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Mathematics · Class 12 Science

Ch 7Linear Programming — Class 12 Mathematics, concept-first.

A straight line (with not both zero) divides the whole coordinate plane into three pieces: the points lying exactly on the line, and two separate half-planes on either side of it.

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Key concepts

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Linear Inequations in Two Variables and Their Graphical Solution

A linear inequation in two variables is an expression such as , , or , where are real numbers and are not both zero.

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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.

7.1

Linear Inequations in two variables

A straight line (with not both zero) divides the whole coordinate plane into three pieces: the points lying exactly on the line, and two separate half-planes on either side of it.

+Exercise 7.1i21 questions
  1. Q1Solve graphically : $x \ge 0$Free
  2. Q2Solve graphically : $x \le 0$Free
  3. Q3Solve graphically : $y \ge 0$Free
  4. Q4Solve graphically : $y \le 0$Preview
  5. Q5Solve graphically : $x \ge 0$ and $y \ge 0$Preview
  6. Q6Solve graphically : $x \le 0$ and $y \ge 0$Preview
  7. Q7Solve graphically : $x \le 0$ and $y \le 0$Preview
  8. Q8Solve graphically : $x \ge 0$ and $y \le 0$Preview
  9. Q9Solve graphically : $2x - 3 \ge 0$Preview
  10. Q10Solve graphically : $2y - 5 \ge 0$Preview
  11. Q11Solve graphically : $3x + 4 \le 0$Preview
  12. Q12Solve graphically : $5y + 3 \le 0$Preview
  13. Q13Solve graphically : $x + 2y \le 6$Preview
  14. Q14Solve graphically : $2x - 5y \ge 10$Preview
  15. Q15Solve graphically : $3x + 2y \ge 0$Preview
  16. Q16Solve graphically : $5x - 3y \le 0$Preview
  17. Q17Solve graphically : $2x + y \ge 2$ and $x - y \le 1$Preview
  18. Q18Solve graphically : $x - y \le 2$ and $x + 2y \le 8$Preview
  19. Q19Solve graphically : $x + y \ge 6$ and $x + 2y \le 10$Preview
  20. Q20Solve graphically : $2x + 3y \le 6$ and $x + 4y \ge 4$Preview
  21. Q21Solve graphically : $2x + y \ge 5$ and $x - y \le 1$Preview
7.1.1

Convex Sets

Definition (Convex set). A set of points in a plane is said to be a convex set if the line segment joining any two points of the set lies entirely within the set.

7.1.2

Graphical representation of linear inequations in two variables

This sub-section fixes the standard shading convention for four increasingly general families of linear inequation in two variables, each illustrated with a small reference graph.

7.1.3

Graphical solution of linear inequation

This sub-section works through six fully solved examples applying the shading method of section 7.1.2.

7.2

Linear Programming Problem (L.P.P.)

Linear Programming is an optimization technique used across management, planning, production, transportation and many other fields.

+Exercise 7.2i8 questions
  1. Q22Find the feasible solution of the following inequations graphically : $3x + 2y \le 18,\ 2x + y \le 10,\ x \ge 0,\ y \ge 0$Free
  2. Q23Find the feasible solution of the following inequations graphically : $2x + 3y \le 6,\ x + y \ge 2,\ x \ge 0,\ y \ge 0$Free
  3. Q24Find the feasible solution of the following inequations graphically : $3x + 4y \ge 12,\ 4x + 7y \le 28,\ y \ge 1,\ x \ge 0$Free
  4. Q25Find the feasible solution of the following inequations graphically : $x + 4y \le 24,\ 3x + y \le 21,\ x + y \le 9,\ x \ge 0,\ y \ge 0$Preview
  5. Q26Find the feasible solution of the following inequations graphically : $0 \le x \le 3,\ 0 \le y \le 3,\ x + y \le 5,\ 2x + y \ge 4$Preview
  6. Q27Find the feasible solution of the following inequations graphically : $x - 2y \le 2,\ x + y \ge 3,\ -2x + y \le 4,\ x \ge 0,\ y \ge 0$Preview
  7. Q28A company produces two types of articles A and B which requires silver and gold. Each unit of A requires 3 gm of silver and 1 gm of gold, wh…Preview
  8. Q29A furniture dealer deals in tables and chairs. He has Rs.1,50,000 to invest and a space to store at most 60 pieces. A table costs him Rs.150…Preview
7.2.1

Meaning of Linear Programming Problem

Meaning of a Linear Programming Problem (L.P.P.). The word "linear" means that every mathematical function involved — the objective function and every constraint — contains its variables raised to at…

7.2.2

Mathematical formulation of L.P.P.

Mathematical formulation of an L.P.P. follows a fixed three-step working rule:

7.2.3

Solution of L.P.P. by graphical methods

Formal definitions related to L.P.P.:

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 11 questions11 questions
  1. Q1Minimize $z = 4x + 5y$ subject to $2x + y \ge 7$, $2x + 3y \le 15$, $x \le 3$, $x \ge 0$, $y \ge 0$. Solve using graphical method.Preview
  2. Q2Solve the following L.P.P. by graphical method: Maximise $Z = 6x + 4y$ subject to $x \le 2$, $x + y \le 3$, $-2x + y \le 1$, $x \ge 0$, $y \…Preview
  3. Q3Minimize $Z = 7x + y$ subject to $5x + y \geq 5$, $x + y \geq 3$, $x \geq 0$, $y \geq 0$.Preview
  4. Q4Maximize: $z = 3x + 5y$ Subject to $x + 4y \leq 24$, $3x + y \leq 21$, $x + y \leq 9$, $x \geq 0, y \geq 0$Preview
  5. Q5Solve the following linear programming problem: Maximise: $z = 150x + 250y$; Subject to: $4x+y \le 40$, $3x+2y \le 60$, $x \ge 0$, $y \ge 0$Preview
  6. Q6Solve the L.P.P. by graphical method, Minimize $z = 8x + 10y$ Subject to $2x+y \ge 7$, $2x+3y \ge 15$, $y \ge 2$, $x \ge 0$Preview
  7. Q7If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0) then the point of minimum $z = 3x+2y$ is ________. (a) (2, 2) (…Preview
  8. Q8Solve the following inequations graphically and write the corner points of the feasible region: $2x+3y \le 6$, $x+y \ge 2$, $x \ge 0$, $y \g…Preview
  9. Q9Solve the following L.P.P. by graphical method: Maximize: $z=10x+25y$ Subject to: $0\le x\le 3$, $0\le y\le 3$, $x+y\le 5$. Also find the ma…Preview
  10. Q10Solve the linear programming problem graphically. Maximize: $z=3x+5y$ Subject to: $x+4y\le 24$, $3x+y\le 21$, $x+y\le 9$, $x\ge 0, y\ge 0$.…Preview
  11. Q11Solve the L.P.P. graphically: Minimize: $z=5x+2y$, Subject to, $5x+y\ge 10$, $x+y\ge 6$, $x\ge 0, y\ge 0$Preview

More questions

60 Q
+Show 9 questions9 questions
  1. Q30A manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry and then sent to machine shop for finis…Free
  2. Q31In a cattle breading firm, it is prescribed that the food ration for one animal must contain 14, 22 and 1 units of nutrients A, B and C resp…Free
  3. Q32A company manufactures two types of chemicals A and B. Each chemical requires two types of raw material P and Q. The table below shows numbe…Free
  4. Q33A printing company prints two types of magazines A and B. The company earns Rs. 10 and Rs. 15 on magazines A and B per copy. These are proce…Preview
  5. Q34A manufacture produces bulbs and tubes. Each of these must be processed through two machines M1 and M2. A package of bulbs require 1 hour of…Preview
  6. Q35A company manufactures two types of fertilizers F1 and F2. Each type of fertilizer requires two raw materials A and B. The number of units o…Preview
  7. Q36A doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carb…Preview
  8. Q37If John drives a car at a speed of 60 kms/hour he has to spend Rs. 5 per km on petrol. If he drives at a faster speed of 90 kms/hour, the co…Preview
  9. Q38The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs Rs.20 per…Preview
+Show 8 questions8 questions
  1. Q39Maximize : $z = 11x + 8y$ subject to $x \le 4,\ y \le 6,\ x + y \le 6,\ x \ge 0,\ y \ge 0$.Free
  2. Q40Maximize : $z = 4x + 6y$ subject to $3x + 2y \le 12,\ x + y \ge 4,\ x, y \ge 0$.Free
  3. Q41Maximize : $z = 7x + 11y$ subject to $3x + 5y \le 26,\ 5x + 3y \le 30,\ x \ge 0,\ y \ge 0$.Free
  4. Q42Maximize : $z = 10x + 25y$ subject to $0 \le x \le 3,\ 0 \le y \le 3,\ x + y \le 5$, also find maximum value of $z$.Preview
  5. Q43Maximize : $z = 3x + 5y$ subject to $x + 4y \le 24,\ 3x + y \le 21,\ x + y \le 9,\ x \ge 0,\ y \ge 0$, also find maximum value of $z$.Preview
  6. Q44Minimize : $z = 7x + y$ subject to $5x + y \ge 5,\ x + y \ge 3,\ x \ge 0,\ y \ge 0$.Preview
  7. Q45Minimize : $z = 8x + 10y$ subject to $2x + y \ge 7,\ 2x + 3y \ge 15,\ y \ge 2,\ x \ge 0,\ y \ge 0$.Preview
  8. Q46Minimize : $z = 6x + 2y$ subject to $x + 2y \ge 3,\ x + 4y \ge 4,\ 3x + y \ge 3,\ x \ge 0,\ y \ge 0$.Preview
+Show 43 questions43 questions
  1. Q47The value of objective function is maximum under linear constraints _______. A) at the centre of feasible region B) at (0, 0) C) at a vertex…Free
  2. Q48Which of the following is correct _______. A) every L.P.P. has an optimal solution B) a L.P.P. has unique optimal solution C) if L.P.P. has…Free
  3. Q49Objective function of L.P.P. is _______. A) a constraint B) a function to be maximized or minimized C) a relation between the decision varia…Free
  4. Q50The maximum value of $z = 5x + 3y$ subjected to the constraints $3x + 5y \le 15,\ 5x + 2y \le 10,\ x, y \ge 0$ is _______. A) 235 B) $\dfrac…Preview
  5. Q51The maximum value of $z = 10x + 6y$ subjected to the constraints $3x + y \le 12,\ 2x + 5y \le 34,\ x \ge 0,\ y \ge 0$ is _______. A) 56 B) 6…Preview
  6. Q52The point at which the maximum value of $x + y$ subject to the constraints $x + 2y \le 70,\ 2x + y \le 95,\ x \ge 0,\ y \ge 0$ is obtained a…Preview
  7. Q53Of all the points of the feasible region, the optimal value of $z$ obtained at the point lies _______. A) inside the feasible region B) at t…Preview
  8. Q54Feasible region is the set of points which satisfy _______. A) the objective function B) all of the given constraints C) some of the given c…Preview
  9. Q55Solution of L.P.P. to minimize $z = 2x + 3y$ s.t. $x \ge 0,\ y \ge 0,\ 1 \le x + 2y \le 10$ is _______. A) $x = 0,\ y = \dfrac{1}{2}$ B) $x…Preview
  10. Q56The corner points of the feasible solution given by the inequation $x + y \le 4,\ 2x + y \le 7,\ x \ge 0,\ y \ge 0$ are _______. A) (0, 0),…Preview
  11. Q57The corner points of the feasible solution are (0, 0), (2, 0), $\left(\dfrac{12}{7}, \dfrac{3}{7}\right)$, (0, 1). Then $Z = 7x + y$ is maxi…Preview
  12. Q58If the corner points of the feasible solution are (0, 0), (3, 0), (2, 1) and $\left(0, \dfrac{7}{3}\right)$, the maximum value of $z = 4x +…Preview
  13. Q59If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0) then the point of minimum $z = 3x + 2y$ is _______. A) (2, 2) B…Preview
  14. Q60The half plane represented by $3x + 2y < 8$ contains the point _______. A) $\left(1, \dfrac{5}{2}\right)$ B) (2, 1) C) (0, 0) D) (5, 1)Preview
  15. Q61The half plane represented by $4x + 3y > 14$ contains the point _______. A) (0, 0) B) (2, 2) C) (3, 4) D) (1, 1)Preview
  16. Q62Solve each of the following inequations graphically using X Y plane : $4x - 18 \ge 0$Preview
  17. Q63Solve each of the following inequations graphically using X Y plane : $-11x - 55 \le 0$Preview
  18. Q64Solve each of the following inequations graphically using X Y plane : $5y - 12 \ge 0$Preview
  19. Q65Solve each of the following inequations graphically using X Y plane : $y \le -3.5$Preview
  20. Q66Sketch the graph of each of following inequations in XOY co-ordinate system : $x \ge 5y$Preview
  21. Q67Sketch the graph of each of following inequations in XOY co-ordinate system : $x + y \le 0$Preview
  22. Q68Sketch the graph of each of following inequations in XOY co-ordinate system : $2y - 5x \ge 0$Preview
  23. Q69Sketch the graph of each of following inequations in XOY co-ordinate system : $1x + 51 \le y$Preview
  24. Q70Find graphical solution for each of the following system of linear inequation : $2x + y \ge 2,\ x - y \le 1$Preview
  25. Q71Find graphical solution for each of the following system of linear inequation : $x + 2y \ge 4,\ 2x - y \le 6$Preview
  26. Q72Find graphical solution for each of the following system of linear inequation : $3x + 4y \le 12,\ x - 2y \ge 2,\ y \ge -1$Preview
  27. Q73Find feasible solution for each of the following system of linear inequations graphically : $2x + 3y \le 12,\ 2x + y \le 8,\ x \ge 0,\ y \ge…Preview
  28. Q74Find feasible solution for each of the following system of linear inequations graphically : $3x + 4y \ge 12,\ 4x + 7y \le 28,\ x \ge 0,\ y \…Preview
  29. Q75Solve each of the following L.P.P. : Maximize $z = 5x_1 + 6x_2$ subject to $2x_1 + 3x_2 \le 18,\ 2x_1 + x_2 \le 12,\ x_1 \ge 0,\ x_2 \ge 0$Preview
  30. Q76Solve each of the following L.P.P. : Maximize $z = 4x + 2y$ subject to $3x + y \ge 27,\ x + y \ge 21$Preview
  31. Q77Solve each of the following L.P.P. : Maximize $z = 6x + 10y$ subject to $3x + 5y \le 10,\ 5x + 3y \le 15,\ x \ge 0,\ y \ge 0$Preview
  32. Q78Solve each of the following L.P.P. : Maximize $z = 2x + 3y$ subject to $x - y \ge 3,\ x \ge 0,\ y \ge 0$Preview
  33. Q79Solve each of the following L.P.P. : Maximize $z = 4x_1 + 3x_2$ subject to $3x_1 + x_2 \le 15,\ 3x_1 + 4x_2 \le 24,\ x_1 \ge 0,\ x_2 \ge 0$Preview
  34. Q80Solve each of the following L.P.P. : Maximize $z = 60x + 50y$ subject to $x + 2y \le 40,\ 3x + 2y \le 60,\ x \ge 0,\ y \ge 0$Preview
  35. Q81Solve each of the following L.P.P. : Maximize $z = 4x + 2y$ subject to $3x + y \ge 27,\ x + y \ge 21,\ x + 2y \ge 30;\ x > 0,\ y > 0$Preview
  36. Q82A carpenter makes chairs and tables. Profits are Rs.140/- per chair and Rs. 210/- per table. Both products are processed on three machines :…Preview
  37. Q83A company manufactures bicycles and tricycles, each of which must be processed through two machines A and B. Maximum availability of Machine…Preview
  38. Q84A factory produced two types of chemicals A and B. The following table gives the units of ingredients P and Q (per kg) of chemicals A and B…Preview
  39. Q85A company produces mixers and food processors. Profit on selling one mixer and one food processor is Rs. 2,000/- and Rs. 3,000/- respectivel…Preview
  40. Q86A chemical company produces a chemical containing three basic elements A, B, C so that it has at least 16 liters of A, 24 liters of B and 18…Preview
  41. Q87A person makes two types of gift items A and B requires the services of a cutter and a finisher. Gift item A requires 4 hours of cutter's ti…Preview
  42. Q88A firm manufactures two products A and B on which profit earned per unit Rs.3/- and Rs.4/- respectively. Each product is processed on two ma…Preview
  43. Q89A firm manufacturing two types of electrical items A and B, can make a profit of Rs.20/- per unit of A and Rs.30/- per unit of B. Both A and…Preview