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Mathematics · Class 12 Science

Ch 13Linear Programming — Class 12 Mathematics, concept-first.

Many everyday decisions involve getting the best possible outcome from a limited supply of something. A factory owner has only so many machine-hours and only so much raw material, and wants to decide how many units of each product to make so as to earn the maximum possible profit.

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Key concepts

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Multiple Optimal Solutions

Imagine climbing to the highest point of a mountain range and finding two peaks of exactly the same height, joined by a flat ridge. You have not found one best spot — you have found many, all equally high.

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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.

1

Introduction to Linear Programming

Many everyday decisions involve getting the best possible outcome from a limited supply of something. A factory owner has only so many machine-hours and only so much raw material, and wants to decide…

2

Related Terminology: Constraints, Objective Function and Optimization

Decision variables. The first step in setting up any LPP is to name the unknown quantities that are actually under the decision-maker's control -- for instance, the number of units of two products to…

3

Graphical Method of Solution

Once an LPP has been formulated with exactly two decision variables, it can be solved by an exact geometric procedure, because every linear constraint corresponds to a straight line dividing the coord…

4

Feasible and Infeasible Regions: Bounded and Unbounded

Feasible region, precisely. The feasible region of an LPP is the set of all points in the plane that satisfy every constraint of the problem simultaneously -- both the non-trivial constraints and the…

5

Feasible and Infeasible Solutions

Feasible solution. A feasible solution to a linear programming problem is any single point that satisfies EVERY constraint of the problem at once -- both the non-trivial constraints and the non-negati…

6

Optimal Feasible Solutions and the Corner Point Method

The Corner Point Theorem. The central theoretical fact underlying the whole graphical method is this: if a linear programming problem has an optimal solution over its feasible region, that optimal val…

Summary

Linear programming. A linear programming problem (LPP) optimizes (maximizes or minimizes) a linear objective function in decision variables , subject to linear constraints (each of or type, arising fr…

Sample & Board Papers

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

+Show 12 questions12 questions
  1. Q1A manufacturer produces two models A and B of a product. Each piece of model A requires 9 labour hours for fabricating and 1 labour hour for…Preview
  2. Q2Solve the following linear programming problem graphically. (Graph sheet is not required): Minimize Z = 200x + 500y subject to the constrain…Preview
  3. Q3A dietician wishes to mix two types of foods in such a way that the mixture contain at least 8 units of Vitamin A and 10 units of Vitamin C.…Preview
  4. Q4Solve the linear programming problem graphically: Maximise Z = 4x + y where x + y ≤ 50, 3x + y ≤ 90, x ≥ 0 and y ≥ 0. (Graph sheet is not re…Preview
  5. Q5A manufacturer produces two models A and B of a product. Each piece of model A requires 9 labour-hours for fabricating and 1 labour-hour for…Preview
  6. Q6Solve the following linear programming problem graphically: (Graph sheet is not required) Minimize Z = 200x + 500y subject to the constraint…Preview
  7. Q7A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods A and B are availab…Preview
  8. Q8Solve the following linear programming problem graphically (Graph sheet is not required): Maximize Z = 4x + 3y subject to x + y ≤ 50, x + 2y…Preview
  9. Q9A farmer purchased some sheep and goats at Rs. 1,500 per sheep and Rs. 2,000 per goat, and makes a profit of Rs. 150 per sheep and Rs. 200 p…Preview
  10. Q10Solve the following linear programming problem by graphical method. (Graph sheet is not required) Minimize Z = 2x - y subject to x+y≤5, x+2y…Preview
  11. Q11In a Linear Programming Problem, the corner points of the feasible convex region are $(0, 0)$, $(0, 4)$, $(2, 3)$ and $(5, 0)$. Find the cor…Preview
  12. Q12Solve the following linear programming problem by graphical method (Graph sheet is not required): Minimize $Z = 3x + 5y$ subject to $x + 3y…Preview

More questions

27 Q
+Show 2 questions2 questions
  1. Q26The corner points of an unbounded feasible region determined by a set of linear constraints are $(0,4)$, $(2,2)$ and $(5,0)$, and the region…Free
  2. Q27Two students solve the linear programming problem: Maximize $Z = 5x + 3y$ subject to $x + y \le 6,\ 2x + y \le 9,\ x, y \ge 0$. Student A ev…Preview
+Show 8 questions8 questions
  1. Example 1A stationery dealer deals in only two items: notebook packs and pen sets. She has Rs 20,000 to invest and storage space for at most 80 piece…Free
  2. Example 2For the linear programming problem Maximize $Z = 3x + 2y$ subject to the constraints $x + y \le 4,\ x \le 3,\ x, y \ge 0$, determine, giving…Free
  3. Example 3A toy company manufactures two types of toys, A and B. Each toy of type A requires 3 hours on machine $M_1$ and 1 hour on machine $M_2$; eac…Free
  4. Example 4Solve the following linear programming problem graphically: Maximize $Z = 4x + 3y$ subject to the constraints $x + 2y \le 10,\ 3x + y \le 15…Preview
  5. Example 5Solve the following linear programming problem graphically: Minimize $Z = 3x + 5y$ subject to the constraints $x \ge 1,\ y \ge 1,\ x + y \le…Preview
  6. Example 6Minimize $Z = 5x + 7y$ subject to the constraints $2x + y \ge 8,\ x + 2y \ge 10,\ x, y \ge 0$. Also determine, with reason, whether $Z$ has…Preview
  7. Example 7A dietician wishes to design a diet using two food items, Food I and Food II, that contains at least 8 units of a certain protein and at lea…Preview
  8. Example 8A small workshop manufactures two types of electronic gadgets, A and B. Each unit of gadget A requires 2 hours of assembly and 1 hour of tes…Preview
+Show 4 questions4 questions
  1. Q9A factory produces two types of packaged snacks, P and Q. Each packet of P requires 3 kg of raw material and 1 labour-hour; each packet of Q…Free
  2. Q10A trust fund has Rs 30,000 to invest in two types of bonds. Bond A yields an annual return of 5\% and Bond B yields 7\%. The trust's rules r…Free
  3. Q11A farmer has 10 hectares of land to allocate between wheat and mustard. Each hectare of wheat requires 6 labour-days and yields a profit of…Preview
  4. Q12A courier company transports packages using two types of vehicles: vans and bikes. Each van can carry 100 packages per trip and costs Rs 800…Preview
+Show 5 questions5 questions
  1. Q13Solve graphically: Maximize $Z = 5x + 4y$ subject to $x + 2y \le 8,\ 3x + 2y \le 12,\ x, y \ge 0$.Free
  2. Q14Solve graphically: Minimize $Z = 2x + 3y$ subject to $x \ge 2,\ y \ge 1,\ x + y \le 8$.Free
  3. Q15Solve graphically: Maximize $Z = 3x + 2y$ subject to $x + y \le 6,\ 2x + y \le 10,\ x, y \ge 0$.Preview
  4. Q16Solve graphically: Minimize $Z = 7x + 5y$ subject to $2x + y \le 10,\ x + 3y \le 15,\ x, y \ge 0$, and explain why the minimum value occurs…Preview
  5. Q17Solve graphically: Maximize $Z = 2x + 3y$ subject to $x + 2y \le 10,\ 2x + y \le 10,\ x + y \le 6,\ x, y \ge 0$.Preview
+Show 4 questions4 questions
  1. Q18Solve graphically: Minimize $Z = 3x + 4y$ subject to $x + 2y \ge 8,\ 3x + 2y \ge 12,\ x, y \ge 0$. State, with reason, whether $Z$ also has…Free
  2. Q19Show that $Z = 4x + 6y$ attains a minimum value, but no maximum value, on the feasible region determined by $x + y \ge 5,\ x + 2y \ge 8,\ x,…Free
  3. Q20Solve graphically: Minimize $Z = 3x + 2y$ subject to $x + y \ge 10,\ 2x + y \ge 15,\ x + 2y \ge 12,\ x, y \ge 0$.Preview
  4. Q21Solve graphically: Minimize $Z = 20x + 10y$ subject to $x + 2y \ge 40,\ 3x + y \ge 30,\ x, y \ge 0$. Does $Z$ have a maximum value on this f…Preview
+Show 4 questions4 questions
  1. Q22A dietician wishes to mix two types of food, $F_1$ and $F_2$, to prepare a diet containing at least 8 units of protein and at least 8 units…Free
  2. Q23A company manufactures two products, X and Y, using two machines. Product X requires 3 hours on machine $M_1$ and 1 hour on machine $M_2$; p…Free
  3. Q24A shopkeeper assembles two types of grocery combo-packs, Combo-A and Combo-B, from stock. Each Combo-A uses 2 kg of rice and 1 kg of dal; ea…Preview
  4. Q25An advertising agency wants to reach at least 60 lakh people using two media, television and newspaper. Each television advertisement reache…Preview