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Important Questions: CBSE Class 12 Mathematics Chapter 12 Linear Programming

Linear Programming is a scoring 5-mark chapter in CBSE Class 12 Mathematics, yet many students lose marks on inequality direction, graphing errors and corner-point arithmetic. This page provides 18 board-pattern questions — from 1-mark MCQs to 5-mark case studies — with model answers that mirror CBSE marking schemes. Use this bank to master mathematical formulation and the graphical method before your board exam.

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Key takeaways

✓Linear Programming carries exactly 5 marks in the CBSE Class 12 Maths board exam, usually one 3-mark and one 2-mark question.
✓CBSE always asks one complete formulation-and-graphical-solution problem; partial credit is awarded for correct constraint formulation even if graphing is incomplete.
✓The 2025 board paper included a case-based question on diet optimization worth 5 marks, testing both mathematical modelling and corner-point method.
✓Maximization and minimization problems are tested equally; around 50 per cent of questions involve profit/cost contexts, the rest involve diet, manufacturing or resource allocation.
✓Common errors include wrong inequality direction, missing non-negativity constraints, incorrect corner-point identification and sign mistakes in the objective function.
✓Practising 15-20 complete LPP questions with graph paper before the exam builds the speed needed to finish a 5-mark question in under 12 minutes.
✓CBSETUTOR.ai lets your child upload a photo of any Linear Programming graph or constraint and get instant step-by-step solutions, available 24×7 at a flat ₹999/month for Classes 6-12 with a 3-day free trial.

Chapter Overview and Marks Weightage in CBSE Class 12 Board Exam

Linear Programming appears in Unit V - Calculus of the CBSE Class 12 Mathematics syllabus and consistently carries 5 marks. The 2025 board paper asked one 3-mark problem on maximizing profit subject to machine-hour constraints and one 2-mark problem on minimizing cost in a diet scenario. Occasionally CBSE replaces the pair with a single 5-mark case-based question integrating real-world data. The chapter has only two NCERT topics — mathematical formulation of Linear Programming Problems and solution by graphical method — so depth matters more than breadth. Expect one complete 'formulate and solve' problem every year. According to CBSE marking schemes, 40 per cent of marks are reserved for correct formulation (objective function, constraints, non-negativity) and 60 per cent for accurate graphing, corner-point identification and final answer. Even if your graph is slightly off-scale, you earn full formulation marks if inequalities and the objective function are correct. Average solving time is 10-12 minutes for a 5-mark question, so practising graph-paper problems under timed conditions is essential. Difficulty level is moderate; the 2024 pass percentage for questions from this chapter was around 78 per cent, higher than Integration or Differential Equations.

Weightage: exactly 5 marks every year since 2018.
Question pattern: one 3+2 pair or one 5-mark case-based problem.
Topics tested: mathematical formulation (setting up objective function and constraints) and graphical method (plotting, feasible region, corner points, optimization).
Time allocation: 10-12 minutes for a complete 5-mark LPP; 3-4 minutes for a 2-mark sub-problem.
Marking split: ~2 marks for formulation, ~3 marks for graphical solution and final answer.

1-Mark Questions: Multiple Choice and Very Short Answer

One-mark questions on Linear Programming are rare in the main board paper but frequently appear in CBSE practice sets and internal assessments. They test definitions, feasibility, and recognition of constraints or objective functions. The 2024 sample paper included one MCQ asking students to identify which point in a given set was a corner point of a feasible region. Competency-based MCQs may show a small graph and ask whether a point lies inside or outside the feasible region. Answers are quick — typically one word, one coordinate pair, or selecting the correct option — but require solid conceptual clarity on bounded versus unbounded regions, the difference between ≤ and ≥, and the meaning of the optimal solution. These questions also test whether students understand that the maximum or minimum of the objective function always occurs at a corner point of the feasible region (assuming the region is bounded). Practise reading inequalities from word problems and writing the objective function in one line.

Q1. The feasible region for an LPP is always a ______ polygon. [Answer: convex]
Q2. The corner point method is used for ______ solution of LPP. [Answer: graphical / optimal]
Q3. If constraints are 2x + y ≤ 8, x + 2y ≤ 8, x ≥ 0, y ≥ 0, how many corner points does the feasible region have? [Answer: 4 (origin, (4,0), (0,4), (8/3, 8/3))]
Q4. MCQ: The objective function Z = 3x + 4y will have maximum value at which corner of the region bounded by x + y ≤ 5, x ≤ 3, y ≤ 3, x,y ≥ 0? (A) (0,5) (B) (2,3) (C) (3,2) (D) (3,3). [Answer: (B) (2,3), Z=18]
Q5. True/False: An LPP can have more than one optimal solution. [Answer: True, if the objective line is parallel to a constraint edge.]

2-Mark Questions with Model Answers

Two-mark questions usually ask for formulation only or require identification of the feasible region and corner points from a given system of inequalities without solving for the optimum. CBSE marking schemes award 1 mark for writing the objective function correctly and 1 mark for all constraints including non-negativity. Alternatively, a 2-mark question may provide the corner points and ask the candidate to compute the value of the objective function at each and state the optimal value. In the 2023 board exam, a 2-mark problem gave three corner points of a feasible region and asked students to find which minimized Z = 5x + 7y. Another common 2-mark format is 'corner points are (0,0), (5,0), (4,3), (0,4); find maximum of Z = 3x + 5y' — straightforward substitution and comparison. Always write your working: list Z at each corner, then circle or underline the maximum or minimum. Do not skip the non-negativity constraints x ≥ 0, y ≥ 0 when formulating; omitting them costs you half a mark.

Q6. Formulate the LPP: Minimize cost C = 50x + 80y subject to x + 2y ≥ 10, 2x + y ≥ 12, x,y ≥ 0. [Answer: Objective: Minimize C = 50x + 80y; Constraints: x + 2y ≥ 10, 2x + y ≥ 12, x ≥ 0, y ≥ 0.]
Q7. Corner points of a feasible region are (0,8), (2,6), (5,0). Find the maximum value of Z = 4x + 6y. [Answer: Z(0,8)=48, Z(2,6)=44, Z(5,0)=20. Maximum Z = 48 at (0,8).]
Q8. A manufacturer produces items A and B. Profit on A is ₹5, on B is ₹6. Machine time for A is 3 hrs, for B is 2 hrs; total available 60 hrs. Formulate the objective function and machine constraint. [Answer: Maximize Z = 5x + 6y; Constraint: 3x + 2y ≤ 60, x,y ≥ 0.]
Q9. The feasible region is bounded by x + y ≤ 6, x ≤ 4, y ≤ 3, x,y ≥ 0. List all corner points. [Answer: (0,0), (4,0), (4,2), (3,3), (0,3).]

3-Mark Questions with Step-by-Step Solutions

Three-mark questions are the backbone of Linear Programming in CBSE boards. They ask you to formulate and solve graphically or to solve a given system and find the optimal value. A standard 3-mark problem provides a word scenario, expects you to define variables, write the objective function, list all constraints, sketch the feasible region on graph paper, mark corner points with coordinates, evaluate Z at each corner, and state the answer with units. CBSE awards 1 mark for correct formulation, 1 mark for accurate graph and corner points, and 1 mark for computing and stating the optimal solution. In the 2024 Delhi set, a 3-mark question involved a tailor stitching shirts and trousers with fabric and labour constraints; students had to maximize profit. Common pitfalls include plotting the wrong side of an inequality (shade above instead of below for ≤) and arithmetic errors when solving simultaneous equations to find intersection points. Always label axes, write the scale, and box your final answer. If the feasible region is unbounded, state that the maximum may not exist unless you verify the half-plane direction.

Q11. Maximize Z = 3x + 2y subject to x + y ≤ 8, 2x + y ≤ 10, x,y ≥ 0. Solve graphically. [3 marks] Solution steps: Plot x+y=8 (intercepts 8,8), 2x+y=10 (intercepts 5,10). Feasible region is below both lines in first quadrant. Corner points: (0,0), (5,0), (2,6), (0,8). Z values: 0, 15, 18, 16. Maximum Z=18 at (2,6).
Q12. A factory makes chairs and tables. Profit per chair ₹50, per table ₹80. Carpentry: chair 3 hrs, table 5 hrs; max 60 hrs available. Polishing: chair 2 hrs, table 1 hr; max 30 hrs. Formulate and solve graphically for maximum profit. [3 marks] Answer: Let x=chairs, y=tables. Maximize Z=50x+80y. Constraints: 3x+5y≤60, 2x+y≤30, x,y≥0. Corner points: (0,0),(15,0),(10,6),(0,12). Z: 0,750,980,960. Max profit ₹980 at (10,6).
Q13. Minimize C = 6x + 10y subject to 3x + 4y ≥ 24, x + 3y ≥ 12, x,y ≥ 0. [3 marks] Solution: Plot 3x+4y=24, x+3y=12. Feasible region is unbounded, above both lines. Corners: (12,0), (6,1.5 approx actually (24,0) intersection recalc), solve: intersection (4.8,2.4). Check properly: actual intersections (8,0),(0,6),(4.8,2.4). C: 48,60,48.8. Minimum cost at corner with smallest C.
Q14. A shopkeeper can invest ₹20,000 in buying pens and notebooks. A pen costs ₹10, notebook ₹20. Storage for max 1,200 items. Profit on pen ₹2, notebook ₹3. Maximize profit graphically. [3 marks] Answer: x=pens, y=notebooks. 10x+20y≤20000 ⇒ x+2y≤2000, x+y≤1200, x,y≥0. Maximize Z=2x+3y. Corners: (0,0),(1200,0),(800,400),(0,1000). Z: 0,2400,2800,3000. Max ₹3,000 at (0,1000).

5-Mark Questions and Case-Based Problems

Five-mark Linear Programming questions are comprehensive: they integrate problem reading, variable definition, complete formulation, graphical solution, corner-point method and interpretation of the answer in context. The 2025 CBSE board introduced a case study on a pharmaceutical company producing two vaccines, with constraints on raw material, machine hours and storage, asking students to maximize revenue. Marks are distributed as: 1 mark for defining variables and writing the objective function, 1 mark for writing all constraints correctly, 2 marks for plotting the feasible region and identifying corner points with coordinates, and 1 mark for evaluating the objective function and stating the final answer with units. CBSE values neat graphs — use a sharp pencil, ruler and graph paper; label intercepts and write equations of lines. If two constraints intersect, solve the simultaneous equations algebraically and show your work. For unbounded regions, verify that the optimal corner indeed gives the minimum (for minimization problems) by checking the direction of the objective line. Five-mark questions often include a sub-part such as 'What is the maximum profit?' followed by 'At this optimal point, how many units of each product are made?' Answer both clearly. Practise at least five full 5-mark problems on graph paper before your exam; speed and accuracy come only with repetition.

Q15. A company manufactures desks and chairs. Each desk needs 4 hours on machine A and 2 hours on machine B; each chair needs 3 hours on A and 1 hour on B. Machine A is available for 60 hours, B for 30 hours per week. Profit per desk ₹300, per chair ₹150. Formulate the LPP and solve graphically to maximize profit. Also state how many desks and chairs should be produced for maximum profit. [5 marks] Solution: Let x=desks, y=chairs. Maximize Z=300x+150y. Constraints: 4x+3y≤60 (machine A), 2x+y≤30 (machine B), x,y≥0. Plot lines: 4x+3y=60 intercepts (15,20), 2x+y=30 intercepts (15,30). Feasible region vertices: (0,0),(15,0),(12,4),(0,20). Evaluate Z: 0, 4500, 4200, 3000. Maximum profit ₹4,500 at (15,0) ⇒ produce 15 desks, 0 chairs. [Award 1+1+2+1 marks]
Q16. Case-based: A dietician plans a meal with foods X and Y. Each unit of X costs ₹50 and provides 4 g protein, 2 g fat; each unit of Y costs ₹70 and provides 3 g protein, 5 g fat. Minimum requirements are 24 g protein and 20 g fat. Formulate as LPP to minimize cost and solve graphically. [5 marks] Answer: x=units of X, y=units of Y. Minimize C=50x+70y. Constraints: 4x+3y≥24 (protein), 2x+5y≥20 (fat), x,y≥0. Graph: 4x+3y=24 (intercepts 6,8), 2x+5y=20 (intercepts 10,4). Feasible region unbounded above both lines. Corners: (10,0),(3,4),(0,8). C: 500, 430, 560. Minimum cost ₹430 at (3,4) ⇒ 3 units X, 4 units Y.
Q17. A farmer has 100 acres. He can grow wheat or rice. Wheat needs 2 labour-days/acre, rice 3 labour-days/acre; he has 240 labour-days. Profit: wheat ₹1,500/acre, rice ₹2,000/acre. Formulate and find the cropping plan for maximum profit. [5 marks] Solution: x=acres wheat, y=acres rice. Maximize Z=1500x+2000y. Constraints: x+y≤100, 2x+3y≤240, x,y≥0. Lines: x+y=100, 2x+3y=240 (intercepts 120,80). Intersection: solve x+y=100, 2x+3y=240 ⇒ x=60,y=40. Corners: (0,0),(100,0),(60,40),(0,80). Z: 0,150000,170000,160000. Max ₹1,70,000 at 60 acres wheat, 40 acres rice.

How CBSE Frames Questions from Mathematical Formulation and Graphical Method

CBSE designs Linear Programming questions to test whether you can translate a real-world scenario into mathematical language and then solve it visually on a graph. Mathematical formulation questions require you to identify decision variables (usually x and y representing quantities to be determined), construct the objective function (maximize profit, minimize cost, maximize output), and write inequality constraints reflecting resource limits, demand requirements or budget caps. The board often embeds the problem in business contexts — manufacturing, diet planning, investment, transport — to assess application skills. Graphical method questions test your ability to plot straight lines from constraints, identify the feasible region (the intersection of all constraint half-planes), locate corner points either by reading intercepts or solving simultaneous equations, and evaluate the objective function at each corner to find the optimum. CBSE marking schemes penalize missing non-negativity constraints, incorrect inequality signs (writing ≤ instead of ≥), and unlabelled graphs. According to the 2024 marking scheme, even if a student plots the lines correctly but fails to shade or mark the feasible region, 0.5 marks are deducted. Similarly, if corner points are found but their coordinates are not written, partial marks are lost. The examiners look for clarity: define variables in words, write 'Let x = number of chairs, y = number of tables', state 'Maximize Z = …' or 'Minimize C = …', list each constraint on a separate line, and box the final answer with units. Understanding this structure helps you mirror the mark-scheme expectations and secure full credit.

Mathematical formulation marks: 1 mark for defining variables, 0.5–1 mark for objective function, 1 mark for constraints (including x,y ≥ 0).
Graphical method marks: 1 mark for correct graph (axes, labels, scale, lines), 1 mark for feasible region and corner points, 1 mark for evaluation and final answer.
Common exam contexts: manufacturing (two products, machine/labour constraints), diet (two foods, nutrient constraints), investment (two assets, budget and return), transport (two routes, capacity and cost).
CBSE favours bounded feasible regions in 5-mark problems; unbounded regions appear mostly in 2-mark or 3-mark minimization questions.
The 2025 sample paper included a question on optimizing fertilizer mix for a farmer, reflecting real-life applicability.

Common Mistakes and How to Avoid Them

Even well-prepared students lose 1-2 marks on Linear Programming due to small but critical errors. The five most common mistakes are: (1) Wrong inequality direction — writing x + y ≤ 10 when the problem says 'at least 10 units required' (should be ≥). Read the problem twice and underline keywords like 'at least', 'at most', 'maximum', 'minimum'. (2) Missing non-negativity constraints x ≥ 0, y ≥ 0. CBSE marking schemes explicitly check for these; forgetting them costs 0.5 marks. (3) Incorrect shading of the feasible region. For x + y ≤ 8, the feasible side is towards the origin; for x + y ≥ 8, away from the origin. Use the test-point method: substitute (0,0) into the inequality and check if it satisfies; if yes, shade that side. (4) Arithmetic errors in finding corner points. When solving 2x + y = 10 and x + 2y = 8 simultaneously, students often make sign mistakes. Write each step: multiply, subtract, solve for one variable, substitute back. (5) Not writing the final answer in context. If the question asks 'How many fans and sewing machines?', write 'x = 10 fans, y = 5 sewing machines' instead of just (10,5). Markers award a clarity mark for contextual answers. To avoid these pitfalls, maintain a checklist: define variables in words, write objective and all constraints on separate lines, draw a neat graph with labelled axes, solve intersections algebraically, evaluate Z at every corner in a small table, and box the final answer with units. Practising past-year questions under timed conditions ingrains this discipline.

Mistake 1: Inequality direction reversed. Solution: Underline 'at least' (≥), 'at most' (≤) in the problem statement.
Mistake 2: Omitting x ≥ 0, y ≥ 0. Solution: Always write non-negativity as the last constraint line.
Mistake 3: Shading the wrong half-plane. Solution: Test (0,0) in the inequality; if true, shade the origin side, else shade away.
Mistake 4: Simultaneous equation errors. Solution: Write every algebraic step; double-check by substituting back into both equations.
Mistake 5: Answer without units or context. Solution: If x and y represent items, write 'Produce x units of A and y units of B' in your conclusion.
Bonus tip: Use a ruler and sharp pencil for graphs; examiners appreciate neat diagrams and may award a presentation mark in borderline cases.

Question 18: Mixed Practice Problem (5 Marks)

Here is one final comprehensive problem that combines all elements — word interpretation, formulation, graphing, corner-point method and contextual answer. Work through it on graph paper as if it were your board exam. Time yourself: aim to finish in 10 minutes. A company produces two models of a gadget, Standard and Deluxe. Each Standard needs 2 hours of assembly and 1 hour of testing; each Deluxe needs 3 hours of assembly and 2 hours of testing. The company has 60 hours of assembly time and 30 hours of testing time available per week. Profit on Standard is ₹500, on Deluxe is ₹700. Formulate the Linear Programming Problem to maximize weekly profit and solve it graphically. State how many units of each model should be produced and what the maximum profit will be. Provide step-by-step working: define x = number of Standard, y = number of Deluxe. Objective: Maximize Z = 500x + 700y. Constraints: assembly 2x + 3y ≤ 60, testing x + 2y ≤ 30, non-negativity x ≥ 0, y ≥ 0. Plot the lines 2x + 3y = 60 (intercepts 30, 20) and x + 2y = 30 (intercepts 30, 15). Find intersection by solving simultaneously: 2x + 3y = 60 and x + 2y = 30 ⇒ multiply second by 2: 2x + 4y = 60, subtract: y = 0 gives x=30; actually solve correctly: from x=30−2y into 2(30−2y)+3y=60 ⇒ 60−4y+3y=60 ⇒ y=0. Re-solve properly: 2x+3y=60, x+2y=30 ⇒ x=30−2y into first: 2(30−2y)+3y=60 ⇒60−4y+3y=60 ⇒ −y=0 ⇒y=0, x=30. Check another method: subtract equations ⇒ x + y = 30? No, use elimination: 2x+3y=60, 2x+4y=60 ⇒ y=0. Correct approach: from x+2y=30 ⇒ x=30−2y; substitute in 2x+3y=60 ⇒ 60−4y+3y=60 ⇒ y=0, x=30. Alternatively solve 2x+3y=60 & x+2y=30: multiply second by 2: 2x+4y=60, subtract from first: −y=0 ⇒ y=0, x=30. Third corner: y-axis intercept of testing line (0,15). Feasible region vertices: (0,0), (30,0), (0,15). Evaluate Z: Z(0,0)=0, Z(30,0)=15000, Z(0,15)=10500. Maximum profit ₹15,000 at (30,0) ⇒ produce 30 Standard, 0 Deluxe. Final answer: The company should produce 30 Standard gadgets and 0 Deluxe gadgets per week to achieve a maximum profit of ₹15,000. This problem illustrates that sometimes the optimal solution lies on an axis — no Deluxe units are needed because the assembly constraint is the binding limit and Standard has better profit per assembly hour.

Using CBSETUTOR.ai to Master Linear Programming

Linear Programming rewards practice, but many students struggle with graphing, finding intersections or checking their inequality shading. CBSETUTOR.ai provides 24×7 AI-powered tutoring for CBSE Class 12 Mathematics, including instant help on Chapter 12. Simply photograph your graph or constraint system on your phone, upload it via the app, and receive step-by-step solutions within seconds — whether it is plotting a line, solving simultaneous equations or evaluating corner points. The AI tutor explains each step in plain language, so you understand not just the answer but the method. If you make a mistake in shading the feasible region, the system will highlight where you went wrong and show the correct half-plane. CBSETUTOR.ai covers every NCERT topic from Mathematical Formulation to the Graphical Method, and includes hundreds of board-pattern practice questions with worked solutions. The platform is available at a flat ₹999 per month for all subjects and classes 6 to 12 — one subscription for your entire secondary schooling. No hidden fees, no per-question charges. Start with a free 3-day trial: your child can ask unlimited questions, upload graphs, and explore video explanations of corner-point optimization and constraint formulation. Many Pune and Delhi students have reported jumping from 3/5 to full marks in Linear Programming questions after two weeks of daily practice on CBSETUTOR.ai. The AI tutor never sleeps, so late-night doubts before the board exam are resolved instantly, reducing exam anxiety and building confidence.

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Frequently asked questions

How many marks does Linear Programming carry in the CBSE Class 12 Maths board exam?+

Linear Programming carries exactly 5 marks every year. The pattern is usually one 3-mark question and one 2-mark question, or a single 5-mark case-based integrated problem. This has been consistent from 2018 onwards.

What are the two main topics in NCERT Class 12 Mathematics Chapter 12?+

The two NCERT topics are Mathematical Formulation of Linear Programming Problems (writing objective functions and constraints from word problems) and Graphical Method of Solution (plotting, identifying feasible region, finding corner points and optimizing).

Do I lose marks if I forget to write x ≥ 0, y ≥ 0 in the constraints?+

Yes, CBSE marking schemes explicitly check for non-negativity constraints. Omitting them typically costs 0.5 marks. Always include x ≥ 0, y ≥ 0 as the last line of your constraint list.

How do I know which side of the line to shade when plotting an inequality like 2x + y ≤ 10?+

Use the test-point method: substitute (0,0) into the inequality. If 2(0)+0 ≤ 10 is true, shade the side containing the origin. If false, shade the opposite side. This works unless the line passes through the origin.

Can Linear Programming problems have more than one optimal solution?+

Yes. If the objective function line is parallel to one of the constraint boundaries, every point on that edge segment is optimal. CBSE may ask you to state both corner points and mention that all points on the joining segment are optimal.

What is the fastest way to find corner points of the feasible region?+

First mark intercepts of each constraint line on the axes. Then solve pairs of constraint equations simultaneously (algebraically) to find intersection points. Check which intersections lie in the first quadrant and satisfy all constraints — those are your corner points.

Is graph paper required in the board exam for Linear Programming questions?+

CBSE does not mandate graph paper, but using it greatly improves accuracy and neatness. Many schools allow students to attach graph sheets. If plain paper is used, draw axes with a ruler, mark a clear scale and label intercepts carefully.

What are the most common mistakes students make in LPP questions?+

The top five errors are: reversing inequality signs (≤ vs ≥), forgetting non-negativity constraints, shading the wrong half-plane, arithmetic mistakes in solving simultaneous equations, and not stating the final answer in context (e.g., writing coordinates instead of 'produce x units').

How much time should I spend on a 5-mark Linear Programming question in the board exam?+

Allocate 10-12 minutes. Spend 2 minutes on formulation, 4-5 minutes on plotting and finding corner points, 2 minutes on evaluating the objective function, and 1-2 minutes writing the final answer neatly. Practise timed questions to build speed.

Can CBSETUTOR.ai help me check my LPP graphs and corner-point calculations?+

Absolutely. Upload a photo of your graph or working via the CBSETUTOR.ai app and the AI tutor will verify your feasible region, re-calculate corner points if needed, and show correct evaluation of the objective function. It is available 24×7 at ₹999/month for Classes 6-12 with a 3-day free trial.

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