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CBSE Class 12 Mathematics Chapter 12 Linear Programming Worksheet with Answers

Linear Programming is a scoring chapter in CBSE Class 12 Mathematics, consistently appearing as one 5-6 mark question in board exams. This printable worksheet provides focused practice on both mathematical formulation and graphical method as outlined in NCERT. Complete all sections within 90 minutes to simulate exam conditions, then verify your work with the detailed answer key provided at the end.

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Key takeaways

✓Worksheet contains 24+ questions covering mathematical formulation and graphical method of linear programming as per NCERT Class 12 Mathematics syllabus.
✓Includes six multiple-choice questions testing objective and corner-point methods with typical board exam scenarios.
✓Five short-answer questions focus on formulating LPP from word problems and identifying feasible regions worth 2-3 marks each.
✓Three long-answer HOTS questions mirror 5-6 mark board questions requiring complete graphical solutions and optimization.
✓Complete answer key with step-by-step explanations helps students verify their solutions and understand common mistakes.
✓Case-study question reflects the new CBSE 2025 pattern, linking linear programming to real manufacturing or diet optimization scenarios.
✓Suggested time of 90 minutes simulates actual exam conditions for Chapter 12, which typically carries 5-6 marks in CBSE Class 12 board papers.

Quick Chapter Recap: Linear Programming Essentials

Chapter 12 of NCERT Class 12 Mathematics introduces Linear Programming Problems (LPP) where we optimize a linear objective function subject to linear constraints. The chapter focuses on two main techniques: mathematical formulation and the graphical method for solving LPPs in two variables. Mathematical formulation involves identifying decision variables, formulating the objective function (to maximize profit, minimize cost, etc.), and writing constraints as linear inequalities. The graphical method plots these constraints on a coordinate plane to identify the feasible region - the set of all points satisfying every constraint simultaneously. The optimal solution lies at one of the corner points (vertices) of this feasible region, which we test in the objective function. Key terminology includes feasible solution, optimal solution, bounded and unbounded regions, and the corner-point method. Mastering these concepts is crucial because board exam questions typically ask students to formulate an LPP from a word problem and solve it graphically, showing all corner points clearly.

Decision variables: quantities we control, usually denoted x and y in two-variable problems
Objective function: linear expression Z = ax + by to maximize or minimize
Constraints: linear inequalities representing limitations on resources, capacity, or requirements
Feasible region: intersection of all constraint half-planes, containing all valid solutions
Corner-point theorem: optimal value occurs at a vertex of the feasible region
Bounded vs unbounded regions: bounded regions are enclosed polygons; unbounded extend infinitely

Worksheet Difficulty Level and Time Guidelines

This worksheet is designed at Medium to Hard difficulty, mirroring the standard expected in CBSE Class 12 board examinations for Linear Programming. Students who have completed NCERT exercises 12.1 and 12.2 should be able to attempt all sections. The MCQs and fill-in-the-blanks test conceptual clarity and quick problem-solving, while short-answer questions require formulation skills and identification of feasible regions. Long-answer questions demand complete graphical solutions with accurate plotting, shading, and corner-point evaluation, exactly as expected in board exams. Allocate 90 minutes total: 15 minutes for Section A and B combined, 5 minutes for Section C, 25 minutes for Section D, 35 minutes for Section E, and 10 minutes for the case study. Keep graph paper, ruler, and pencil ready. Students scoring above seventy percent on this worksheet demonstrate strong readiness for board exams. Those scoring below sixty percent should revisit NCERT examples and solved problems before attempting the worksheet again.

Difficulty: Medium to Hard, aligned with CBSE board exam standards
Suggested total time: 90 minutes under exam conditions
Prerequisites: completion of NCERT exercises 12.1 and 12.2
Materials needed: graph paper, ruler, pencil, eraser, and calculator
Target score: 70% indicates board-exam readiness; below 60% requires concept revision
Sections A-C (objective): 20 minutes; Section D (short): 25 minutes; Section E (long): 35 minutes; Case study: 10 minutes

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions designed to test your conceptual understanding of linear programming fundamentals, corner-point method, and feasible region properties. Each question mirrors the objective-type questions that occasionally appear in CBSE board papers and are common in school pre-board exams. Read each question carefully and select the single best answer. In board exams, these questions typically assess whether students can identify optimal solutions from given corner points, recognize bounded versus unbounded regions, or understand the conditions under which maximum or minimum values occur. Pay special attention to the wording - 'maximize' versus 'minimize' and whether constraints use ≤ or ≥ inequality signs. These questions also test your ability to eliminate incorrect options quickly, a vital skill when managing time in the actual board examination where every mark counts toward your final percentage.

Q1. The corner points of the feasible region for an LPP are (0,0), (4,0), (3,2), and (0,3). If Z = 5x + 3y, the maximum value of Z occurs at: (a) (0,0) (b) (4,0) (c) (3,2) (d) (0,3)
Q2. The feasible region for a system of linear constraints is shown to be unbounded. The optimal solution: (a) always exists (b) never exists (c) may or may not exist (d) exists only for minimization
Q3. In an LPP, if the objective function is parallel to a constraint line, then: (a) there is no solution (b) there is a unique solution (c) there are infinitely many solutions (d) the solution is always at the origin
Q4. The constraint x ≥ 0, y ≥ 0 represents: (a) first quadrant including axes (b) first quadrant excluding axes (c) entire coordinate plane (d) only positive x-axis
Q5. For the constraints x + y ≤ 5, x ≥ 0, y ≥ 0, the feasible region is: (a) unbounded (b) bounded forming a triangle (c) bounded forming a rectangle (d) empty
Q6. If an LPP has a unique optimal solution, it must occur at: (a) any point in feasible region (b) a corner point of feasible region (c) the centre of feasible region (d) the origin only

Section B: Fill in the Blanks (1 mark each)

Complete each statement by filling in the blank with the most appropriate mathematical term or value from Chapter 12 Linear Programming. This section tests your recall of definitions, theorems, and standard procedures from NCERT Class 12 Mathematics. Pay attention to exact terminology as used in your textbook - for instance, 'feasible region' not just 'solution area', or 'corner point' rather than 'vertex' unless specified. In board exams, fill-in-the-blank questions demand precision; marks are awarded only when the answer matches the expected term exactly. These five questions cover the spectrum from basic definitions to recognition of optimization conditions and properties of constraint inequalities. Before writing your answer, read the complete sentence to ensure grammatical and mathematical coherence. If a numerical value is required, show brief working in the margin even though only the final answer goes in the blank, as this helps you verify correctness during self-checking with the answer key provided later in this worksheet.

Q7. The set of all points that satisfy all constraints of an LPP simultaneously is called the __________ region.
Q8. In the graphical method of solving LPP, the optimal value always occurs at a __________ point of the feasible region.
Q9. If the feasible region of an LPP is __________, the maximum or minimum value of the objective function may or may not exist.
Q10. The linear function to be optimized in an LPP is called the __________ function.
Q11. For the constraint 2x + 3y ≥ 12, the boundary line passes through points (6, 0) and (0, __________).

Section C: True or False (1 mark each)

Determine whether each statement about linear programming is True or False. If a statement is False, you should be able to explain why or provide a counterexample, though for this worksheet you need only write True or False as your answer. This question type appears frequently in school-level assessments and helps reinforce critical concepts from NCERT Class 12 Mathematics Chapter 12. Understanding the truthfulness of these statements indicates deep conceptual clarity rather than rote memorization. For example, knowing whether optimal solutions must always exist for bounded regions, or whether constraints can ever be strict inequalities in the NCERT formulation, reveals whether you have internalized the theory behind the graphical method. When reviewing your answers against the answer key, if you marked any statement incorrectly, revisit the corresponding NCERT section to understand the underlying principle. These four statements touch upon corner-point method, existence of solutions, and properties of constraints that frequently confuse students during board exam preparation.

Q12. Every linear programming problem has a unique optimal solution. (True / False)
Q13. If the feasible region is bounded, both maximum and minimum values of the objective function exist. (True / False)
Q14. The constraints in an LPP can include strict inequalities like x < 5 or y > 3. (True / False)
Q15. The optimal value of the objective function in an LPP can occur at any point inside the feasible region. (True / False)

Section D: Short Answer Questions (2-3 marks each)

Answer each of the following questions in 30 to 50 words, showing essential steps and reasoning. These five questions typically carry 2 to 3 marks each in CBSE board exams and test your ability to formulate problems, identify feasible regions, and apply the corner-point method. Write clear mathematical statements, define your decision variables explicitly, and ensure all constraints are written as linear inequalities. For graphical sketches, indicate key intercepts and label axes even in a rough diagram. Board examiners award partial marks for correct formulation even if the final numerical answer contains a minor calculation error, so show all working. Each question here is modeled on past CBSE board papers and NCERT solved examples, ensuring you practice exactly what will appear in your actual examination. Read the problem statement twice to identify all given information and what is being asked before you begin writing. Remember that clarity and logical presentation earn marks just as much as the correct final answer does.

Q16. Formulate the following as a linear programming problem: A manufacturer produces two products A and B. Each unit of A requires 2 hours on machine M1 and 1 hour on machine M2. Each unit of B requires 1 hour on M1 and 2 hours on M2. Machine M1 is available for 10 hours and M2 for 8 hours. Profit per unit of A is ₹30 and per unit of B is ₹20. Formulate the LPP to maximize profit. (Do not solve.)
Q17. Find the feasible region graphically for the constraints: x + y ≤ 6, x ≥ 0, y ≥ 0. Identify whether it is bounded or unbounded.
Q18. The corner points of a feasible region are (0,4), (2,3), (5,0), and (0,0). Find the maximum and minimum values of Z = 3x + 4y.
Q19. Explain the corner-point method for solving a linear programming problem graphically.
Q20. A dietary problem requires at least 8 units of vitamin A and 10 units of vitamin B daily. One unit of food F1 contains 2 units of A and 1 unit of B; one unit of F2 contains 1 unit of A and 2 units of B. If F1 costs ₹5 per unit and F2 costs ₹4 per unit, formulate the LPP to minimize cost. (Do not solve.)

Section E: Long Answer and HOTS Questions (5-6 marks each)

Solve the following three questions completely, showing all steps including graphical representation, identification of corner points, and evaluation of the objective function at each corner point. Each question is worth 5 to 6 marks and reflects the most common long-answer format in CBSE Class 12 board examinations for Linear Programming. Use graph paper for accurate plotting; mark intercepts clearly and shade the feasible region distinctly. Label each corner point with its coordinates and show the calculation of Z at each vertex in a tabular format for clarity. These questions test Higher Order Thinking Skills by requiring you to integrate mathematical formulation, graphical solution, and logical reasoning to identify optimal solutions. In board exams, examiners give step-wise marking, so even if your final answer is incorrect due to a plotting error, you can still earn marks for correct formulation, proper graph, and systematic corner-point evaluation. Allocate approximately 10 to 12 minutes per question. These problems are slightly harder than standard NCERT exercises to prepare you thoroughly for board exam challenges.

Q21. (6 marks) Maximize Z = 3x + 5y subject to constraints: x + 2y ≤ 10, 3x + y ≤ 12, x ≥ 0, y ≥ 0. Solve graphically and find the maximum value of Z.
Q22. (6 marks) A furniture dealer deals in tables and chairs. He has ₹15,000 to invest and a storage space of 60 sq. meters. A table costs ₹750 and occupies 10 sq. meters, while a chair costs ₹250 and occupies 2 sq. meters. He can sell a table at a profit of ₹100 and a chair at a profit of ₹50. Formulate and solve the LPP graphically to maximize profit.
Q23. (5 marks) Minimize Z = 5x + 10y subject to constraints: x + 2y ≥ 10, 3x + y ≥ 12, x ≥ 0, y ≥ 0. Solve graphically and determine whether the minimum value exists. If yes, find it.

Section F: Case-Study Based Question (4 marks)

Read the following case study carefully and answer the sub-questions that follow. Case-study questions were introduced in the CBSE board exam pattern to test application of mathematical concepts in real-world scenarios. This question carries 4 marks and typically includes two sub-parts of 1 mark each and one sub-part of 2 marks. Such questions have appeared consistently in CBSE Class 12 Mathematics board papers since 2021 and test your ability to extract mathematical information from descriptive text, formulate constraints, and solve optimization problems. Read the case study twice, underline key numerical data, and identify the decision variables before attempting the sub-questions. Even if you find the context unfamiliar, the underlying mathematics remains the same linear programming you have practiced. Show all steps for the 2-mark sub-question to earn full credit. This case study focuses on a manufacturing scenario, one of the most common real-life applications of linear programming in NCERT Class 12 Mathematics syllabus.

Complete Answer Key with Explanations

Use this answer key to verify your responses after completing the entire worksheet. Each answer includes a brief explanation to help you understand the reasoning or method. If your answer differs, review the corresponding NCERT section or solved example before moving forward. For numerical answers, minor calculation variations are acceptable if your method is correct, but conceptual errors indicate gaps that need targeted revision. Students preparing for CBSE board exams should aim for at least eighty-five percent accuracy on objective sections (A, B, C) and seventy percent on subjective sections (D, E, F). If you score lower, identify patterns in your mistakes - are they conceptual misunderstandings, calculation errors, or graphing inaccuracies? - and address them specifically. Self-assessment using this answer key is more valuable than passive reading of solutions; try every question sincerely before checking answers. Remember that CBSE awards partial marks for method and steps, so comparing your working with these solutions helps you learn where marks might be lost even when your final answer is incorrect.

A1. (b) (4,0) — Evaluate Z at each corner: (0,0)→0, (4,0)→20, (3,2)→21, (0,3)→9. Maximum is 21 at (3,2). Wait, recalculating: at (4,0) Z=20, at (3,2) Z=15+6=21. Actually maximum 21 at (3,2). Correction: answer is (c) (3,2).
A2. (c) may or may not exist — For unbounded regions, maximum may not exist if objective function can increase infinitely; minimum can exist.
A3. (c) there are infinitely many solutions — All points on the overlapping line segment are optimal solutions.
A4. (a) first quadrant including axes — These constraints define all points with non-negative coordinates including the axes themselves.
A5. (b) bounded forming a triangle — Corner points are (0,0), (5,0), (0,5), forming a triangular bounded region.
A6. (b) a corner point of feasible region — Corner-point theorem states optimal solution occurs at a vertex of the feasible region.
A7. feasible — Standard NCERT terminology for the solution set satisfying all constraints.
A8. corner — The corner-point method evaluates Z at all vertices to find the optimum.
A9. unbounded — Unbounded feasible regions may have no maximum (or minimum for some objectives).
A10. objective — The objective function is the expression we seek to maximize or minimize.
A11. 4 — Substitute x=0 in 2x+3y=12 to get 3y=12, so y=4. Point is (0,4).
A12. False — Multiple optimal solutions can exist if the objective function is parallel to a constraint; some LPPs have no solution.
A13. True — Bounded feasible regions guarantee both maximum and minimum exist by the extreme point theorem.
A14. False — NCERT formulation uses non-strict inequalities (≤, ≥) to ensure boundary points are included in the feasible region.
A15. False — Optimal value occurs at a corner point of the feasible region, not in the interior (unless all points are optimal).

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Frequently asked questions

How many marks does Linear Programming carry in CBSE Class 12 board exams?+

Linear Programming typically carries 5 to 6 marks in the CBSE Class 12 Mathematics board exam, appearing as one long-answer question requiring graphical solution and corner-point evaluation. Occasionally, 1-mark MCQ or case-study sub-parts also test LPP concepts.

Is the graphical method the only technique in NCERT Class 12 Chapter 12?+

Yes, NCERT Class 12 Mathematics Chapter 12 focuses exclusively on the graphical method for solving linear programming problems in two variables. Advanced methods like Simplex are beyond the CBSE syllabus and are introduced only in higher courses.

What is the most common mistake students make in LPP board questions?+

The most frequent error is incorrect shading of the feasible region, especially when constraints use ≥ inequalities. Students often shade toward the origin by default instead of testing a point. Another common mistake is forgetting to check all corner points systematically before concluding the optimal solution.

Can an LPP have multiple optimal solutions?+

Yes, if the objective function is parallel to one of the boundary lines of the feasible region, then every point on that boundary segment is an optimal solution, resulting in infinitely many optimal solutions all yielding the same maximum or minimum value of Z.

How do I know if a feasible region is bounded or unbounded?+

A feasible region is bounded if it forms a closed polygon that can be enclosed within a circle of finite radius. It is unbounded if it extends infinitely in one or more directions. Graphically, if you can draw a large enough circle containing the entire feasible region, it is bounded; otherwise unbounded.

Do I need to memorize the corner points or can I derive them in the exam?+

You must derive corner points during the exam by solving pairs of boundary line equations simultaneously. Memorization is neither required nor useful, as each problem has unique constraints. Practice solving systems of two linear equations quickly and accurately to save time in the board exam.

What if my graph is slightly inaccurate - will I lose all marks?+

CBSE awards step-wise marks, so even with a minor graphing error, you earn marks for correct formulation, labeling axes, identifying the correct boundary lines, and systematic corner-point evaluation. However, a significantly wrong graph that leads to incorrect corner points will cost you marks in the final answer.

How should I present the solution to an LPP in the board exam?+

Start with clear formulation: define variables, write the objective function, list all constraints. Draw a neat graph on the provided graph paper, label intercepts and corner points. Create a small table evaluating Z at each corner point. Conclude by stating the optimal solution clearly with the maximum or minimum value of Z.

Are case-study questions on Linear Programming difficult?+

Case-study questions in CBSE board exams present real-world scenarios but the underlying mathematics remains standard LPP formulation and solution. Read the case carefully, extract numerical data, identify decision variables, and proceed as you would with any NCERT exercise. The 4 marks are distributed across easier sub-parts, making them scoring if approached systematically.

How much time should I spend on the LPP question in the board exam?+

Allocate 10 to 12 minutes for a 5-6 mark LPP question. Spend 2 minutes on formulation, 4-5 minutes on accurate graphing and identifying corner points, 2-3 minutes on evaluating Z and writing the conclusion, and keep 1-2 minutes as buffer for rechecking calculations and labeling.

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