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CBSE Class 12 · Mathematics

Linear Programming

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Linear Programming” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Linear Programming Problem and its Mathematical Formulation

An LPP (school): maximise or minimise z=ax+by subject to linear inequalities as taught. The feasible region is the overlap of those half-planes. Corner-point theorem: the optimum sits at a vertex of that region as framed. LPP is a region-plus-corners job, not a single equation.

Solving z=3x+2y as one line with no inequalities is a miss.

Figure. A linear program is constraints plus an objective. The feasible set is the polygon they cut in the first quadrant: here O, (3,0), (2,3), (0,4).

How it works

  1. Write z and the inequality-listThe LPP.
  2. Sketch the feasible overlapThe region.
  3. Evaluate z at each vertex; pick max or minThe corner theorem.

Two-corner compare

Max z=3x+2y on vertices (0,0), (4,0), (0,3).

  • At (0,0)z=0
  • At (4,0)z=12
  • At (0,3)z=6 so max 12 at (4,0)

Pro tip. Only vertices, then compare z.

2A definition is a test you can run

LPP-word is a test: an objective, a feasible overlap, or a corner evaluation. If you only say “programming”, you have a heading.

A computer-sticker is not the test.

Figure. A point is feasible only if every inequality holds. (1,1) sits inside; (4,2) fails 3x+y<=9, so it is out — that is the test.

How it works

  1. Name z, the region, or a vertex-zThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

3Name the given before the unknown

The given is z and the inequalities. The unknown is the corner that wins. Copy z=3x+2y before you treat it as 2x+3y.

Evaluating z inside the region and skipping vertices is a silent skip of the theorem.

Figure. Name the decision variables and the inequalities before you write Z. The objective is the last given, not the first doodle.

How it works

  1. Copy z and the constraintsThe given.
  2. Name the winning vertexThe unknown.
  3. Then compare vertex-zGiven first.

4One worked case is enough at this class

One worked case is enough: max 12 at (4,0). Do not stack four farms. The case teaches vertex-compare.

A poster of ten polygons is not more science.

Figure. One corner walk is the whole method: Z=3x+2y is 0, 9, 12, 8 at the four vertices. The maximum 12 sits at (2,3).

How it works

  1. Take one z and a small vertex-listThe case.
  2. Compare z at those cornersThe teach.
  3. Stop — one caseThis class.

5A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Feasible region” is the overlap-test — a green blob is a setting.

A pretty polygon without inequalities is a mute picture.

Figure. A constraint is a fence, not the thing you maximise. Swapping x+2y=8 for the objective Z=3x+2y is the picture-for-name miss.

How it works

  1. Keep feasible/objective as the namesThe science.
  2. Use a shade only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.

6Check by the opposite action or the opposite test

Check by the opposite: a point outside a constraint cannot be feasible; a claimed max should beat the other vertices. If a leftover vertex has larger z, the claim failed.

A pretty interior point as “the max” is a miss.

Figure. The opposite check is to evaluate Z at every corner. 12 beats 9 and 8, so (2,3) is the maximum — a missed corner would hide it.

How it works

  1. Test each vertex against the constraintsFeasible?
  2. Compare z valuesThe opposite of a missed vertex.
  3. If a better vertex sits, redoThe check.

7Keep the claim at this chapter, not the next

Keep the claim at this chapter: linear objective, linear constraints, corners. It does not start a non-linear dump or a computer-simplex essay.

A quadratic objective here is a chapter-swap.

Figure. This chapter’s claim is the two-variable graph. Simplex for more variables is a later course — do not import it here.

How it works

  1. Stay on linear z and linear inequalitiesThis chapter.
  2. Leave later methodsThe boundary.
  3. Refuse a computer-manualThis class.

8The story is the hook; the test is the idea

The story is the hook; the test is the idea: a factory-hours story is a setting. The idea is still z at the vertices of the overlap.

A story with no inequalities is a mute hook.

Figure. The factory story is the hook: chairs and tables eat hours. The test is the feasible region those hours cut, not the story itself.

How it works

  1. Keep the story as a hookA setting.
  2. Run the corner evaluationThe idea.
  3. Refuse a story as the whole scienceThe test.
The school optimum of a linear LPP sits
  1. At a vertex of the feasible region as taught
  2. Always at the origin
  3. At any interior point

Corner-point theorem.

Notes

  • The official chapter title is “Linear Programming”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • optimum at a vertex of the feasible region
  • z=ax+by

Recap

Hold these pegs from the official chapter “Linear Programming”. The wording is ExamMaster’s teaching, not a textbook recap.

Linear Programming Problem and its Mathematical Formulation
An LPP (school): maximise or minimise z=ax+by subject to linear inequalities as taught.
A definition is a test you can run
LPP-word is a test: an objective, a feasible overlap, or a corner evaluation.
Name the given before the unknown
The given is z and the inequalities. The unknown is the corner that wins. Copy z=3x+2y before you treat it as 2x+3y.
One worked case is enough at this class
One worked case is enough: max 12 at (4,0).
A miss: swapping the school name for the picture
A miss: swapping the school name for a picture.
Check by the opposite action or the opposite test
Check by the opposite: a point outside a constraint cannot be feasible; a claimed max should beat the other vertices.

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