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NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

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NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

Edited By Ramraj Saini | Updated on Sep 19, 2023 09:36 AM IST | #CBSE Class 12th

NCERT Linear Programming Class 12 Questions And Answers

NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming provided here. These NCERT solutions are prepared by expert team at Careers360 considering the latest syllabus of CBSE 2023-24. This ch 12 maths class 12 helps solving and finding optimal solutions to problems related to maximising profit or minimising cost. NCERT linear programming class 12 solutions will help in formulating these real life problems into a mathematical model. You should practise linear programming problems (lpp class 12) for getting command of concepts and in depth understanding of this linear programming class 12 chapter.

In this linear programming class 12 solutions, you are going to deal with problems on linear programming like maximisation and minimization of equations, mathematical and graphical methods to solve problems of linear programming. You can also refer to linear programming class 12 ncert solutions for better understanding of concepts. Below we have given complete Class 12 maths chapter 12 NCERT solutions. Check all NCERT solutions from class 6 to 12 at a single place to prepare better for exams.

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NCERT Class 12 Maths Chapter 12 Question Answer - Important Formulae

>> Feasible Region: The feasible region, or solution region, of a linear programming problem is the common area determined by all the constraints, including the non-negativity constraints (x ≥ 0, y ≥ 0).

>> Infeasible Solution: Any point within or on the boundary of the feasible region represents a feasible solution to the constraints. Points outside the feasible region are considered infeasible solutions.

>> Optimal Solution: An optimal solution is any point within the feasible region that provides the optimal value (maximum or minimum) of the objective function.

Fundamental Theorems in Linear Programming:

>> Optimality at Corner Points: For a linear programming problem with a feasible region represented as a convex polygon, if the objective function Z = ax + by has an optimal value, this optimal value must occur at one of the corner points (vertices) of the feasible region.

>> Existence of Maxima and Minima: If the feasible region R is bounded, then the objective function Z has both a maximum and a minimum value on R, and each of these values occurs at a corner point (vertex) of R. If R is unbounded, a maximum or minimum may not exist. However, if it does exist, it must occur at a corner point of R.

>> Corner Point Method: The corner point method is used to solve a linear programming problem and consists of the following steps:

Find the feasible region of the linear programming problem and determine its corner points (vertices).

Evaluate the objective function Z = ax + by at each corner point. Let M and m represent the largest and smallest values obtained at these points.

If the feasible region is bounded, M and m respectively represent the maximum and minimum values of the objective function.

If the feasible region is unbounded, then:

  • M is the maximum value of the objective function if the open half-plane determined by ax + by > M has no points in common with the feasible region.

  • m is the minimum value of the objective function if the open half-plane determined by ax + by < M has no points in common with the feasible region.

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NCERT Linear Programming Class 12 Questions And Answers (Intext Questions and Exercise)

NCERT linear programming class 12 solutions - Exercise: 12.1

Question:1 Solve the following Linear Programming Problems graphically: Maximise Z = 3x + 4y Subject to the constraints x+y\leq 4,x\geq 0,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+y\leq 4,x\geq 0,y\geq 0. is as follows,

1627031435613

The region A0B represents the feasible region

The corner points of the feasible region are B(4,0),C(0,0),D(0,4)

Maximize Z = 3x + 4y

The value of these points at these corner points are :

Corner points
Z = 3x + 4y

B(4,0)
12

C(0,0)
0

D(0,4)
16
maximum

The maximum value of Z is 16 at D(0,4)

Question:2 Solve the following Linear Programming Problems graphically: Minimise z=-3x+4y Subject to . x+2y\leq 8,3x+2y\leq 12,x\geq 0,y\geq 0. Show that the minimum of Z occurs at more than two points

Answer:

The region determined by constraints, x+2y\leq 8,3x+2y\leq 12,x\geq 0,y\geq 0. is as follows,

1627031511366

The corner points of feasible region are A(2,3),B(4,0),C(0,0),D(0,4)

The value of these points at these corner points are :

Corner points
z=-3x+4y

A(2,3)
6

B(4,0)
-12
Minimum
C(0,0)
0

D(0,4)
16

The minimum value of Z is -12 at B(4,0)

Question:3 Solve the following Linear Programming Problems graphically: Maximise Z = 5x + 3y Subject to 3x + 5y \leq 15 , 5x+2y\leq 10 , x\geq 0,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, 3x + 5y \leq 15 , 5x+2y\leq 10 , x\geq 0,y\geq 0 is as follows :

1627031555890

The corner points of feasible region are A(0,3),B(0,0),C(2,0),D(\frac{20}{19},\frac{45}{19})

The value of these points at these corner points are :

Corner points
Z = 5x + 3y

A(0,3)
9

B(0,0)
0

C(2,0)
10

D(\frac{20}{19},\frac{45}{19})
\frac{235}{19}
Maximum

The maximum value of Z is \frac{235}{19} at D(\frac{20}{19},\frac{45}{19})

Question:4 Solve the following Linear Programming Problems graphically: Minimise Z = 3x + 5y Such that x+3y\geq 3,x+y\geq 2,x,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x+3y\geq 3,x+y\geq 2,x,y\geq 0. is as follows,

1627031646530

The feasible region is unbounded as shown.

The corner points of the feasible region are A(3,0),B(\frac{3}{2},\frac{1}{2}),C(0,2)

The value of these points at these corner points are :

Corner points
Z = 3x + 5y

A(3,0)
9

B(\frac{3}{2},\frac{1}{2})
7
Minimum
C(0,2)
10


The feasible region is unbounded, therefore 7 may or may not be the minimum value of Z .

For this, we draw 3x + 5y< 7 and check whether resulting half plane has a point in common with the feasible region or not.

We can see a feasible region has no common point with. Z = 3x + 5y

Hence, Z has a minimum value of 7 at B(\frac{3}{2},\frac{1}{2})

Question:5 Solve the following Linear Programming Problems graphically: Maximise Z = 3x + 2y Subject to x+2y\leq 10,3x+y\leq 15,x,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+2y\leq 10,3x+y\leq 15,x,y\geq 0 is as follows,

1627031733350

The corner points of feasible region are A(5,0),B(4,3),C(0,5)

The value of these points at these corner points are :

Corner points
Z = 3x + 2y

A(5,0)
15

B(4,3)
18
Maximum
C(0,5)
10


The maximum value of Z is 18 at B(4,3)

Question:6 Solve the following Linear Programming Problems graphically: Minimise Z = x + 2y Subject to 2x+y\geq 3,x+2y\geq 6,x,y\geq 0.

Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints 2x+y\geq 3,x+2y\geq 6,x,y\geq 0. is as follows,

1627031776022

The corner points of the feasible region are A(6,0),B(0,3)

The value of these points at these corner points are :

Corner points
Z = x + 2y
A(6,0)
6
B(0,3)
6

Value of Z is the same at both points. A(6,0),B(0,3)

If we take any other point like (2,2) on line Z = x + 2y , then Z=6.

Thus the minimum value of Z occurs at more than 2 points .

Therefore, the value of Z is minimum at every point on the line Z = x + 2y .

Question:7 Solve the following Linear Programming Problems graphically: Minimise and Maximise z=5x+10y Subject to x+2y\leq 120,x+y\geq 60,x-2y\geq 0,x,y\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints, x+2y\leq 120,x+y\geq 60,x-2y\geq 0,x,y\geq 0 is as follows,

1627031823610

The corner points of feasible region are A(40,20),B(60,30),C(60,0),D(120,0)

The value of these points at these corner points are :

Corner points
z=5x+10y

A(40,20)
400

B(60,30)
600
Maximum
C(60,0)
300
Minimum
D(120,0)
600
maximum

The minimum value of Z is 300 at C(60,0) and maximum value is 600 at all points joing line segment B(60,30) and D(120,0)

Question:8 Solve the following Linear Programming Problems graphically: Minimise and Maximise z=x+2y Subject to x+2y\geq 100,2x-y\leq 0,2x+y\leq 200,x,y,\geq 0 Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x+2y\geq 100,2x-y\leq 0,2x+y\leq 200,x,y,\geq 0 is as follows,

1627031869634

The corner points of the feasible region are A(0,50),B(20,40),C(50,100),D(0,200)

The value of these points at these corner points are :

Corner points
z=x+2y

A(0,50)
100
Minimum
B(20,40)
100
Minimum
C(50,100)
250

D(0,200)
400
Maximum

The minimum value of Z is 100 at all points on the line segment joining points A(0,50) and B(20,40) .

The maximum value of Z is 400 at D(0,200) .

Question:9 Solve the following Linear Programming Problems graphically: Maximise Z = -x+2y Subject to the constraints: x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0. is as follows,

1627032034587

The corner points of the feasible region are A(6,0),B(4,1),C(3,2)

The value of these points at these corner points are :

Corner points
Z = -x+2y

A(6,0)
- 6
minimum
B(4,1)
-2

C(3,2)
1
maximum

The feasible region is unbounded, therefore 1 may or may not be the maximum value of Z.

For this, we draw -x+2y> 1 and check whether resulting half plane has a point in common with a feasible region or not.

We can see the resulting feasible region has a common point with a feasible region.

Hence , Z =1 is not maximum value , Z has no maximum value.

Question:10 Solve the following Linear Programming Problems graphically: Maximise Z = x + y, Subject to x-y\leq -1,-x+ y\leq 0,x,y,\geq 0. Show that the minimum of Z occurs at more than two points.

Answer:

The region determined by constraints x-y\leq -1,-x+ y\leq 0,x,y,\geq 0. is as follows,

1627032109317

There is no feasible region and thus, Z has no maximum value.


NCERT linear programming class 12 solutions - Exercise: 12.2

Question:1 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.

Answer:

Let mixture contain x kg of food P and y kg of food Q. Thus, x\geq 0,y\geq 0 .

The given information can be represented in the table as :


Vitamin A
Vitamin B
Cost
Food P
3
5
60
Food Q
4
2
80
requirement
8
11

The mixture must contain 8 units of Vitamin A and 11 units of Vitamin B.

Therefore, we have

3x+4y\geq 8

5x+2y\geq 11

Total cost is Z. Z=60x+80y

Subject to constraint,

3x+4y\geq 8

5x+2y\geq 11

x\geq 0,y\geq 0

The feasible region determined by constraints is as follows:

1627041171227

It can be seen that a feasible region is unbounded.

The corner points of the feasible region are A(\frac{8}{3},0),B(2,\frac{1}{2}),C(0,\frac{11}{2})

The value of Z at corner points is as shown :

corner points
Z=60x+80y

A(\frac{8}{3},0)
160
MINIMUM
B(2,\frac{1}{2})
160
minimum
C(0,\frac{11}{2})
440

Feasible region is unbounded, therefore 160 may or may not be the minimum value of Z.

For this, we draw 60x+80y< 160\, \, or \, \, \, 3x+4y< 8 and check whether resulting half plane has a point in common with the feasible region or not.

We can see a feasible region has no common point with. \, \, 3x+4y< 8

Hence, Z has a minimum value 160 at line segment joining points A(\frac{8}{3},0) and B(2,\frac{1}{2}) .


Question:2 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:

Let there be x cakes of first kind and y cakes of the second kind.Thus, x\geq 0,y\geq 0 .

The given information can be represented in the table as :


Flour(g)
fat(g)
Cake of kind x
200
25
Cake of kind y
100
50
Availability
5000
1000

Therefore,

200x+100y\leq 5000

\Rightarrow \, \, \, \, 2x+y\leq 50

. \, \, 25x+50y\leq 10000

\Rightarrow \, \, x+2y\leq 400

The total number of cakes, Z. Z=X+Y

Subject to constraint,

\Rightarrow \, \, \, \, 2x+y\leq 50

\Rightarrow \, \, x+2y\leq 400

x\geq 0,y\geq 0

The feasible region determined by constraints is as follows:

1627041262324

The corner points of the feasible region are A(25,0),B(20,10),C(0,20),D(0,0)

The value of Z at corner points is as shown :

corner points
Z=X+Y

A(25,0)
25

B(20,10)
30
maximum
C(0,20)
D(0,0)
20
0

minimum

The maximum cake can be made 30 (20 of the first kind and 10 of the second kind).


Question:3 A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’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?

Answer:

Let number of rackets be x and number of bats be y.

the machine time availability is not more than 42 hours.

i.e. 1.5x+3y\leq 42

craftsman’s time availability is 24 hours

i.e. 3x+y\leq 24

The factory has to work at full capacity.

Hence, 1.5x+3y= 42...............1

3x+y= 24...............2

Solving equation 1 and 2, we have

x=4\, \, and\, \, \, y=12

Thus, 4 rackets and 12 bats are to be made .

Question:3 A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’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.

(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.

Answer:

Let the number of rackets is x and the number of bats is y.

the machine time availability is not more than 42 hours.

craftsman’s time availability is 24 hours

The given information can be repreented in table as shown :


racket
bat
availability
machine time
1.5
3
42
craftman's time
3
1
24

1.5x+3y\leq 42

3x+y\leq 24

x,y\geq 0

The profit on the bat is 10 and on the racket is 20.

Z=20x+10y

The mathematical formulation is :

maximise Z=20x+10y

subject to constraints,

1.5x+3y\leq 42

3x+y\leq 24

x,y\geq 0

The feasible region determined by constraints is as follows:

1627370575766

The corner points are A(8,0),B(4,12),C(0,14),D(0,0)

The value of Z at corner points is as shown :

CORNER POINTS
Z=20x+10y

A(8,0)
160

B(4,12)
200
maximum
C(0,14)
140

D(0,0)
0

Thus, the maximum profit of the factory when it works at full capacity is 200.

Question:4 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 Rs17.50 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 maximise his profit, if he operates his machines for at the most 12 hours a day?

Answer:

Let packages of nuts be x and packages of bolts be y .Thus, x\geq 0,y\geq 0 .

The given information can be represented in table as :


bolts
nuts
availability
machine A
1
3
12
machine B
3
1
12




Profit on a package of nuts is Rs. 17.5 and on package of bolt is 7.

Therefore, constraint are

x+3y\leq 12

3x+y\leq 12

x\geq 0,y\geq 0

Z= 17.5x+7y

The feasible region determined by constraints is as follows:

1627370632779

The corner points of feasible region are A(4,0),B(3,3),C(0,4),D(0,0)

The value of Z at corner points is as shown :

Corner points
Z= 17.5x+7y

A(4,0)
70

B(3,3)
73.5
maximum
C(0,4)
28

D(0,0)
0

The maximum value of z is 73.5 at B(3,3) .

Thus, 3 packages of nuts and 3 packages of bolts should be manufactured everyday to get maximum profit.

Question:5 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 Rs 10. 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 maximise his profit? Determine the maximum profit.

Answer:

Let factory manufactures screws of type A and factory manufactures screws of type B. Thus, x\geq 0,y\geq 0 .

The given information can be represented in the table as :


screw A
screw B
availability
Automatic machine
4
6
4\times 60=240
hand operated machine
6
3
4\times 60=240




Profit on a package of screw A is Rs.7 and on the package of screw B is 10.

Therefore, the constraint is

4x+6y\leq 240

6x+3y\leq 240

x\geq 0,y\geq 0

Z= 7x+10y

The feasible region determined by constraints is as follows:

1627370732345

The corner points of the feasible region are A(40,0),B(30,20),C(0,40),D(0,0)

The value of Z at corner points is as shown :

Corner points
Z= 7x+10y

A(40,0)
280

B(30,20)
410
maximum
C(0,40)
400

D(0,0)
0

The maximum value of z is 410 at B(30,20) .

Thus, 30 packages of screw A and 20 packages of screw B should be manufactured every day to get maximum profit.

Question:6 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 maximise his profit?

Answer:

Let the cottage industry manufactures x pedestal lamps and y wooden shades. Thus, x\geq 0,y\geq 0 .

The given information can be represented in the table as :


lamps
shades
availability
machine (h)
2
1
12
sprayer (h)
3
2
20




Profit on a lamp is Rs. 5 and on the shade is 3.

Therefore, constraint is

2x+y\leq 12

3x+2y\leq 20

x\geq 0,y\geq 0

Z= 5x+3y

The feasible region determined by constraints is as follows:

1627370794134

The corner points of the feasible region are A(6,0),B(4,4),C(0,10),D(0,0)

The value of Z at corner points is as shown :

Corner points
Z= 5x+3y

A(6,0)
30

B(4,4)
32
maximum
C(0,10)
30

D(0,0)
0

The maximum value of z is 32 at B(4,4) .

Thus, 4 shades and 4 pedestals lamps should be manufactured every day to get the maximum profit.

Question:7 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 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?

Answer:

Let x be Souvenirs of type A and y be Souvenirs of type B .Thus, x\geq 0,y\geq 0 .

The given information can be represented in table as :


Type A
Type B
availability
cutting
5
8
(3\times 60)+20=200
asembling
10
8
4\times 60=240




Profit on type A Souvenirs is Rs. 5 and on type B Souvenirs is 6.

Therefore, constraint are

5x+8y\leq 200

10x+8y\leq 240

x\geq 0,y\geq 0

Z=5x+6y

The feasible region determined by constraints is as follows:

1627370879345

The corner points of feasible region are A(24,0),B(8,20),C(0,25),D(0,0)

The value of Z at corner points is as shown :

Corner points
Z=5x+6y

A(24,0)
120

B(8,20)
160
maximum
C(0,25)
150

D(0,0)
0

The maximum value of z is 160 at B(8,20) .

Thus,8 Souvenirs of type A and 20 Souvenirs of type B should be manufactured everyday to get maximum profit.

Question:8 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 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

Answer:

Let merchant plans has personal computers x desktop model and y portable model

.Thus, x\geq 0,y\geq 0 .

The cost of desktop model is cost Rs 25000 and portable model is Rs 40000.

Merchant can invest Rs 70 lakhs maximum.

25000x+40000y\leq 7000000

5x+8y\leq 1400

the total monthly demand of computers will not exceed 250 units.

x+y\leq 250

profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

Total profit = Z , Z=4500x+5000y

The mathematical formulation of given problem is :
5x+8y\leq 1400

x+y\leq 250

x\geq 0,y\geq 0

Z=4500x+5000y

The feasible region determined by constraints is as follows:

1627377288762

The corner points of feasible region are A(250,0),B(200,50),C(0,175),D(0,0)

The value of Z at corner points is as shown :

Corner points
Z=4500x+5000y

A(250,0)
1125000

B(200,50)
1150000
maximum
C(0,175)
875000

D(0,0)
0

The maximum value of z is 1150000 at B(200,50) .

Thus, merchant should stock 200 desktop models and 50 portable models to get maximum profit.

Question:9 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 food 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. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

Answer:

Let diet contain x unit of food F1 and y unit of foof F2 .Thus, x\geq 0,y\geq 0 .

The given information can be represented in table as :


Vitamin
minerals
cost per unit
foof F1
3
4
4
food F2
6
3
6

80
100

Cost of food F1 is Rs 4 per unit and Cost of food F2 is Rs 6 per unit

Therefore, constraint are

3x+4y\geq 4

6x+3y\geq 6

x\geq 0,y\geq 0

Z= 4x+6y

The feasible region determined by constraints is as follows: 1627377385273

We can see feasible region is unbounded.

The corner points of feasible region are A(\frac{80}{3},0),B(24,\frac{4}{3}),C(0,\frac{100}{3})

The value of Z at corner points is as shown :

Corner points
Z= 4x+6y

A(\frac{80}{3},0)
106.67

B(24,\frac{4}{3}),
104
minimum
C(0,\frac{100}{3})
200
maximum


Feasible region is unbounded , therefore 104 may or may not be minimum value of Z .

For this we draw 4x+6y< 104 or 2x+3y< 52 and check whether resulting half plane has point in common with feasible region or not.

We can see feasible region has no common point with 2x+3y< 52 .

Hence , Z has minimum value 104.

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 atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costs Rs 5/kg, 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:

Let farmer buy x kg of fertilizer F1 and y kg of F2 .Thus, x\geq 0,y\geq 0 .

The given information can be represented in table as :


Nitrogen
phosphoric acid
Cost
F1
10
6
6
F2
5
10
5
requirement
14
14

F1 contain 10% nitrogen and F2 contain 5% nitrogen .Farmer requires atleast 14 kg of nitrogen

10\%x+5\%y\geq 14

\frac{x}{10}+\frac{y}{20}\geq 14

2x+y\geq 280

F1 contain 6% phophoric acid and F2 contain 10% phosphoric acid .Farmer requires atleast 14 kg of nitrogen

6\%x+10\%y\geq 14

\frac{6x}{100}+\frac{y}{20}\geq 14

3x+56y\geq 700

Total cost is Z . Z=6x+5y

Subject to constraint ,

2x+y\geq 280

3x+56y\geq 700

x\geq 0,y\geq 0

Z=6x+5y

The feasible region determined by constraints is as follows:

1627377453836

It can be seen that feasible region is unbounded.

The corner points of feasible region are A(\frac{700}{3},0),B(100,80),C(0,280)

The value of Z at corner points is as shown :

corner points
Z=6x+5y

A(\frac{700}{3},0)
1400

,B(100,80)
1000
minimum
C(0,280)
1400

Feasible region is unbounded , therefore 1000 may or may not be minimum value of Z .

For this we draw 6x+5y< 1000 and check whether resulting half plane has point in common with feasible region or not.

We can see feasible region has no common point with 6x+5y< 1000 .

Hence , Z has minimum value 1000 at point ,B(100,80)

Question:11 The corner points of the feasible region determined by the following system of linear inequalities:

2x+y \leq 10,x+3y \leq 15,x,y\geq 0 are (0,0),(5,0),(3,4) and (0,5) . Let Z=px+qy, where p,q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is

(A) p=q

(B)p=2q

(C)p=3q

(D)q=3p

Answer:

The maximum value of Z is unique.

It is given that maximum value of Z occurs at two points (3,4)\, \, and\, \, \, (0,5) .

\therefore Value of Z at (3,4) =value of Z at (0,5)

\Rightarrow \, \, \, p(3)+q(4)=p(0)+q(5)

\Rightarrow \, \, \, 3p+4q=5q

\Rightarrow \, \, \, q=3p

Hence, D is correct option.


NCERT solutions for class 12 maths chapter 12 linear programming-Miscellaneous Exercise

Question:1 Reference of Example 9 (Diet problem): 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 atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol.

How many packets of each food should be used to maximise the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

Answer:

Let diet contain x packets of food P and y packets of food Q. Thus, x\geq 0,y\geq 0 .

The mathematical formulation of the given problem is as follows:

Total cost is Z . Z=6x+3y

Subject to constraint,

4x+y\geq 80

x+5y\geq 115

x\geq 0,y\geq 0

The feasible region determined by constraints is as follows:

1627377546036

The corner points of feasible region are A(15,20),B(40,15),C(2,72)

The value of Z at corner points is as shown :

corner points
Z=6x+3y

A(15,20)
150
MINIMUM
B(40,15)
285
maximum
C(2,72)
228

Hence, Z has a maximum value of 285 at the point B(40,15) .

to maximise the amount of vitamin A in the diet, 40 packets of food P and 15 packets of food Q should be used. The maximum amount of vitamin A is 285 units.

Question:2 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 element B and 2 units of element C. Brand Q costing Rs 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?

Answer:

Let farmer mix x bags of brand P and y bags of brand Q. Thus, x\geq 0,y\geq 0 .

The given information can be represented in the table as :


Vitamin A
Vitamin B
Cost
Food P
3
5
60
Food Q
4
2
80
requirement
8
11

The given problem can be formulated as follows:

Therefore, we have

3x+1.5y\geq 18

2.5x+11.25y\geq 45

2x+3y\geq 24

Z=250x+200y

Subject to constraint,

3x+1.5y\geq 18

2.5x+11.25y\geq 45

2x+3y\geq 24

x\geq 0,y\geq 0

The feasible region determined by constraints is as follows:

1627377686090

The corner points of the feasible region are A(18,0),B(9,2),C(3,6),D(0,12)

The value of Z at corner points is as shown :

corner points
Z=250x+200y

A(18,0)
4500

B(9,2)
2650

C(3,6)
1950
minimum
D(0,12)
2400

Feasible region is unbounded, therefore 1950 may or may not be a minimum value of Z. For this, we draw 250x+200y< 1950 and check whether resulting half plane has a point in common with the feasible region or not.

We can see a feasible region has no common point with 250x+200y< 1950 .

Hence, Z has a minimum value 1950 at point C(3,6) .

Question:3 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 contents of one kg food is given below:

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?

Answer:

Let mixture contain x kg of food X and y kg of food Y.

Mathematical formulation of given problem is as follows:

Minimize : z=16x+20y

Subject to constraint ,

x+2y\geq 10

x+y\geq 6

3x+y\geq 8

x,y\geq 0

The feasible region determined by constraints is as follows:

1627377734165

The corner points of feasible region are A(10,0),B(2,4),C(1,5),D(0,8)

The value of Z at corner points is as shown :

corner points
z=16x+20y

A(10,0)
160

B(2,4)
112
minimum
C(1,5)
116

D(0,8)
160

The feasible region is unbounded , therefore 112 may or may not be minimum value of Z .

For this we draw 16x+20y< 112 and check whether resulting half plane has point in common with feasible region or not.

We can see feasible region has no common point with 16x+20y< 112 .

Hence , Z has minimum value 112 at point B(2,4)

Question:4 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:

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.

Answer:

Let x and y toys of type A and type B.

Mathematical formulation of given problem is as follows:

Minimize : z=7.5x+5y

Subject to constraint ,

2x+y\leq 60

x\leq 20

2x+3y \leq 120

x,y\geq 0

The feasible region determined by constraints is as follows:

1627377785282

The corner points of feasible region are A(20,0),B(20,20),C(15,30),D(0,40)

The value of Z at corner points is as shown :

corner points
z=7.5x+5y

A(20,0)
150

B(20,20)
250

C (15,30)
262.5
maximum
D(0,40)
200

Therefore 262.5 may or may not be maximum value of Z .

Hence , Z has maximum value 262.5 at point C (15,30)

Question:5 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. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

Answer:

Let airline sell x tickets of executive class and y tickets of economy class.

Mathematical formulation of given problem is as follows:

Minimize : z=1000x+600y

Subject to constraint ,

x+y\leq 200

x\geq 20

y-4x\geq 0

x,y\geq 0

The feasible region determined by constraints is as follows:

1627377831247

The corner points of feasible region are A(20,80),B(40,160),C(20,180)

The value of Z at corner points is as shown :

corner points
z=1000x+600y

A(20,80)
68000

B(40,160)
136000
maximum
C (20,180)
128000


therefore 136000 is maximum value of Z .

Hence , Z has maximum value 136000 at point B(40,160)

Question:6 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:

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

Answer:

Let godown A supply x and y quintals of grain to shops D and E respectively. Then , (100-x-y) will be supplied to shop F. Requirements at shop D is 60 since godown A supply x .Therefore remaining (60-x) quintals of grain will be transported from godown B.

Similarly, (50-y) quintals and 40-(100-x-y)=(x+y-60) will be transported from godown B to shop E and F respectively. The problem can be represented diagrammatically as follows:

1627377923876

x,y\geq 0 and 100-x-y\geq 0

x,y\geq 0 and x+y\leq 100

60-x\geq 0,50-y\geq 0\, \, \, and\, \, x+y-60\geq 0

\Rightarrow \, \, \, \, x\leq 60,y\leq 50,x+y\geq 60

Total transportation cost z is given by ,

z=6x+3y+2.5(100-x-y)+4(60-x)+2(50-y)+3(x+y-60)

z=2.5x+1.5y+410

Mathematical formulation of given problem is as follows:

Minimize : z=2.5x+1.5y+410

Subject to constraint ,

x+y\leq 100

x\leq 60

y\leq 50

x+y\geq 60

x,y\geq 0

The feasible region determined by constraints is as follows:

1627377885300

The corner points of feasible region are A(60,0),B(60,40),C(50,50),D(10,50)

The value of Z at corner points is as shown :

corner points
z=2.5x+1.5y+410

A(60,0)
560

B(60,40)
620

C(50,50)
610

D(10,50)
510
minimum

therefore 510 may or may not be minimum value of Z .

Hence , Z has miniimum value 510 at point D(10,50)

Question:7 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 distances (in km) between the depots and the petrol pumps is given in the following table:

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?

Answer:

Let x and y litres of oil be supplied from A to petrol pump,D and E. Then , (7000-x-y) will be supplied from A to petrol pump F.

Requirements at petrol pump D is 4500 L. since x L A are transported from depot A,remaining 4500-x L will be transported from petrol pump B

Similarly, (3000-y)L and 3500-(7000-x-y)=(x+y-3500) L will be transported from depot B to petrol E and F respectively.

The problem can be represented diagrammatically as follows:

1627377981082

x,y\geq 0 and 7000-x-y\geq 0

x,y\geq 0 and x+y\leq 7000


4500-x\geq 0,3000-y\geq 0\, \, \, and\, \, x+y-3500\geq 0

\Rightarrow \, \, \, \, x\leq 4500,y\leq 3000,x+y\geq 3500

Cost of transporting 10 L petrol =Re 1

Cost of transporting 1 L petrol =\frac{1}{10}

Total transportation cost z is given by ,

z=\frac{7}{10}x+\frac{6}{10}y+\frac{3}{10}(7000-x-y)+\frac{3}{10}(4500-x)+\frac{4}{10}(3000-y)+\frac{2}{10}(x+y-3500)

z=0.3x+0.1y+3950

Mathematical formulation of given problem is as follows:

Minimize : z=0.3x+0.1y+3950

Subject to constraint ,

x+y\leq 7000

x\leq 4500

y\leq 3000

x+y\geq 3500

x,y\geq 0

The feasible region determined by constraints is as follows:

1627378013511

The corner points of feasible region are A(3500,0),B(4500,0),C(4500,2500),D(4000,3000),E(500,3000)

The value of Z at corner points is as shown :

corner points
z=0.3x+0.1y+3950

A(3500,0)
5000

B(4500,0)
5300

C(4500,2500)
5550

E(500,3000)
4400
minimum
D(4000,3000)
5450

Hence , Z has miniimum value 4400 at point E(500,3000)

Question:8 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 minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?


Answer:

Let fruit grower use x bags of brand P and y bags of brand Q.

Mathematical formulation of given problem is as follows:

Minimize : z=3x+3.5y

Subject to constraint ,

x+2y\geq 240

x+0.5y\geq 90

1.5x+2y\geq 310

x,y\geq 0

The feasible region determined by constraints is as follows:

1627378058480

The corner points of feasible region are A(140,50),C(40,100),B(20,140)

The value of Z at corner points is as shown :

corner points
z=3x+3.5y

A(140,50)
595

B(20,140)
550

C(40,100)
470
minimum

Therefore 470 is minimum value of Z .

Hence , Z has minimum value 470 at point C(40,100)

Question:9 Reference of Que 8 : 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 maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Kg per bag

Brand A
Brand P
Nitrogen
3
3.5
Phosphoric Acid
1
2
Potash
3
1.5
Chlorine
1.5
2


Answer:

Let fruit grower use x bags of brand P and y bags of brand Q.

Mathematical formulation of given problem is as follows:

Maximize : z=3x+3.5y

Subject to constraint ,

x+2y\geq 240

x+0.5y\geq 90

1.5x+2y\geq 310

x,y\geq 0

The feasible region determined by constraints is as follows:

1627378216485

The corner points of feasible region are B(20,140),A(140,50),C(40,100)

The value of Z at corner points is as shown :

corner points
z=3x+3.5y

A(140,50)
595
maximum
B(20,140)
550

C(40,100)
470
minimum

therefore 595 is maximum value of Z .

Hence , Z has minimum value 595 at point A(140,50)

Question:10 A toy company manufactures two types of dolls, A and B. Market research 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 Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

Answer:

Let x and y be number of dolls of type A abd B respectively that are produced per week.

Mathematical formulation of given problem is as follows:

Maximize : z=12x+16y

Subject to constraint ,

x+y\leq 1200

y\leq \frac{x}{2}\Rightarrow x\geq 2y

x-3y\leq 600

x,y\geq 0

The feasible region determined by constraints is as follows:

1627378272830

The corner points of feasible region are A(600,0),B(1050,150),C(800,400)

The value of Z at corner points is as shown :

corner points
z=12x+16y

A(600,0)
7200

B(1050,150)
15000

C(800,400)
16000
Maximum

Therefore 16000 is maximum value of Z .

Hence , Z has minimum value 16000 at point C(800,400)

If you are looking for ncert exercise solutions of linear programming class 12 then they are listed below.

More about NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

In this article, you will get NCERT solutions for class 12 and you can also use the webpage pdf downloader tool to download NCERT Solutions for Class 12 maths chapter 12 PDF from here. Generally, one five marks question is asked from Linear programming class 12 in the final examination. This chapter has around 21 lpp class 12 questions, here you will find all solutions very easily. These NCERT Class 12 maths solutions chapter 12 are explained in a detailed manner that will help you in scoring high scores.

Let's take an NCERT problem - A furniture dealer deals in only two items–chairs and tables. Has storage space of at most 60 pieces and He has Rs 50,000 to invest. A chair costs Rs 500 and A table costs Rs 2500. He estimates that from the sale of one chair, he can make a profit of Rs 75 and that from the sale of one table a profit of Rs 250. He wants to know how many chairs and tables he should buy from the available money so as to maximise his total profit, assuming that he can sell all the items which he buys.

The problems that involve maximising cost and minimising profit are called optimization problems. This ch 12 maths class 12 is quite important because it includes such concepts. The above given problem is an example of linear programming.

Also read,

Linear Programming Class 12 - Topics

12.1 Introduction

12.2 Linear Programming Problem and its Mathematical Formulation

12.2.1 Mathematical formulation of the problem

12.2.2 Graphical method of solving linear programming problems

12.3 Different Types of Linear Programming Problems

NCERT solutions for class 12 maths - Chapter Wise

Linear Programming Class 12 NCERT Solutions - Key Features

  1. Detailed explanations: The class 12 maths ch 12 question answer provided in NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming are explained in a detailed and step-by-step manner. This helps students to understand the concepts better and makes it easy for them to solve similar problems.

  2. Easy to understand: The maths chapter 12 class 12 solutions are written in simple language, making it easy for students to understand and learn the concepts. The solutions are designed to cater to the needs of students of all learning levels.

  3. Covers all the topics: The class 12 linear programming solutions cover all the topics in Chapter 12 Linear Programming of Class 12 Maths. This helps students to have a comprehensive understanding of the chapter.

JEE Main Important Physics formulas

As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters

Exercise-wise solutions: The maths chapter 12 class 12 solutions are provided exercise-wise, which helps students to focus on specific problems and concepts that they find difficult.

NCERT solutions for class 12 subject wise

class wise NCERT Solutions

Benefits of NCERT Solutions for Class 12 maths chapter 12

  • NCERT Class 12 maths solutions chapter 12 are prepared and explained in detailed form. It makes these Class 12 maths chapter 12 NCERT solutions easy to understand.

  • NCERT Solutions for Class 12 maths chapter 12 PDF is very helpful for the preparation of this chapter.

  • These NCERT Solutions for Class 12 maths chapter 12 will give you a new way to solve questions.

  • Miscellaneous exercise is quite important, so to develop a grip on the concepts. In NCERT Class 12 maths solutions chapter 12 linear programming, you will get solutions for miscellaneous exercise too.

NCERT Books and NCERT Syllabus

Happy Reading!

Frequently Asked Question (FAQs)

1. How can NCERT solutions be helpful in CBSE board exam ?

As most the questions in CBSE board exam are asked directly from NCERT textbook, so one must know the NCERT well but only knowing the answer is not guaranteed to score good marks in the exam. One should know how to answer in board exams in order to get good marks. NCERT solutions are provided by the experts who know how best to write answers in the board exam in order to get good marks. Interested students can study linear programming class 12 solutions pdf both online and offline.

2. How is the Linear Programming chapter helpful in solving real-life problems?

Yes, linear programming is useful in formulating these real-life problems into a mathematical model and then solving them. Some optimisation problems like maximizing the profits and minimizing the cost can be solved by linear programming. concepts of liner programming important in board and engineering exam therefore this chapter is very important. you should solve ncert exercise to get command on concepts.

3. Does CBSE provide the solutions of NCERT for class 12 maths ?

No, CBSE doesn’t provide NCERT solutions for any class or subject. but you can find solutions free from careers360 official website. these solutions are explained in details by our expert team in very simple format so that students can understand these very easily.

4. Where can I find the complete solutions of NCERT for class 12 maths ?

Here, you will get the detailed NCERT solutions for class 12 maths  by clicking on the links that are listed above in this article, for comfortability of students these are listed subjects as well as chapter wise. or you can find these solutions from careers360 official website.

5. What is the weightage of the chapter Linear Programming for CBSE board exam ?

Generally, one question of 5 marks is asked from this chapter in the CBSE 12th board final exam. if you want to score five out of five then it demand practice and in depth understanding of concepts therefore it ncert solutions and ncert exercise are recommended to students.

6. What are the important topics in chapter Linear Programming ?

Linear programming problem and its mathematical formulation, graphical method of solving linear programming problems are the important topics of this chapter. you can refer NCERT solutions to get in depth understanding and getting hold on concepts.

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Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

hello,

Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

I hope this was helpful!

Good Luck

Hello dear,

If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.


As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.


Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.


Believe in Yourself! You can make anything happen


All the very best.

Hello Student,

I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects  and  we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.

You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.

All the best.

If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.

Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary. 

4 Jobs Available
Data Administrator

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available
Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available
Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available
Remote Sensing Technician

Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive. 

Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.

3 Jobs Available
Geotechnical engineer

The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction. 

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions. 

3 Jobs Available
Cartographer

How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.

3 Jobs Available
Budget Analyst

Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.

4 Jobs Available
Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Investment Banker

An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

3 Jobs Available
Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

3 Jobs Available
Finance Executive
3 Jobs Available
Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available
Plumber

An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.

2 Jobs Available
Construction Manager

Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.

2 Jobs Available
Urban Planner

Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities. 

Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.

2 Jobs Available
Highway Engineer

Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.

2 Jobs Available
Environmental Engineer

Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems. 

2 Jobs Available
Naval Architect

A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability. 

2 Jobs Available
Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available
Veterinary Doctor
5 Jobs Available
Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available
Speech Therapist
4 Jobs Available
Gynaecologist

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth. 

4 Jobs Available
Oncologist

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available
Audiologist

The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.

3 Jobs Available
Hospital Administrator

The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.

2 Jobs Available
Actor

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs. 

4 Jobs Available
Acrobat

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.

Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available
Radio Jockey

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
Choreographer

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

2 Jobs Available
Videographer
2 Jobs Available
Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications. 

2 Jobs Available
Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

5 Jobs Available
Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

3 Jobs Available
Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
Linguist

Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning). 

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian

2 Jobs Available
Public Relation Executive
2 Jobs Available
Travel Journalist

The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.

2 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
QA Manager
4 Jobs Available
Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Production Manager
3 Jobs Available
Merchandiser
2 Jobs Available
QA Lead

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

2 Jobs Available
Metallurgical Engineer

A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment. 

2 Jobs Available
Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

4 Jobs Available
AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

4 Jobs Available
QA Manager
4 Jobs Available
Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
ITSM Manager
3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

3 Jobs Available
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