “linear programming class 12 exercise 12.1”. You’ll use each solved example to clear confusion around constraint-making and practical problem-solving, with visual cues that support faster and more accurate answers in exams.
Vedantu’s reliable learning path emphasizes graphical representation, exam patterns, and practical strategies for time management. If you need a broader context or extra revision, refer to the Class 12 Maths NCERT Solutions hub and always stay mapped to the latest Class 12 Maths syllabus for 2025.
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:
The values of $Z$at the corner points $O(0,0),A(4,0),B(0,4)$ of the feasible region are:
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:
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:
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:
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:
Hence, the maximum value of $Z$ is $\frac{235}{19}$ at the point$C(\frac{20}{19},\frac{45}{19})$.
4: Minimise $Z=3x+5y$
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:
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:
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:
The values of $Z$at the corner points $A(5,0),B(4,3)$ and $C(0,5)$ of the feasible region are:
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:
The values of $Z$at the corner points $A(6,0)$ and $B(0,3)$ of the feasible region are:
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:
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:
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:
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:
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:
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:
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:
It can be seen that there is no feasible region. Thus, $Z$ does not have any maximum value.
Conclusion
Exercise 12.1 of Chapter 12 in Class 12 Maths is essential for mastering linear programming. This exercise focuses on formulating linear programming problems, using the graphical method to solve them, and identifying feasible and optimal solutions. Understanding objective functions, constraints, and the feasible region is crucial. Vedantu's NCERT Solutions offer clear, step-by-step guidance, helping students grasp these concepts effectively. Practicing these solutions will enhance problem-solving skills and exam readiness. Concentrate on the graphical method and evaluating objective functions at corner points for optimal solutions.
CBSE Class 12 Maths Chapter 12 Other Study Materials
NCERT Solutions for Class 12 Maths | Chapter-wise List
Given below are the chapter-wise NCERT 12 Maths solutions PDF. Using these chapter-wise class 12th maths ncert solutions, you can get clear understanding of the concepts from all chapters.
Related Links for NCERT Class 12 Maths in Hindi
Explore these essential links for NCERT Class 12 Maths in Hindi, providing detailed solutions, explanations, and study resources to help students excel in their mathematics exams.
Important Related Links for NCERT Class 12 Maths
FAQs on CBSE Class 12 Maths Chapter 12 Linear Programming – NCERT Solutions 2025-26
1. What are the main steps to solve Exercise 12.1 of Class 12 Maths NCERT?
The main steps to solve Class 12 Maths Chapter 12 Exercise 12.1 (Linear Programming) are as follows:
1. Carefully read and interpret each word problem.
2. Formulate the constraints as linear inequalities.
3. Plot all constraints on a graph to determine the feasible region.
4. Identify and list the corner points of the feasible region.
5. Write down the objective function to be maximized or minimized.
6. Evaluate the objective function at each corner point to find the optimal solution.
These process steps are essential for scoring high marks in CBSE board exams and mastering graphical solutions for linear programming problems (LPP).
2. How do you identify the feasible region in a linear programming problem?
The feasible region in a linear programming problem is the shared area that satisfies all linear inequalities (constraints).
To identify it:
- Plot each linear inequality on the graph as a half-plane (one side of its line).
- Shade the region that fulfills each constraint.
- The overlapping (common) shaded area for all inequalities is the feasible region.
- Check whether the region is bounded (closed) or unbounded (open).
- Corner points of the feasible region are critical for finding the optimum value of the objective function.
3. Where can students download free PDF solutions for Chapter 12 Exercise 12.1?
You can download free PDF solutions for Class 12 Maths Chapter 12 Exercise 12.1 (Linear Programming) directly from trusted educational websites like Vedantu.
Benefits include:
- Stepwise solutions aligned with the NCERT syllabus
- Printable format for offline revision
- Supports exam preparation with explained answers
4. What are the most common mistakes made in CBSE board exams for LPP?
The most common mistakes in Linear Programming Problems (LPP) during CBSE board exams are:
- Incorrect formulation of constraints from word problems.
- Improper or missing graph plotting, leading to wrong feasible regions.
- Confusing maximization and minimization in the objective function.
- Forgetting to evaluate the objective function at all corner points of the feasible region.
- Not shading the correct region for inequalities, especially in complex diagrams.
5. How to graph inequalities accurately for Exam 12.1?
To accurately graph inequalities in Class 12 Maths Exercise 12.1:
- Rearrange each inequality into the standard form (e.g., ax + by ≤ c).
- Draw the boundary line (ax + by = c); use a solid line for ≤/≥, and a dashed line for < or >.
- Choose a test point (often origin (0,0)) to determine which side to shade.
- Shade the region representing the solution set for each constraint.
- The intersection of all shaded regions gives the feasible region.
6. What is the exam pattern for Class 12 Maths Chapter 12?
Chapter 12 (Linear Programming) in Class 12 Maths is usually tested as follows:
- CBSE Board: Generally 1 long answer question of 5 marks
- Type of Questions: Formulating LPPs from word problems, plotting constraints, identifying feasible regions, and maximizing or minimizing objective functions
- Difficulty: Moderate, with a focus on proper graphing and correct formulation
- Frequency: Appears every year and is also relevant for JEE Main (1–2 questions expected)
7. How to solve a linear programming problem by graphical method in Class 12?
To solve a linear programming problem graphically in Class 12 Maths:
Summary: Formulate all constraints as inequalities, plot them, find the feasible region, and evaluate the objective function at each corner.
Stepwise method:
- Formulate all given constraints as linear inequalities.
- Plot each constraint as a straight line on the graph.
- Shade the required region for each inequality.
- Locate the feasible region that satisfies all constraints.
- Find the corner points (vertices) of the feasible region.
- Evaluate the objective function at each vertex to identify the maximum or minimum value as required.
8. What is a constraint and objective function in Linear Programming?
In Linear Programming:
- Constraint: A mathematical inequality (e.g., 2x + 3y ≤ 6) that limits possible solutions; constraints define the feasible region on the graph.
- Objective Function: An expression (e.g., Z = 5x + 2y) to be maximized or minimized depending on the problem goal (like maximizing profit or minimizing cost).
9. What is meant by bounded and unbounded feasible region in LPP?
A bounded feasible region is entirely enclosed by the constraints; it forms a closed shape (like a polygon), ensuring optimal solution always lies at a vertex.
An unbounded feasible region is open and extends infinitely in at least one direction; in this case, extra checking is needed to verify the existence of an optimal value.
Understanding these is vital for graphical method in Class 12 Linear Programming and helps prevent common exam mistakes.
10. How can I quickly check if my solution to a linear programming problem is correct?
To quickly check your solution in Exercise 12.1 (Linear Programming):
- Verify each corner point of the feasible region against all constraints.
- Ensure the objective function value matches your maximization/minimization goal.
- Double-check your graph for proper intersection points.
- If possible, compare your answer with NCERT solutions or teacher-verified PDFs.
