Linear Programming – Chapter 12

📘 Linear Programming Class 12 – Complete 2000+ Word Worksheet Practice Guide

Linear Programming, often abbreviated as LPP, is a practical branch of mathematics in which a linear objective is optimized subject to a collection of linear restrictions. In Class 12, the graphical approach is especially important because students learn how to translate a word problem into variables, an objective function, constraints, non-negativity conditions and a feasible region. This DigiSadhan worksheet is designed as a repeated practice resource for these skills. It uses fresh parameter combinations, selectable topics and difficulty levels so learners can practise recognition as well as calculation.

The supplied NCERT URL opens the official NCERT textbook portal. The page identifies itself as “Textbooks PDF (I-XII)”; it is a textbook access page rather than a chapter-text extraction page. citeturn0view0 The practice material below therefore follows the standard Class 12 Linear Programming terminology and method areas rather than claiming to reproduce unseen text from that page.

1. Meaning of Linear Programming

A linear programming problem involves optimizing a linear objective function subject to linear constraints. The word “linear” means that variables occur only to the first power and are not multiplied by one another. The “programming” part refers to planning or deciding how limited resources should be used to obtain the best possible result. The goal can be maximization, such as maximizing profit or production, or minimization, such as minimizing cost.

Here x and y represent decision variables, while a and b are fixed coefficients. A complete LPP also contains constraints and non-negativity conditions.

2. Decision Variables

The first step in a word problem is to decide what the unknown quantities represent. For example, if a factory makes two products, x may represent the number of units of product A and y the number of units of product B. Clearly defined variables make the later formulation much easier. State units when appropriate and remember that school-level graphical LPP generally works with two decision variables.

3. Objective Function

The objective function describes what has to be maximized or minimized. If each unit of product A contributes a profit of p and each unit of product B contributes q, then the profit function can be written as Z = px + qy. If the question asks for minimum cost, the same idea is used with the word “minimize”.

Read the question carefully to determine whether the objective is profit, revenue, production, cost, time or another quantity. A common error is to build correct constraints but optimize the wrong expression.

4. Constraints

Constraints represent limitations such as raw material, machine hours, labour, storage or minimum requirements. A phrase such as “at most” usually leads to a less-than-or-equal-to inequality, while “at least” usually leads to a greater-than-or-equal-to inequality. Equality can arise when a resource is required to be used exactly or a quantity is fixed.

Translate every resource statement separately. Check coefficients against the units in the original problem before moving to the graph.

5. Non-negativity Restrictions

In ordinary production and allocation problems, negative quantities are not meaningful. Therefore x ≥ 0 and y ≥ 0 are commonly included. These conditions restrict the feasible region to the first quadrant. Do not forget them when drawing or interpreting the feasible region.

6. Feasible Region

The feasible region is the set of points satisfying all constraints and non-negativity conditions simultaneously. To construct it graphically, first draw the boundary line of each inequality. Decide which side of the boundary satisfies the inequality, then take the common region. The final common region is the feasible region.

For a bounded feasible region, the boundary usually forms a polygon. Its vertices, also called corner points, are particularly important because they are candidates for the optimum of a linear objective function.

7. Graphing a Linear Constraint

To graph an equation such as ax + by = c, two convenient points can be found using intercepts. Put x = 0 to obtain the y-intercept and y = 0 to obtain the x-intercept, provided those values are defined. Plot the points and draw the boundary line. For inequalities, test a point such as the origin when it is not on the boundary to determine the required side.

8. Corner Point Method

Once the feasible region has been obtained, identify all its corner points. Substitute each corner point into the objective function. Compare the resulting objective values. For a maximization problem, the largest value is selected; for a minimization problem, the smallest value is selected.

This method is central to the graphical solution of a two-variable LPP. It is important not to evaluate only the visually obvious corners. Every vertex of the feasible polygon should be checked.

9. Optimal Value

If the objective function has different values at the feasible corner points, the greatest or least value gives the optimum depending on the problem. The corresponding coordinates give the optimal values of the decision variables. State both the variables and the optimized objective whenever the question asks for a complete solution.

10. Multiple Optimal Solutions

Sometimes the objective function has the same optimum value at more than one corner. In graphical problems, this can occur when an objective-function line is parallel to a boundary edge of the feasible region and that edge gives the optimum. In such a situation, every point on the relevant segment can provide the same optimal objective value. Students should distinguish this from a case where only one corner gives the optimum.

11. Unbounded and Infeasible Cases

A feasible region may be unbounded, meaning it extends indefinitely in at least one direction. An LPP can still have a finite optimum depending on the objective direction, but an optimization problem may also fail to have a finite maximum or minimum. An infeasible problem has no common point satisfying all constraints. In school exercises, diagrams and inequality checks help identify these cases.

12. Standard Question Types in This Worksheet

This DigiSadhan practice engine includes more than seven related question formats: formulation from word statements, identifying decision variables, writing objective functions, converting statements into constraints, non-negativity questions, feasible-region identification, boundary/intercept questions, corner-point calculations, graphical-solution concepts, maximum/minimum evaluation, and formula/concept MCQs. This variety is intended to practise both interpretation and calculation.

13. How to Use the Interactive Worksheet

Select the number of questions, question mode, topic, difficulty, timer and sound. Click Start New Test. Before starting, the worksheet remains locked and answers cannot be attempted. After the test starts, every question has its own checking control. MCQs display four options; short-answer questions provide a textbox. Press Check Answer after responding.

Correct responses are highlighted in green, while incorrect responses are highlighted in red. The answer is kept hidden until the checking step. Once a question has been checked, the Show Answer control becomes available. The progress bar, score and accuracy update as you work.

14. Whole Worksheet Shuffle

The Shuffle Whole Worksheet button changes the order of the generated question cards. It uses a Fisher-Yates-style random shuffle, so repeated shuffles can produce a new arrangement. MCQ options are also independently shuffled when questions are generated, reducing the chance that the correct choice always appears in the same position.

15. Timer and Exam Practice

For relaxed revision choose No Timer. For exam-style practice select 5, 10, 20, 30 or 60 minutes before starting the test. The timer begins with the new test and the worksheet reports time remaining. When time expires, unanswered questions are closed and the final result is displayed. Short timed tests can help students improve speed, while untimed practice is useful when first learning the graphical method.

16. Sound and Animation

DigiSadhan includes optional feedback sounds and visual animations. A successful answer receives a positive animation and sound, while an incorrect answer receives a different feedback effect. Sound can be switched off from the controls. This makes the worksheet suitable for both classroom-style practice and quiet individual study.

17. Progress, Score and Final Result

Every checked question contributes to progress. The score counts correct checked responses, and the accuracy percentage is calculated from the total worksheet size. After all questions have been checked, the tool displays the final score and percentage. Students should use the result as a revision signal: a lower score indicates which topic areas need more practice rather than simply representing a final judgment.

18. PDF and Print Options

The worksheet includes two separate output controls. Save as PDF creates a downloadable PDF containing the generated questions and their answers. Print opens the browser's printing workflow through the normal print command. These options are kept separate so that students can choose a digital copy or a paper practice sheet.

19. Formula Quick Revision

20. Common Mistakes to Avoid

• Forgetting x ≥ 0 and y ≥ 0.
• Using the wrong inequality direction for “at most” and “at least”.
• Drawing the boundary line correctly but shading the wrong side.
• Missing a corner point of the feasible polygon.
• Optimizing the wrong expression.
• Reporting only the objective value without the corresponding decision variables.
• Using a point that is outside the feasible region when evaluating an LPP.
• Ignoring the possibility of multiple optimal solutions.

21. Best Study Strategy

Start with formulation questions. Practise identifying variables and translating each sentence into an objective or constraint. Next, practise intercepts and feasible-region questions. Then move to corner-point calculations and full graphical problems. Once the method becomes comfortable, use Mixed mode and a timer. Repeated shuffled tests are particularly useful because the numerical values and question order change, encouraging method recognition rather than memorisation.

22. Final Revision Plan

Before an examination, revise the meanings of decision variables, objective function, constraints, feasible region and corner points. Practise at least one example involving maximization and one involving minimization. Review how inequality signs determine the shaded region. Finally, check every corner point systematically. The fastest reliable approach is not to guess the optimum from the picture but to calculate the objective function at all relevant feasible vertices.

DigiSadhan's Linear Programming worksheet is intended to make this revision process convenient on mobile and desktop devices. With up to 2,000 questions, topic filters, difficulty levels, randomized ordering, instant checking, progress tracking and PDF/print support, students can create short revision sessions or longer practice tests whenever needed.