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Edited By Kuldeep Maurya | Updated on Jan 25, 2022 02:53 PM IST

The RD Sharma class 12th exercise 29.4 book is one of the best NCERT solutions that students will ever find. So far, hundreds of high school students have placed their trust in RD Sharma Solutions and have been able to experience the endless benefits of the book. The RD Sharma class 12 chapter 29 exercise 29.4 book is easily the most informative NCERT solution that will be an excellent guide for school students.

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Linner Programming Exercise 29.4 Question 1

Distance travelled at speed of and at a speed of

Man spend when speed is 23km/hr and Rs 5/km when speed Rs 50 Km/hr.

Let he drives x Km at sped of 25 Km/hr. and y Km/hr. at a speed of 40 Km/hr.

Let Z be total distance travelled by him so,

Since he spend Rs 2 per Km on petrol when speed is 25 Km/hr. and Rs5 per Km on petrol when speed is 40 Km/hr, So, expense on x Km and y Km or Rs2x and Rs2y respectively. But he has only Rs100, 50.

2x+5y≤100 (1st constraint)

Time taken to travel

Given he has 1 hr. to travel, so,

Hence, mathematical formulation of LPP is find x and y which

Maximum

Subject to constraint

since distance cannot be less than zero

The corresponding line is

....(i)

…(ii)

Point is obtain by solving

Points | |

And

Distance travelled at speed of and at a speed of

Maximum distance

Linear Programming Exercise 29.4 Question 2

Maximum Profit = Rs 40 (A = 4, B = 4)

Let and y are required quantity of x and y.

He makes a profit of ?6.00 on item A and ?4.00 on item B.

So, profits of one X item =s of type A and Y item of type B are 6x and Rs 4y respectively.

Maximum (Where x and y are required quantity)

Given machine I works 1 hour and 2 hours on item A and B respectively., so x number of item A and y number of item B need x hour and 2y hours on machine I respectively, but machine I works at most 12 hours so,

( first constraint)

Given, machine II works 2 hours and 1 hour on item A and B respectively, but machine II works maximum 12 hours, So

(second constraint)

Given, machine III works 1 hour and hour on one item A and B respectively, but machine III works maximum 5 hours, So

(third constraint)

The required LPP

Subject to constraint

since the number of item A and B not be less than zero

The corresponding line is

Shaded region represent feasible region

Points | |

Hence z is maximum at x = 4 and y = 4

Required number of product A = 4, B = 4

Maximum profit=Rs 40

Linear Programming Exercise 29.4 Question 3

A should work for 5 days and B should for 3 days.

Let tailor A and B work for x and y days respectively.

Tailor A and B earn Rs15 and Rs20 respectively.

Min

Since tailor A and B stitch 6 and 10 shirts respectively in a day. But it is desired to produce 60 shirts at least. So,

(First constraint)

Since tailor A and B stitch 4pants per day but it is desired to produce at least 32 pants so

(Second constraint)

Hence the required LPP .

Min

Subject to constraints

Since x and y not be less than zero.

The corresponding equation is

Point P(5,3) obtain by solving (i) and (ii)

Unbounded shaded region represents feasible region with corner points

Points | |

Tailor A should work for 5 days and B should for 3 days.

Linear Programming exercise 29.4 question 4

The factory produces 30 and 20 packages of screw A and screw B to get the maximum profit.

let factory manufacture x screws of type A and y screw of type B.

The information can be competed in table follow

Screw A | Screw B | Availability | |

Automatic machine (min) | |||

Hand operated machine (min) |

Let the factory manufacture x screws of type A and y screws of type B on each day.

The profit on a package of screws A is Rs 7 and on the package of screws B is Rs 10.

The constraints are

Total profit

The required LPP

Max

Subject to constraints

....(i)

....(ii)

The feasible region determined by the system of constraints is

Points (30,20) solution obtain by solving (i) and (ii). The corner points A(40,0), B(30,20) and C(0,40).

Points | |

Thus, the factory should produce 30 packages of screw A and 20 packages of screw B to get the maximum profit of Rs41.

Linear Programming exercise 29.4 question 5

Required number belt A is 200, while B is 600, maximum profit Rs1300.

Let required number of belt A and B be x and y.

Profit on belt A and B be Rs2 and Rs1.50

Let z be total profit

Where x and y be required number of belt A and belt B.

Since each belt of type A required twice as much time as belt B. let each belt B require to make, so A requires 2 hours but total time available is equal to production 1000 belt B that is 1000 hours, so

(first constraint)

Given supply of lather only for 800 belts per day both A and B combined, so

(second constraint)

Buckets available for A is only 400 and for B only 700, so

(third constraint)

(forth constraint)

Hence the required LPP

Subject to constraints

....(i)

....(ii)

.....(iii)

...(iv)

,number of belt cannot less than zero

The feasible region determined by the system of constraints

Obtain point Q(200,600) by solving (i) and (ii) and P(400,200) by solving (i) and (iii) and R(100,700) by solving (ii) and (iv)

The shading region is

Corner Points | Value of |

0 | |

Linear Programming exercise 29.4 question 6

Maximum profit =350

Deluxe model = 10

Ordinary model = 20

Let required number of deluxe and ordinary model be x and y.

Since, Profit on each model of deluxe and ordinary type of Rs15 and Rs10 respectively.

Let z be total profit then

Where x and y are required deluxe and ordinary model

Since each deluxe and ordinary model required 2 and 1 hour of skilled men, but twice available skilled men is 5×8 = 4 hours so,

(first constraint)

Given each deluxe and ordinary model require 2 and 3 hour of semi-skilled men, but total ratable by semi -skilled men is 100×8 = 80 hours so

(second constraint)

Hence the required LPP

Subject to constraints

,since number of ordinary model cannot less than zero

The feasible region obtain by the system of constraint

Point (10,20) obtain by solving (i) and (ii).

The corner point of feasible region is

Corner Points | Value of |

Required number of deluxe model = 10 and required number of ordinary model = 20

Maximum profit = 350

Linear Programming Exercise 29.4 Question 7

15 tea cups of type A and 30 tea cups of type B.

Let required number of tea cups of type A and B are x and y.

Profit on each tea cups of type A and B are 75 paisa and 50 paisa.

Let the total profit of on tea cups be z.

Where x and y are the required number of tea cups.

Since each tea cup of type A and B require the work machine 1 for 12 and 6 min but total time available on machine I is 6×60 = 360min

Since each tea cup of type A and B require to work machine II for 6 and 0 min but total time available for machine II is 6×60=360min.

Since each tea cup of type A and B require the work machine III for 6 and 9 min. but total time available for machine III is 6×60=360min.

The required LPP is

Max

Subject to constraints

...(i)

....(ii)

...(iii)

The feasible region obtains by the system of constraint

Point (20,20) and Q(15,30) is obtain by solving (ii) and (iii) and (i) and (iii).

Corner Points | Value of |

15 tea cups of type A and 30 tea cups of type B, we need to maximize the profit.

Linear Programming Exercise 29.4 Question 8

Output is maximum when type A = 4, type B = 3 or type A = 6, type B = 0.

Let required number of machine A and B are x and y.

Production of each machine A and B are 60 and 40 units daily.

Let z donate total output daily, so

Since each machine of type A and type B require 100sq.m and 1200sq.m but total area available for machine is 76000sq.m

Each machine of type A and B require 12 men and 8 men to work respectively. But total 72 men available for work so

The required LPP is

Max

Subject to constraints

The feasible region obtains by the system of constraint

P(4,3) is obtain by solving (i) and (ii).

are the shaded region

Corner Points | Value of |

Output is maximum when 4 machine of type A and 3 machines of type B or 6 machine of type A and no machine of type B.

Linear Programming Exercise 29.4 Question 9

Max profit =230 at type A=2 , type B=3.

Let number of goods A and B are x and y respectively.

Profit of each A and B are Rs40 and Rs50

Max

Since each A and B require 3gm and 1gm of silver but total silver available are 9gm.

Since each A and B require 1gm and 2gm of gold but total gold available are 8gm.

The required LPP is

Max

Subject to constraints

The feasible region obtains by the system of constraints

P(2,3) is obtain by solving (i) and (ii).

are the shaded region

Corner Points | Value of |

Hence maximum profit = 230, number of goods of type A =2, type B =3

Linear Programming exercise 29.4 question 10

Maximum profit Rs22.2.

Let daily production of chairs and table be x and y.

Profit on each chair and table are Rs3 and Rs5.

Let 2 be total profit on table and chair

Max [when x and y are daily production on table and chair]

Since each chair and table require 2hrs and 4hrs on a machine A but maximum time available on machine A be 16 hrs.

Since each chair and table require 6hrs and 2hrs on machine B, but maximum time available on machine B be 30 hrs.

The required LPP is

Max

Subject to constraints

The feasible region obtains by the system of constraints

obtain by solving equation (i) and (ii).

are the shaded region

Corner Points | Value of |

Linear Programming exercise 29.4 question 11

Maximum profit = Rs. 2025 could be obtained if 45 units of chairs and no units of table are produce.

Use graph and simultaneous equation

Resources available 400square feet of teak wood and 450 man hours.

Let required production of chairs and tables be z=x and y respectively.

Since, profits of each chair and table is Rs45 and Rs80 respectively

So, profit on x number of type A and y number of type B are 45x and 80y respectively.

Let z denotes total output daily so,

Since each chair and table require 5 sq. and 80sq.ft of wood respectively. So, x number of chair and y number of table require 5x and 80y sq. of wood respectively. But 400sq.ft of wood available

So,

.

(first constraints)

Since, each chair ad table requires 10 and 25 man hours respectively, so, x number of chair and y number of tables are require 10x and 25y men hours respectively. But, only 450 hours are available.

So,

(second constraints)

Hence mathematical formulation of the given LPP is

Max

Subject to constraints

[since production of chair and table can not be less than 0]

Region : line meets the axes at A980,0), B(1,20) respectively.

Region containing the origin represents as origin satisfies

Region : line meets the axes at C(45,0), D(0,20) respectively.

Region containing the origin represents as origin satisfies

Regionx, : it represents the first quadrant.

The corner points are 0(0,0), D(1,18),C(45,0).

The value of z at these corner points are as follows.

Corner Points | Value of |

Thus, maximum profit of Rs2025 is obtained when 45 units of chairs and no units of tables are produced.

Linear Programming exercise 29.4 question 12

could be obtained when no units of product A and 125 units of product B are manufactured.

Form Equation and solve graphically.

A firm manufactures two products A and B. Each product is processed on two machines. M1 and M2 . Product A requires 4 minutes of processing time on M1 and 8 min on M2 ; Product B requires 4 min on M1 and 4 min on M2

Let required production of product A and B be x and y respectively.

Since profit on each product A and B are Rs. 3 and Rs. 4 respectively. So, profits on x number of type A and y number of type B are 3x and 4y respectively.

Let Z denotes total output daily, so,

Since, each A and B requires 4 minutes each on machine M1 . So, x of type A and y of type B require 4x and 4y minutes respectively, But total number available on machine M1 is 8 hours 20 minutes is equal to 500 minutes.

So,

{ first constraint}

Since each A and B requires 8 minutes and 4 minutes on machine M2 So , 2x of type A and y of type B require 8x and 4y minutes respectively. But

Total time available on Machine M1 is 10 hours = 600 minutes

So,

{Second constraint}

Hence, mathematical formulation of the given L.P.P is,

Max

Subject to constraints,

[Since production of chairs and tables cannot be less than 0]

Region: : Line meets the axes at respectively.

Region containing the origin represents as origin satisfies

Region : Line meets the axes at respectively.

Region containing the origin represents as origin satisfies

Region : it represents the first quadrant.

The corner points are 0(0,0),B(0,125), ?(25,100),C(75,0).

The value of z at these corner points are as follows.

Corner Points | |

Thus, maximum profit is Rs500 obtained when no units of product A and 125 units of product B are manufactured.

Linear Programming Exercise 29.4 Question 13

when 10 units of item A and 65 units of item B are manufactured.

Form Linear Equations and solve graphically.

A firm manufacturing two types of electric items A and B can make a profit of Rs.20 per unit of A and E 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month.

Let x units of item A and y units of item B are manufactured.

Numbers of items cannot be negative.

Therefore,

The given information can be tabulated as follows:

Product | Motors | Transformers |

Availability |

Therefore, the constraints are

A and B make profit of Rs20 and Rs30 per unit respectively.

Therefore, Profit gained from x units of item A and y units od Item B is Rs. 20 x and 30uy respectively.

which according to question is to be maximized

Thus, mathematical formulation of the given L.P.P is,

Max

Subject to constraints,

; Region represented by .

The line meets the axes at A(70,0) , B(0,105) respectively

Region containing the origin represents as origin satisfies

Region : The line meets the axes at C75,0, D0,75 respectively.

Region containing the origin represents as origin satisfies

Y=65 is the line passing through the point ?(0,65) and is parallel to x axis

Region it represents the first quadrant.

Scale: On x-axis, 1 big division =20 units

On y-axis, 1 big division =20 units

The corner points are 0(0,0),?(0,65),G(10,65),F(60,15) and A(70,0).

The value of z at these corner points are as follows.

Corner Points | |

Thus, maximum profit is Rs 2150 obtained when 10 units of item A and 65 units of item B are manufactured.

Linear Programming Exercise 29.4 Question 14

when two units of first product and 4 units of second product were manufactured.

Form Linear Equation and solve graphically.

A factory uses three different resources for the manufacture of two different products , 20 units of the resources A , 12 units of B and 16 units of C. 1 unit of the first Product requires 2, 2 and 4 units of the respective resources and 1 unit of the second product requires 4,2 and 0 units of respective resources. It is known that the first product gives a profit of 2 monetary units per unit and the second 3.

Let number of product I and product II are x and y respectively.

Since profit on each product I and II requires 2 an 4 units of resources A:50 , x units of product I and y units of product II requires 2x and 4y minutes respectively. But maximum available quantity of resources A is 20 units.

So,

{ first constraint}

Since each I and II requires 2 and 2 units of resources B . So, x units of product I and y units of product II requires 2x and 2y minutes respectively. But maximum available quantity of resources A is 12 units

So ,

{Second constraint}

Since each units of product I requires 4 units of resources C . It is not required product II. So x units of product I require 4x units of resource C . But maximum available quantity of resource C is 16 units.

So ,

{Third constraint}

Hence mathematical formulation of the given L.P.P is,

Max

Subject to constraints,

[Since production of I AND II cannot be less than 0]

Region represented by The line meets the axes at A(10,0) , B(0,5) respectively.

Region containing the origin represents as origin satisfies

Region represented by Line meets the axes at C(6,0), D(0,6) respectively.

Region containing the origin represents as origin satisfies

Region it represents the first quadrant.

The corner points are 0(0,0),B(0,5), G(2,4),?(4,0).

The value of z at these corner points are as follows.

Corner Points | |

O | 0 |

B | 15 |

G | 16 |

F | 14 |

? | 8 |

Thus, maximum profit is 16 monetary units obtained when 2 units of the first product and 4 units of the second product were manufactured.

Linear Programming Exercise 29.4 Question 15

which is obtained by selling 360 copies of hardcover edition and 600 copies paperback edition.

Form Linear Equation and solve graphically.

A publisher sells a hard cover edition of a text 600K for Rs.72.00 and a paperback edition of the same text for Rs.240.00. Costs of the publisher are Rs.56.00 and Rs.28.00 per book respectively in addition to weekly costs of Rs.9600.00. Both types require 5 minutes of printing time, although hard cover requires 10 minutes binding time and the paperback requires only 2 minutes. Bothe the printing back and binding operations have 4800 min available each week.

Let the sale of hard cover edition be h and that of paperback editions be t

S.P. of a hard cover edition of the textbook

S.P. of a paperback edition of the textbook

Cost to the publisher for a paperback edition

weakly cost to the publisher

Profit to bae maximized,

The corner points are 0(0,0),B(0,960), ?(360,600),F(480,0).

The value of z at these corner points are as follows.

Corner Points | |

O | -9600 |

B | 1920 |

? | 3360 |

F | -1920 |

Thus, maximum profit is 3360 which is obtained by selling 360 copies of hardware edition and 600 copies paperback edition.

Linear Programmig Excercise 29.4 Question 16

Codeine Quantity = 24(approx.) and least quantity of pill A=5.86 and B =0.27

Form Linear Equation and solve graphically.

A firm manufactures headache pills in two sizes A and B. Size A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine, Size B contains 1 grain of aspirin and 18 grains of bicarbonate and 66 grains od codeine. It has been found by users that it requires at least 12 grains of aspirin. 7.4 grains of bicarbonate and 24 grains of codeine for providing immediate effect.

The above LPP can be presented in a table below,

Aspirin | |||

Bicarbonate | |||

Codeine | |||

Relief | minimize |

Hence mathematical formulation of the given L.P.P is,

Max z = x + y

Subject to constraints,

[Since production cannot be less than 0]

The corner points are B(0,12), P(5.86,0.27),?(24,0).

The value of z at these corner points are as follows.

Corner Points | |

(0,12) | 12 |

(24,0) | 24 |

(5.86,0.27) | 6.13 |

Clearly it can be seen that it does not has any common region

So,

This is the least quantity of Pill A and B

Codeine quantity

Linear Programmig Excercise 29.4 Question 17

when 14 units of compound A and 33 unit compound B are produced.

Form Linear Equation and solve graphically.

A chemical company produces two compounds A and B . The following table gives the units of ingredients C and D per kg of Compounds A and B as well as minimum requirement of C and D and costs per kg of A and B

compound | Minimum | ||

A | B | requirement | |

Ingredient C | 1 | 2 | 80 |

Ingredient D | 3 | 1 | 75 |

Cost(inE) perkg | 4 | 6 |

Let required quantity of compound A and B are x and y kg. Since, cost of 1 kg of compound A and B are Rs.4 and Rs.6 per kg. So, Cost of x kg compound A and y kg of compound B are Rs.4x and Rs.6 respectively.

Let z be the total cost of compounds.

So,

Since compound A and B contain 1 and 2 units of ingredient C per kg respectively , So x kg of Compound A and y kg of Compound B contain x and 2y units of ingredient C respectively but minimum requirement of ingredient C is 80 units.

So ,

{first constraint}

Since, compound A and B contain 3 and 1 units of ingredient D per kg respectively.

So x kg of compound A and y kg of compound B contain 3x and y units of ingredient D respectively but minimum requirement of ingredient C is 75 units

So ,

{Second constraint}

Hence mathematical formulation of the given L.P.P is,

Min

Subject to constraints,

[Since production cannot be less than 0]

Region represented by : The line meets the axes at A(80,0) , B(0,40) respectively.

Region not containing represents as (0,0) does not satisfy satisfies

Region : line meets axes at respectively.

Region not containing origin represents as (0,0) does not satisfy .

Region it represents first quadrant.

The corner points are .

The value of z at these corner points are as follows.

Corner Points | |

D | 450 |

ε | 254 |

A | 320 |

Thus, the minimum cost is Rs. 254 obtained when 14 units of compound A and 33 units compound B are produced.

Linear Programmig Excercise 29.4 Question 18

Max Profit = Rs.16 when 8 souvenirs of Type A and 20 Souvenirs of Type B is produced.

Form Linear Equation and solve graphically.

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 hours 20 min available for cutting and 4 hours available for assembling. The profit is 50 paise each of type A and 60 paise each of type B souvenirs.

Let the company manufacture x souvenirs of Type A and y souvenirs of Type B

Therefore,

The given information can be compiled in a table as follows:

Type A | Type B | Availability | |

Cutting(min) | 5 | 8 | 3×60+20=200 |

Assembling (min) | 10 | 8 | 4×60=240 |

The profit on type A souvenirs is 50 paise and on Type B souvenirs is 60 paise. Therefore, profit gained on x souvenirs of type A and y souvenirs of type B is Rs.0.50 x and Rs. 0.60 y respectively

Total Profit,

The mathematical formulation of the given problem is,

max: , Subject x constraint,

Region : The line meets the axes at A(40,0) , B(0,25) respectively.

Region containing origin represents the solution of the in equation as (0,0) satisfy satisfies

Region : line meets axes at respectively.

Region containing origin represents the solution of in equation as (0,0) satisfies

Region : it represents first quadrant.

The corner points are

The value of z at these corner points are as follows.

Corner Points | |

O | 0 |

B | 1500 |

1600 | |

C | 1200 |

Linear Programming Exercise 29.4 Question 19

Max Profit = Rs1,440 when 48 units of product A and 16 units of product B are manufactured

Form Linear Equation and solve graphically.

A manufacturer makes two products A and B. Product A sells at Rs.200 each and takes 2hrs to make. Product B sells at rs.300 each and rakes 1 hr. to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hrs. of production and weekly turn over must not be less that Rs. 10000. If the profit on product A is 20Rs and on Product B is Rs.300

Let x be units of product A and y be units of product B are manufactured.

Number of units cannot be negative

Therefore,

According to question, the given information can be tabulated as:

Selling Price (Rs) | Manufacturing time (hrs.) | |

Product A(x) | 200 | 0.5 |

Product B(y) | 300 | 1 |

Also, the availability of time is 40 hrs and the revenue should be at least Rs.10000

Further, it is given that there is a permanent order for 14 units of Product A and 16 units of Product B

Therefore, the constraints are,

If the profit on each of product A is Rs,20 and on product B is Rs,30. Therefore, profit gained on x units of product A and y units of Product B is Rs. 20x and Rs.30y resepectively.

Total Profit=20x+30y which is to be maximized.

Thus, the mathematical formulation of the given LPP is,

Max:

Subject to constraints,

Region meets the areas at A(50,0) , respectively.

Region not containing origin represents as (0,0) does not satisfy

Region meets the area at C(80,0) D(0,40) respectively.

Region containing origin represents

Region represented by

x=14 is the line passes through (14,0) and is parallel to the X-axis.

The region to the right of the line y=14 will satisfy the in equation

Region it represents first quadrant.

The corner points of the feasible region are E(26,16), F(48,16), G(14,33) and H(14,24)

The value of Z at the corner points are as follows:

Corner Points | |

E | 1000 |

F | 1440 |

G | 1270 |

H | 1000 |

Thus, the maximum profit is Rs.1440 obtained when 48 units of Product A and 16 units of Product B are manufactured.

Linear Programming Exercise 29.4 Question 20

Max Profit = Rs.165 when 3 units of each type of trunk is manufactured.

Form Linear Equation and solve graphically.

A manufacturer produces two types of steel trunks. He has two machines A and B. For completing the first type of the trunk requires 3 hours on machine A and 3 hours on Machine B. whereas the second type of the trunk requires 3 hours on Machine A and 2 hours on Machine B. Machine A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs.30 and Rs.25 per tank of the first type and the second type respectively.

Let x be trunks of first type and y trunks of second type were manufactured. Number of trunks cannot be negative.

Therefore,

According to the question, the given information can be tabulated as

Machine A (hours) | Machine B (hours) | |

First type (x) | 3 | 3 |

Second type (y) | 3 | 2 |

Availability | 18 | 15 |

Therefore, the constraints are,

He earns a profit of Rs.30 and Rs.25 per trunk of the first type and second type respectively. Therefore, profit gained by him from x trunks of first type and y trunks of second type is Rs.30x and Rs.25y respectively.

Total Profit:

Subject to

Region : line meets areas at A(6,0), B(0,6) respectively. Region containing origin represents the solution of the in equation as (0,0) satisfies

Region : line meets areas at respectively. Region containing origin represents the solution of the in equation as (0,0) satisfies

Region : it represents first quadrant.

The corner points are O(0,0), B(0,6), E(3,3) and C(5,0)’

The value of Z at the corner points are as follows.

Corner Points | |

O | 0 |

B | 150 |

E | 165 |

C | 150 |

Thus, the maximum profit is of Rs.165 obtained when 3 units of each type of trunk is manufactured.

Linear Programming Exercise 29.4 Question 21

Max Profit = Rs.3, 25,500 when 10500 bottles of A and 34500 bottles of B are manufactured.

Form Linear Equation and solve graphically.

A manufacturer of patent medicines is preparing a production plan on medicines A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B. But there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation.

Let production of each bottle of A and B are x and y respectively.

Since profits on each bottle of A and B are Rs.8 and Rs.7 per bottle respectively. So, profit on x bottles of A and y bottles of B are 8x and 7y respectively. Let Z be total profit on bottles so,

Z = 8x + 7y

Since, it takes 3 hours and 1 hour to prepare enough material to fill 1000 bottles of Type A and Type B respectively, so x bottles of A and y bottles of B are preparing is hours and hours respectively, about only 66 hours are available, so,

Since raw materials available to make 2000 bottles of A and 4000 bottles of B but there are 45000 bottles in which either of these medicines can be put so,

[Since production of bottles cannot be negative]

Hence mathematical formulation of the given LPP is,

Max Z = 8x + 7y

Subject to constraints,

Region : line meets the axes at A (22000,0), B(0,66000) respectively.

Region containing origin represents as (0,0) satisfy

Region : line meets the axes at C (45000,0) and D(0,45000) respectively.

Region towards the origin will satisfy the in equation as (0,0) satisfies the in equation.

Region represented by

is the line passes through (20000,0) and is parallel to the y-axis. The region towards the origin will satisfy the in equation.

Region represented by ,

Y=40000 is the line passes through (0, 40000) and is parallel to the x- axis. The region towards the origin will satisfy the in equation.

Region it represents first quadrant.

Scale: On y-axis, 1 Big division=20000 units

On x-axis, 1 Big division=10000 units

The corner points are O(0,0), B(0,40000), G(10500,34500), H(20000,6000), A(20000,0)

The value of Z at these corner points are,

Corner Points | |

O | 0 |

B | 280000 |

G | 325500 |

H | 188000 |

A | 160000 |

Thus the maximum profit is Rs.325500 obtained when 10500 bottles of A and 34500 bottles of B are manufactured.

Linear Programming exercise 29.4 question 22

Max Profit = Rs.116000 when 20 first class tickets and 180 economy class tickets are sold.

Form Linear Equation and solve graphically.

An aero plane can carry a maximum of 200 passengers. A profit of Rs.400 is made on each first class ticket and a profit of Rs.600 is made on each economy class ticket. The airline reserves at least 20 seats of first class. However, at least 4 times as many passengers prefer to travel by economy class to the first class.

Let required number of first class and economy class tickets be x and y respectively.

Each ticket of first class and economy class make profit of Rs.400 and Rs.600 respectively.

So, x ticket of first class and y tickets of economy class make profit of Rs.400x and Rs.600y respectively.

Let total profit be Z=400x+600y

Given, aero plane can carry a minimum of 200 passengers, so

Given, airline reserves at least 20 seats for first class, so

Also, at least 4 times as many passengers prefer to travel by economy class to the first class, so

Hence the mathematical formulation of the LPP is

Max Z=400x+600y

Subject to constraints

{Seats in both the classes cannot be 0}

Region represented by : the line x + y = 200 meets the axes at A(200,0), B(0,200). Region containing origin represents as (0,0) satisfies

Region represented by : line x=20 passes through (20,0) and is parallel to y-axis. The region to the right of the line x=20 will satisfy the in equation

Region represented by : line y=4x passes through (0,). The region above the line y=4x will satisfy the in equation

Region : it represents the first quadrant.

Scale: On y-axis, 1 big Division=100 units

On x-axis, 1 big Division=50 units

The corner points are C(20,80), D(40,160), E(20,180)

The values of Z at these corner as follows

Corner Points | |

O | 0 |

C | 56000 |

D | 112000 |

E | 116000 |

Thus, the max profit is Rs.116000 obtained when 20 first class tickets and 180 economy class tickets sold.

Linear Programming exercise 29.4 question 23

Minimum cost = Rs.92 when 100kg of type I fertilizer and 80 kg of Type II fertilizer is supplied.

Form Linear Equation and solve graphically.

A gardener supply fertilizer of type I which consists of 10% of nitrogen and 6% phosphoric acid and Type II fertilizer which consist of 5% nitrogen and 10% of phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14kg of phosphoric acid for two crop of the type I fertilizer cost 60P/Kg and types II fertilizer 40P/Kg

Let x kg of Type I fertilizer and y kg of Type II fertilizers are supplied.

The quantity of fertilizers cannot be negative.

So,

A gardener has a supply of fertilizer of type I which consists of 10% nitrogen and type II consists of 5% nitrogen, and he needs at least 14kg for his crop.

So,

Or

A gardener has a supply of fertilizer of type I which consists of 6% phosphoric acid and type II consists of 10% phosphoric acid, and he needs at least 14 kg of phosphoric acid for his crop.

Sp,

Or

Therefore, A/Q, constraint is,

If the type I fertilizer costs 60 paise per kg and Type II fertilizer costs 40 paise per kg. Therefore, the cost of x kg of Type 1 fertilizer and y kg of Type II fertilizer is Rs.0.60x and Rs.0.40y respectively.

Total cost=Z(let)=0.6x + 0.4y is to be minimized.

Thus the mathematical formulation of the given LPP ism,

Min Z=0.6 + 0.4y

Subject to the constraints,

Region represented by : the line passes through , B(0,140). The region which does not contains origin represents solutions of the in equation as (0,0) doesn’t satisfy the in equation

Region represented by : the line passes through C(140,0) and D(0,280). The region which does not contains origin represents solutions of the in equation as (0,0) doesn’t satisfy the in equation

The region : represents the first quadrant.

The corner points are D(0,280),E(100,80), A(700/3,0)

The values of Z at these points are as follows:

Corner Points | |

O | 0 |

D | 112 |

E | 92 |

F | 140 |

Thus, the minimum cost is Rs.92 obtained when 100kg of Type I fertilizer and 80 kg of Type II fertilizer is supplied.

Linear Programming exercise 29.4 question 24

Maximum Earning Rs.1160 when Rs.2000 was invested in SC and Rs.10000 in NSB

Form Linear Equation and solve graphically.

Anil wants to invest at most Rs.12000 in saving certificates and National saving Bonds. According to rules, he has to invest at least Rs.2000 in saving certificate and at least Rs.4000 in National saving Bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on National Saving Bond is 10% per annum.

Let Anil invests Rsx and Rs.y in saving certificate (SC) and National Saving Band (NSB) respectively.

Since, the rate of interest on SC is 8% annual and on NSB is 10% annual. So, interest on Rs.x of SC is and Rs.y of NSB is per annum.

Let Z be total interest earned so,

Given he wants to invest Rs.12000 is total

According to the rules he has to invest at least Rs.2000 in SC and at least Rs.4000 in NSB

Region represented by : the line x=2000 is parallel to the y-axis and passes through (2000,0).

The region which does not contains origin represents as (0,0) doesn’t satisfy the in equation

Region represented by : the line y=4000 is parallel to the x-axis and passes through (0,4000).

The region which does not contains origin represents as (0,0) doesn’t satisfy the in equation

Region represented by : the line meets axes at A(12000,0) and B(0,12000)respectively. The region which contains origin represents the solution set of as (0,0) satisfies the in equation

Region is represented by the first quadrant.

Scale: On x-axis, 1 Big Division=2000 units

On y-axis, 1 Big Division=2000 units

The corner points are E(2000,10000), C(2000,4000),D(8000,4000)

The values of Z at these corner points are as follows

Corner Points | |

O | 0 |

E | 1160 |

D | 1040 |

C | 560 |

Thus the maximum earning is Rs.1160 obtained when Rs.2000 were invested in SC and Rs.10000 in NSB.

Linear Programming Exercise 29.4 Question 25

Maximum Profit Rs.1800 when Rs.20 were involved in Type A and Rs.40 were involved in Type B.

Form Linear Equation and solve graphically.

A men owns a field of area 1000 m2. He wants to plant fruit trees in it. He has a sum of Rs.1400 to purchase young trees. He has the choice of two types of trees. Type A require 10 m2 of ground per tree and costs Rs.20 per tree and type B requires 20m2 of ground per tree and costs Rs.25 per tree. When fully grown type A produces an average of 20 kg of fruit which can be sold at a profit of Rs.2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of Rs.1.50 per kg.

Let the required number of trees of Type A and B be Rs.x and Rs.y respectively.

Number of trees cannot be negative.

To plant tree of Type A requires 10sq.m and Type B requires 20sq.m of ground per tree. And it is given that a man owns a field of are 1000sq.m.

Therefore,

Type A costs Rs.20 per tree and type B costs Rs.25 per tree. Therefore, x trees of type A and y trees of type B cost Rs.20x and Rs.25y respectively. A man has a sum of Rs.1400 to purchase young trrs

Thus the mathematical formulation of the given LPP is

Max Z = 40x -20x + 60y - 25y = 20x + 35y

Subject to,

Region : the line meets axes at A1(70,0) and B1(0,56)respectively.

The region which contains origin represents as (0,0) satisfies .

Region : the line meets axes at A2(70,0) and B2(0,56)respectively.

The region which contains origin represents as (0,0) satisfies .

Region : It represents by first quadrant.

The maximum value of Z is 1800 which is attained at P(20,40) as

The value of Z at these corner points are as follows

Corner Points | |

O | 0 |

A1 | 1750 |

P | 1800 |

B2 | 1400 |

Linear Programming Exercise 29.4 Question 26

4 pedestal lamps and 4 wooden shades

By using the mathematical formulation of the given Linear programming is Max Z= ax+by

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/ cutting machine and a sprayer. It takes 2 hours on the grinding/ cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp while it takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, line sprayer is available for at most 20 hours and the grinding/ cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs.5.00 and a shade is Rs.3.00. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?

Let the cottage industry manufactures x pedestal lamps & y wooden shades

Therefore,

The given information is as follow:

Lamps | Shades | Availability | |

Grinding machine | 2 | 1 | 12 |

Sprayer | 3 | 2 | 20 |

The profit on a lamp is Rs.5 and on the shades is Rs.3

Max Z = 5x + 3y … (i)

Subject to constraints:

… (ii)

… (iii)

… (iv)

The feasible region is as follows:

Corner points | |

Linear Programming Exercise 29.4 Question 27

Maximum Revenue is Rs.1260 obtained when 3 units of x and 8 units of y were produced is as follows.

By using the mathematical formulation of the given Linear programming is Max Z= ax+by

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods x and y. To produce one unit of x, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and I unit of capital is required to produces one unit of y

Let x1 and y1 units of goods x and y were produced respectively.

Number of units of goods cannot be negative.

Therefore,

To produce one unit of x, 3 units of Capital is required and 1 unit of capital is required to produce one unit of y.

If x and y are priced at Rs.100 and Rs.120 per unit respectively. Therefore, cost of x1 and y1 units of goods x and y is Rs.100 x1 and Rs.120 y1

Total revenue = Z = 100 x1 +120 y1 which B to be maximized.

Thus the mathematical formulation of the given linear programming problem is

Max Z = 100 x1 +120 y1

Subject to

First, we will convert in equation into equations as follows

Region represented by : the line meets axes at A1(15,0) and B1(0,10)respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

the region which contains origin represents the solution set of the in equation .

Region represented by : the line meets axes at respectively. By joining these points we obtain the line . Clearly (0,0) satisfies the in equation . So the region which contains origin represents the solution set of the in equation .

Region represented by : Since, every point in the first quadrant satisfies these in equations. So, the first quadrant is the region represented by the in equation

The feasible region determined by the system of constraints as follows.

The corner points are B(0,10), E(3,8) and C(17/3,0)

The values of Z at these corner points.

Corner Points | |

B | 1200 |

E | 1260 |

C |

Thus, the maximum revenue is Rs.1260 obtained when 3 units of x and 8 units of y were produces are as follows.

Linear Programmig Excercise 29.4 Question 28

Maximum Value of Z is Ra.1020 which is attained at . Thus the maximum profit is Rs.1020 obtained when 60 units of product A and 240 units of product B were manufactured.

By using the mathematical formulation of the given Linear programming is Max Z=ax+by

Firm manufacturer’s two types of Products A and B and sells them at a profit of Rs.5 per unit of Type A and Rs.3 per unit of Type B.

Let x units of Product A and y units of Product B were manufactured.

Number of products cannot be negative.

Therefore,

According to question, the given information can be tabulated as

Time on M1(Minutes) | Time on M2(Minutes) | |

Product A(x) | 1 | 2 |

Product B(y) | 1 | 1 |

Availability | 300 | 360 |

Firm manufactures two types of Products A and B and sells them at a profit of Rs.5 per unit of type A and Rs. 3 per unit of the type B. Therefore x unit of product A and y units of product B costs Rs.5x and Rs.3y respectively.

Total Profit = Z = 5x + 3y which is to be maximized

Thus, the mathematical formulations of the given linear programming problem is,

Max Z = 5x + 3y

Subject to

First we will convert in equations as follows

Region represented by : the line meets the coordinate axes at A1(300,0) and B1(0,300)respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

the region which contains origin represents the solution set of the in equation .

Region represented by : the line meets the coordinate axes at C1(180,0) and D1(0,360) respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

the region which contains origin represents the solution set of the in equation .

Region represented by : Since, every point in the first quadrant satisfies these in equations. So, the first quadrant is the region represented by the in equation

The feasible region determined by the system of constraints are as follows.

Scale: On x-axis: 1 Big Division = 100 units

On y- axis: 1 Big Division = 100 units

The corner points are

The values of Z at these corner points are as follows

Corner Points | |

0 | |

900 | |

1020 | |

900 |

Thus, the maximum profit is Rs.1020 obtained when 60 units of Product A and 240 units of Product B were manufactured.

Linear Programmig Excercise 29.4 Question 29

The answer of the given question B that the firm should produce 8A products and 16B products to earn maximum profit of Rs.4960

By using the mathematical formulation of the given Linear programming is Max Z=ax + by

A small firm manufactures item A and B- The total number of items that it can manufacture in a day B at most 24. Item A takes one hour while items B takes only half an hour.

Let the firm manufactures x members of A and y members of B products.

According to the question

Maximize Z = 300x + 160y

The feasible region determined by given by

The corner points of feasible region are A(0,0), B(0,24), C(8,16) and D(16,0)

The value of Z at corner point is

Corner Points | Z=300x+160y | |

A(0,0) | 0 | |

B(0,24) | 3840 | |

C(8,16) | 4960 | Maximum |

D(16,0) | 4800 |

The firm should produce 8A products and 16 B products to earn maximum profit of Rs.4960.

Linear Programmig Excercise 29.4 Question 30

The answer of the given question is that the maximum profit is Rs.1500 at E(12,15)

By using the mathematical formulation of the given Linear programming is Max Z=ax + by

Two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling.

Toy A | Toy B | Times in a day | |

Cutting time | 5 min | 8 min | 180 min |

Assembly time | 10 min | 8 min | 240 min |

Profit | 50 | 60 | |

Assumed Quantity | x | y |

Profit function Z=50x+60y

A | B | |

x | 0 | 36 |

y | 22.5 | 8 |

C | D | |

x | 0 | 24 |

y | 30 | 0 |

Corner Points | |

At O(0,0) | 0 |

At D(24,0) | 1200 |

At E(12,15) | 1500 |

At A(0,22.5) | 1350 |

Linear Programming Exercise 29.4 Question 31

The answer of the given question B that the maximum profit is Rs.120 obtained when 12 units of articles A and 6 units of articles B were manufactured.

By using the mathematical formulation of the given Linear programming is Max Z=ax + by

The maximum capacity of first department is 60 hours a week and that of other department is 48 hours per week. The product of each unit of article A requires 4 hours in assembly and 2 hours in finishing and that of each unit of B requires 2 hours in assembly and 4 hours in finishing.

Let x units and y units of articles A and B are produced respectively.

Number of articles can’t be negative.

Therefore,

The product of each unit of article A requires 4 hours in assembly and that of articles B requires 2 hours in assembly and the maximum capacity of the assembly department in 60 hours a week.

The product of each unit of article A requires 2 hours in finishing and that of articles B requires 4 hours in assembly and the maximum capacity of the finishing department in 48 hours a week.

If the profit is Rs.6 for each unit of A and Rs.8 for each unit of B. Therefore, profit gained from x units and y units of articles A and B respectively is Rs.6x and Rs.8y respectively.

Total revenue = Z = 6x + 8y which is to be maximized.

Thus, the mathematical formulation of the given linear programming problem is

Max Z = 6x + 8y

Subject to

First, we will convert in in equations into equations as follows:

Region represented by : the line meets the coordinate axes at A1(24,0) and B1(0,12)respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

The region which contains origin represents the solution set of the in equation .

Region represented by : the line meets the coordinate axes at C1(15,0) and B1(0,30)respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

The region which contains origin represents the solution set of the in equation .

Region represented by : Since, every point in the first quadrant satisfies these in equations. So, the first quadrant is the region represented by the in equation

The feasible region determined by the system of constraints are as follows.

The corner points are O(0,0), B(0,12),E1(12,6) and C1(15,0)

The values of Z at these corner points as follows.

Corner Points | |

O | 0 |

B1 | 96 |

E1 | 120 |

C1 | 90 |

Thus, the maximum profit is Rs. 120 obtained when 12 units of articles A and 6 units of Article B were manufactured.

Linear Programming Exercise 29.4 Question 32

The answer of the given questions is that 8 items of type A and 16 of type B should be produced for max profit.

By using the mathematical formulation of the given Linear programming is Max Z=ax + by

A firm makes items A and B and the total number of items it can make in a day is m. It takes one hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours.

Let the number of items of type A and B produce be x and y respectively.

The LPP is maximize Z=300x+160y

Subject to the constraints.

Draw the lines

… (i)

… (ii)

These meet at P (8, 16)

The feasible region is OCPB

The value of Z = 300x + 160y at 0 is zero

At C(16,0) is 4800

At B(0,24) is 3840

At P(8,16) is 4960

Clearly, value is max at P(8,16)

8 items type A and 16 type B should be produced for max. Profit

Linear Programming Exercise 29.4 Question 33

By using the mathematical formulation of the given Linear programming is Max Z=ax + by

Total capacity of 500man-hour. It takes 5 hours to produce unit A and 3 hours to produce unit B.

Let x units of Product A and y units of Product B were manufactured.

Clearly,

It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The two products are produced in a common production process, which has total capacity of 500 man-hours.

The maximum number of unit of A that can be sold is 70 and that for B is 125.

If the profit is Rs.20 per unit for the product A and Rs.15 per unit for the product B. Therefore, profit x units of product A and y units of product B is Rs.20x and 15y respectively.

Total Profit

The mathematical formulation of the given problem is

Max

Subject to

First we will convert in equation into equations as follows:

Region represented by : the line meets the coordinate axes at A1(100,0) and B1 respectively.

By joining these points we obtain the line . Clearly (0,0) satisfies the . So

The region which contains origin represents the solution set of the in equation .

Region represented by .

The line x = 70 is the line passes through C1(70,0) and is parallel to y-axis. The region to the left of the line x = 70 will satisfy the in equation .

Region represented by

The line y = 125 is the line passes through D1(0,125) and is parallel to x-axis. The region below the line

y=125 will satisfy the in equation .

Region represented by

Since, every point in the first quadrant satisfies these in equations. So the first quadrant is the region represented by the in equation

The feasible region determined by the system of constraints are as follows:

The corner points are O(0,0), D(0,125), E(25,125), F(70,50) and C(70,0). The values of Z at the corner points are:

Corner Points | |

O | 0 |

1875 | |

2375 | |

2150 | |

1400 |

Thus, the maximum profit is Rs.2375. 25 units of A and 125 units of B should be manufactured.

Linear Programming exercise 29.4 question 34

No. of box=6, small box=12

Maximum profit=Rs.42

Let required quantity of large and small boxes are x and y respectively.

Profits on each unit of large and small boxes are Rs.3 and Rs.2 respectively.

Let Z be total profit

Z=3x+2y [Where x and y are the quantity of large and small boxes]

Since each large and small box require 4sq.m and 3sq.m cardboard, but only 60sq.m of cardboard is available

Since manufacture is required to make at least three large boxes.

So,

Since manufacture is required to make at least twice as many small boxes as large boxes.

So,

The required LPP is Max Z = 20x + 5y

Linear Programming exercise 29.4 question 35

Let x be the number of units A and y be the number of units of B

The given data can be written in the tabular form

Product | A | B | Working week | Turn over |

Time | 0.5 | 1 | 40 | |

Price | 200 | 300 | 10000 | |

Profit | 20 | 30 | ||

Permanent order | 14 | 16 |

The mathematical model of the LPP as follows.

Max Z = 20x + 30y

Subject to:

Subject to constraint:

… (i)

… (ii)

… (iii)

The feasible region determined by the subject of constraints.

No of large box=6, small box=12

Maximum profit=Rs.42

The coordinates of the vertices (corner points) of shaded region ABCD are A(26,16),B(48,16),C(14,33) and D(14,24)

Points | Z=20x+30y |

A(26,16) | Z=1000 |

B(48,16) | Z=1440 |

C(14,33) | Z=1270 |

D(14,24) | Z=60 |

48 units of Product A and 16 units of Product B should be produced to earn the maximum profit of Rs.1440.

Linear Programming exercise 29.4 question 36

Let us assume that the man travels xkm/hr. and y km when the speed is 40km/hr.

Speed of vehicle=25km/hr.

Per km cost on petrol=2Rs/km

Young man carries only =100 Rs.

Let us assume that the man travels x km when the speed is 25km/hour and y km when the speed is 40km/hour.

Thus, the total distance travelled is (x+y) km.

Now, it’s given that the man has 100Rs to spend on petrol.

Total cost of petrol=

Now, Time taken to travel x km = hour

Time taken to travel y km = hour

Now it’s given that the maximum time is 1 hour. So,

Thus, the given linear programming problem is

Maximize Z=x+y

Subject to the constraints

The feasible region determined by the given constraints can be diagrammatically represented as,

The coordinates of the corner points of region are O(0,0), A(25,0), B(50/3,40/3) and C(0,20).

The value of objective function at these points are given in the following

Corner Points | Z=x+y |

(0,0) | 0+0=0 |

(25,0) | 25+0=25 |

(50/3,40/3) | 50/3+50/3=30 |

(0,20) | 0+20=20 |

So the maximum value of Z is 30 at

Thus, the maximum distance that the man can travel in one hour 30 km

Hence, the distance travelled by the man at the speed of 225 km/hr is 50/3 km and the distance travelled by him at the speed of the 40km/hour is 40/3 km.

Linear Programming Exercise 29.4 Question 37

The minimum transportation cost is 4400.

Assuming that the transportation cost of 50 liters of oil is Rs.1 per km.

Distance | (in km) | |

To/From | A | B |

D | 7 | 3 |

E | 6 | 4 |

F | 3 | 2 |

Let x and y liters of oil be supplied from A to the petrol pumps, D and E. Then, (7000-x-y) will be supplied from A to petrol pump F. The requirement at petrol pump D is 4500l since xl are transported from depot A, the remaining (7000-x) will be transported from petrol pump B.

Similarly, (3000-y)l and 35000-(7000-x-y)=(x+y-3500)l will be transported from depot B to petrol pump E and F respectively. The given problem can be represented diagrammatically as follows.

Cost of transporting 10 litres of petrol = Rs.1

Cost of transporting 1 litre of petrol =

Therefore, total transportation cost is given by,

= 0.3x + 0.1y + 3950

The problem can be formulated as follows.

Minimize Z=0.3x + 0.1y + 3950 …(i)

Subject to the constraints

… (ii)

… (iii)

… (iv)

… (v)

… (vi)

The feasible region determined by the constraints as follows

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

The values at the corner points as follows:

Corner Points | Z=0.3x+0.1y+3950 | |

A(3500,0) | 5000 | |

B(4500,0) | 5300 | |

C(4500,2500) | 5550 | |

D(4000,3000) | 0+20=20 | |

E(500,3000) | 4400 | Minimum |

The minimum value of Z is 4400 at (500, 3000)

Thus the oil supplied from depot A is 500L, 3000L and 3500L and from depot B is 400L, 0L and 0L petrol pumps D,E and F respectively.

The minimum transportation cost is Rs.4400.

Linear Programming Exercise 29.4 Question 38

Let, gold ring manufactured per day= x

Chains manufactured per day= y

Total number of rings and chain manufactured per day is almost 24.

Time taken to make a ring=1 hour

Time taken to make a chain=30 minutes

Maximum number of hours available per day=16 hrs

Profit on a chain =190 and profit on ring=300

Let gold rings manufactured per day=x

Chain manufacture per day=y

LPP is:

Maximize Z= 300x + 190y

Subject to

Possible point for maximum Z is (16, 0), (8, 16) and (0, 24)

Hence Z MAXIMUM at (8, 16)

8 gold rings and 16 chains must be manufactured per day

Linear Programming Exercise 29.4 Question 39

Let, two types of books be x and y.

Thickness of the books=6cm and 4 cm

Weight of the books=1 kg and kg each

Shelf is 9 cm and at most can support a weight of 21 kg

Let two types of books be x and y respectively.

The required LPP is maximized.

Z=x+y

Subject to the constraints,

And

On considering the inequalities as equations,

We get,

…(i)

… (ii)

Table for line is

x | 0 | 16 |

y | 24 | 0 |

So, it passes through (0, 24) and (16,0)

On putting (0, 0) in ,

We get [Which is true]

Table for

x | 0 | 21 |

y | 14 | 0 |

So it passes through (0, 14) and (21, 0)

On putting (0, 0) in ,

We get [Which is true]

On solving equation (i) and (ii), we get

X = 12 and y = 6

Thus, the intersection point is B (12, 6)

From the graph, OABCD is the feasible region which is bounded. The corner points are O(0,0) A(0,14), B(12,6) and C(16,0)

The values of Z at corner points are as follows.

Corner Points | Z=x+y |

O(0,0) | Z=0+0=0 |

A(0,14) | Z=0+14=14 |

B(12,6) | Z=12+6=18 |

C(16,0) | Z=16+0=16 |

From the table, the maximum value of Z is 18 at B(12,6)

Hence, the maximum number of books is 18 and number of books of I type is 12 and books of II type is 6.

Linear Programming Exercise 29.4 Question 40

Let the number of tennis rackets and cricket bats be x and y.

A tennis racket takes 1.5 hours if machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day factory has availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. If the profit on a rackets and bat is 20Rs and 10Rs.

Let the number of tennis racket and cricket bats manufactured by factory be x and y.

Hence, Profit is the objective function Z.

Z = 20x + 10y … (i)

We have to maximize Z subject to the constraints.

… (ii) [Constraints for machine hour]

… (iii) [Constraints for craftman’s hour]

Graph of x = 0 and y = 0 is the y-axis and x-axis.

Graph of is the 1st quadrant.

Graph of

x | 0 | 28 |

y | 14 | 0 |

Graph for is the part of 1st quadrant which contains the origin.

Graph for

x | 0 | 8 |

y | 24 | 0 |

Graph for is the part of st quadrant in which origin lies.

Hence, shaded area OACB is the feasible region for coordinate of C equation

… (iv)

… (v)

X = 4 (Substituting y = 12 in (iv))

Now the value of objective function Z at each corner of feasible region is

Corner Points | |

O(0,0) | |

A(8,0) | |

B(0,14) | |

C(4,12) |

Therefore, maximum profit is Rs.200 when factory makes 4 tennis rackets and 12 cricket bats.

Linear Programming Exercise 29.4 Question 41

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

Cost of desktop model computer=25000

Cost of portable model computer=40000

Total monthly demand will not exceed 250 units.

Profit on desktop model=4500Rs.

Profit on portable model=5000Rs.

Let merchant plans has personal computers x desktops model and y portable model.

Thus,

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

Merchant can invest Rs.70 lakhs maximum

The total monthly demand will not exceed 250 units.

Profit on desktop model is 4500 and on portable model is Rs.5000

Total Profit = Z,

The feasible region determined by constraints is as follows.

The corner points of feasible region are A(25,0), B(250,50), C(0,175) , D(0,0)

The value of Z corner points is as shown

Corner Points | Z=4500x+500y |

A(250,0) | 1125000 |

B(200,50) | 1150000(Maximum) |

C(0,175) | 875000 |

D(0,0) | 0 |

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

Linear Programming Exercise 29.4 Question 42

Let x hectare of land be allocated to crop x and y hectare to crop y.

Profit per hectare on crop x=Rs.10500

Profit per hectare on crop y = Rs.9000

By the given profit on x crop and y crop.

Total profit=Rs.(10500x+9000y)

The mathematical formulation of the problem is as follows.

Maximize Z = 10500x + 9000y

Subject to the constraints

(Constraint related to land) … (i)

(Constraint related to use of herbicide) i.e … (ii)

(Non –negative constraint) … (iii)

Let us draw the graph of the system of inequalities (i) to (iii).

The feasible region ABC is shown (shaded)

Corner Points | Z=10500x+9000y |

O(0,0) | 0 |

A(40,0) | 420000 |

B(30,20) | 495000(Maximum) |

C(0,50) | 450000 |

The coordinate of the corner points O, A, B and C are (0,0), (45,0),(30,20) and (0,50) respectively.

Let us evaluate the objection function Z = 10500x + 9000y at these vertices to find on gives the maximum profit.

Hence, the society will get the maximum profit of Rs.495000 by allocating 90 hectares for crop x and 20 hectares for crop y.

Linear Programming exercise 29.4 question 43

Let x hectare of land be allocated to crop x and y hectare to crop y.

Profit per hectare on crop x=Rs.10500

Profit per hectare on crop y = Rs.9000

By the given profit on x crop and y crop.

Total profit=Rs.(10500x+9000y)

The mathematical formulation of the problem is as follows.

Maximize Z = 10500x + 9000y

Subject to the constraints

(Constraint related to land) … (i)

(Constraint related to use of herbicide) i.e … (ii)

(Non –negative constraint) … (iii)

Let us draw the graph of the system of inequalities (i) to (iii).

The feasible region ABC is shown (shaded)

Corner Points | Z=10500x+9000y |

O(0,0) | 0 |

A(40,0) | 420000 |

B(30,20) | 495000(Maximum) |

C(0,50) | 450000 |

The coordinate of the corner points O, A, B and C are (0,0), (45,0),(30,20) and (0,50) respectively.

Let us evaluate the objection function Z = 10500x + 9000y at these vertices to find on gives the maximum profit.

Hence, the society will get the maximum profit of Rs.495000 by allocating 90 hectares for crop x and 20 hectares for crop y.

Linear Programming exercise 29.4 question 44

- 4 Tennis Racket Bats must be made so that factory runs at all capacity.
- Maximum profit is 200 when 4 tennis and 12 cricket balls are produced.

Let the number of tennis racket be x number of cricket bat be y

Item | Number | Machine hours | Crafts man hours | Profit |

Tennis Racket | x | 1.5 | 3 | Rs.20 |

Cricket Bats | y | 3 | 1 | Rs.10 |

Maximum time available | 42 | 24 |

Let the number of Tennis Racket be x.

Number of cricket bats be y.

According to the question,

Machine hours Craftsman’s hours

Tennis Racket – 1.5 hours Tennis Racket requires -3 hours

Cricket bat requires – 3 hours Cricket Bat requires – 1 hour

Maximum time available – 42 hours Max time available – 24 hour

Also

As we want maximize the profit.

Hence function used here will be maximize

profit on each tennis rocket à Rs.20

Profit on each Bat à Rs.10

Max Z = 20x + 10y

Combining all the constraints:

Maximize Z = 20x + 10y

Subject to the constraints:

Now:

x | 0 | 14 |

y | 14 | 7 |

x | 2 | 8 |

y | 18 | 0 |

Corner Points | Value of Z |

(0,4) | 14 |

(4,12) | 16 |

(8,0) | 8 |

Thus,

- 4 Tennis Racket Bats must be made so that factory runs at all capacity.
- Maximum profit is 200 when 4 tennis and 2 cricket balls are produced.

Linear Programming exercise 29.4 question 45

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

Cost of desktop model computer=25000

Cost of portable model computer=40000

Total monthly demand will not exceed 250 units.

Profit on desktop model=4500Rs.

Profit on portable model=5000Rs.

Let merchant plans has personal computers x desktops model and y portable model.

Thus,

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

Merchant can invest Rs.70 lakhs maximum

The total monthly demand will not exceed 250 units.

Profit on desktop model is 4500 and on portable model is Rs.5000

Total Profit = Z,

The feasible region determined by constraints is as follows.

The corner points of feasible region are A(25,0), B(250,50), C(0,175) , D(0,0)

The value of Z corner points is as shown

Corner Points | Z=4500x+500y |

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.

Linear Programming Exercise 29.4 Question 46

Use properties of LPP

A toy company manufacturers two types of dolls A and B and if the company makes profit of Rs.12 and Rs.16 per doll respectively.

Let x units of doll A and y units of doll B are manufactured to obtain the maximum profit.

The mathematical formulation of the above problem as follows.

Maximize Z = 12x + 16y

Subject to

The shaded region represents the set of feasible solutions.

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

=12(0) + 16(0) = 0

The value of Z at A(800,400)

=12(800) + 16(400) = 16000

Maximum value of Z at B(1020,150)

=12(1050) + 16(150) = 15000

The value of Z at C(600,0)

=12(600) + 6(0) = 7200

Therefore, 800 units of doll A and 400 units of doll B should be produced weekly to get the maximum profit of Rs.16000.

Linear Programming Exercise 29.4 Question 47

Use properties of LPP

Two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid if F1 consists of Rs.6/kg and F2 costs Rs.5/Kg

Suppose x kg of fertilizer F1 and y kg of fertilizer F2 is used to meet the nutrient requirements.

F1 consists of 10% nitrogen and F2 consists of 5% nitrogen. Bu the farmer needs at least 14 kg of nitrogen for the crops.

Similarly, F1 consists of 6% phosphoric acid and F2 consists of 10% phosphoric acid. But the farmer need at least 14 kg of phosphoric acid for the crops.

The cost of fertilizer F1 is Rs.6/kg and Fertilizer F2 in Rs.5/Kg, therefore total cost of x kg of fertilizer F1 and y kg of fertilizer F2 is Rs.(6x+5y)

Thus, the given linear programming problem is

Minimize Z = 6x + 5y

Subject to the constraints

The feasible region determined by the given constraints can be diagrammatically represented as,

The coordinates of the corner points of the feasible region are

The value of the objective function at these points are given in the following table.

Corner Points | Z=6x+5y |

(minimum) | |

The smallest value of Z is 1000 which is obtained at x = 100, y = 80.

It can be sure that the open half-plane represented by 6x + 5y < 1000 has no common points with the feasible region.

So, the minimum value of Z is 1000.

Hence, 100 kg of fertilizer F1 and 80 kg of fertilizer F2 should be used so that the nutrient requirements are met at minimum cost.

The minimum cost is Rs.1000

Linear Programming Exercise 29.4 Question 48

Use properties of LPP

Item | Number of hours required on machines |

I II III | |

M | 1 2 1 |

N | 2 1 1.25 |

She makes a profit of Rs.600 and Rs.400 on items M and N respectively.

Suppose x units of item M and y units of item N are produced to maximize the profit. Since each unit of item M require hour on machine I and each unit of item N require hours on Machine I, therefore, the total hours required for producing x units of item M and y units of item N on machine I are (2x +y). But machines I is capable of being operated for at most 12 hours.

Similarly, each unit of item M require 2 hours on machine II and each unit of item N require 1 hour on machine II, therefore, total hours required for producing x units of item M and y units of item N on machine II are (x + 2y). But machines II is capable of being operated for at most 12 hours.

Also, each unit of item M require 1 hour on machine III and each unit of item N require 1.25 hour on machine III, therefore, the total hours required for producing x units of item M and y units of item N on machine III are (x + 1.25y). But, machines III must be operated for at least 5 hours.

The profit from each unit of item M is Rs.600 and each unit of item N is Rs.400, Therefore the total profit from x units of item M and y units of item N is (600x + 400y).

Thus, the given linear programming problem is

Maximize Z = 600x + 400y

Subject to the constraints,

The feasible region determined by the given constraints can be diagrammatically represented as,

The coordinates of the corner points of the feasible region are A(5,0), B(6,0), C(4,4), D(0,6) and E(0,4)

The value of the object function at these points are given in the following table.

Corner Points | Z=600x+400y |

(5,0) | 3000 |

(6,0) | |

(4,4) | (maximum) |

(0,6) | |

(0,4) |

The maximum value of Z is 4000 at x=4,y=4.

Hence, 4 units of item M and 4 units of item N should be produced to maximize the profit.

The maximum profit of the manufacturer is Rs.4000.

Linear Programming Exercise 29.4 Question 49

Use properties of LPP

From To | Cost (in Rs.) |

A B C | |

P | 160 100 150 |

Q | 100 120 100 |

Here, demand of the commodity (5 + 5 +4 = 14 units) is equal the supply of the commodity (8 + 6 = 14 units). So, no commodity could be left at the two factories.

Let x units and y units of the commodity be transported from the factory P to the depots A and B respectively.

Then (8-x-y) units of the commodity will be transported from the factory P to the depot C.

Now, the weekly requirement of depot A is 5 units of the commodity. Now, x units of the commodity are transported from factory P so the remaining (5-x) units of the commodity are transported from the factor Q to the depot A.

The weekly requirement of depot B is 5 units of the commodity. Now, y units of the commodity are transported from factory P. So the remaining (5-y) units of the commodity are transported from the factory Q to the depot B.

Similarly, units of the commodity will be transported from the factory Q to the depot C.

Since the number of the units of commodity transported are from the factories to the depots are non-negative, therefore,

Total transportation cost =

Thus, the given linear programming problem is

Minimize Z = 10x – 70y + 1900

Subject to constraints:

The feasible region determined by the given constraints can be diagrammatically represented as,

The coordinates of the corner points of the feasible region are A(4,0),B(5,0),C(5,3), D(3,5), E(0,5) and F(0,4).

The value of the objective function at these points is given in the follow table.

Corner Points | Z=10x-70y+1900 |

(4,0) | |

(5,0) | |

(5,3) | |

(3,5) | |

(0,5) | (minimum) |

(0,4) |

The minimum value of Z is 1550 at x = 0, y = 5

Hence, for minimum transportation cost factory P should supply 0, 5, 3 units of commodity to depots A,B,C respectively and factory Q should supply 5,0,1 units of commodity to depots A,B and C respectively.

The minimum transportation cost is Rs.1550.

Linear Programming Exercise 29.4 Question 50

Use properties of LPP

Types of toys | Machines |

I II III | |

A | 12 18 6 |

B | 16 0 9 |

Suppose the manufacturer makes x toys of type A and y types of toy B.

Since each toy of type A requires 12 minutes on machine I and each toy of type B require 6 minutes on machine I, therefore, x toys of type A and y toys of type B require (12x + 6y) minutes on machine I.But, machines I is available for at most 6 hours.

Similarly, each toy of type A requires 18 minutes on machine II and each toy of type B require 0 minutes on machine II, therefore, x toys of type A and y toys of type B require (18x + 0y) minutes on machine II. But, machines II is available for at most 6 hours.

Also, each toy of type A requires 6 minutes on machine III and each toy of type B require 9 minutes on machine III, therefore, x toys of type A and y toys of type B require (6x + 9y) minutes on machine III. But, machines III is available for at most 6 hours.

The profit on each toy of type A is Rs.7.50 and each toy of type B is Rs.5. Therefore, the total profit from x toys of type A and y toys of type B is Rs.(7.50x+5y)

Thus the given linear programming problem is

Maximize Z=7.5x+5y

Subject to the constraints:

The feasible region determined by the given constraints can be diagrammatically represented as.

The coordinates of the corner points of the feasible region are O(0,0), A(20,0), B(20,20), C(15,30) and D(0,40).

The value of the object function at these points is given in the following table.

Corner Points | Z=7.5x+5y |

(0,0) | |

(20,0) | |

(20,20) | |

(15,30) | (maximum) |

(0,40) |

The maximum value of Z is 262.5 at x = 15, y = 30.

Hence, 15 toys of type A and 30 toys of type B should be manufactured in a day to get maximum profit.

The maximum profit is Rs.262.5

Linear Programming exercise 29.4 question 51

Max Profit = Rs.1, 36,000 when 40 first class tickets and 160 economy class tickets are sold.

Form Linear Equation and solve graphically.

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

Let required number of first class and economy class tickets be x and y respectively.

Each ticket of first class and economy class make profit of Rs.1000 and Rs.600 respectively.

So, x ticket of first class and y tickets of economy class make profit of Rs.1000x and Rs.600y respectively.

Let total profit be Z=1000x+600y

Given, aero plane can carry a minimum of 200 passengers, so

Given, airline reserves at least 20 seats for first class, so

Also, at least 4 times as many passengers prefer to travel by economy class to the first class, so

Hence the mathematical formulation of the LPP is

Max Z = 1000x+600y

Subject to constraints

{Seats in both the classes cannot be 0}

The feasible region determined by the given constraints can be diagrammatically represented as.

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

The values of Z at these corner as follows

Corner Points | z=1000x+600y |

O | 0 |

A | 68000 |

B | 136000(maximum) |

C | 128000 |

Thus, the max profit is Rs.136000 obtained when 40 first class tickets and 160 economy class tickets sold.

Linear Programming exercise 29.4 question 52

Use property of LPP

That men and women workers are equally efficient and So, he pays them at the same rate.

Let x unit of A and y units of B produced by the manufacturer.

The price of one unit of A is Rs.100 and the price of one unit of B is Rs.120. Therefore, the total price of x unit of A and y units of B. The total revenue is Rs. (100x+120y) one unit of A requires 2 workers are one unit of B requires 3 workers. Therefore x unit of A and y units of B requires (2x + 3y) workers. But, the manufacturer has 30 workers.

Similarly, one unit of A requires 3 units of capital. Therefore, x unit of A and y unit of B requires (3x+y) units of capital. Therefore, x unit of A and y unit of B requires (3x + y) units of capital. But the manufacturer has 17 units of capital.

The feasible region determined by the given constraints can be diagrammatically represented as.

The coordinates of the corner points of the feasible region are O(0,0), A(0,10), B(17/3,0) and C(3,8)

The value of the objective function at these points is given in the following table.

Corner Points | |

O | |

A | |

B | |

C | (maximum) |

Hence the maximum total revenue is Rs.1260 when 3 units of A and 8 units of B are produced.

Yes, because the efficiency of a worker does not depend on whether the worker is a male or female.

Linear Programming exercise 29.4 question 53

Use property of LPP

First machine is 12 hours and second machine is 9 hours per day.

Let x units of product A and y units of Product B be manufactured by the manufacturer per dat.

It is given that one unit of product A requires 3 hours of processing time on first machine, while one unit of product B requires 2 hours of processing time on first machine.

It is also given that first machine is available for 12 hours per day.

Also, one unit of product A requires 3 hours of processing time on second machine, while one unit of product B requires 1 hour of processing time on second machine.

It is also given that second machine is available for 9 hours per day.

The profits on one unit each of Product A and B are Rs.7 and Rs.4 respectively.

So the objective function is given by,

Hence the mathematical formulation of the LPP is

Maximize Z=7x+4y

Subject to the constraints

… (i)

… (ii)

… (iii)

The feasible region determined by constraints (1) and (2) is graphically represented as

Here it is seen that OABCO is the feasible region and it is bounded. The value of Z at the corner points of the feasible region are represented in tabular form as follows.

Corner Points | |

O(0,0) | |

A(3,0) | |

B(2,3) | (maximum) |

C(0,6) |

Thus, 2 units of Product A and 3 units of product B. Should be manufactured by the manufacturer per day in order to maximize the profit.

Also, the maximum daily profit of the manufacturer is Rs.26.

Linear Programming exercise 29.4 question 54

Use property of LPP

A consists of 12% of nitrogen and 5% of phosphoric acid at costs Rs.10/kg and B consists of 4% of nitrogen and 5% of phosphoric acid at costs Rs.8/kg

The given information can be tabulated as follows.

Fertilizer | Nitrogen | Phosphoric Acid | Cost/Kg |

A | 12% | 5% | 10 |

B | 4% | 5% | 8 |

Let the requirement of fertilizer A by the farmer be x kg and that of B be y kg

It is given that farmer requires at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops.

The in equations thus formed based on the given information are as follows.

Also,

Total cost of the fertilizer Z = Rs.(10x+8y)

Therefore, the mathematical formulation of the given LPP is

Minimize Z=10x+8y

Subject to the constraints

… (i)

… (ii)

… (iii)

The feasible region determined by constraints (1) to (3) is graphically represented as

Hence, it is seen that the feasible region is unbounded. The value of Z at the corner points of the feasible region are represented in tabular form as

Corner Points | |

(0,300) | |

(30,210) | (minimum) |

B(240,0) |

The minimum value of Z is 1980, which is obtained at x = 30 and y = 210.

Thus, the minimum requirement of fertilizer of type A will be 30kg and that of type B will be 210 kg.

Also, the total minimum cost of the fertilizers is Rs.1980.

Linear Programming exercise 29.4 question 55

Use property of LPP

The maximum number of hours available per day is16, If the profit on a necklace is Rs.100 and the bracelet is Rs.300

Let the number of necklaces manufacturer be x and the number of bracelets manufacture be y.

Since the total number of item are at most 24.

… (i)

Bracelets take 1 hour to manufacture and necklaces take half an hour to manufacture.

X item take x hour to manufacture and y item take y/2 hours to manufacture and maximum time available is 16 hours

Therefore,

… (ii)

The profit on one necklace is Rs.100 and the profit on one bracelet is Rs.300.

Let the profit be Z. Now we wish to maximize the profit.

So,

Maximize Z=100x + 300y … (iii)

So,

Maximize Z = 100x + 300y is required LPP

X+y=24

Corner points | Max Z |

(0,16) | 1800 |

(24,0) | 2400 |

(16,18) | 7000 |

The maximum value of Z = 7000 at (16,18)

RD Sharma class 12 solutions Linear Programming 29.4 is the ideal NCERT solutions to have for board exam preparations. Chapter 29 of the NCERT is titled Linear Programming and the concepts covered are formulating problems with different conditions, objective function, constraints, optimisation problem, feasible region, Bounded and unbounded region and are based on diet problems, manufacturing problems, and transportation problems. Exercise 29.4 has 55 questions that cover concepts from the entire chapter. The RD Sharma class 12th exercise 29.4 will help you solve all these questions and improve your performance.

For students in class 12, the class 12 RD Sharma chapter 29 exercise 29.4 solution will be indispensable. If students practice the book well, they will be able to avail the benefits of the book which are:-

The RD Sharma class 12th exercise 29.4 has its answers crafted by experts in maths. These experts are mindful about implementing new and improved calculations to help students solve questions faster.

The RD Sharma class 12 solutions Linear Programming 29.4 has answers to all questions from the NCERT book as it's updated with every new edition of the NCERT textbooks.

The RD Sharma class 12th exercise 29.4 contains some important answers which can be used for self-study and to check student's performance at home.

RD Sharma class 12 solutions Linear Programming 29.4 has a ton of questions and answers which might appear in the board exams. Hence, if students practice the book well, they will find common questions.

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The RD Sharma class 12 solutions chapter 29 ex 29.4 pdf can be availed for free from the Career360 website. There is no need for any financial investment.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

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The RD Sharma class 12th exercise 29.4 book is highly beneficial for students who are appearing for board exams. Maths can stress them out, so students can use these solutions to test themselves at home and compare answers to record their daily performance.

2. What are the concepts covered in chapter 29 of the NCERT maths book?

The 29th Chapter of the maths book in NCERT contains the chapter Linear Programming. The concepts covered are bounded linear equations, not feasible linear equations, unbounded linear equations, diet problems, manufacturing problems, and transport problems.

3. Can I use class 12 RD Sharma chapter 29 exercise 29.4 solutions for exam preparations?

Students can use the class 12 RD Sharma chapter 29 exercise 29.4 solution for their exam preparations. The syllabus followed by these books covers exams like school tests, boards, and JEE mains.

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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.

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A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

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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.

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Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

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Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

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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.

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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.

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A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

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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.

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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.

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A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

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A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

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A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

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A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

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Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.

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Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.

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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.

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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.

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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.

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A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

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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.

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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.

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When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

**Also Read:** Career as Nurse

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A career as Critical Care Specialist is responsible for providing the best possible prompt medical care to patients. He or she writes progress notes of patients in records. A Critical Care Specialist also liaises with admitting consultants and proceeds with the follow-up treatments.

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Individuals in the ophthalmologist career in India are trained medically to care for all eye problems and conditions. Some optometric physicians further specialize in a particular area of the eye and are known as sub-specialists who are responsible for taking care of each and every aspect of a patient's eye problem. An ophthalmologist's job description includes performing a variety of tasks such as diagnosing the problem, prescribing medicines, performing eye surgery, recommending eyeglasses, or looking after post-surgery treatment.

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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.

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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.

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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.

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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.

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The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

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Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

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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.

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Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

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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.

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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.

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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.

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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 cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

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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.

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Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

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Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

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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.

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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.

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Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

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A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

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A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

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The procurement Manager is also known as Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

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Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

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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

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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.

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ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

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.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of creating, designing and developing applications using .NET languages such as VB and C#.

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Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

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A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.

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Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

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