Class 12 Maths Chapter 12 Linear Programming – Exercise 12.2 NCERT Solutions

Introduction

Exercise 12.2 focuses on formulating and solving Linear Programming Problems (LPPs). Students learn how to identify constraints, represent feasible regions graphically, and optimize objective functions. This exercise is crucial for practical applications in business, economics, and operations research.

Key Concepts

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

Z=3x+4y

Constraints: Linear inequalities representing conditions, e.g.,

x+2y≤10,x≥0,y≥0

Feasible Region: Common region satisfying all constraints.
Corner Point Method: The maximum or minimum value of $Z$ occurs at a corner point of feasible region.

Students Frequently Make Mistakes

Forgetting to check all corner points.
Misinterpreting inequality signs.
Errors in plotting feasible region.
Skipping non‑negative conditions.
Confusing maximization with minimization.

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

Q1. Maximize $Z = 3 x + 2 y$ subject to $x + y \leq 4, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (4, 0), (0, 4)$ . Values: $Z = 0, 12, 8$ . Maximum $Z = 12$ at $(4, 0)$ .

Q2. Maximize $Z = 5 x + 3 y$ subject to $x + 2 y \leq 10, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (10, 0), (0, 5)$ . Values: $Z = 0, 50, 15$ . Maximum $Z = 50$ at $(10, 0)$ .

Q3. Maximize $Z = 2 x + 5 y$ subject to $2 x + y \leq 6, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (3, 0), (0, 6)$ . Values: $Z = 0, 6, 30$ . Maximum $Z = 30$ at $(0, 6)$ .

Q4. Maximize $Z = 4 x + 3 y$ subject to $x + 3 y \leq 9, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (9, 0), (0, 3)$ . Values: $Z = 0, 36, 9$ . Maximum $Z = 36$ at $(9, 0)$ .

Q5. Maximize $Z = 6 x + 2 y$ subject to $x + 4 y \leq 12, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (12, 0), (0, 3)$ . Values: $Z = 0, 72, 6$ . Maximum $Z = 72$ at $(12, 0)$ .

Q6. Maximize $Z = 3 x + 4 y$ subject to $3 x + 2 y \leq 12, x \geq 0, y \geq 0$ . Vertices: $(0, 0), (4, 0), (0, 6)$ . Values: $Z = 0, 12, 24$ . Maximum $Z = 24$ at $(0, 6)$ .

Q7. Minimize $Z = 2 x + 3 y$ subject to $x + 2 y \geq 4, x \geq 0, y \geq 0$ . Vertices: $(0, 2), (4, 0)$ . Values: $Z = 6, 8$ . Minimum $Z = 6$ at $(0, 2)$ .

Q8. Minimize $Z = 5 x + 2 y$ subject to $2 x + 3 y \geq 6, x \geq 0, y \geq 0$ . Vertices: $(0, 2), (3, 0)$ . Values: $Z = 4, 15$ . Minimum $Z = 4$ at $(0, 2)$ .

Q9. Minimize $Z = 3 x + 4 y$ subject to $x + y \geq 3, x \geq 0, y \geq 0$ . Vertices: $(0, 3), (3, 0)$ . Values: $Z = 12, 9$ . Minimum $Z = 9$ at $(3, 0)$ .

Q10. Minimize $Z = 2 x + 5 y$ subject to $x + 2 y \geq 5, x \geq 0, y \geq 0$ . Vertices: $(0, 2.5), (5, 0)$ . Values: $Z = 12.5, 10$ . Minimum $Z = 10$ at $(5, 0)$ .

FAQs (10)

FAQ1. What is objective function? Function to be optimized.

FAQ2. What is feasible region? Region satisfying all constraints.

FAQ3. What is corner point method? Method to find optimum value at vertices.

FAQ4. What is maximization problem? Finding maximum value of objective function.

FAQ5. What is minimization problem? Finding minimum value of objective function.

FAQ6. What is solution of Q1? Maximum $Z = 12$ at $(4, 0)$ .

FAQ7. What is solution of Q2? Maximum $Z = 50$ at $(10, 0)$ .

FAQ8. What is solution of Q3? Maximum $Z = 30$ at $(0, 6)$ .

FAQ9. What is solution of Q7? Minimum $Z = 6$ at $(0, 2)$ .

FAQ10. Why is Exercise 12.2 important? It builds mastery of solving Linear Programming Problems using graphical method.

Conclusion

Exercise 12.2 has 10 solved questions and 10 FAQs that strengthen your understanding of solving Linear Programming Problems using graphical method. This builds the foundation for optimization in Class 12 Maths.

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