Graphical Method of Solving Linear Programming Problems

Q. In a bolt factory, three machines $A,B$ and $C$ produce $25\%,35\%$ and $40\%$ of total output respectively. It was found that $5\%,4\%$ and $2\%$ are defective bolts in the production by machines $A,B,C$ respectively. If a bolt is chosen at random from the total output and is found to be defective, then the chance that the bolt comes from the machine:

1. $A\text{is}\frac{125}{449}$
2. $B\text{is}\frac{125}{449}$
3. $C\text{is}\frac{128}{449}$
4. $A\text{is}\frac{128}{449}$

Q. A manufacturing company makes two types of television sets; one is black and white and the other is colour. The company has resources to make at most 350 sets a week. It takes Rs 1800 to make a black and white set and Rs 2700 to make a coloured set. The company can spend not more than Rs 648000 a week to make television sets. If it makes a profit of Rs 500 per black and white set and Rs 600 per coloured set. Formulate this problem as an LPP given that the objective is to maximise the profit. what is the maximum profit ?

1. 175000
2. 177000
3. 188000
4. 144000

Q. The number of point(s) of discontinuity of $f\left(x\right)=\left[2\cos x\right],x\in [0,2\pi ]$, is (where $[.]$ represents Greatest Integer function)

Q. The corner points of the feasible region determined by the system of linear constraints are $(0,10),(5,5),(15,15),(0,20).$ Let $z=px+qy$ where $p,q>0.$ Condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,15)$ and $(0,20)$ is

1. $q=3p$
2. $p=2q$
3. $q=2p$
4. $p=q$

Q.

What is the dual simplex method?

Q.

One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g 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. If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

1. the required optimal solution is at the midpoint of the line joining two points.
2. the optimal solution occurs at every point on the line joining these two points
3. the LPP under consideration must be reconstructed
4. the LPP under consideration is not solvable

Q.

Maximum value of $Z=12x+3y$, subject to constraints $x\ge 0$, $y\ge 0$, $x+y\le 5$ and $3x+y\le 9$ is

1. $15$

2. $36$

3. $60$

4. $40$

Q.

The output quality of a printer is measured by:

1. Dots per line

2. Dots printed per unit time

3. Dots per inch

4. All of the above

Q.

Find the maximum value of z = 3x + 4y subject to constraints x + y ≤ 4, x ≥ 0 and y ≥ 0

1. 12

2. 14

3. 16

4. 7

Q.

The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in column A and column B.

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{Maximum of Z}& 325\end{array}$
(a) The qunatity in column A is greater
(b) The qunatity in column b is greater
(c) The two quantities are equal
(d) The relationship cannot be determined on the basis of the information supplied.

Q.

A diet is to contain atleast 80 units of vitamin A and 100 units of minerals. Two foods $F_{1}and{F}_2$ are available. Food $F_1$ costs Rs. 4 per unit and food $F_2$ costs Rs. 6 per unit. One units of food $F_1$ contains at 3 units of vitamin A and 4 units of minerals. One unit of food $F_2$ contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

Q.

Four fifth of a number is more than three fourth of the number by $4$. Find the number.

Q.

A manufacturer produces nuts and bolts. It takes 1 h of work on machine A and 3 h on machine B to produce a package of nuts. It takes 3 h on machine A and 1 h on maching 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 package of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 h a day ?

Q.

The constraints $-x_1+x_2\le 1,-x_1+3x_2\le 9$ and $x_1,x_2\ge 0$ defines on

1. bounded feasible space

2. unbounded feasible space

3. both bounded and unbounded feasible space

4. None of the above

Q.

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A^4$is equal to ___________

Q. An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table: Distance in (km) From/To A B D E F 7 6 3 3 4 2 Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?

Q.

he expenditure of a family on different heads in a month is given below. Draw a bar graph to represent the data above.

Head	Food	Education	Clothing	House Rent	Others	Saving
Exenditure	4000	2500	1000	3500	2500	1500

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 of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resitors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60, formulate this problem as a LPP, so that the manufacturer can maximise his profit.

Q.

The default alignment for paragraph is

Q.

A plumber can be paid under two schemes as given below:

$$\begin{gathered} \mathrm{I}:\mathrm{Rs}.600\mathrm{and}\mathrm{Rs}.50\mathrm{per}\mathrm{hour}. \\ \mathrm{II}:\mathrm{Rs}.170\mathrm{per}\mathrm{hour}. \end{gathered}$$
If the job takes $n$ hours, for how many values of $n$ does the Scheme I give the plumber the better wages?

Q.

What is called the simplex method?

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.

Solve the following systems of inequations graphically:

(i) $2x+y\ge 8,x+2y\ge 8,x+y\le 6$

(ii) $12x+12y\le 840,3x+6y\le 300,8x+4y\le 480x\ge 0,y\ge 0$

(iii) $x+2y\le 40,3x+y\ge 30,4x+3y\ge 60,x\ge 0,y\ge 0$

(iv) $5x+y\ge 10,2x+2y\ge 12,x+4y\ge 12,x\ge 0,y\ge 0$

Q.

Solve the following systems of linear inequations graphically :

$\left(i\right)2x+3y\le 6,3x+2y\le 6,x\ge 0,y\ge 0\left(ii\right)2x+3y\le 6,x+4y\le 4,x\ge 0,y\ge 0\left(iii\right)x+y\ge 1,x+2y\le 8,2x+y\ge 2,x\ge 0,y\ge 0\left(iv\right)x+y\ge 1,7x+9y\le 63,y\le 5,x\ge 0,y\ge 0\left(v\right)2x+3y\le 35,y\ge 3,x\ge 2,x\ge 0,y\ge 0$

Q. Solve the following linear programming problem graphically: Maximise $Z=7x+10y$ subject to the constraints
$4x+6y\le 2406x+3y\le 240x\ge 10x\ge 0,y\ge 0$

Q.

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of elements B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional elements 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 length of the rectangular field is thrice of its breadth. If the perimeter of this field is $800m$, what is the length of the field.

Q.

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 min each for cutting and 10 min each for assembling. Souvenirs of type B require 8 min each for cutting and 8 min each for assembling. There are 3 h 20 min available for cutting and 4 h for 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 maximise the profit?