Mathematics helps students make logical, data-driven decisions, and Chapter 12: Linear Programming is a highly application-oriented and scoring chapter in Class 12 Maths. This chapter introduces students to Linear Programming Problems (LPPs), where real-life situations are modelled using linear inequalities and solved using the graphical method to find the maximum or minimum value of an objective function.
NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming are prepared strictly according to the CBSE syllabus and board exam pattern. These solutions include important CBSE board questions asked between 2018 and 2025 and are explained in a clear, step-by-step manner using simple language and neat graphical representation. This helps students understand concepts easily, practise effectively, and score full marks in board examinations.
NCERT Solutions For Class 12 Maths Chapter 12 Linear Programming
Q.
MaximiseZ=3x+4ySubjecttotheconstraints:x+y≤4,x≥0,y≥0.
Q.
One kind of cake requires 200g flour and 25g of fat, and another kind of cake requires100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes?
Q.
A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the maximum amount of nitrogen added in the garden?
| Kg per bag |
| | Brand P | Brand Q |
| Nitrogen | 3 | 3.5 |
| Phosphoric acid | 1 | 2 |
| Potash | 3 | 1.5 |
| Chlorine | 1.5 | 2 |
Q.
A merchant plans to sell two types of personal computers − a desktop model and a portable model that will cost ₹ 25,000 and ₹ 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is ₹ 4500 and on portable model is ₹ 5000.
Q.
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours of assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?
Q.
A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?
Q.
A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.
Q.
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit, of Rs 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
Q.
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.
Q.
Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B. Determine the minimum cost of the mixture?
Q.
MinimiseZ=−3x+4ySubjecttox+y≤,x+y≤,x≥0,y≥0.
Q.
MaximiseZ=x+y,subjecttothecontraints:x−y≤−,−x+y≤;x,y≥0.
Q.
MaximiseZ=−x+2y,subjecttothecontraints:x≥,x+y≥,x+2y≥;y≥0.
Q.
MinimiseandMaximiseZ=x+2ySubjecttox+2y≤100,2x−y≤,2x+y≤00;x,y≥0.
Q.
MinimiseandMaximiseZ=5x+10ySubjecttox+2y≤120,x+y≥,x−2y≥0;x,y≥0.
Q.
MinimiseZ=x+2ySubjectto2x+y≥,x+2y≥,x,y≥0.
Q.
MaximiseZ=3x+2ySubjecttox+2y≤,x+y≥,x,y≥0.
Q.
MinimiseZ=3x+5ySubjecttox+3y≤,x+y≥,x,y≥0.
Q.
MaximiseZ=5x+3ySubjectto3x+y≤,x+y≤,x≥0,y≥0.
Q.
A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of ₹ 12 and ₹ 16 per doll respectively on dolls A and B, How many of each should be produced weekly in order to maximize the profit?
Question
Maximise Z = 3x + 4y subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0.
Answer
The maximum value of Z is 16 at the point (0, 4).
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Easy
Question
One kind of cake requires 200 g of flour and 25 g of fat, and another kind requires 100 g of flour and 50 g of fat. Find the maximum number of cakes that can be made from 5 kg of flour and 1 kg of fat.
Answer
The maximum number of cakes that can be made is 30.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Medium
Question
A fruit grower uses fertilizers of brand P and Q. The garden requires at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. Find the maximum amount of nitrogen added.
Answer
The maximum amount of nitrogen added is 595 kg by using 140 bags of brand P and 50 bags of brand Q.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Difficult
Question
A company manufactures two types of novelty souvenirs. Type A and B require different cutting and assembling times. Find the production to maximize profit.
Answer
The maximum profit is ₹160 at the point (8, 20). Hence, produce 8 souvenirs of type A and 20 of type B.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Difficult
Question
A factory manufactures two types of screws A and B using automatic and hand-operated machines. Determine the number of packages to maximize profit.
Answer
The maximum profit is ₹410 at the point (30, 20). Thus, 30 packages of screw A and 20 packages of screw B should be produced.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Difficult
Question
Reshma mixes food P and Q such that the mixture contains at least 8 units of vitamin A and 11 units of vitamin B. Find the minimum cost.
Answer
The minimum cost of the mixture is ₹160.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Difficult
Question
A company manufactures dolls of type A and B with given constraints. Find the production that maximizes profit.
Answer
The maximum profit is ₹16000 at (800, 400). Hence, 800 dolls of type A and 400 dolls of type B should be produced.
Subject: Mathematics | Chapter: Linear Programming | Difficulty: Difficult
FAQs: NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming
Q1. Why is Linear Programming important for Class 12 Maths?
Linear Programming helps students apply mathematics to real-life decision-making problems and is one of the most scoring chapters in the CBSE board exams.
Q2. How many marks are usually asked from Chapter 12 in board exams?
Generally, 5 to 6 marks worth of questions are asked from Linear Programming, often as a complete long-answer question.
Q3. Are NCERT Solutions enough to score full marks in Linear Programming?
Yes. NCERT textbook questions and NCERT Solutions are sufficient, as board exam questions are mostly asked directly from NCERT.
Q4. Which topics are most important in Linear Programming?
The most important topics include:
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Formulation of LPP
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Graphical solution
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Feasible region
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Finding optimal values
Q5. Why do students find Linear Programming confusing?
Students often face difficulty in:
Regular practice with graphs and careful step-by-step solving helps overcome these issues.
Q6. Is Linear Programming useful beyond board exams?
Yes. Linear Programming is widely used in economics, business management, operations research, and engineering, making it useful for higher studies as well.