Linear Programming — Class 12 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Linear Programming" — 6 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Linear Programming" — 6 important questions with detailed answers for CBSE board e…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

Solve the linear programming problem: Maximize z = 3x + 4y subject to x + y ≤…
Minimize z = 2x + 3y subject to x + y ≥ 5, 2x + y ≥ 8, x ≥ 0, y ≥ 0.
A factory produces chairs and tables. Each chair takes 2 hours and each table…
A diet must contain at least 400 units of calcium and 900 units of protein. F…
A company makes two products A and B. Each unit of A requires 3 hours of skil…
State the Fundamental Theorem of Linear Programming.

Solutions Summary:

Question	Status
Solve the linear programming problem: Maximize z = 3x + 4…	✓ Solved
Minimize z = 2x + 3y subject to x + y ≥ 5, 2x + y ≥ 8, x …	✓ Solved
A factory produces chairs and tables. Each chair takes 2 …	✓ Solved
A diet must contain at least 400 units of calcium and 900…	✓ Solved
A company makes two products A and B. Each unit of A requ…	✓ Solved
State the Fundamental Theorem of Linear Programming.	✓ Solved

Showing 6 of 6 questions

Q1: Solve the linear programming problem: Maximize z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0.

To solve the LPP: Step 1: Identify constraints. (1) x + y ≤ 4 (2) x ≥ 0 (3) y ≥ 0 Objective: Maximize z = 3x + 4y Step 2: Find corner points of feasible region. Constraint (1): x + y = 4 Intersection with x = 0: (0, 4) Intersection with y = 0: (4, 0) Origin: (0, 0) Corner points: (0, 0), (0, 4), (4, 0) Step 3: Evaluate objective function at each corner. At (0, 0): z = 3(0) + 4(0) = 0 At (0, 4): z = 3(0) + 4(4) = 16 At (4, 0): z = 3(4) + 4(0) = 12 Step 4: Identify optimal solution. Maximum v...

Q2: Minimize z = 2x + 3y subject to x + y ≥ 5, 2x + y ≥ 8, x ≥ 0, y ≥ 0.

To minimize z = 2x + 3y: Step 1: Identify constraints and find boundary lines. (1) x + y = 5 (2) 2x + y = 8 (3) x = 0 (y-axis) (4) y = 0 (x-axis) Step 2: Find intersection points. Line 1 ∩ Line 2: x + y = 5 ... (i) 2x + y = 8 ... (ii) Subtracting: x = 3, y = 2 Point: (3, 2) Line 1 ∩ y-axis (x = 0): 0 + y = 5, so y = 5 Point: (0, 5) Line 2 ∩ x-axis (y = 0): 2x + 0 = 8, so x = 4 Point: (4, 0) Line 1 ∩ x-axis: (5, 0) Line 2 ∩ y-axis: (0, 8) Step 3: Check feasibility and find corner points. Te...

Q3: A factory produces chairs and tables. Each chair takes 2 hours and each table takes 3 hours. The factory has 36 hours available. Profit per chair is Rs 100 and per table is Rs 150. Maximize profit.

LPP formulation and solution: Step 1: Define variables. Let x = number of chairs Let y = number of tables Step 2: Formulate constraints. Time constraint: 2x + 3y ≤ 36 Non-negativity: x ≥ 0, y ≥ 0 Step 3: Formulate objective function. Maximize P = 100x + 150y Step 4: Find corner points of feasible region. Constraint boundary: 2x + 3y = 36 Intercepts: When x = 0: 3y = 36, y = 12, Point: (0, 12) When y = 0: 2x = 36, x = 18, Point: (18, 0) Origin: (0, 0) Corner points: (0, 0), (0, 12), (18, 0)...

Q4: A diet must contain at least 400 units of calcium and 900 units of protein. Food A contains 100 units calcium and 300 units protein per kg, costing Rs 50/kg. Food B contains 200 units calcium and 150 units protein per kg, costing Rs 60/kg. Find minimum cost diet.

LPP formulation: Step 1: Define variables. x = kg of Food A y = kg of Food B Step 2: Formulate constraints. Calcium: 100x + 200y ≥ 400 Protein: 300x + 150y ≥ 900 x ≥ 0, y ≥ 0 Simplify: Constraint 1: x + 2y ≥ 4 Constraint 2: 2x + y ≥ 6 Step 3: Objective function. Minimize Cost C = 50x + 60y Step 4: Find corner points. Line 1: x + 2y = 4 Line 2: 2x + y = 6 Intersection: From Line 1: x = 4 - 2y Substitute in Line 2: 2(4 - 2y) + y = 6 8 - 4y + y = 6 -3y = -2 y = 2/3 x = 4 - 2(2/3) = 4 - 4/3 =...

Q5: A company makes two products A and B. Each unit of A requires 3 hours of skilled labor and 2 units of raw material. Each unit of B requires 4 hours and 3 units. Available: 120 hours labor and 90 units material. Profit per A is Rs 40, per B is Rs 50. Maximize profit.

LPP formulation: Step 1: Define variables. x = units of product A y = units of product B Step 2: Formulate constraints. Labor: 3x + 4y ≤ 120 Raw material: 2x + 3y ≤ 90 x ≥ 0, y ≥ 0 Step 3: Objective function. Maximize P = 40x + 50y Step 4: Find corner points. Line 1: 3x + 4y = 120 Line 2: 2x + 3y = 90 Intercepts of Line 1: x = 0: y = 30, Point: (0, 30) y = 0: x = 40, Point: (40, 0) Intercepts of Line 2: x = 0: y = 30, Point: (0, 30) y = 0: x = 45, Point: (45, 0) Intersection of Lines 1 an...

Q6: State the Fundamental Theorem of Linear Programming.

Fundamental Theorem of Linear Programming: Theorem Statement: If a linear programming problem has an optimal solution, then it occurs at one of the corner (vertex) points of the feasible region. Key Points: 1. Existence of Optimal Solution: If the feasible region is non-empty and bounded, an optimal solution exists. 2. Location of Optimal Solution: The optimal solution lies at a corner point of the feasible region (not in the interior). 3. Multiple Optimal Solutions: If the objective functi...

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