Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step Class 12 Maths NCERT Solutions Chapter 12 for the one exercise that makes up this entire chapter — 10 Linear Programming Problems, every one of them solved graphically using the Corner Point Method, with the feasible region, corner points and objective-function table worked out in full for each question.
Linear Programming is where all the graphing of linear inequalities from Class 11 finally earns its keep. A Linear Programming Problem (LPP) asks you to maximise or minimise a linear objective function — profit, cost, distance, whatever the real-world question is about — subject to a set of linear constraints that box in the possible values of the decision variables. Unlike most Class 12 Maths chapters, this one is refreshingly short: there's exactly one exercise, and everything you need is the graphical Corner Point Method.
The chapter walks through the same short story every time: translate the word problem (or the given inequalities) into a mathematical formulation, sketch the feasible region those constraints carve out, mark its corner points, and evaluate the objective function at each one. What makes Exercise 12.1 worth taking slowly is the handful of edge cases it deliberately builds in — unbounded feasible regions, problems with no feasible region at all, and situations where more than one corner point gives the same optimal value. Recognising which case you're in in a live board exam is genuinely most of the skill here.
Write the objective function Z = ax + by, and translate every restriction into a linear inequality — plus the non-negativity constraints x ≥ 0, y ≥ 0.
Graph every constraint. The region common to all of them — the set of points satisfying every inequality at once — is the feasible region.
Find the vertices of the feasible region and evaluate Z at each one. For a bounded region, the largest and smallest values are the true max and min.
Check for an unbounded region, multiple optimal solutions along an edge, or no feasible region at all — these are exactly what Exercise 12.1 tests.
Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below have the full printable version for the whole syllabus.
The linear function to be maximised or minimised; a, b are constants, x and y are the decision variables.
Linear inequalities (or equations) restricting the decision variables, including the non-negative restrictions x ≥ 0, y ≥ 0.
An LPP is a special, particularly important type of optimisation problem.
Every point in it is a feasible solution; every point outside it is infeasible.
Found either by inspection or by solving the two boundary equations simultaneously.
This distinction decides which version of the Corner Point Method rule you apply next.
M is the maximum and m is the minimum of Z — guaranteed to exist (Theorem 2).
Same idea in reverse for a minimum, using ax+by\lt m. If the half-plane does share a point, Z has no max (or min) at all.
Then every point on the line segment joining them gives that same optimal value too.
No point satisfies every constraint simultaneously, so the LPP has no feasible solution at all.
Once you've graphed the feasible region and listed its corner points, use this to decide what happens next.
Drawn from where students actually lose marks on graphical LPP questions.
Every Linear Programming Problem in the chapter, solved graphically with the Corner Point Method — including unbounded regions, multiple optimal solutions and infeasible cases · 10 questions
Solve Exercise 12.1 →Every formula for Linear Programming — plus every other chapter — in one printable set of Formula Cards.
Get Formula Cards →Quick answers about Chapter 12, Linear Programming.
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