Class 12 Maths Chapter 12 Linear Programming – Exercise 12.3 NCERT Solutions

Introduction

Exercise 12.3 focuses on real‑life applications of Linear Programming Problems (LPPs). Students learn how to formulate word problems into mathematical models, identify constraints, and solve optimization problems using the graphical method. This exercise is essential for applying mathematics to practical scenarios in economics, business, and resource management.

Key Concepts

1. Formulation of LPP: Translate word problems into objective function and constraints.

2. Objective Function: Function to be maximized or minimized, e.g.,

$$Z=3x+4y$$

3. Constraints: Conditions expressed as inequalities, e.g.,

$$x+2y\le 10,x\ge 0,y\ge 0$$

4. Corner Point Method: Optimum value of $Z$ occurs at a corner point of feasible region.

Students Frequently Make Mistakes

• Misinterpreting word problems into inequalities.

• Forgetting non‑negative conditions.

• Errors in identifying feasible region.

• Skipping evaluation of all corner points.

• Confusing maximization with minimization.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. A factory produces two products A and B. Profit per unit is ₹3 and ₹5 respectively. Constraints: $x+2y\le 10,x+y\le 6,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(6,0),(0,5),(4,2)$. Values: $Z=0,18,25,22$. Maximum $Z=25$ at $(0,5)$.

Q2. A company makes two items P and Q. Profit per unit is ₹20 and ₹30. Constraints: $x+2y\le 40,3x+y\le 60,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(20,0),(0,20),(10,15)$. Values: $Z=0,400,600,650$. Maximum $Z=650$ at $(10,15)$.

Q3. A manufacturer produces two products A and B. Profit per unit is ₹50 and ₹60. Constraints: $x+2y\le 40,3x+y\le 45,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(15,0),(0,20),(5,15)$. Values: $Z=0,750,1200,1150$. Maximum $Z=1200$ at $(0,20)$.

Q4. A farmer grows wheat and barley. Profit per acre is ₹200 and ₹300. Constraints: $x+2y\le 100,3x+y\le 90,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(30,0),(0,50),(20,30)$. Values: $Z=0,6000,15000,13000$. Maximum $Z=15000$ at $(0,50)$.

Q5. A company produces two types of toys A and B. Profit per toy is ₹5 and ₹7. Constraints: $x+2y\le 20,2x+y\le 20,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(10,0),(0,10),(5,7.5)$. Values: $Z=0,50,70,72.5$. Maximum $Z=72.5$ at $(5,7.5)$.

Q6. A firm produces two products X and Y. Profit per unit is ₹40 and ₹50. Constraints: $x+2y\le 40,2x+y\le 50,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(25,0),(0,20),(10,15)$. Values: $Z=0,1000,1000,1150$. Maximum $Z=1150$ at $(10,15)$.

Q7. A company produces two items A and B. Profit per unit is ₹100 and ₹120. Constraints: $x+2y\le 60,2x+y\le 60,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(30,0),(0,30),(20,20)$. Values: $Z=0,3000,3600,4400$. Maximum $Z=4400$ at $(20,20)$.

Q8. A farmer grows rice and maize. Profit per acre is ₹150 and ₹250. Constraints: $x+2y\le 60,3x+y\le 60,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(20,0),(0,30),(15,15)$. Values: $Z=0,3000,7500,6000$. Maximum $Z=7500$ at $(0,30)$.

Q9. A company produces two items A and B. Profit per unit is ₹10 and ₹15. Constraints: $x+2y\le 30,3x+y\le 30,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(10,0),(0,15),(5,10)$. Values: $Z=0,100,225,200$. Maximum $Z=225$ at $(0,15)$.

Q10. A manufacturer produces two products A and B. Profit per unit is ₹8 and ₹10. Constraints: $x+2y\le 40,2x+y\le 40,x,y\ge 0$. Maximize profit. Vertices: $(0,0),(20,0),(0,20),(10,15)$. Values: $Z=0,160,200,230$. Maximum $Z=230$ at $(10,15)$.

FAQs (10)

FAQ1. What is Linear Programming Problem (LPP)? Optimization problem with linear constraints and objective function.

FAQ2. What is objective function? Function to be maximized or minimized.

FAQ3. What is feasible region? Region satisfying all constraints.

FAQ4. What is corner point method? Optimum value occurs at corner points.

FAQ5. What is solution of Q1? Maximum profit ₹25 at $(0,5)$.

FAQ6. What is solution of Q2? Maximum profit ₹650 at $(10,15)$.

FAQ7. What is solution of Q3? Maximum profit ₹1200 at $(0,20)$.

FAQ8. What is solution of Q4? Maximum profit ₹15000 at $(0,50)$.

FAQ9. What is solution of Q7? Maximum profit ₹4400 at $(20,20)$.

FAQ10. Why is Exercise 12.3 important? It builds mastery of real‑life applications of Linear Programming.

Conclusion

Exercise 12.3 has 10 solved questions and 10 FAQs that strengthen your understanding of real‑life applications of Linear Programming Problems. This completes the Linear Programming chapter in Class 12 Maths.

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