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Mathematics · Class 12 Science

Ch 12Linear Programming — Class 12 Mathematics, concept-first.

In earlier classes you solved systems of linear equations, and in Class XI you studied linear inequalities and systems of linear inequalities in two variables, solving them graphically. Many real situations involve exactly this kind of system of inequalities or equations.

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Key concepts

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Linear Programming Graphical Method

When a linear programming problem has just two decision variables, and , you can solve it by drawing a picture. This is the graphical method, and it is the technique the CBSE Class-12 course expects you to use.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

12.1

Introduction

In earlier classes you solved systems of linear equations, and in Class XI you studied linear inequalities and systems of linear inequalities in two variables, solving them graphically.

12.2

Linear Programming Problem and Its Mathematical Formulation

A dealer can invest in two products: tables and chairs. Each table costs ₹2500 and yields ₹250 profit; each chair costs ₹500 and yields ₹75 profit.

12.2.1

Mathematical Formulation of the Problem

The core of any linear programming problem is translating a real-world situation into precise mathematical language.

12.2.2

Graphical Method of Solving Linear Programming Problems

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In Class XI you learned to graph a system of linear inequalities in two variables and find the solution region.

+Worked Examplesi5 questions
  1. Example 1Solve the following linear programming problem graphically: Maximise $Z = 4x + y$ subject to the constraints: $x + y \le 50$, $3x + y \le 90…Free
  2. Example 2Solve the following linear programming problem graphically: Minimise $Z = 200x + 500y$ subject to the constraints: $x + 2y \ge 10$, $3x + 4y…Free
  3. Example 3Solve the following problem graphically: Minimise and Maximise $Z = 3x + 9y$ subject to the constraints: $x + 3y \le 60$, $x + y \ge 10$, $x…Preview
  4. Example 4Determine graphically the minimum value of the objective function $Z = -50x + 20y$ subject to the constraints: $2x - y \ge -5$, $3x + y \ge…Preview
  5. Example 5Minimise $Z = 3x + 2y$ subject to the constraints: $x + y \ge 8$, $3x + 5y \le 15$, $x \ge 0,\ y \ge 0$. ![Fig 12.6 — no feasible region](/f…Preview
+Exercise 12.1i10 questions
  1. Q1Find the value of the following: Maximise $Z = 3x + 4y$ subject to the constraints : $x + y \le 4, x \ge 0, y \ge 0$.Free
  2. Q2Find the value of the following: Minimise $Z = -3x + 4y$ subject to $x + 2y \le 8$, $3x + 2y \le 12$, $x \ge 0$, $y \ge 0$.Free
  3. Q3Find the value of the following: Maximise $Z = 5x + 3y$ subject to $3x + 5y \le 15$, $5x + 2y \le 10$, $x \ge 0$, $y \ge 0$.Free
  4. Q4Find the value of the following: Minimise $Z = 3x + 5y$ such that $x + 3y \ge 3$, $x + y \ge 2$, $x, y \ge 0$.Preview
  5. Q5Find the value of the following: Maximise $Z = 3x + 2y$ subject to $x + 2y \le 10$, $3x + y \le 15$, $x, y \ge 0$.Preview
  6. Q6Minimise $Z = x + 2y$ subject to $2x + y \ge 3$, $x + 2y \ge 6$, $x, y \ge 0$. Show that the minimum of $Z$ occurs at more than two points.Preview
  7. Q7Find the value of the following: Minimise and Maximise $Z = 5x + 10y$ subject to $x + 2y \le 120$, $x + y \ge 60$, $x - 2y \ge 0$, $x, y \ge…Preview
  8. Q8Find the value of the following: Minimise and Maximise $Z = x + 2y$ subject to $x + 2y \ge 100$, $2x - y \le 0$, $2x + y \le 200$; $x, y \ge…Preview
  9. Q9Find the value of the following: Maximise $Z = -x + 2y$, subject to the constraints: $x \ge 3$, $x + y \ge 5$, $x + 2y \ge 6$, $y \ge 0$.Preview
  10. Q10Find the value of the following: Maximise $Z = x + y$, subject to $x - y \le -1$, $-x + y \le 0$, $x, y \ge 0$.Preview

Summary

- Linear Programming Problem (LPP): Optimizing (maximizing or minimizing) a linear objective function subject to linear constraints (inequalities) and non-negativity restrictions .

Miscellaneous Exercise on Chapter 12

Miscellaneous Examples

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 45 questions45 questions
  1. Q1Determine the maximum value of $Z = 11x + 7y$ subject to the constraints: $2x + y \le 6$, $x \le 2$, $x \ge 0$, $y \ge 0$.Free
  2. Q2Maximise $Z = 3x + 4y$, subject to the constraints: $x + y \le 1$, $x \ge 0$, $y \ge 0$.Free
  3. Q3Maximise the function $Z = 11x + 7y$, subject to the constraints: $x \le 3$, $y \le 2$, $x \ge 0$, $y \ge 0$.Free
  4. Q4Minimise $Z = 13x - 15y$ subject to the constraints: $x + y \le 7$, $2x - 3y + 6 \ge 0$, $x \ge 0$, $y \ge 0$.Preview
  5. Q5The feasible region of a linear programming problem is the bounded region in the first quadrant ($x \ge 0$, $y \ge 0$) satisfying $x + 2y \l…Preview
  6. Q6The feasible region of a linear programming problem is the bounded quadrilateral in the first quadrant with corner points $O(0, 0)$, $(0, 2)…Preview
  7. Q7The feasible region of a linear programming problem is the triangle bounded by the $y$-axis and the lines $x + y = 5$ and $x + 3y = 9$. Expl…Preview
  8. Q8Refer to Exercise 7 above. Find the maximum value of $Z$.Preview
  9. Q9The feasible region of a linear programming problem is the unbounded region in the first quadrant ($x \ge 0$, $y \ge 0$) satisfying $x + y \…Preview
  10. Q10The feasible region of a linear programming problem is a bounded quadrilateral with corner points $P\left(\dfrac{3}{13}, \dfrac{24}{13}\righ…Preview
  11. Q11A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types…Preview
  12. Q12A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The…Preview
  13. Q13A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of…Preview
  14. Q14A company manufactures two types of sweaters: type A and type B. It costs Rs 360 to make a type A sweater and Rs 120 to make a type B sweate…Preview
  15. Q15A man rides his motorcycle at the speed of 50 km/hour. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of 80 km/hour…Preview
  16. Q16Refer to Exercise 11. How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximise his profit? Det…Preview
  17. Q17Refer to Exercise 12. What will be the minimum cost?Preview
  18. Q18Refer to Exercise 13. Solve the linear programming problem and determine the maximum profit to the manufacturer.Preview
  19. Q19Refer to Exercise 14. How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit?Preview
  20. Q20Refer to Exercise 15. Determine the maximum distance that the man can travel.Preview
  21. Q21Maximise $Z = x + y$ subject to $x + 4y \le 8$, $2x + 3y \le 12$, $3x + y \le 9$, $x \ge 0$, $y \ge 0$.Preview
  22. Q22A manufacturer produces two Models of bikes - Model X and Model Y. Model X takes 6 man-hours to make per unit, while Model Y takes 10 man-ho…Preview
  23. Q23In order to supplement daily diet, a person wishes to take some X and some Y tablets. The contents of iron, calcium and vitamins in X and Y…Preview
  24. Q24A company makes 3 models of calculators: A, B and C at factory I and factory II. The company has orders for at least 6400 calculators of mod…Preview
  25. Q25Maximise and Minimise $Z = 3x - 4y$ subject to $x - 2y \le 0$, $-3x + y \le 4$, $x - y \le 6$, $x, y \ge 0$.Preview
  26. Q26In a LPP, the linear inequalities or restrictions on the variables are called _________.Preview
  27. Q27In a LPP, the objective function is always _________.Preview
  28. Q28If the feasible region for a LPP is _________, then the optimal value of the objective function $Z = ax + by$ may or may not exist.Preview
  29. Q29In a LPP if the objective function $Z = ax + by$ has the same maximum value on two corner points of the feasible region, then every point on…Preview
  30. Q30A feasible region of a system of linear inequalities is said to be _________ if it can be enclosed within a circle.Preview
  31. Q31A corner point of a feasible region is a point in the region which is the _________ of two boundary lines.Preview
  32. Q32The feasible region for an LPP is always a _________ polygon.Preview
  33. Q33State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective f…Preview
  34. Q34State whether the following statement is True or False: Maximum value of the objective function $Z = ax + by$ in a LPP always occurs at only…Preview
  35. Q35State whether the following statement is True or False: In a LPP, the minimum value of the objective function $Z = ax + by$ is always $0$ if…Preview
  36. Q36State whether the following statement is True or False: In a LPP, the maximum value of the objective function $Z = ax + by$ is always finite…Preview
  37. Q37The corner points of the feasible region determined by the system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(6…Preview
  38. Q38The feasible region of a linear programming problem is the bounded polygon with corner points $(0, 0)$, $(0, 8)$, $(4, 10)$, $(6, 8)$, $(6,…Preview
  39. Q39Refer to Exercise 27. Maximum of $Z$ occurs at (A) $(5, 0)$ (B) $(6, 5)$ (C) $(6, 8)$ (D) $(4, 10)$Preview
  40. Q40Refer to Exercise 27. (Maximum value of $Z$ + Minimum value of $Z$) is equal to (A) $13$ (B) $1$ (C) $-13$ (D) $-17$Preview
  41. Q41The feasible region of a linear programming problem is an unbounded region lying between two parallel straight lines that are both parallel…Preview
  42. Q42Refer to Exercise 30. Minimum value of $F$ is (A) $0$ (B) $-16$ (C) $12$ (D) does not existPreview
  43. Q43Corner points of the feasible region for an LPP are $(0, 2)$, $(3, 0)$, $(6, 0)$, $(6, 8)$ and $(0, 5)$. Let $F = 4x + 6y$ be the objective…Preview
  44. Q44Refer to Exercise 32. Maximum of $F$ - Minimum of $F =$ (A) $60$ (B) $48$ (C) $42$ (D) $18$Preview
  45. Q45Corner points of the feasible region determined by the system of linear constraints are $(0, 3)$, $(1, 1)$ and $(3, 0)$. Let $Z = px + qy$,…Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 7 questions7 questions
  1. Q1A student of class XII studying Mathematics comes across an incomplete question in a book. Maximise 𝑍 = 3𝑥 + 2𝑦 + 1 Subject to the constrain…Preview
  2. Q2For Visually Impaired: If 𝑍 = 𝑎𝑥 + 𝑏𝑦 + 𝑐, where 𝑎, 𝑏, 𝑐 > 0, attains its maximum value at two of its corner points (4,0) and (0,3) of the f…Preview
  3. Q3The feasible region of a linear programming problem is bounded but the objective function attains its minimum value at more than one point.…Preview
  4. Q4For the linear programming problem (LPP), the objective function is $Z=4x+3y$ and the feasible region determined by a set of constraints is…Preview
  5. Q5A linear programming problem (LPP) along with the graph of its constraints is shown below. The corresponding objective function is: $Z=18x+1…Preview
  6. Q6Consider the following Linear Programming Problem: Minimise $Z = x + 2y$ Subject to $2x + y \geq 3$, $x + 2y \geq 6$, $x, y \geq 0$. Show gr…Preview
  7. Q7A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes…Preview