Class 12 Math Chapter 12 Linear Programming – Complete NCERT Solutions

Introduction

Linear programming is a powerful mathematical technique used to optimise resources. It helps in maximising profit or minimising cost subject to given constraints. In this chapter, we learn how to formulate linear programming problems and solve them using the graphical method.

Key Concepts and Formulas

• Objective Function:

$$\mathrm{<b><span\; style="font-family:\; verdana;">Z</span></b>}<b><span\; style="font-family:\; verdana;">=</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">a</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">x</span></b>}<b><span\; style="font-family:\; verdana;">+</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">y</span></b>}$$

where $a,b$ are constants.

• Constraints: Linear inequalities such as

$$x\ge 0,y\ge 0,5x+y\le 100$$

• Feasible Region: The common region satisfying all constraints.

• Corner Point Method:

  • Find feasible region.

  • Identify corner points.

  • Evaluate objective function at each corner point.

  • Maximum or minimum occurs at a corner point.

Solved Examples from NCERT

Example 1: Maximise $Z=4x+y$ subject to $x+y\le 50,3x+y\le 90,x,y\ge 0$. Solution: Corner points are (0,0), (30,0), (20,30), (0,50). Values of Z: 0, 120, 110, 50. Maximum = 120 at (30,0).

Example 2: Minimise $Z=200x+500y$ subject to $x+2y\ge 10,3x+4y\le 24,x,y\ge 0$. Solution: Corner points (0,5), (4,3), (0,6). Values of Z: 2500, 2300, 3000. Minimum = 2300 at (4,3).

Example 3: Minimise and Maximise $Z=3x+9y$ subject to $x+3y\le 60,x+y\ge 10,x\le y,x,y\ge 0$. Solution: Corner points (0,10), (5,5), (15,15), (0,20). Values of Z: 90, 60, 180, 180. Minimum = 60 at (5,5). Maximum = 180 at (15,15) and (0,20).

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 12.1 (Q1): Maximise $Z=3x+4y$ subject to $x+y\le 4,x,y\ge 0$. Solution: Corner points (0,0), (4,0), (0,4). Values of Z: 0, 12, 16. Maximum = 16 at (0,4).

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

1. Q: What is linear programming? A: A method to optimise a linear objective function subject to linear constraints.

2. Q: What is an objective function? A: The function to be maximised or minimised, e.g., $Z=ax+by$.

3. Q: What are constraints? A: Linear inequalities restricting values of variables.

4. Q: What is feasible region? A: The region satisfying all constraints.

5. Q: What is feasible solution? A: Any point inside or on boundary of feasible region.

6. Q: What is infeasible solution? A: A point outside feasible region.

7. Q: What is corner point method? A: Evaluating objective function at vertices of feasible region.

8. Q: When does maximum occur? A: At a corner point of feasible region.

9. Q: When does minimum occur? A: At a corner point of feasible region.

10. Q: What if feasible region is unbounded? A: Maximum or minimum may not exist.

11. Q: What is bounded region? A: Region enclosed within a circle.

12. Q: What is unbounded region? A: Region extending indefinitely.

13. Q: What is optimisation problem? A: Problem seeking maximum or minimum value of objective function.

14. Q: What are decision variables? A: Variables like x and y representing choices.

15. Q: What is simplex method? A: An iterative method to solve LPP.

16. Q: What is transportation problem? A: A type of LPP dealing with distribution of goods.

17. Q: What is diet problem? A: LPP to determine optimal diet at minimum cost.

18. Q: What is multiple optimal solution? A: When more than one corner point gives same optimal value.

19. Q: What is historical origin of LPP? A: Developed during World War II for resource optimisation.

20. Q: Why study LPP? A: It has wide applications in industry, commerce, and management.

Conclusion

Linear programming is a practical tool for solving optimisation problems. By mastering graphical methods and corner point analysis, students can confidently solve NCERT problems and apply these concepts in real‑life scenarios.

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