Constraints

Q.

If $y=(x+1)(x+2)(x+3)(x+4)(x+5)$, then the value of $\frac{dy}{dx}$ at $x=0$is equal to

1. $742$

2. $472$

3. $374$

4. $274$

Q.

The linear inequalities or equations or restrictions on the variables of a linear programming problem are called...... The conditions x ≥ 0, y ≥ 0 are called.......

1. Objective functions, non-negative restrictions

2. Constraints, non-negative restrictions

3. Objective functions, optimal value

4. Constraints, negative restrictions

Q.

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 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 lakh and if his profit on the desktop model Rs. 4500 and on portable model is Rs. 5000.

Q. What is linear programming?

Q.

Why is it called linear programming?

Q.

The feasible region for an LPP is shown in the following figure. Let F=3x-4y be the objective function. Maximum value of F is

(a)0
(b)8
(c)12
(d)-18

Q. The corner points of the feasible region for an LPP are $(0,2),$ $(3,0),$ $(6,0),$ $(6,8)$ and $(0,5).$ Let $Z=4x+6y$ be the objective function. Then minimum value of $Z$ occurs at

1. $(0,2)$ only
2. $(3,0)$ only
3. the mid-point of the line segment joining the points $(0,2)$ and $(3,0)$ only
4. any point on the line segment joining the points $(0,2)$ and $(3,0)$

Q. For the objective function $Z=x+2y,$ subject to constraints : $x-y\le -1,-x+y\le 0,$ which of the following(s) is not correct:

1. $Z_{max}=1$
2. $Z_{min}=-1$
3. $Z_{min}=0$
4. No feasible solution is possible.

Q. The number of constraints allowed in a linear program is which of the following?

1. Less than 5
2. Less than 72
3. Less than 512
4. Unlimited
5. Less than 1, 024

Q. If $Z=5x+3y,$ subject to $3x+5y\le 15,5x+2y\le 10,x\ge 0,y\ge 0,$ then $Z_{max}$ is equal to

1. $10$
2. $9$
3. $\frac{235}{19}$
4. $\frac{255}{19}$

Q. If $Z=3x+4y,$ subject to the constraints: $x+y\le 4,$ $x\ge 0,$ $y\ge 0,$ then $Z_{max}$ is equal to

1. $12$
2. $16$
3. $20$
4. $24$

Q. The corner points of the feasible region determined by the system of linear constraints are $(0,10),(5,5),(25,20)$ and $(0,30)$. Let $Z=px+qy$, where $p,q>0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both the points $(25,20)$ and $(0,30)$ is

1. $5p=2q$
2. $2p=5q$
3. $p=2q$
4. $q=3p$

Q.

Which of the term is not used in a linear programming problem?

1. Optimal solution

2. Feasible solution

3. Concave region

4. Objective function

Q. A vanilla cake requires $200\text{g}$ of flour and $25\text{g}$ of fat, and a strawberry cake requires $100\text{g}$ of flour and $50\text{g}$ of fat. From $5\text{kg}$ of flour and $1\text{kg}$ of fat, the maximum number of cakes that can be made is

Q. Check the differentiability of the y=Cosx , at x=2 both graphically as well as analytically?

Q. The corner points of the feasible region determined by a system of linear constraints are $(0,10),(5,5),(15,15),(0,20).$ Let $z=px+qy$ where $p,q>0.$ If the maximum of $z$ occurs at both the points $(15,15)$ and $(0,20),$ then which of the following is true:

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

Q.

Minimum z = x + 2y subject to x ≤ 20, y ≥ 10, x ≥ 0, y ≥ 0

1. 40

2. 20

3. 30

4. 10

Q. A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1 kg of food X and 1 kg of food Y are given below:

	Vitamin A	Vitamin B	Vitamin C	Vitamin D
Food X Food Y	1 2	1 1	1 3	2 1

One kg food X costs Rs 5, whereas one kg of food Y costs Rs 8. Find the least cost of the mixture which will produce the desired diet.

Q. Solve the following linear programming problem graphically:

Maximise $Z=34x+45y$

under the following constraints

$x+y\le 3002x+3y\le 70x\ge ,y\ge 0$

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.Let the mixture contain x kg of food P and y kg of food Q. Which of the following are the constraints of this linear programming problem?

1. x ≥ 0 and y ≥ 0

2. 3x + 4y ≥ 8

3. 5x + 2y ≥ 11

4. x+y < 20

Q. Minimise $Z={\sum }_{j=1}^n{\sum }_{i=1}^m{c}_{ij}.x_{ij}$
Subject to ${\sum }_{i=1}^m{x}_{ij}=b_j,j=1,2,......n$
${\sum }_{j=1}^n{x}_{ij}=b_j,j=1,2,......,m$ is a LPP with number of constraints

1. $m-n$
2. $mn$
3. $m+n$
4. $\frac{m}{n}$

Q. A chemical company produces two products, $X$ and $Y$.Each unit of product $X$ requires $3$hrs on operation $I$ and $4$hrs on operation $II$, while each unit of product $Y$ requires $4$hrs on operation $I$ and $5$hrs on operation$II$.
Total available time for operations $I$ and $II$ is $20$hrs and $26$ hrs respectively.The production of each unit of product $Y$ also results in two units of a by-product $Z$ at no extra cost.
Product $X$ sells at profit of Rs.$10$/unit, while $Y$ sells at profit of Rs.$20$/unit.By-product $Z$ brings a unit profit of Rs.$6$ if sold;in case it cannot be sold, the destruction cost is Rs.$4$ per unit.Forecasts indicate that not more that $5$ units of $Z$ can be sold.Formulate the L.P model to determine the quantities of $X$ and $Y$ to be produced, keeping $Z$ in mind, so that the profit earned is maximum.

Q.

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg food is given below:

Food	Vitamin A	Vitamin B	Vitamin C
X	1	2	3
Y	2	2	1

One kg of food X costs Rs 16 and one kg of food Y costs Rs 20. Find the least cost of the mixture which will produce the required diet?

Q. A cooperative society of farmers has 50 hectare 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 wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so at to maximise the total profit of the society?

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. What is linear programming?

Q. The corner points of the feasible region for optimization problem determined by a system of linear constraints are $(6,5),(9,7),(15,0),(0,10).$ Let $z=px+qy$ and the maximum of $z$ occurs at both the points $(6,5)$ and $(9,7),$ then which of the following is true:

1. $3p+2q=0$ and $p>0,q>0$
2. $3p+2q=0$ and $p>0,q<0$
3. $3p+2q=0$ and $p<0,q<0$
4. $3p+2q=0$ and $p<0,q>0$

Q. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available. Food F₁ costs Rs 4 per unit and F₂ costs Rs 6 per unit one unit of food F₁ contains 3 units of vitamin A and 4 units of minerals. One unit of food F₂ contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements.

Q. Which of the following is an essential condition in a situation for linear programming to be useful?

1. Linear constraints
2. Bottlenecks in the objective function
3. Non-homogeneity
4. Uncertainty
5. None of the above

Q. Shaded region is represented by

1. $2x+5y\ge 80,x+y\le 20,x\ge 0,y\le 0$
2. $2x+5y\ge 80,x+y\ge 20,x\ge 0,y\ge 0$
3. $2x+5y\le 80,x+y\le 20,x\ge 0,y\ge 0$
4. $2x+5y\le 80,x+y\le 20,x\le 0,y\le 0$