CBSE Ch 12 Important Questions Class 12 Maths 2025 PDF

Linear Programming (LP) is a vital topic in Class 12 Mathematics, focusing on optimisation problems. It helps in maximising profit or minimising costs while following a set of constraints, usually written as inequalities.

LP is widely used in:
Economics – Managing resources efficiently
Business – Planning production and minimising costs
Agriculture – Deciding how much crop to grow for maximum profit
Engineering – Optimising processes and logistics

To do well in Class 12 exams, students must understand the concepts of LP and practise solving important questions from different types of problems.

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In the table below, we have provided the links to the Linear Programming Class 12 Maths Ch 12 Important Questions PDFs. You can download them without having to share any login info.

Graphical Method

The graphical method is the most commonly used technique for solving linear programming problems (LPPs) with two variables. It involves:

• Plotting constraints as lines on a graph.
• Identifying the feasible region (where all constraints are satisfied).
• Finding the optimal solution at the corner points of the feasible region.

Important Questions

Q1: Maximise Z = 3x + 2y subject to:

• x + y ≤ 4
• 2x + y ≤ 5
• x ≥ 0, y ≥ 0

Q2: Maximise Z = 2x + 3y subject to:

• 4x + y ≤ 12
• x + 2y ≤ 8
• x ≥ 0, y ≥ 0

Q3: Maximise Z = 3x + 5y subject to:

• x + y ≤ 5
• 2x + 3y ≤ 12
• x ≥ 0, y ≥ 0

Q4: Minimise Z = 5x + 4y subject to:

• x + y ≥ 6
• 3x + 2y ≥ 12
• x ≥ 0, y ≥ 0

Q5: Minimise Z = 4x + 3y subject to:

• 2x + y ≥ 8
• x + 3y ≥ 10
• x ≥ 0, y ≥ 0

Important Points

• The graphical method works only for two-variable problems.
• The feasible region must be bounded for a solution to exist.
• The optimal solution is found at the corner points of the feasible region.

Simplex Method

For problems with more than two variables, the graphical method is not practical. Instead, we use the Simplex method, which involves:

• Creating a tableau to organise constraints.
• Performing step-by-step row operations to reach an optimal solution.

Important Questions

Q1: Maximise Z = 3x + 4y subject to:

• 2x + y ≤ 8
• 4x + 3y ≤ 12
• x ≥ 0, y ≥ 0

Q2: Maximise Z = 3x + 2y subject to:

• 4x + y ≤ 10
• 2x + 3y ≤ 12
• x ≥ 0, y ≥ 0

Q3: Maximise Z = 2x + 3y subject to:

• x + 2y ≤ 6
• 2x + y ≤ 6
• x ≥ 0, y ≥ 0

Q4: Minimise Z = 7x + 5y subject to:

• x + y ≥ 3
• 2x + 3y ≥ 9
• x ≥ 0, y ≥ 0

Q5: Minimise Z = 4x + 3y subject to:

• x + 2y ≥ 5
• 2x + y ≥ 6
• x ≥ 0, y ≥ 0

Important Points

• Used for more than two variables.
• Requires setting up a tableau and performing calculations.
• Guarantees an optimal solution if one exists.

Word Problems (Real-Life Applications)

LP word problems involve converting real-world situations into mathematical models and solving them using LP techniques.

Important Questions

Q1: A factory produces two types of chairs (A and B):

• Chair A requires 2 hours of labour and 3 units of wood.
• Chair B requires 3 hours of labour and 2 units of wood.
• The factory has 12 hours of labour and 10 units of wood.
• Profit: £40 (A), £50 (B).

Find the optimal number of chairs to maximise profit.

Q2: A dietician is preparing a diet using Food X and Food Y:

• Food X: 5g protein, 10g carbohydrates per unit.
• Food Y: 10g protein, 5g carbohydrates per unit.
• Minimum requirements: 50g protein, 40g carbohydrates.
• Cost: £20 (X), £30 (Y) per unit.

Find the optimal mix to minimise cost while meeting nutritional needs.

Q3: A company manufactures Product A and Product B:

• Product A requires 3 hours of labour and 2 units of raw material.
• Product B requires 2 hours of labour and 3 units of raw material.
• Available: 18 hours of labour, 24 units of raw material.
• Profit: £50 (A), £60 (B).

Find the optimal production strategy to maximise profit.

Duality in Linear Programming

Each LP problem (called the primal problem) has a corresponding dual problem, which provides useful insights into the original problem.

Important Questions

Q1: Consider the primal problem:
Maximise Z = 3x + 4y subject to:

• 2x + y ≤ 8
• 4x + 3y ≤ 12
• x, y ≥ 0

Find the dual problem.

Q2: Minimise Z = 2x + 3y subject to:

• x + y ≥ 3
• 2x + y ≥ 4
• x ≥ 0, y ≥ 0

Formulate the dual problem.

Q3: Maximise Z = 3x + 4y subject to:

• 2x + y ≤ 8
• 4x + 3y ≥ 10
• x ≥ 0, y ≥ 0

Check if the feasible region is bounded or unbounded. If unbounded, find the maximum value of Z.

Q4: Minimise Z = x + y subject to:

• x + y ≤ 4
• x + y ≥ 5
• x ≥ 0, y ≥ 0

Determine if the problem is feasible.

Important Points

• The primal and dual problems offer different perspectives on the same issue.
• The Strong Duality Theorem links the optimal values of both problems.

Linear Programming is a critical topic in Class 12 Mathematics, with applications in economics, business, engineering, and computer science.

To succeed in exams:

✔ Master graphical and Simplex methods.
✔ Practise word problems to improve real-world application skills.
✔ Understand duality to solve LP problems efficiently.

📌 Tip: Solve at least 10 LP problems before the exam to boost your confidence!

We hope that you practice the above Class 12 Maths Linear Programming Extra Questions and achieve your dream marks.

All the best!