Feasible Solution

Q.

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Q. A manufacturing company makes two models A and B of a product. Each piece of model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of ₹8000 on each piece of model A and ₹12000 on each piece of model B. How many pieces of model A and model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Q.

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹100 and ₹120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Q. Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$. [CBSE 2015]

Q.

logx+ log( xy⁸)/(log x)²+(log y)²=2 & logy+log(x⁸/y)/(log x)²+(log y)²=0( where base of log is 10 ) compute the product of(xy).

Q. Determine graphically the minimum value of the objective function
Z = 50x + 20y
subject to the constraints:
2x – y ≥ – 3
3x + y ≥ 3
2x – 3y ≤ 12
x ≥ 0, y ≥ 0

1. 100
2. 300
3. 60
4. 50

Q.

If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.

Q.

One kind of cake requires 200g 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 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. For the LPP; maximize $z=5x-2y$ subject to the constraints $3x+2y\le 6,10x+9y\ge 45,x,y\ge 0$

1. $z_{min}=-15$
2. $z_{min}=0$
3. $z_{min}=-6$
4. The LPP has no feasible solution.

Q. Find the equation of the plane which passes through the point (3, 4, −1) and is parallel to the plane 2x − 3y + 5z + 7 = 0. Also, find the distance between the two planes.

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. The no.of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty?

Q. A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as ₹10, 500 and ₹9, 000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at the rate of 20 litres and 10 litres per hectare, respectively. Further not 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? [CBSE 2013]

Q.

Maximise Z = 3x + 2y

subject to.

Q. A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 9000 on each piece of model A and Rs 11000 on each piece of Model B.How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit?

1. 20 pieces of model A & 0 pieces of model B
2. 12 pieces of model A & 6 pieces of model B
3. 0 pieces of model A & 10 pieces of model B
4. 10 pieces of model A & 10 pieces of model B

Q.

Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.

Q.

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Type of toys	Machines
	I	II	III
A	12	18	6
B	6	0	9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Q. A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹250 per bag, contains 2 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Q. The feasible solutions for a linear programming problem with constraints
x ≥0, y ≥ 0
3x+5y ≤ 15
5x+2y ≤ 10
is equal to

1. $(1,-1),(0,0)$
2. $(-3,5),(1,-1)$
3. $(0,0),(-3,5)$
4. $(2,0),(0,0)$

Q.

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in Rs)
From/To	A	B
D E F	6 3 2.50	4 2 3

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

Q. Minimize Z = 30x + 20y
Subject to
$$\begin{gathered} x+y\le 8 \\ x+4y\ge 12 \\ 5x+8y=20 \\ x,y\ge 0 \end{gathered}$$

Q. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/cutting machine and sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp while 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 most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is ₹5.00 and a shade is ₹3.00. Assuming that the manufacturer sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? [NCERT, CBSE 2013]

Q. Make a sketch of the region {(x, y) : 0 ≤ y ≤ x² + 3; 0 ≤ y ≤ 2x + 3; 0 ≤ x ≤ 3} and find its area using integration.

Q. The feasible region for an LPP is shown in the Figure.

Let F = 3x – 4y be the objective function. Maximum value of F is.

1. 6
2. -12
3. 8
4. 0

Q. Solved the following linear programming problem graphically:
Maximize Z = 60x + 15y
Subject to constraints
$$\begin{gathered} x+y\le 50 \\ 3x+y\le 90 \\ x,y\ge 0 \end{gathered}$$

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.

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. The maximum profit that can be made is equal to

1. 142000
2. 124000
3. 163000
4. 136000

Q. Maximize Z = x + y
Subject to
$$\begin{gathered} -2x+y\le 1 \\ x\le 2 \\ x+y\le 3 \\ x,y\ge 0 \end{gathered}$$

Q. A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 9000 on each piece of model A and Rs 11000 on each piece of Model B.How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit?

1. 20 pieces of model A & 0 pieces of model B
2. 0 pieces of model A & 10 pieces of model B
3. 12 pieces of model A & 6 pieces of model B
4. 10 pieces of model A & 10 pieces of model B

Q. A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 sq.m of ground per tree and costs Rs 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs 25 per tree. When fully grown, type A produces an average of 20 kg of fruit which can be sold at a profit of Rs 2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?