RG's E-Learning Centre · Book a free demo class · WhatsApp +91-9379416374

Linear programming by corner points: Class 12 practice

The maximum or minimum of Z = ax + by over a bounded feasible region occurs at a corner point. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Solving pairs of equations
  • Inequalities

The idea behind linear programming by corner points

The maximum or minimum of Z = ax + by over a bounded feasible region occurs at a corner point. Find the corners, evaluate Z at each and pick the best.

Step-by-step method

  1. Draw the constraints and find the feasible region.
  2. Find the corner points (intersections).
  3. Evaluate Z at each corner.
  4. Choose the largest (max) or smallest (min).

Worked example 1 (Basic)

Maximise Z = 1x + 6y subject to x + y ≤ 6, x ≤ 2, x ≥ 0, y ≥ 0.

  1. Corners: (0, 0), (2, 0), (2, 4), (0, 6)
  2. Z values: 0, 2, 26, 36
  3. Maximum 36 at (0, 6)
  4. Answer: Z = 36 at (0, 6)

Worked example 2 (Standard)

Maximise Z = 6x + 4y subject to x + y ≤ 6, x ≤ 5, x ≥ 0, y ≥ 0.

  1. Corners: (0, 0), (5, 0), (5, 1), (0, 6)
  2. Z values: 0, 30, 34, 24
  3. Maximum 34 at (5, 1)
  4. Answer: Z = 34 at (5, 1)

Worked example 3 (Challenge)

Maximise Z = 6x + 6y subject to x + y ≤ 4, x ≤ 3, x ≥ 0, y ≥ 0.

  1. Corners: (0, 0), (3, 0), (3, 1), (0, 4)
  2. Z values: 0, 18, 24, 24
  3. Maximum 24 at (3, 1)
  4. Answer: Z = 24 at (3, 1)

Common mistakes and misconceptions

  • Missing a corner
  • Evaluating Z at points outside the region

Practice questions

  1. (Basic) Maximise Z = 2x + 5y subject to x + y ≤ 6, x ≤ 3, x ≥ 0, y ≥ 0.
  2. (Basic) Maximise Z = 3x + 2y subject to x + y ≤ 6, x ≤ 2, x ≥ 0, y ≥ 0.
  3. (Standard) Maximise Z = 3x + 4y subject to x + y ≤ 8, x ≤ 4, x ≥ 0, y ≥ 0.
  4. (Standard) Maximise Z = 6x + 4y subject to x + y ≤ 3, x ≤ 2, x ≥ 0, y ≥ 0.
  5. (Challenge) Maximise Z = 3x + 4y subject to x + y ≤ 6, x ≤ 2, x ≥ 0, y ≥ 0.
  6. (Challenge) Maximise Z = 6x + 2y subject to x + y ≤ 3, x ≤ 2, x ≥ 0, y ≥ 0.

Answer key with full working

  • Q1: Corners: (0, 0), (3, 0), (3, 3), (0, 6) → Z values: 0, 6, 21, 30 → Maximum 30 at (0, 6) → Answer: Z = 30 at (0, 6)
  • Q2: Corners: (0, 0), (2, 0), (2, 4), (0, 6) → Z values: 0, 6, 14, 12 → Maximum 14 at (2, 4) → Answer: Z = 14 at (2, 4)
  • Q3: Corners: (0, 0), (4, 0), (4, 4), (0, 8) → Z values: 0, 12, 28, 32 → Maximum 32 at (0, 8) → Answer: Z = 32 at (0, 8)
  • Q4: Corners: (0, 0), (2, 0), (2, 1), (0, 3) → Z values: 0, 12, 16, 12 → Maximum 16 at (2, 1) → Answer: Z = 16 at (2, 1)
  • Q5: Corners: (0, 0), (2, 0), (2, 4), (0, 6) → Z values: 0, 6, 22, 24 → Maximum 24 at (0, 6) → Answer: Z = 24 at (0, 6)
  • Q6: Corners: (0, 0), (2, 0), (2, 1), (0, 3) → Z values: 0, 12, 14, 6 → Maximum 14 at (2, 1) → Answer: Z = 14 at (2, 1)

Syllabus note

This skill is listed under "Linear Programming" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is linear programming by corner points in?

"Linear Programming", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Book a free demo class