The CBSE Class 12th Mathematics Examination 2020 will be held on March 17, 2020. The students appearing in the CBSE Class 12th Mathematics Board Examination 2020 can go through the below-mentioned questions for Chapter 12 - Linear Programming. These Questions are based on the latest syllabus prescribed by the CBSE Board.
Question 1- Maximise and minimise Z = 3x - 4y subject to x - 2y < 0, - 3x + y ≤ 4, x - y < 6 and x, y≥ 0.
Answer: From the graph, we know that there are common points with the feasible region and therefore the equation does not have any minimum value. For the maximum value, there are no common points with the feasible region and for the given equation maximum value is 12.
Question 2- A company makes 3 models of calculators i.e., A, B and C at the factories 1 and 2. The company has orders for at least:
6400 calculators of model A,
4000 calculators of model B and
4800 calculators of model C.
Factory |
Model A |
Model B |
Model C |
Cost (in Rs) |
Factory 1 |
50 |
50 |
30 |
12000 |
Factory 2 |
40 |
20 |
40 |
15000 |
Find the number of days each factory should operate to minimise the operating costs and still meet the demand.
Answer: For factory 1 should be operated for 80 days and factory 2 should be operated for 60 days to minimise the operating cost while meeting the demand.
Question 3- A man rides his motorcycle at a speed of 50 km/h. He has to spend Rs 2 per km on petrol. If he rides it at a speed of 80 km/h, the petrol cost increases to Rs 3 per km. He has almost Rs 120 to spend on petrol and one hour's time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.
Answer: Maximise Z = x + y, subject to
2x + 3y ≤ 120
8x + 5y ≤ 400
x ≥ 0, y ≥ 0
Question 4- Solve the following problem graphically: Minimise and Maximise Z = 3x + 9y. subject to the constraints:
x + 3y ≤ 60
x + y ≥ 10
x ≤ y
x ≥ 0, y ≥ 0
Answer: The minimum value of Z is 60 at point B (5, 5) of the feasible region. The maximum value of Z on the feasible region occurs at the two corner points C (15, 15) and D (0, 20) and it is 180 in each case.
Question 5- A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A?
Answer: The amount of vitamin A will be minimum if 15 packets of food P and 20 packets of food Q are used in the special diet. The minimum amount of vitamin A will be 150 units.
Question 6- A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?
Answer: The society will get the maximum profit of Rs. 4,95,000 by allocating 30 and 20 hectares for Crop X and Y respectively.
Question 7- A company manufactures two types of sweaters A and B. It costs Rs 360 to make a sweater A and Rs 120 to make a sweater B. The company can make at most 300 sweaters and spend at most Rs 72,000 a day. The number of sweaters B cannot exceed the number of sweaters A by more than 100. The company makes a profit of Rs 200 for each sweater A and Rs 120 for every sweater B. Express this problem as an LPP to maximise the profit to the company.
Answer: Required LPP to Maximise profit is Z = 200x + 120y is subject to constraints.
3x + y ≤ 600
x + y ≤ 300
x – y ≥ -100
x ≥ 0, y ≥ 0
Question 8- A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3,000 is to be spent on the job and the number of large vans cannot exceed the number of small vans. Express this problem as a linear programming problem given that the objective is to minimise cost.
Answer: Required LPP to minimise cost is Z = 400x + 200y, subject to 5x + 2y ≥ 30.
2x + y < 15
x ≤ y
x ≥ 0, y ≥ 0
Question 9- One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
Answer: Maximum number of cakes = 30 of kind one and 10 cakes of another kind.
Question 10- There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 and F2 cost Rs 6/kg and Rs 5/kg respectively. Determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
Answer: 100 kg of fertiliser F1 and 80 kg of fertiliser F2; Minimum cost = Rs 1000
The above-mentioned questions are strictly based on the latest CBSE pattern prescribed by the CBSE Board. The students appearing for the CBSE Class 12th Mathematics Examination 2020 will find these questions helpful while preparing for the upcoming Examination.