Class 12 Maths Chapter 12 Linear Programming – Exercise 12.1 NCERT Solutions

Introduction

Exercise 12.1 introduces the basic concepts of Linear Programming (LP). Students learn how to formulate linear inequalities, represent them graphically, and identify feasible regions. This exercise builds the foundation for solving optimization problems in later sections.

Key Concepts

Linear Inequality: An inequality involving linear expressions in two variables. Example: $x + 2 y \leq 10$ .
Feasible Region: The common region satisfying all inequalities.
Constraints: Conditions expressed as inequalities.
Objective Function: Function to be optimized (maximized or minimized). Example: $Z = 3 x + 4 y$ .

Students Frequently Make Mistakes

Forgetting to draw boundary lines correctly.
Misinterpreting inequality signs ( $\leq, \geq$ ).
Errors in shading feasible region.
Skipping non‑negative conditions ( $x \geq 0, y \geq 0$ ).
Confusing feasible region with solution set of single inequality.

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

Q1. Represent graphically: $x + y \leq 5, x \geq 0, y \geq 0$ . Boundary: $x + y = 5$ . Feasible region: Triangle with vertices $(0, 0), (5, 0), (0, 5)$ .

Q2. Represent graphically: $x + 2 y \leq 10, x \geq 0, y \geq 0$ . Boundary: $x + 2 y = 10$ . Feasible region: Triangle with vertices $(0, 0), (10, 0), (0, 5)$ .

Q3. Represent graphically: $2 x + y \leq 6, x \geq 0, y \geq 0$ . Boundary: $2 x + y = 6$ . Feasible region: Triangle with vertices $(0, 0), (3, 0), (0, 6)$ .

Q4. Represent graphically: $x + 3 y \leq 9, x \geq 0, y \geq 0$ . Boundary: $x + 3 y = 9$ . Feasible region: Triangle with vertices $(0, 0), (9, 0), (0, 3)$ .

Q5. Represent graphically: $x + 4 y \leq 12, x \geq 0, y \geq 0$ . Boundary: $x + 4 y = 12$ . Feasible region: Triangle with vertices $(0, 0), (12, 0), (0, 3)$ .

Q6. Represent graphically: $3 x + 2 y \leq 12, x \geq 0, y \geq 0$ . Boundary: $3 x + 2 y = 12$ . Feasible region: Triangle with vertices $(0, 0), (4, 0), (0, 6)$ .

Q7. Represent graphically: $x + 2 y \geq 4, x \geq 0, y \geq 0$ . Boundary: $x + 2 y = 4$ . Feasible region: Above line, bounded by axes.

Q8. Represent graphically: $2 x + 3 y \geq 6, x \geq 0, y \geq 0$ . Boundary: $2 x + 3 y = 6$ . Feasible region: Above line, bounded by axes.

Q9. Represent graphically: $x + y \geq 3, x \geq 0, y \geq 0$ . Boundary: $x + y = 3$ . Feasible region: Above line, bounded by axes.

Q10. Represent graphically: $x + 2 y \geq 5, x \geq 0, y \geq 0$ . Boundary: $x + 2 y = 5$ . Feasible region: Above line, bounded by axes.

FAQs (10)

FAQ1. What is linear inequality? An inequality involving linear expressions.

FAQ2. What is feasible region? Common region satisfying all inequalities.

FAQ3. What is objective function? Function to be optimized.

FAQ4. What is boundary line? Equation obtained by replacing inequality sign with equality.

FAQ5. How to represent inequality graphically? Draw boundary line, shade feasible region.

FAQ6. What is condition for feasible region? $x \geq 0, y \geq 0$ .

FAQ7. What is solution of $x + y \leq 5$ ? Triangle with vertices $(0, 0), (5, 0), (0, 5)$ .

FAQ8. What is solution of $x + 2 y \leq 10$ ? Triangle with vertices $(0, 0), (10, 0), (0, 5)$ .

FAQ9. What is solution of $2 x + y \leq 6$ ? Triangle with vertices $(0, 0), (3, 0), (0, 6)$ .

FAQ10. Why is Exercise 12.1 important? It builds foundation for optimization problems in Linear Programming.

Conclusion

Exercise 12.1 has 10 solved questions and 10 FAQs that strengthen your understanding of graphical representation of linear inequalities in 3D geometry. This begins the Linear Programming chapter in Class 12 Maths.

visit:www.fuzymathacademy.com