Knowledge organisers

Linear Programming: CBSE Class 12 Maths knowledge organiser

Everything to know about linear programming on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Objective function: The linear function Z = ax + by to be maximised or minimised.
Constraints: The linear inequalities the variables must satisfy.
Feasible region: The region where all the constraints hold; the optimum is at a corner point.

Key formulas

Maximise/minimise \(Z=ax+by\) subject to linear constraints, \(x,y\ge0\)
Corner point method: the optimum occurs at a corner of the feasible region. Bounded region: both max and min exist
Unbounded: \(M\) is the max only if \(ax+by>M\) has no point in the feasible region (use \(<m\) for the min)

Worked example

Maximise \(Z = 2x + 5y\) with \(x + y \le 5,\ y \le 3,\ x, y \ge 0\): corners \((0, 0), (5, 0), (2, 3), (0, 3)\), \(Z = 0, 10, 19, 15\); maximum \(19\) at \((2, 3)\).

Common mistakes

Missing a corner (e.g. where a constraint meets an axis) or including an intersection point that violates another constraint.
Declaring a maximum/minimum on an unbounded region without the half-plane check.
Shading the wrong side of a line — always test a point.
Forgetting the non-negativity constraints, or units and the final interpretation in word problems.

You should be able to…

Use LP vocabulary, test points against constraints, formulate a simple LPP and read optimal values from given corner points.
Draw feasible regions, find corner points algebraically and solve bounded and unbounded problems, including the half-plane check.
Write full-mark 5-mark LPP answers from a word problem: formulation, labelled graph, corner-point table and conclusion.
Handle parameters in the objective function, multiple optimal solutions and existence questions for unbounded regions.

The printable sheet

Linear Programming knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Linear Programming knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Probability · All CBSE Class 12 Maths organisers