Linear Programming — Class 12 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Linear Programming, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Linear Programming in one look
shade the feasible region, then check every corner: the optimum always sits at one
Linear Inequations and Half-Planes
- ax+by+c=0 splits the plane into the line, and two open half-planes, < 0 and > 0
- the half-plane with the origin is the origin side; the line is the boundary of both closed half-planes
- to solve ax+by≤ c: draw ax+by=c, test the origin off the line, shade its side if TRUE
- a line needs two points: y=0 for the x-intercept, x=0 for the y-intercept, then join
Shading a region from a test point
Q. represent the solution set of 2x+3y≤6 graphically
- draw 2x+3y=6: y=0 gives x=3, so (3,0); x=0 gives y=2, so (0,2)
- the origin is off this line, so use it as the test point
- substitute (0,0): 2(0)+3(0)=0, and 0≤6 is TRUE
- the region is the side of the line containing the origin
Answer: shade the closed half-plane on the origin side of the line joining (3,0) and (0,2)
Convex Sets
- a set is convex if the segment joining ANY two of its points stays inside it
- a disc, a triangle's interior, a half-plane are convex; a crescent or star is not
- convex sets may be bounded (a polygon's interior) or unbounded (a half-plane)
- Theorem: the feasible region of a linear programming problem is always convex
Formulating a Linear Programming Problem
- decision variables: the unknown quantities being decided, usually x and y
- objective function: the linear expression in x, y being optimised, e.g. profit or cost
- constraints: linear inequations or equations the variables must satisfy, built from the resources stated
- non-negativity: x≥0 and y≥0, added whenever negative values make no physical sense
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Linear Programming sheet covers
- Linear Inequations and Half-Planes
- Shading a region from a test point
- Convex Sets
- Two lines, the pictures the rest of the chapter leans on
- Formulating a Linear Programming Problem
- Working Rule to Formulate an L.P.P.
- step 1
- step 2
- step 3
- Formulating an L.P.P. from a word problem
- Feasible Region and the Corner Point Method
- Corner-Point Method, Step by Step
- step 4
- Special cases of an L.P.P.
- Solving an L.P.P. by the corner point method
- What does the feasible region look like?
- When Two Corners Tie
- Bounded vs unbounded feasible region
- Bounded
- Unbounded
- Reading the graphical method
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Linear Programming the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.