CBSE Class 12 Linear Programming PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1 MCQ/assertion-reason (1 mark), 1 short answer on formulation (2-3 marks), 1 long answer on graphical solution (4-6 marks).
Repeated Topics: Formulation of LPP (objective function + constraints), graphical method (feasible region, corner points), maximization/minimization at vertices, diet problem, manufacturing problem, transportation problem, unbounded/infeasible cases.
Difficulty: Easy to medium; plotting accurate graph & evaluating at corners is key. Formulation questions increased in recent years.
Changes: More real-life word problems (manufacturing, diet, profit maximization) and case-based in 2023–2025.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
A manufacturer produces two types of products A and B requiring 2 machines. Formulate LPP to maximize profit and solve graphically (typical: profit 40x + 60y, constraints on machine hours, raw material).	Long Answer (Formulation + Graphical) (5-6 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Find feasible region → evaluate objective at corners → max/min value.
Solve graphically: Maximize Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0, 2x + y ≤ 6 (or similar 2-variable LPP).	Long Answer (Graphical) (4-5 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Plot constraints → feasible region polygon → test corners (0,0), (0,4), etc. → Z max at corner point.
A diet problem: Person needs at least 200 units vitamin A, 150 units B from two foods X (₹20/kg) and Y (₹30/kg). Formulate & solve to minimize cost.	Long Answer (Formulation + Solution) (5 marks)	2022, 2023, 2024, 2025	Repeated 4x; Minimize cost → constraints vitamin ≥, x,y ≥0 → graphical or corner method.
Assertion: In LPP, optimal value occurs at one of the vertices of feasible region. Reason: By corner point theorem.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Identify feasible/infeasible/unbounded region from given graph or constraints (e.g., no common region or extends to infinity).	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Explain reason: no intersection, or objective can increase infinitely.
Maximize Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x,y ≥0. Find maximum value.	Short Answer (Graphical) (3 marks)	2022, 2023, 2024	Repeated 3x; Corners (0,0), (0,3), (2,0), (1,2) → Z max = 13 at (1,2) or similar.
MCQ: The feasible region for LPP is always: (a) convex (b) concave (c) unbounded (d) empty	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (a) convex (intersection of half-planes).
A company makes two products using two resources. Formulate LPP to maximize profit given constraints on resources.	Formulation (2-3 marks)	2023, 2025	Repeated 2x; Define variables → objective → inequalities.
Case-based: Given word problem (manufacturing/diet), formulate LPP, solve graphically, interpret result.	Case-Based (4-6 marks)	2023, 2025	Repeated 2x; Full process: formulation → graph → optimum.
Find minimum value of Z = 4x + 6y subject to constraints (often unbounded or infeasible variant).	Short Answer (3 marks)	2021 Term 2, 2024	Repeated 2x; Check region → if unbounded in direction of decrease → no minimum.

Best Format for Preparation and Answering

Study Format: Practice graphing inequalities quickly (shade feasible region). Solve 10–12 PYQs daily on graphical method & formulation. Memorize corner point theorem & steps for max/min.
Answer Format: 1. Define variables, 2. Write objective & constraints, 3. Draw graph (label axes, shade region), 4. List corner points coordinates, 5. Evaluate Z at each → state optimum value & point.
Tips: Formulation (word to constraints) & finding optimum at corners are highest marks. Always check if region bounded. Time: 10–12 mins for 5–6 mark questions. Very scoring if graph is neat & accurate.

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