Skip to content
← Mathematics and Statistics

Mathematics and Statistics · Class 12 Commerce

Ch 14Linear Programming — Class 12 Mathematics and Statistics, concept-first.

This Maharashtra Std XII (Commerce) Mathematics and Statistics chapter studies Linear Programming — a technique for getting the best possible outcome (largest profit, smallest cost) when limited resources must be shared between competing activities.

28

Q&A

3

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Graphical Solution by the Corner-Point Method

By the Corner-Point Theorem, an LPP's optimum (if it exists on the feasible region) occurs at at least one corner point, because is constant along parallel iso-profit/iso-cost lines whose last contact with a polygonal re…

Most relevant Q&A

Start with this concept →

Chapter contents

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

1

Formulation of a Linear Programming Problem (LPP)

This Maharashtra Std XII (Commerce) Mathematics and Statistics chapter studies Linear Programming — a technique for getting the best possible outcome (largest profit, smallest cost) when limited resou…

2

Feasible Region — Bounded and Unbounded

Once an LPP is formulated, its constraints are graphed exactly as in Std XI: each inequation is a half-plane, and the region satisfying all of them at once is the set of allowable plans.

3

Graphical Solution by the Corner-Point Method

For an LPP in two variables the feasible region is a flat region of the plane, and the optimum can be found graphically. The method rests on one key result.

4

Maximisation and Minimisation Applications

The corner-point method turns real commercial questions into a short, repeatable calculation. Two families of problem recur throughout the Std XII (Commerce) course.

Exercises

+Show 5 questions5 questions
  1. Q9The optimal value of the objective function of an LPP (when it exists on the feasible region) always occurs: (A) at the centre of the feasib…Free
  2. Q10The feasible region of an LPP is the set of all points that: (A) satisfy at least one constraint (B) satisfy the objective function (C) sati…Free
  3. Q11A furniture trader has ₹210 hundred (₹21{,}000) to invest and space for at most 40 pieces. A table costs ₹5 hundred and a chair ₹3 hundred.…Preview
  4. Q12For the LPP: Maximise $Z=4x+3y$ subject to the feasible region having corner points $(0,0),\ (6,0),\ (4,3)$ and $(0,5)$, find the maximum va…Preview
  5. Q13Which of the following is NOT a component of a Linear Programming Problem? (A) An objective function (B) Linear constraints (C) Non-negativi…Preview

Sample & Board Papers

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

+Show 15 questions15 questions
  1. Q1In a cattle breeding firm, it is prescribed that the food ration for one animal must contain 14, 22, and 1 unit of nutrients A, B, and C res…Preview
  2. Q2Solve the following LP.P. Maximize $z = 13x + 9y$, Subject to $3x + 2y \leq 12$, $x + y \geq 4$, $x \geq 0$, $y \geq 0$.Preview
  3. Q3If the corner points of the feasible region are (0, 0), (3, 0), (2, 1) and $\left(0, \frac{7}{3}\right)$ the maximum value of $z = 4x + 5y$…Preview
  4. Q4Graphical solution set of the inequations $x \geq 0$ and $y \leq 0$ lies in ______ quadrant.Preview
  5. Q5Solve the following L.P.P. by graphical method: Maximize: $Z = 4x + 6y$ Subject to $3x + 2y \leq 12$, $x + y \geq 4$, $x, y \geq 0$.Preview
  6. Q6Objective function of LPP is ______. (a) A constraint (b) A function to be maximised or minimised (c) A relation between the decision variab…Preview
  7. Q7State whether the following is True or False: The optimum value of the objective function of LPP occurs at the centre of the feasible region…Preview
  8. Q8Solve the following L.P.P. by graphical method: Minimize: $Z = 6x + 2y$ subject to $x + 2y \geq 3$, $x + 4y \geq 4$, $3x + y \geq 3$, $x \ge…Preview
  9. Q9The 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 ₹ 20 per…Preview
  10. Q10Solve the following L.P.P. by graphical method: Maximize: $Z = 4x + 6y$ Subject to $3x + 2y \le 12$, $x + y \ge 4$, $x, y \ge 0$.Preview
  11. Q11If the corner points of the feasible region are $(0, 10)$, $(2, 2)$ and $(4, 0)$ then the point of minimum $z = 3x + 2y$ is ______. (a) $(2,…Preview
  12. Q12A dish washing machine holds up to 40 pieces of large crockery ($x$). This constraint is given by ______.Preview
  13. Q13Solve the following L.P.P. by graphical method: Minimize: $z = 8x + 10y$ Subject to: $2x + y \geq 7$, $2x + 3y \geq 15$, $y \geq 2$, $x \geq…Preview
  14. Q14Show the solution set for the following inequation $x + 4y \leq 0$ graphically.Preview
  15. Q15Solve the following L.P.P. graphically: Minimize: $Z = 3x + 2y$, Subject to the constraints $x - y \leq 1$, $x + y \geq 3$, $x \geq 0$, $y \…Preview

More questions

+Show 8 questions8 questions
  1. Example 1A firm manufactures two products A and B. Each unit of A requires 2 hours on machine I and 1 hour on machine II; each unit of B requires 1 h…Free
  2. Example 2Solve the LPP graphically: Maximise $Z=5x+4y$ subject to $2x+y\le 8,\ x+3y\le 9,\ x\ge 0,\ y\ge 0$.Free
  3. Example 3Maximise $Z=3x+2y$ subject to $x+2y\le 10,\ 3x+y\le 15,\ x\ge 0,\ y\ge 0$ by the corner-point method.Free
  4. Example 4A company produces two goods A and B. Making one unit of A uses 3 units of raw material and 2 labour-hours; making one unit of B uses 1 unit…Preview
  5. Example 5Minimise $Z=3x+5y$ subject to $x+3y\ge 3,\ x+y\ge 2,\ x\ge 0,\ y\ge 0$.Preview
  6. Example 6Maximise $Z=3x+4y$ subject to $x+y\le 4,\ x\ge 0,\ y\ge 0$.Preview
  7. Example 7Show that the LPP: Maximise $Z=x+y$ subject to $x-y\ge -1,\ -x+y\le 1$ (equivalently $y-x\le1$), $x\ge 0,\ y\ge 0$ has no maximum value.Preview
  8. Example 8A dietician wishes to mix two foods F1 and F2 so that the mixture contains at least 8 units of vitamin A and at least 10 units of vitamin C.…Preview