MIXTURE (C)

                    MIXTURE

                     + + ** ++

1) One vessel A contains mixture of milk and water in the Proportion 4:5 and in another vessel B they are mixed in the Proportion 5:1. In what Proportion should quantities be taken from the two vessels so as to form a mixture in which milk and water will be in the Proportion of 5:4.

2) A cask is completely filled with spirit. Four litres are drawn off and four litres of water is added. The operation is repeated for second time. If now the ratio of spirit and water in the cask is 4:5, find the capacity of the cask.

3) A mixture of 30 litres contains milk and water in the ratio of 7:3; how much water must be added to the mixture so that the ratio of milk to water may be 3:7.

4) A mixture of 30 litres contains milk and water in the ratio 7:3. How much mixture must be replaced by water to make the

ratio 3:7

5) Water and milk are mixed in the ratio 2:7 in one vessel and in the ratio 2:9 in another vessel. Determine the volume of the mixture in the second vessel that should be mixed with 225c.c. of the mixture in the first vessel to form a new mixture containing milk and water in the ratio 4:1.

6) Two mixture contains milk and honey with ratios of 7:2 and 5:1. In what ratio these two mixtures should be mixed so that the resulting mixture may contain milk and honey in the ratio 9:2.

7) Two vessels contain mixture of milk and water in the Proportion 2:3 and 4:3 respectively. In what Proportion should these two mixtures be mixed so as to form a new mixture containing equal proportions of milk and water.

8) Three equal glasses are filled with mixture of spirit and water, the Proportion of spirit to water being as 3:1, 5:3, 9:7. The context of three glasses are poured into a bigger vessel. What is the Proportion of spirit and water in the final mixture.

9) Two varieties rice at ₹10 per kg and ₹12 per kg are mixed together in the ratio 1:2. Find the average price of resulting mixture.

A) 10.33 B) 11.33 C) 12 D) 13.33 E) n

10) On combining two groups of students having 30 and 40 marks respectively in an exam, the resultant group has an average score of 34. Find the ratio of the number of students in the first group to the number of students in the second group.

A) 2/3 B) 3/2 C) 3/4 D) 4/3 E) n

11) On mixing 2 classes of students having average marks 25 and 40 respectively, the overall average obtained is 30 marks. find

i) the ratio of student in the classes.

A) 1:2 B) 2:1 C) 3:2 D) 2:3 E) n

ii) the number of students in the first class if the second class had 30 students.

A) 30 B) 50 C) 60 D) 80 E) n

12) 4 kg rice at ₹5 per kg is mixed with 8kg of rice at ₹6 per kg. Find the average price of the mixture.

A) ₹5 B) 5.66₹ C) ₹ 6 D) ₹ 6.32 E) n

13) 5kg of rice at₹6 per kg is mixed with 4 kg of rice to get a mixture costing ₹7 per kg. Find the price of the costlier rice.

A) ₹6.25  B) ₹7.25 C) ₹8.25 D) ₹ 9.25 E) n

14) On mixing two classes of students having average marks 25 and 40 respectively, the overall average obtained is 30 marks. Find

i) the ratio in which the classes were mixed.

A) 2:1 B) 3:1 C) 1:2 D) 1:3 E) none

ii) the number of students in the first class if the second class had 30 students.

A) 50 B) 60 C) 79 D) 80 E) n

15) 4kg of rice at₹5 per kg is mixed with 8 kg of rice at ₹6 per kg. Find the average price of the mixture.

A) 5.66 B) 5 C) 6.66 D) 7.66 E) n

16) 5 kg price at ₹6 per kg is mixed with 4 kg of rice to get a mixture coting ₹7 per kg. Find the price of the costlier rice.

A) 6.25 B) 7.25 C) 8.25 D) 9 D) n

17) a man buys 40 kg of rice at ₹20/kg and 60 kg of rice at ₹30/kg. Find the average price.

A) 23/kg B) 25/kg C) 26/kg D) 27/kg E) n

18) Pradeep mixes two mixtures of milk and water. He mixes 40 litres of the first containing 20% water and 60 litres of the second containing 30% water. Find the percentage of water in the final mixture.

A) 24% B) 25% C) 26% D) 27% E) n

19) Two classes are combined to form a larger class. The first class having 40 students scored an average of 20 marks on a test while the second having 60 students scored an average of 30 marks on the same test. What was the average score of the combined class on the test.

A) 24 B) 25 C) 26 D) 27 E) n

20) A trader earns a profit of 20% on 40% of his goods sold, while he earns a profit of 30% on 60% of his goods sold. Find his percentage profit on the whole.

A) 24 B) 25 C) 26 D) 27 E) n 

21) A car travels 20 km/h for 40 minutes and 30 km/h for 60 minutes. Find the average speed of the car for the journey.

A) 24 B) 25 C) 26 D) 27 E) n

22) 40% of the revenues of a school came from the junior classes while 60% of the revenues of the school came from the senior classes. If the school raises its fees by 20% for the junior classes and by 30% for the senior classes, find the percentage increase in the revenues of the school.

A) 24 B) 25 C) 26 D) 27 E) n

23) A merchant Mixes 3kgs of tea costing ₹50 per kg and 5kgs of another variety costing ₹34 per kg. At what price per kg should sell the mixture so as to make a profit of 10% on the cost price?

A) 34 B) 44 C) 54 D) 64 E) n

24) If 5 kg of salt consisting of 5/kg and 3 kg of salt consting ₹ 4/kg are mixed, find the average cost of the mixture per kilogram.

A)₹ 4.5 B) ₹4.625 C) ₹4.75 D) ₹ 4.125 E) ₹4.25

25) A vessel A contains milk and water in the ratio 4:5 and the vessel B contains the same in the ratio 5:1. In what proportion should the quantities to be taken from A and B to form a mixture in which milk and water are in the ratio of 5:4 ?

A)5:2 B) 2:5 C) 3:5 D) 5:3  E) n

26) Two types of oils having the rates of  ₹4/kg and ₹5/kg respectively are mixed in order to produce a mixture having the rate of ₹4.60/kg. What should be the amount of the second type of oil if the amount of the first type of the oil in the mixture is 40 kg 

A)75kg B) 50kg C) 60kg D) 40kg E) n

27) A container has a capacity of 80 litres and is full of milk, 5 litres of milk is taken out of it and the container is filled with water. This process is repeated 4 times. How much milk is left in the container?

A) 61.8 B) 62.5 C) 72.3 D) 83.5 E) n

28) how many kilograms of sugar worth ₹ 3.60 per kg should be mixed with 8 kg of sugar worth ₹4.20 per kg. such that by selling the mixture at ₹4.40 per kg, there may be a gain of 10%

A) 6 B) 3 C) 2 D) 5 E) 4kg

29) A mixture of 125 gallons of wine and water contains 20% water. How much water must be added to the mixture in order to increase the percentage of water to 25% of the new mixture?

A) 10gals B) 8.5gals C) 8gals D) 6.66gals E) 8.33gals

30) Ram lends ₹3600 on simple interest to Harsh for a period of 5 years. He lends a part of the amount at 4% interest and the rest at 6% and receives ₹960 as the amount of interest. How much money did he lend on 4% intrest rate?

A) 2800 B) 2100 C) 2400 D) 1200 E) 1000

31) A mixture of 40 litres of milk and water cont610% water. How much water must be added to make 20% in the new mixture.

A) 3 B) 5 D) 7 D) 10 E) n

32) 400 students took the exam. 60% of the boys and 80% of the girls cleared the cut off in the examination. If the total percentage of student qualifying is 65%, how many girls appeared in the examination 

A)100 B) 120 C)150 D) 300 E) 175

33) A man purchased a cow and a calf for ₹1300. He sold the calf at a profit of 20% in the cow at the profit of 25%, in this way, his total profit was 300/13% . Find the cost price of the cow.

A) 1100 B) 600 C) 500 D) 400 E) 800

33) The average salary per head of all employees of a company is ₹600. The average salary of 120 officers is₹4000. If the average salary per head of the rest of employees is ₹ 560, find the total number of workers in the company.

A) 1020 B) 1032 C) 1050 D) 1068 E) 1080 

34) In what ratio should silver at ₹5 per gram be mixed with silver at ₹4.75 per gram to obtain a mixed variety of silver worth ₹4.92 per gram?

A) 8:1 B) 8:23 C) 8:15 D) 8:17 E) n

35) In what ratio must water be added to the milk to gain 25/2% by selling the diluted milk at the cost price?

A) 1:8 B) 8:1 C) 2:3 D) 4:5 E) 5:6

36) A dishonest milkman purchased milk at 10 per litre and mixed of water in it. By selling the mixture at the rate of ₹10 per litre earns profit of 25%. The quantity of the amount of the mixture that he had was?

A) 15 B) 20  C) 25 D) 30 E) n

37) A cistern contains 50 litres of water. 5 litres of water is taken out of it and replaced by wine. the process is repeated again. Find the proportion of wine and water in the resulting mixture

A) 1:4 B) 41:50 C) 19: 81 D) 81:19 E) 50:41 

38) A container has a capacity of 20 gallons and is full of spirit. 4 gallons of spirit is drawn and the container is again filled with water. this process is repeated 5 times. Find out how much spirit is left in the resulting mixture finally

A) 257 B) 6257 C) 564 D) 2130 E) n

39) A vessel is full of refined oil. 1/4 of the refined oil is taken out and the vessel is filled with mustard oil. If the process is repeated 4 times and 10 litres of refined oil is finally left in the vessel, what is the capacity of the vessel?

A) 33 B) 2460/81 C) 2560/81 D) 30 E) n

40) how many kilograms of sugar worth ₹6.50 per kg must be mixed with 80kg of sugar worth ₹8.00 per kg so that the mixture is worth ₹7.50 per kg

A) 40 B) 60 C) 80 D) 100 E) n

41) A sum of ₹39 was divided among 45 boys and girls. Each girls gets 50 paise,.whereas a boy gets one rupee. Find the number of boys and girls.

A) 12,33 B) 33,12 C) 15, 60 D) 60,15 E) n

42) In what ratio should qualities of  coffee powder having the rates of ₹47 per kg and ₹ 32 per kg be mixed in order to get a mixture that would have a rate of 37 per kg

A) 1: 2 B) 2:1 C) 1:3 D) 3:1 E) 4:1

43) An alloy of two metals weighing 18gms is worth ₹87 but if their composition weights be interchanged, it would worth ₹78.60. if the price of one metal be ₹6.70 per gm. Find the weight of the other metal in the mixture.

A) 8 B) 10 C) 22 D) 15 E) n

44) A butler stole wine from a butt of sherry which contained 32% spirit and then replaced what he stole by wine containing 18% spirit. The butt was then of 24% strength only. How much butt did he steal ?

A) 4/7 B) 3/7 C) 2/7 D) 6 D) n

45) In what ratio should water be mixed with soda costing ₹12 per litres so as to make a profit of 25% by selling the dilute liquid at 13.75 per litre ?

A) 10:1 B) 11:1 C) 1:11 D) 12:1 E) 1:12

46) A sum of ₹36.90 is made up of 90 coins that are either 20 paise coins or 50 paise coins. Find out how many 20 paise coins are there in the total amount ?

A) 47 B) 43 C) 27 D) 63 E) 70

47) A  dishonest grocer professes to sale pure butter at cost price but he mixes it with adulterated fat and thereby gains 25%. Find the percentage of adulterated fat in the mixture assuming that adulterated fat is really available.

A) 20% B) 25% C) 33.33% D) 40% E)  35% 

48) A mixture of 70 litres of alcohol and water contains 10% of water. How much water must be added to the above mixture to make the water 12.5% of the resulting mixture ?

A)1 lt. B) 1.5 litres C) 2 lt D)  2.5 lt E) 3 lt

49) A mixture of 20 litres of brandy and water content 10% water. How much water should be added to it to increase the percentage of water 25% ?

A) 2lt B) 3lt C) 2.5lt D) 4lt E) 5 lt

50) A merchant purchased two qualities a pulse rate of 200 per quintal and ₹ 260 per quintal. In 52 quintal of the second quality, how much pulse of first quality should be mixed so that by selling the resulting mixture at ₹300 per quintal, he gains a profit of 25%?

A) 100 B) 104 C) 26  D) none

51) A man buys milk at 8.5 per litre and dilutes it with water. He sells the mixture at the same rate and thus gains 11.11%. Find the quantity of water mixed by him in every litres of milk.

A)0.111 lt B) 0.909 lt C) 0.1 lt D)0.125 lt

52) There are two mixtures of honey and water the quantity of honey in them being 25% and 75% of the mixture. If two gallons of the first are mixed with three gallons of the second, what will be the ratio honey to water in the new mixture?

A) 11:2 B) 11: 9 C) 9:11 D) 2:11

53) There are two kinds of alloys of tin and copper. The first alloy contains tin and copper such that 93.33% of it is tin. In the second alloy there is 86.66% tin. What weight of the first alloy should be mixed some weight of the second alloy so to make a 50 kg of mass contains 90% of tin?

A) 15kg b) 30kg c) 20 kg d) 25kg

54) Two containers of equal capacity are full of mixture of oil and water. In the first, the ratio of oil to water is 4:7 and in the second it is 7:11. Now both the mixture and mixed in a bigger container. What is the resulting ratio of oil to water?

A) 149:247 B) 247:149 C) 143: 241 D) 241:143

55) Two vessels contain spirit and water mixed respectively in the ratio 1:3 and 3:5. Find the ratio in which these are to be mixed to get a new mixture in which the ratio of spirit to water is 1:2.

A) 2:1 B) 3:1 c) 1:2 d) 1:3 

56) The price of pen and a pencil is ₹35. The pen was sold at a 20% profit and the Pencil at a 10% loss. If in the transaction a man gains ₹4, how much is cost price of the pen ?

A) 10 b) 25 c) 20 D) none 

57) A person purchased a cupboard and a cot for ₹18000. He sold the cupboard at a profit of 20% and the cot at a profit of 30%. If his Total profit was 25.833%, find the cost price of the cupboard.

A) 10500 B) 12000 c) 7500 d) 10000

58) A vessel is of a mixture of kerosene and petrol in which there is 18% kerosene. 8 liters are drawn off and then the vessel is filled with petrol. If the kerosene is now 15%, how much does the vessel hold ?

A) 40 litres b) 32 c) 36 litres d) 48 

59) two solutions of 90% 97% purity are mixed resulting in 21 l of mixture of 94% of purity. how much is the quantity of the first solution in the resulting mixture?

A) 15 litres b) 12 litres c) 9 liter d) 6

60) in the Singapore zoo,  there are deers and there are ducks. If the heads are counted, there are 180, while the legs are 448. What will be the number of deers in the zoo?

A) 136 b) 68 c) 44 d) 22

61) A bonus of ₹985000 was divided among 300 workers of factory. Each male worker get 5000 rupees and each family workers gets 2500 rupees. Find the number of male workers in the factory.

A) 253 b) 47 c) 94 d) 206 

62) What will be the ratio of petrol and kerosene in the final solution formed by mixing petrol and kerosene that are present in 3 vessels in the ratio 4:1, 5:2,  6:1 respectively?

A) 166 :22 b) 83 :22 c) 83 :44  d) n

63) A mixture worth ₹ 3.25 a kg is formed by mixing two types of flour, 1 costing of ₹ 3.1 0 per kg while the other ₹3.60 per kg, In what proportion must they have been mixed ?

A) 3:7 b) 7:10 c) 10:3 d) 7:3 

64) A gain percentage of 20 is made by selling a mixture of two types of ghee at ₹480 per kg. If the type costing 610 per kg was mixed with 126 kg of the other, how many kilograms of the former was mixed?

A)139 B) 34.5 c) 69 d) cannot be determined

65) In what proportion must water be mixed with milk so as to gain 20% by selling the mixture at the cost price of the milk ?(assume that water is free available)

A) 1:4 b) 1:5 c) 1:6 d) 1:12

66) A bartender stole champagne from a bottle that contained 50% of spirit and he replaced what he had stolen with champagne having 20% spirit . The bottle then contained only 25% spirit. How much of the bottle did he steal?

A) 80% b) 83.3 c) 85.71% d) 88.88% 

67) A bag contains a total 105 coins of ₹1.50p and 25p to denominations. Find the total number of coins of ₹one if there are a total of 50.5 rupees in the bag and it is known that the number of 25 paise coins are 133.33% more than the number of 1 rupees coins.

A) 56  b) 25 c) 24 d) none

68) A man possessing ₹6800, lent a part of it at 10% simple interest and the remaining at 7.5% simple interest. His total income after 7/2 years was ₹ 1904. Find the sum lent at 10% rates.

A) 1260 b) 1700 C) 1360 d) none

69) A man decides to travel 80 kilometres in 8 hours partly by foot and partly on a bicycle, his speed on foot being 8 km/hr and that on bicycle being 16 km/hr. What distance would he travel on foot?

A) 20 b) 30 c) 48 d) 60 

70) Two vessels contain a mixture of spirit and water. In the first vessel the ratio of spirit to water is 8:3 and in the second vessel the ratio is 5:1. A 35 litres cask is filled from these vessels so as to contain a mixture of spirit and water in the ratio 4:1. How many litres are taken from the first vessel ?

A) 11 b) 22 c) 16.5 d) 17.5

71) Rakshit bird 19 eraser for rupees 1025 more for each wise Razer them for each brownie razor what could be the price of a white eraser and how many white eraser producer he have but 65 865 12:55 855 10 after paying all your beans you find that you have Rupees 7.20 in your pocket you are equal number of 55 and 10 piece coins but no other coins no rainy other currency notes how many coins do you have 82427 30 Suresh Kumar went to the market with rupees 100 if he buy 3 pens and 6 pencil heuses of all his money on the other any few by three pencils and 6 would fall sort by 20% if you wants to buy equal number Parde work questions what is the amount of money he would save if you had to buy 3 pegs and 3 pencils 50 25 7540 Abdul goes to the market to buy bananas if he can burger in the price but doesn't by 2 he can buy 3 doesn't banana Institute of two doesn't with the money has how much money does he have Rupees 612 1824 inches 3 bananas and 4 apples cost 153 bananas and three apples how much did I pay 10 8:15 cannot be determine John Word 5 mangoes in orange together rupees 1234 00 km about school is to be built for hundred students up down B and 30 students up down A the expenditure on transport age 1.2 0 per kilometre per person if the total expenditure and transport by all 13 students is to be at the small as possible then the school should be built at 33 km from Townhall 33 km from town B Town a town b person who has certain amount with he he goes to the market you can buy 50 oranges are 40 mangoes heritance 10% of the amount for taxi fare and by 20 mangoes and the banana balance the Purchase oranges number of oranges chicken purchase 36 40 15 20 72 hands cost Rupees 96.7 then what does each hand cost were the number at or not divisible or written in illegal hand 3.435.31516.2 class this students are divided into three groups A B and C of 1525 students is the group D what is the average weight of the student in group D more than the average weight of a more than the average weight of she less than the average weight of C cannot be determine student from group B which of the following will be true the average weight of the both groups increases average decreases the average weight of the class remains the same cannot be to my all the students of the class the total weight of P and seastwise the total wet a b the average weight of these greater than the average weight of a the average weight of all the groups remains the same even if a number of students are shifted from one group two another the bridge marks of student in 10 papers are 80 if the highest and the lowest score are not considered the average is 81 it is highest score in 92 and find the lowest 55 662 cannot be to mind a shipping clerk has five boxes of different but unknown weights each wedding less than 100 kg the clockwid in the boxes in pair the words obtain are 102 and 13 10718 120 and 121 kg what is the weight of the having heaviest box 6062 64 cannot be determine the total expenses of a boarding house are partly fix and partly brain linearly with the number of borrowed the average expenses for border is 700 when there are 25 words and Rs 600 when there are 50 border what is the average expense for border when they are 100 border 5:5585 45 70 yearly payment to a servant is 90 + 1 Turban the servant leaves the job after 9 months and receive is 65 and the term then find the price of the turbo 10 15 7.5 0 cannot be determine a leather factory locks the profit margin is 20 on a standard bag and rupees 30 on a Deluxe bag every bag time required hours bags machinen machinery standard back 46 Deluxe 500 10 the total time available on machine A700 hours non machine beast 1250 hours among the following products plans which one made the machine availability constants and maximum the profit standard 75 best Deluxe 80 bags standard 10050 algebra test the average score of class 10 is 83 the average class score of class Y 76 the average score of class Jet is 85 what is the average score of class 6yz at 1.58.583 cannot be determine the order 4 and some additional Park Avenue shirts the price the order was executed it was found that the number of the two brands have been interchange this increase the bill by 40% the ratio of the number of Arrow shirts to the number of Arc Avenue shirts in the original order was 13 14 12 15 3 groups of companies data Birla and Reliance announces the average of the annual profit for all your sins there establishment the average profit of Tata is 75000 the average profit of Billa is 64000 the average profit of Reliance is 73000 the average profit of all results of Tata and Birla together is 70000 the average profit award is a bitland Reliance together 9000 aproximately what is the average profit for all the three groups 7000086

River water consist of 0.5% salt made Lake water consist of 1% Sir Three water consist of 5% of salt in it this three are mixed in the ratio 4:2 and boiled such that concentration of salt in the club eating water from the solution from Hyderabad 55% 45% 75% 50% 60% agrocel mix 3 varieties of Dal consisting of 25 3033 respectively packaging in the ratio 1 decided to say the mixture at rupees 35 per kg and other browser mixes the three variety of dull equal propose save the mixture of 35 kg 300 then find the ratio of 3 variety respectivali because that is brought is mixed up 8615 question what is the ratio of the first process there at 10 best solution capacity 10 litre the first battle content 10% pure milk the third has 100% pure milk 1 l from the first battle 2 l from the second basal and so on uncle 10 litre from the 10th what is the percentage of your name 55% 60% 33% 80% 70% is taken labour problem the 10th percentage of the solution will be pure milk grocer wanted to mixi varieties flower consisting him 15-1821 so that the resultant price of the mixture is 20.90 when sold at 10% was mixed the first one the last variety in the ratio 1 ratio the same 10% profit when the sold at the same price of 20.90 12 13 14 23 24 there are two bases A and B both containing universalism of 30% concentration I add some pure whenever to a to bring the concentration in the Basil b i take out some quantity of the solution and replace it with an equilate quantity of your University A and B if the quantity of initial solution in A and B are in the Rati 12 23 25 in a drum pull up your milk 15th of the milk is drawn a place to the water 15th of the solution is drawn and replace to the water disapporation is performed 5 times all together find the capacity of the drum the difference in the quantity of your mail can the solution wrong out one Drone from the third timing bronze is an alloy of Copper and Tin in the ratio 5 rational lawyer components in consist of these elements in the ratio 3 a new alloy is on the by making bronze and bra such that competition the popular such that contribution of copper in the new alloy equals interest by what percentage is the quantity of zinc present more than the quantity of teenager new alloy 750% 6650 in Homeopathy School of Medicine potential to indicate one part of your medicine mixed with 100 parts of alcohol and so on what is the ratio of pure medicine to the total mixture 2 unit substances to our mixed with one unit of potential one and three units 300 in the world questions important parts of the alcohol medicine potential 2 is 1 part of your medicine 100 parts of the alcohol medicine population so one then find the answer of the above question 3 21 23 Rajat the chemical analyst has got two solution of ferric sulphate in water the concentration of ferric sulphatever 60% then one solution and 50% another solution mixed up the solution and continuous solution the concentration of the new solution is 52% water was added to 100 l of the solution to obtain the solution of 40% concentration what fraction 14 15 413 16 alcohol 10 litres of alcohol and operation is defined as taking out 5 letters of what is represent In The vessel in 10 l water to it what is the ratio of alcohol to water after 2 operation 14 23 16 32 50 the contents of a 750 bottle filled with methanel of 75% concentration along with that of methanol are amted into a vessel with the present concentration of methanol In The vessel is equals to 60% then find the volume of the second bottle 1125 1215 875 150

PYTHAGORAS (C)

1) Length of sides of triangles are 7cm, 24cm, 25cm. Triangle is

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

2) Length of sides of triangles are 50cm, 80cm, 100cm. Triangle is

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

3) Length of sides of triangles are 3cm, 8cm, 6cm. Triangle is

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

4) Length of sides of triangles are 13cm, 12cm, 5cm. Triangle is

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

5) Length of sides of triangles are 1.4cm, 4.8cm, 5cm. Triangle is

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

6) An aeroplane leaves an airport and flies due to North at a speed of 1000km per hour. At the same time, another aeroplane leaves the same airport and flies due to West at a speed of 1200 kmph. How far apart will be the two planes after 3/2 hours?

A) 100 B) 100√6 C) 100/√6 D) 300√61

7) For going to a city B from city A, there is route via city C such that AC perpendicular to CB, AC= 2x km and CB = 2(x+7) km. It is proposed to construct a 26km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of highway. 

A)  8 B) 10 C) 20 D) 12

8) A 5m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4m high. If the foot of the ladder is moved 1.6m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.

A) 0.4 B) 0.6 C) 0.8 D) 0.10

9) ABC is an equilateral triangle of side 2a. Find each of its altitude. 

A) a B) √a C) 2√a D) √3 a

10) Foot of a 10m long ladder leaning against a vertical wall is 6m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches. 

A) 4m B) 6m C) 8m D) 10m

11) A guy attached a wire 24m long to a vertical pole of height 18m and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taught? 

A) 6 B) 7 C) 7√6 D) 6√7

12) Two poles of height 6m and 11m stand on a plane ground. If the distance between their feet is 12m, find the distance between their tops. 

A) 11 B) 12 C) 13 D) 14

13) In a right angled triangle, if hypotenuse is 20cm and the ratio of the other two sides is 4:3, find the sides. 

A) 16, 12 B) 12, 14 C) 14, 18 D) 12, 18

14) If the sides of a triangle are in 3:4:5, then the triangle is;

A) right angled triangle

B) isosceles triangle

C) Equilateral triangle

D) scalene triangle

15) The hypotenuse of a right angled triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

A)  10,24,26 B) 12,24,26 C) 12,24,28 D) none 

16) In ∆PQR PD perpendicular to QR, such that D lies on QR. If PQ= a, PR= b, QD= c and DR= d then find the value of (a+ b)(a - b). 

A)  (c+ d)(c - d) B) (c- d)(c + d)

17) ABC is an isosceles with AB= AC= 12 and BC= 8cm. Find the altitude on BC and hence calculate its area. 

A) 8, 32 B) 8√2, 32 C)  8, 32√2 D)  8√2, 32√2

18) Find the area and the perimeter of a square whose diagonals is 10cm long. 

A) 50, 20 B) 50√2, 20 C)  50, 20√2 D) 50√2, 20√2

19) In ∆ABC, AB= AC=x, BC= 10cm and the area of ∆ABC is 60cm². Find x. 

A) 10 B) 12 C)  13 D) 14

20) In a rhombus, if diagonals are 30cm and 40cm, find its perimeter.

A) 100 B) 8√2 C) 32√2 D) 110

21) In a ∆ABC, AB= 6√3, BC= 6 and AC= 12cm, then angle B is

A) 120° B) 90° C) 60° D) 45°

22) If the sides of a rectangular plot are 15m and 8m, then the length of its diagonal is

A) 17m B) 23m C) 21m D) 17cm

23) If a side of a rhombus is 10cm and one of the diagonals is 16cm, then the length of the other diagonals is

A) 6cm B) 12cm C) 21m D) 17cm

24) If a ladder 10m long reaches a window 8m above the ground, then the distance of the foot of the ladder from the base of the wall is

A) 18m B) 8m C) 6m D) 4m

25) A girl walks 200 towards East and then she walks 150m towards North. The distance of the girl from the starting point is

A) 350m B) 250m C) 300m D) 225m

26) A ladder reaches a window 12m above the ground on one side of the street, keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 9m high. If the length of the ladder is 15m, then the width of the street is

A) 30m B) 24cm C) 21m D) 18m

27) In a triangle ABC, AD perpendicular to BC, AB= 25cm, AC = 17cm and AD= 15cm. Find the length of BC. 28cm

28) ∆ABC is a right angled triangle at B. Given that AB= 9cm, and AC = 15cm and D,E are the midpoint of the sides AB and AC respectively, Calculate

i) the length of BC. 

A) 12cm B) 14cm C) 16cm D) 20cm

ii) the area of ∆ADE. 

A) 12m² B) 13.5cm² C) 21.5m² D) n

29) Which of the triangle whose sides are given below are right angled?

a) 5,12,13

b) 6,8,10

c) 24,10,26

d) 7,24,25

e) 8,15,17

f) 8,10,11

g) 8,40,41

h) 36,15,39

i) 16,63,65

30) A right angled triangle has hypotenuse of length p cm and one side of length q cm. If p - q=1,  find the length of the third side of the triangle.

31) In ∆ ABC, BC= 12cm, CA=16cm and AB= 20 cm. Find the value of angle C.

32) In ∆ ABC, AD perpendicular to BC, AD= BD= 5cm, BC= 17cm. Find the value of AC.

33) The foot of a ladder is at a distance of 8 m from the wall of a house and the top of the ladder is at the height of 15m from the ground. Find the length of the ladder .

34) A ladder 13m long reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building.

35) The heights of two vertical pillars are 7m and 12m. The distance between their feet is 12m. Find the distance between their tops.

36) A man goes 15m due east and then 8m due North. How far is he from the starting point ?

37) In ∆ ABC is right angled at A. AD perpendicular to BC. If BC=1.25m, AB= 1m, Find AD.

38) In the following figure, ABC is a right angled triangle at B. AD and CE are the two median drawn from A and C respectively.

If AC= 5cm and AD=3√5/2 cm,. find the length of CE.

AVERAGE (C)

EXERCISE -1

1) Find the average of first 7 even natural numbers.                      

A) 8 B) 9 C) 10 D) 11

2) The number of children in 25 families of a locality are recorded as follows: 3,1,4,0, 2,2, 1,1, 2, 3, 3, 2, 2, 2, 5, 0, 1, 4, 1, 2, 1, 2, 3, 0, 1, 4. Find the average of number of children per family.                        

A) 2 B) 3 C) 4 D) 5

3) The marks obtained by by 10 students in a test are 56, 54, 71, 60, 62, 72, 59, 64, 70, 52. Find

i) the mean marks of the student

A) 62 B) 63 C) 64 D) 65

ii) The average marks of the student if 5 extra marks are given to each student.        

A) 67 B) 69 C) 70 D) 72

4) A student obtained 60, 75 and 85 marks, respectively in three monthly examination in Physics and 95 marks in the final examination. The three monthly examination are of equal weightage whereas the final examination is waghted twice as much as a monthly examination. Find his average marks for Physics.

A) 82 B) 85 C) 86 D) 88

5) Find the average of

i) first five natural natural numbers.

A) 3 B) 5 C) 6 D) 9

ii) first five positive odd integers

A) 5 B) 6 C) 9 D) 4.5

iii) first five multiples of 3 multiples of 3.

A)  6 B) 9 C) 4.5 D) none

iv) all factors of 10.        

A) 5 B) 6 C) 9 D) 4.5

6) find the average of x, x+2, x+4, x+6, and x+8. 

A) x B) x+3  C) x+4 D) x + 5

7) If the average of 6, 8, 5, 7, x and 4 is 7, find the value of x. 

A)12 B) 14 C) 20 D) 24

8) If the average of 6, 4, 7, p and 10 is 8, find the value of p.               

A) 13 B) 14 C) 15 D) 16

9) If the average of 5 numbers is 7, find the average of another 5 numbers numbers which are obtained by adding 2 to each case of the 5 numbers.

A) 9 B) 10 C) 12 D) 15

10) The average of 8 numbers is 15. If each number is multiplied by 2, what will be the new average.     

A) 30 B) 32 C) 35 D) 40

11) The average is 21 numbers is 15. If each number is multiplied by 2, what will be the new average.

A) 30 B) 20 C) 25 D) 222

12) There are 45 students In a  class, of which 15 are girls. The average weight of 15 girls is 45 kg and that of 30 boys is 52kg. Find the average weight in kg of the entire class.  

A) 45   B) 47 C) 49.67 D) 50.21

13) Average of 25 observations was found to be 78.4. but later on it was discovered that 96 was misread as 69. Find the correct average.   

A) 88.9 B) 90.7 C)  79.48 D) 97

14) Average of 40 observations was 160. It was detected on rechecking that the value 165 was wrongly occupied as 125 for computation of average. Find the correct average

A) 16.1 B) 17 C) 18.2 D) 19

15) The average weight per student in a group of seven students is 55 kg. The individual weight(in kg) of 6 of them are 52, 58, 55, 53, 56, and 54. Find the weight of the 7th student.                                    

A) 55.5 B) 57 C) 65.2 D) 70

16) The average weight of 150 students in a certain class is 60 kg. The average weight of boys in the class of 70 kg and that of the girls is 55kg. Find the number of boys and the number of girls in the class.                                    

A) 50, 100 B) 60, 200 C) 75, 100 D) n

17) In a class, the average score of girls in an examination in 73 and that of a boy is 71. The average score for the whole class 71.8. Find the percentage of girls and the boys in the boys in the class.    

A) 20, 30% B) 30, 40% C) 40, 60% D) 70, 80%

18) A train travels between two stations A and B, while going from A to B, its average speed is 100 km/hr, and when coming back from B to A, its average speed is 150 km/hr. Find the average speed during the whole journey.            

A) 120 B) 140 C) 150 D) 200

19) The average height of 10 students is 153 centimetre. it is discovered later on that while calculating the reading 151cm was wrongly read as as 141cm. Find the correct average.                         

A) 144 B) 150 C) 154 D) 160

20) The average of five number is 27. If one number is excluded, their average is 25. Find the excluded number.                                       

A) 25 B) 30 C) 35 D) 40

21) There are 50 students In a class of which 40 are and the rest girls. The average weight of the the class is 44 kg and the average weight of the girls is 40 kg. Find the average weight of the the boys.                

A) 35 B) 30 C) 40 D) 45

22) There are 120 boys in a class in which twenty of them are girls and the rest boys. If the average marks in Maths of the boys is 65% and that of the girl is 80%. Find the average mark of the class.     

A) 56.83 B) 65.6 C) 76.8 C) 88.8 D) none 

23) The average score of boys in an examination of a school is 71 and that of girls is 73. The average of the school in that examination in 71.8. Find the ratio of the number of boys to the number of girls appeared in the examination.      

A) 3:2 B) 2:3 C) 4: 5 D) 5: 4

24) The average marks secured by 25 students of of section A of class 47, that of 35 students of section B is 51 and that of 30 students of Section C is 53  students of Section C is 53 . Find the average of the marks of students of three sections of class sections of class X.  

A) 50.55 B) 65 C) 65.6 C) 70 D) n

25) The average monthly wage of a group of 10 persons is Rs.1500. One member of the group whose monthly wage is Rs1350, left the group and is replaced by a new member whose monthly wage is Rs.1200. Find the new monthly wage.                                       

A) 1145 B) 1245 C) 1345 D) 1485

26) The average monthly salary of 10 members of a group is Rs1445. One more member whose monthly salary is Rs1500 has joined the group. Find the average monthly salary of 11 members of the group.

A) 1150 B) 1245 C) 2350 D) 1450

27) The average marks secured by hundred students was found to be 40. later on it was discovered that score of 53 was misread 83. Find the correct average.        

A) 30.9 B) 32.7 C) 35.7  D) 39.7

28) The aggregate monthly expenditure of a family was Rs 4050 during the first three months. Rs 4260 during the next four months and Rs4326 during the last five months of a year. If the total saving during the year be Rs 6120, find the average average monthly income.                      

A) 4050 B) 4678 C) 4690 D) n

29) the average of 11 results is 50. If the average of the first six results is 49 and that of last is 52, find the sixth sixth result.                       

A) 50 B) 52 C) 55 D) 56

30) A batsman in his 12th innings makes a score 63 runs and thereby increases his average score by 2. What is his average after the 12th innings.                                       

A) 41 B) 43 C) 44 D) 46

31) The average age of 8 persons in a committee is increased by 2 years, when 2 men aged 35 years and 45 years are substituted by two women. Find the average age of these women.                              

A) 40 B) 42 C) 45 D) 48

32) A Car owner buys petrol at Rs 17, Rs19 and Rs 20 per litre for 3 consecutive years. compute the average cost pee litre, If he spends Rs 6460 per year.                    

A) 18.58 B) 20.20 C) 21.30 D) 19.92

EXERCISE -2

1) The average of a batsman after 25 innings was 56 runs per innings. If after the 26th innings his average increases by 2 runs, then what was his score in the 26th innings?

A) 100 B) 102 C) 104 D) 108

2) The average age of class of 30 students and a teacher reduces by 0.5 years if we exclude the teacher. If the initial average is 14 years, find the age of the class teacher.

A) 25 B) 27 C) 29 D) 30

3) the average marks of a group of 20 student on a test is reduced by 4 when the topper who scored a 90 marks is replaced by a new student. How many marks did the new student have ?

A) 10 B) 60 C) 80 D) 40

4) The average age of 3 students A, B and C is 48 marks. Another student D joins the group and the new average becomes 44 marks. If another student E, who has 3 more marks more than D, joins the group,  the average of the 4 students B, C, D and E becomes 43 marks. Find how many marks A got in the exam.

A) 38 C) 39 C) 40 D) 43

5) the average temperature of Monday to Wednesday 27°C and of Tuesday to Thursday was 24°C. if the temperature on Thursday was 2/3rd of the temperature on Monday what was the temperature on a Monday, what was the temperature on Thursday

A) 40 B) 37 C) 24 D) 20 E) none

6) A person covers half his journey by train at 60 kmph, remainder half by bus at 30 kmph and the rest by cycle at 10 kmph. Find his speed during the entire journey

A) 20 B) 24 C) 30 D) 18 E) none

7) A school has only three classes that contains 10, 20 and 30 students respectively. The pass percentage of thse classes at 20%, 30% then 40% respectively. Find the pass percentage of the entire school.

A) 2 B) 6 C) 12 D) 20 E) none

8) P, Q and R are paid ₹6300 for doing a piece of work together. P and Q start the work and continue to work together for 10 days. After that P quits while, R joins Q and they complete the remaining work. How much does Q receive for his work, given that P can do the work in 30 days, Q in 60 days and R in 90 days?

A) ₹2940 B) ₹3000 C) Rs3500 D)₹ 4000 E) ₹3600

9) The average age of 24 students and the principle is 15 years. When were the principal's age is excluded, the average age decreased by 1 year. What is the age of the principal?

A) 38 B) 40 C) 39 D) 37 D) none

10) the average weight of 3 men A, B and C is 84kg. Another man D joins the group and the average now becomes 80 kg. If another man E, whose weight is 3 kg more than that of D, replaces A then the average weight of B, C, D and E becomes 78kg. The weight of A is

A) 70 kg B) 72kg C) 79kg D) 78 kg E) 80 kg

11) Taps P and Q can fill up a Jug in 50 seconds and 40 seconds respectively. Tap R(which is attached to the jug) can empty the jug in 30 seconds. If taps P, Q and R are opened in turn, in that order, for 3 seconds each, then in how many seconds will the empty jug be filled?

A) 600/7s  B) 256 C) 253 D) 230.6 E) 225

12) the mean temperature of Monday to Wednesday was 37°C and Tuesday to Thursday was 34°C, if the temperature on Thursday was 4/5 that of Monday, the temperature on Thursday was

A) 38°C B) 36°C C) 40°C D) 39°C E) none

13) 3 years ago, the average age of A, B and C was 27 years and that of B and C 5 years ago was 20 years. A's present age is 

A) 30 years B) 35 years C) 40yrs D) 48yrs E) n

14) Sameer can do a piece of work in 9 hours. Rahul can do the same piece of work in 8 hours. Their boss asks them to work along the Ronak and hence they finish the work in 3 hours. If their Boss offers them ₹ 7200 for the work, how much does Ronak earn ?

A) 6900 B) 5000 C) 2100 D) 8000 E) 4200

15) An empty barrel of 240 litres capacity can be filled with beer by 2 pipes P and Q, in 30 minutes and 60 minutes respectively. Pipe R can empty the full barrel in 120 minutes. if the 3 pipes are opened simultaneously, how much beer can accumulate in 15 minutes?

A) 14 litre B) 150 litre C) 60 litre D) 10 litre E) 90 litre

16) Tendulkar has a certain average for 9 innings,. In the 10th innings, he scores 100 runs thereby increasing his average by 8 runs. His new average is

A) 20 B) 24  C) 28 E) 32 F) 36 

17) The average of the first 5 multiples of 7 is 

A) 20 B) 21 C) 28 D) 30 E) 31 

18) There are three fractions A, B and C. If A= 1/4 and B= 1/6 and the average of A,B and C is 1/12. What is the value of C ?

A) -1/2 B) -1/6 C) -1/3 D) -1/4 E) none

19) The marks obtained by Ram in Mathematics, English and Biology are respectively 93 out of 100, 78 out of 150 and 177 out of 200. Find his average score in percentage.

A) 87.83 B) 86.83 C) 76.33 D) 75.33 E) 7u.83

20) The average monthly expenditure of a family was ₹2750 for the first three months, ₹3150 for the next three months and ₹6750 for the next 3 months. Find the average income of the family for the 9 months, if they save ₹650 per month

A) 4866.66 B) 5123.33 C) 4666.66 D) 4216 66 E) none

21) the average age of a family of 6 members is 22 yrs. If the age of the youngest member is 7 years, what was the average age of the family at the birth of the Youngest member?

A) 15 B) 18 C) 21 D) 12 E) n

22) The average age of 8 person in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. Find the average age of the two women.

A) 48 B) 45  C) 51 D) 42 E) 46

23) the average temperature for Wednesday ,Thursday and Friday was 40 degree. the average for Thursday, Friday and Saturday was 41°C. If the temperature on Saturday was 42°C, what was the temperature on Wednesday?

A) 39 B) 44 C) 38 D) 41 E) n

24) the speed of the train is going from Nagpur to Allahabad is 100 km/hr while when coming back from Allahabad to Nagpur, its speed is 150 km/hr. Find the average speed during the whole journey.

A) 125 B) 75 C) 135 D) 141 E) 120

25) 

TIME AND WORK (C)

EXERCISE -1

1) A can do a piece of work in 6 days and B can do the same work in 12 days. How many days will they take to do it together. 

A) 4 B) 6 C) 8 D) 10

2) A can do a piece of work in 24 days and B can do the same work in 12 days. They work together for 6 days and then B leaves. In how many days, A alone finish the remaining work. 

A) 10 B) 8 C) 6 D) 4

3) A and B can do a piece of work in 10 days, B and C can do the same work together in 12 days, while A and C can do together in 15 days. In how many days will they take to do it together. 

A) 4 B) 6 C) 8 D) 10

4) A can do a piece of work in 10 days and B can do the same work in 16 days. With the help of C, they can do the work in 4 days. In how many days can C do the work alone? 

A) 3 B) 16 C) 3/16 D) 16/3

5) Ram can do a piece of work in 6 days and Shyam can do the same work in 4 days. C started the work, worked at it for 2 days and then was B joined. Find the total time taken to finish the work?

A) 11 B) 12 C) 11/12 D) 12/11

6) A can do a piece of work in 12 days and B can do the same work in 15 days and A, B and C together can complete the work in 5 days. In how many days will C alone finish the work? 

A) 20 B) 15 C) 10 D) 4

7) Together, A and B plough a field in 4 days, B alone finish in 6 days to plough the same field. In how many days A alone plough the field? 

A) 12 B) 15 C) 20 D) none

8) A can do a piece of work in 10 days and B can do the same work in 15 days. How many days will they take to do it together.

A) 6 B) 10 C) 12 D) 15

9) A can do a piece of work in 24 days and B can do the same work in 15 days. They work together for 6 days and then B leaves. In how many days will, A alone finish the remaining work? 

A) 5 B) 42/5 C) 5/42 D) 42

10) A can do three times as much work as B. They together finish a work in 14 days. In how many days can A alone finish the work ? 

A) 3 B) 56 C) 3/56 D) 56/3

11) A can do a piece of work in 6 days and B can do the same work in 8 days. How many days will they take to do it together. 

A) 7 B) 24 C) 7/24 D)  24/7

12) A can do a piece of work in 25 days and B can do the same work in 20 days. They work together for 5 days and then A goes away. In how many days will B complete the remaining work? 

A) 12 B) 10 C) 24 D) 4

13) A and B can do a piece of work in 8 days, B and C can do the same work in 6 days, C and A in 4 days. In how many days can A, B and C finish together? Also find the number of days in which B alone can do the work. 

A) 24 B) 48 C) 42 D) 36

14) Ramesh, Dinesh and Suresh can together do a piece of work in 4 days. Ramesh can do the same work in 12 days and Suresh can do it in 20 days. How many days will Dinesh take to do the work by himself. 

A) 10 B) 12 C) 15    D) 20

15) Two inlet pipes can fill up a cistern in 4 hours and 5 hours, repetitively. If both pipes start filling the cistern together, how long will it take for the cistern to get filled?

A) 20 B) 9 C)  20/9 D) 9/25

16) A tap A can fill a tank in 4 hours and tap B can empty the full tank in 6 hours. If both the taps are opened together in the empty tank, in how much time will the tank be filled up ? 

A) 12 B) 15 C) 20 D) 24

17) Two inlet pipes can fill up a cistern in 12 hours and 16 hours, repetitively. The full cistern can be emptied by a third tap in 8 hours. If all pipes start at the same time, in how much time will the empty tank be filled up comptely. 

A) 48 B) 42 C) 20 D) 20

18) Two inlet pipes can fill up a cistern in 3 hours and 5 hours, respectively. If both pipes start filling the cistern together, how long will it take for the cistern to get filled? 

A) 15 B) 8 C) 8/15 D) 15/8

19) Tap A and tap B can together fill an empty cistern in 6 hours. Tap A alone can fill the cistern in 10 hours. Find the time taken by tap B alone to fill the cistern. 

A) 10 B) 15 C) 20 D) 24

20) Tap A and tap B can fill an empty cistern in 12 hours and 15 hours respectively. The full cistern can be emptied by a third tap C in 10 hours. If all the taps are opened on at the same time, in how many much time will the empty tank be filled up comptely? 

A) 48 B) 42 C) 36 D) 30

21) Three pipes A, B and C can fill an empty cistern in 15 hours, 10 hours and 6 hours respectively. In how much time will the cistern be filled, if all the three pipes work together. 

A) 6 B) 5 C) 4 D) 3

22) A pipe can fill a tank in 10 hours. Due to a leak in the bottom it is filled in 12 hours. When the tank in full, in how much time will be emptied by the leak ? 

A) 54 B) 60 C) 42 D) 15

23) Tap A and tap B can empty the cistern in 3 and 5 hours. Tap C alone can empty the cistern in 15/2 hours. If all three taps are open at the same time, in how many hours will the cistern be full? 

A) 5 B) 2 C) 2/5 D)  5/2

24) Tap A and tap B can together fill an empty cistern in 48/5 hours. If Tap A alone can fill the cistern in 16 hours. Find the time taken by tap B alone to fill the cistern. 

A) 24 B) 36 C) 42 D) none

Answers

1A 2D 3A 4D 5D 6B 7A 8A 9B 10D 11D 12A 13B 14C 15C 16A 17A 18D 19B 20A 21D 22B 23D 24A

H.C.F & L.C.M - A to Z

Greatest Common Divisor(G.C.D) Or H.C.F

To find H.C.F follow , following Steps:

A) Obtain the polynomials. Lat the polynomials be p(x) and q(x).

B) Factorise the polynomials p(x) and q(x).

C) Express p(x) and q(x) as a product of powers of irreducible factors. Also, express the numerical factor as a product of powers of primes.

D) Identify common irreducible divisors and find the smallest (least) exponents of these common divisors in the given polynomials.

E) Raise the common irreducible divisors to the smallest exponents found in step D and multiply them to get the GCD(HCF). If there is no common divisors, the GCD (HCF) is 1.

The following formulae will be helpful in factorising polynomials

1) (x+ y)²= x² + 2xy + y²

2)  (x- y)²= x² - 2xy + y²

3)  (x+ y+ z)²= x² + y²+ z²+ 2xy+ 2yz+ 2zx.

4) (x+ y)³ = x³ + 3xy(x+ y) + y³

5) (x- y)³ = x³ - 3xy(x- y) - y³

6) x²- y² = (x+ y)(x-y)

7) x³+ y³ = (x+ y)(x² - xy +y²)

8) x³- y³ = (x- y)(x² + xy +y²)

9) x³+ y³ + z³ - 3xyz= (x+ y+ z)(x² +y²+ z²- xy - yz - zx)

10) x⁴- y⁴ = (x²+ y²)(x² - y²)

                = (x+ y)(x - y)(x² +y²)

11) x⁸- y⁸ = (x⁴- y⁴)(x⁴  +y⁴)

                = (x+ y)(x- y)(x² +y²)(x⁴ + y⁴)

12) x⁴+ y⁴+ x²y² = (x⁴+ y⁴+ 2x²y²) - x²y²

                          = (x² +y²)² -(xy)²

                         = (x²++ xy y²)(x² - xy +y²)

Find HCF of the following:

1) (x -2)²(x-3)²(x+1)² and (x +1)³(x+2)³(x- 3).                                   (x+2)²(x- 3)(x+1)²

2) 30(x²- 3x+2)(x -2) & 50(x²- 2x+1)(x²- 5x+6).                                 10(x- 1)(x -2)

3) (x -1)²(x+ 2)(x+3)³ and (x -1)(x-2)³(x+ 3)².                                 (x- 1)(x +3)²

4) (2x² -18) and (x² - 2x -3).        x - 3

5) 2x² - x -1 and 4x² + 8x +3.         2x+1

6) ⁸(x³ -x²+ x) and 28(x³ +1).        4(x- 2).

7) 24(6x⁴- x³ -2x²) and 20(2x⁶+ 3x⁵+ x⁴).           4x²(2x+1)

8) x² + x-2 and x³ + 4x²+ x - 6.      (x-1)(x +2)

9) 4x³ -16x² + 20x-8 and 8x³ - 48x²+ 72x - 32.                       4(x -1)² 

10) 33(2x+3)(3x -4)³(5- 4x)⁴ and 22(x +1) (2x +3)(5- 4x)³(4x- 9).             11(2x+3)²(4x -5)³

11) 10(x+1)²(x -3)³(x-2) and 15(x -2)³(x -3)²(x+5) and 75(x+5)(x-3)³(x-2)².      5(x -2)(x -3)²

12) 8x⁴ - 16x³ - 40x² + 48x and 16x⁵ + 64x⁴ + 80x³ + 32x².                 8x(x +2)

13) 2x⁴ - 2y⁴ and 3x³ + 6x²y - 3xy² - 6y³.     (x - y)(x + y).

14) 4(x - 3)²(x -1)(x+1)³ and 6(x- 1)²(x +1)²(x+7).                            2(x-1)(x+1)²

15) 16 - 4x² and x² + x- 6.              x-2

16) xy - y and x⁴y - xy.                  (x-1)y

17) x³ + 64 and x² - 16.                   x+4

18) 3 + 13x - 30x² and 25x² - 30x +9.    5x -3

19) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³ + 2y⁷x).                                     8xy²(x²+ y²)

20) (x - 2)²(x +3)(x-4) and (x- 2)(x +2)(x -5).                                                    (x-2)

21) (2x - 7)(3x +4) and (2x- 7)²(x +3).     (2x -7)                     

22) (3x - 2)²(2x+3)³(x -1) and (3x- 2)³(2x +3)(x-1)³.                       (2x+3)(x-1)(3x-2)²

23) (x +1)³(x -1) and (x- 1)³(x +1).         (x-1)(x+1)

24) (x +4)²(x -3)³ and (x- 1)(x -3)²(x+4).                  (x+4)(x-3)²

25) 24(x - 3)(x -2)² and 15(x- 2)(x -3)³.                      3(x-3)(x-2)

26) 2x² - 7x +3 and 3x²-7x - 6.         (x-3)

27) x³+ 2x²- 3x and 2x³ + 5x² -3x.    x(x +3)

28) 22x(x+1)² and 36x²(2x²+ 3x+1).   2x(x +1).                       2x(x+1)

29) 3+ 13x - 50x² and 25x²-30x +9.    (5x-3)

30) x³ - y³ and x⁴+ x²y² + y⁴.      (x² + xy+ y²)

31) 2x² + 7xy + 3y² and 2x²+ 6xy + x + 3y.      x + 3y

32) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³+ 2y⁷x).                                      8xy²(x²+ y²)

33) 54(x³+ 8y³) and 90(x³ + 7x²y+ 16xy²+ 12y³).                                         18(x+2y)

34) (4x⁴+ y⁴) and (2x³ - xy²+ y³).   2x² + 2xy + y²

35) x³ - y³ , x³y - y⁴, y²(x - y)²(x² + xy + y²).    x³ - y³

36) (2x² - 3xy)², (4x - 6y)³ and (8x³ - 27y³).                   2x - 3y

37) (4x⁴ + y⁴), (2x³ - xy² - y³) and (2x² + 2xy + y²).                           2x² + 2xy + y²

38) x⁴ - x³ + x -1 and x⁴+ x² +1.       x² - x +1

39) (8x⁶ - 32x⁵+ 128x³ - 128x²) and (12x⁶ - 36x⁵ + 48x³).        4x²(x -2)

40) (8x³ - x²- x +1) and (x⁴ - 2x³ + 2x -1).                       (x- 1)²(x+1)

41) 2x²y(x²- y²) and 35xy²(x - y).      xy(x- y)

42) (x² + 4x -21) and (x³+ 7x² - 9x -63).     (x+7)(x-3)

43) (6x⁴- 13x³ +6x²) and (8x⁴- 3x³+ 54x² - 27).                            x(2x-3)

44) (4x⁵+ 16x⁴ - 44x² - 24x) and (2x⁵ - x³ - 2x).                  2x(x² - x -1)

45) (3x³ - 14x² + 9x+ 10) and 15(x³-34x²+ 21x-10).                           3x -5

46) 4x²(x² - a²) and 9x²(x³ - a³).     x²(x - a)

47) 2(a² - b²) and 3(a³ - b³).           a-- b

48) 45(2x⁴ - x³ - x²) and 75(8x⁵ + x²).

49) (x²+3x-4) and (x³- 2x²- 2x+3).      x-1

50)12(x⁴ - 25) and 8(x⁴+4x²- 5).   4(x²+5) 

) If x- a is the HCF of x² - x-6 and x²+ 3x - 18 find the value of a.                          3

) For what value of a of x² - 2x -24 and x²- ax - 6 is x-6.                                       5

) x² + x- 2 is the HCF of (x -1)(2x²+ ax +2) and (x+2)(3x²+ bx +1). Find the value of a and b.                                            5, -4

) If x- k is the HCF of x² + x- 12 and 2x²- kx - 9. find the value of k.                    3

) Find the value of a and b so that the polynomials p(x) and q(x) have (x-1)(x +4) as their HCF.                             12,4

) If x- a is the HCF of ax² + bx + c and cx²+ ax + b such that c ≠ 0, then show that a³+ b³ + c³ = 3abc.

) If x- a is the HCF of x² + ax + b= 0 and x²+ bx + a= 0, then find the value of a+ b+ 1.                                                       

) If x- a is the HCF of x² + mx +1 and x²- 3x +2 find the value of m.          -5/2, -2

EXERCISE 

Find the LCM of following:

1) 12(x- 1)³(x³+ 6x + 8) and 150(x- 1)(x +2)(x² + 7x +10).                300(x-1)³ (x+2)²(x+4)(x+5)

2) 18x⁴ - 36x³+ 18x² and 45x⁶ - 45x³.     90x³(x -1)²(x²+ x+1)

3) 15(4x³-4x²+x) and 352x²- 7x+3).       105x(2x -1)²(x-3)

4) x³+ x²+x+1 and x³ +2x²+ x+2    (x+1)(x +2)(x²+1)

5) 7x³+ 2x²- 16x- 31 and x³+6x²+11x+ 6.     (x+1)(x +2)(x-2)(x+3)(7x²+ 16x+16)

6) 3(x⁴- x²y²), 6xy(x²- y²) and 96(x³y + y⁴).    90x²y(x - y)(x+ y)(x²- xy+ y²)

7)  a² - b² - c²+ 2bc, (a+ b+ c)², a² - b² + c² + 2ac.                 (a+ b+ c)(A-- b+ c)(a+ b- c)²

8) - x² - x+6 and -x²+ x +2.      (x-2)(x+1)(x+3)

9) x³+ x²+ x+1, x³+2x² +x+2.       -(x+3)²(x-5)²

10) 13(x -1)(x-2)², 7(x-2)²(x+3)² and (x-1)²(x+3).                    91(x -1)²(x-2)²(x+3)²

11) 8(1- x)³+(x+2)(x-3) and 12(x+2)² (x+1) (x+3).         

12) 12(x⁴+324x), 36x³+ 90x²- 54x.    36x(2x-1)(x+3)(x²- 3x+9)

13) 18x³+45x²-27x, 15x⁴ - 135x².      45x²(x-3)(x+3)(2x-1)

14) 20(2x³+3x² - 2x), 45(x⁴+ 8x).      180(x+2)(2x-1)(x²- 2x+4)

15) 35(x⁴- 27x), 40(2x³ - 5x² -3x).      280x(x-3)(2x-1)(x² + 3x+9)

16) 22x(x+1)², 36x²(2x²+ 3x+1).    396x² (x+1)²(2x+1)    

17) x³ - 1, x⁴+x²+ 1, x⁶ - 1.     (x³ - 1)(x³+ 1)

18) x³ - 8, x²-5x+ 6, x³ - 4x² + 4x.       x(x - 2)²(x-3)(x² + 2x+4)

19) x² - 3x+2, (x-1)², (x⁴ - 1).      (x+1)(x-1)²(x-2)(x²+1)

20) 15x³ - 75x²- 90x, 6x⁴ - 18x³ - 108x².      30x²(x-6)(x+1)(x+3)

21) a³+ 2a²-5a-6, a³- 3a²+4.       (a+1)(a+3)(A-- 2)²

22) a² - 8a + 7, a² - 12a + 35, a³ - 5a² - a +5.                          (a-1)(a+1)(a-5)(a-7)

23)  x² + 2xy + y², xy²+ x²y.     xy(x - y)²

24) (x³- y³), x² - y².          (x² - y²)(x² +xy+ y²)

25) x² + 2xy + y², x³+ y³, xy²+ x²y.     xy(x + y)²(x²- xy+ y²)

26) 6x² - 13xa + 6a², 6x²+11xa-10a², 6x²+ 2ax - 4a².        2(x+a)(3x - 2a)(2x+5a)(2x - 3a).

27) x² + 2xy + y²- z², y²+z ²- x²+ 2yz, z²+ x² - y²+ 2zx.               (x+ y+z)(x+y-z)(y+ z- x)(z+ x - y)

28) 6a³b²- 6ab⁴, 8a³b + 16a²b²+ 8ab³, 4a³- 2a²b - 6ab².          24ab²(a+b)(a-b)( 2a- 3b) 

29) 4(a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²), 6(a⁴ - 3a²b² - 4b²).                 12(a + b)²(a- b)²(a² +b²)(a+ 2b)(a- 2b).

30) 11x³(x+ 1)³, 121x(2x² + 3x+1).      121x³(x+ 1)³(2x+1).

31) 25(x²+7x+ 12), 15x(x² - 16).    75x (x+ 3)(x+4)(x-4).

32) x(8x³- 27), 2x²(2x² +9x+9).    2x²(8x³+27)(x+3).

33) 6(x +2)²(x²-x+1) and 15(x+1)(x+2)².         30(x+1)(x+2)⁴(x²-x+1)

34) (x -3)²(x+2), (x-3)⁴, (x+2)³, (x+1)(x-3).        (x - 3)⁴(x+2)³(x+1)

35) (1- x²), (x³- 1) and (x⁴-1).        (x⁴-2)(x²+x+1)

Misc-

1) The LCM of two polynomial (x+ 3)(- 6x² + 5x+ 4) and (2x²+ 7x +3)(x+3) is (x+3)(2x+1)(4- 3x). Find their HCF.

2) The HCF and LCM of two polynomial are (x+ a) and 12x²(x+ a)(x² - a²) respectively. The first polynomial is 4x(x+a)². Find the other polynomials.

3) The HCF of two polynomials is (x -3)(x² +x -2) and (x²- 5x +6) is (x -3). Find their HCF.

4) The LCM of two polynomial 10 (x² - 9)(x - 1) and 10x(x - 1)(x+3) (x - 2) is 20x(x² - 9)(x² - 3x+2). Find their HCF.

5) Find the LCM and HCF of the polynomial (2x² -8) and (x² - 5x+ 6). Verify that the product of the two polynomials is equal to the product of their HCF and LCM.

6) The HCF of two polynomials is (x+ 1) and their LCM is (x⁶ -1). If one of them is (x³+1), find the other.

7) The LCM of two polynomials is (x +3)(x -2)²(x -6) and their HCF is (x -2). If one of them is (x+3)(x -2)², find the other.

8) Find GCD (x⁶y² - x²y⁶) and 72(x⁵ + 3y⁵x³+ 2y⁷x) 

9) Find the LCM of (4a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²) and 6(a⁴ - 3a²b² - 4b²).

10) The HCF and LCM of two polynomials are 5(x+3)(x -1) and 20(x² -9)(x² - 3x+2) respectively. If one polynomial is 10(x² -9)(x-1), Find thei other polynomial.

SET THEORY (2)

1) Express each of the following statements in set theoretical notation:

A) x belongs to the set A.

B) Set A is a subset of set B.

C) P is a proper subset of Q.

D) a is not an element of set X

E) Y is not a proper subset of Z.

F) A Set whose elements are contained in U but not contained in B.

2) Suppose P and Q are two given sets and the corresponding Universal set is U; Express each of the following statement in symbols :

A) A set whose elements are contained either in P or in Q.

B) A set which contains the common elements of P and Q.

C) A set whose elements belongs to U but do not belong to Q.

D) A set whose elements belongs to Q but do not belong to P.

E) A set whose elements do not belong to any of the set P and Q.

F) A set which does not contain the common elements of P and Q.

3) Represent the following sets in Roster form :

A) set of vowels in English alphabet.

B) set a factors of the number 36.

C) A={x: x is a prime integer and 6< x ≤ 29}.

D) P={x| x ∈ N and x≤ 12}.

E) set of letters in the word 'statistics'.

4) Rewrite the following set in set builder notation form:

A) A={......, -3,-2,-1,0,1,2,3}

B) P={1,3,5,7,9,11}

C) set of roots of the equation x⁴ - 13x²+ 36= 0.

D) set of even positive integers greater than 4 and less than or equals to 19

5) State weather each of the following sets is infinite or finite:

A) D={x: x is the number of people living on the earth}

B) W={x:x is the time a person waits for a bus}

C) V={x:x is an odd integer exceeding 889}.

6) State with reasons whether the sets defined in each of the following are equal sets:

A) X= {∅} ; Y= ∅.

B) A={-2, -5}.

     B={x:x is a root of the equation x²+ 7x + 10= 0}.

C) P={b, i, e, n, s, u}

     Q={x:x is a letter in the word ' Business'.

D) A={x:x is a digit in the number 30255}

    B={x:x is an integer and 0≤x≤5}

    C={1,0,2,3,4,5}.

7) Some well-defined sets are given below; identify the null sets:

A) A={0}

B) B={∅}

C) C= ∅

D) Set of boy students in a girls' college.

E) P={x:x is an integer and 1<x<2}

F) Q={x:x is an integer and 1<x≤2}

G) R={x:x is positive and x² + 7x +12= 0}

H) A={x:x is an integer and 6x² - 5 +1= 0}

I) {x: 3x² -4= 0, x is an integer}

M) {x: (x+3(x+3)= 9, x is a real number}

N) A∩ B - A.

8) State with reasons which of the following statements are correct/ incorrect:

A) If P U Q={a, b, c, d}, then a ∈ P And a ∈ Q

B) If P∩ Q={a,b,c,d}, then a ∈ P and a ∈ Q.

C) If A∩ B=∅, then A and B are disjoint sets.

D) If A is super subset of B, then B is super subset of A.

E) If A is super subset of B, and B is super subset of C, then A is the super subset of C. 

F) x ∈ A U B => x ∈ A.

G) If A is super subset of B, and B is super subset of A, then A = B.

G) If A= {2,4, 6, 8}, then {2,4}∈A.

H) If A= {2,4, 6, 8}, then {2,6,8} is subset of A.

I) If A= {2,4, 6, 8}, then ∅ is the subset of A.

M) If A= {2,4, 6, 8}, then {2,4}is the subset of P(A).

N) (A - B), A ∩ B and (B - A) are mutually disjoint.

O) If A= {2,4,6,8}, then {2,6,8} ∈ P(A).

9) If a ∈ A and a ∈ B, does it follow A is super subset of B ? Give reasons.

10) State with the reasons, which of the following statements are true or false :

A) {a} ∈ {a, b, c}

B) a ∈ {a, b, c}.

C) a⊂ {a, b, c}

D) a not belong to {a, b, c}.

E) 3 ⊂ {1,3,5}

F) 3 ∈ {1,3,5}

G) {3} ⊂ {1,3,5} 

H) {3} ∈ {1,3,5}

11) If A = {a,b,c} Name

A) the subsets of A

B) the proper subsets of A.

12) Define power set of a set A. Find the power state of a A{a, b c}, If B be the power set of A, state with the reason, which of the following statement is correct:

A> B, A ∈ B, A⊂B, A= B, A is not subset of B.

13) Fill in the gaps:

A) The number of subsets in a set consisting of four distinct elements are -----.

B) The number of proper subsets in set consisting of n district elements are----.

C) If x ∈ A => x ∈ B then ----.

D) if A is super subset of B and B is super subset of A then-----.

E) the set of P whose elements are all subsets of the set {1,2} is given by

  P={__, ___, ___, ___}

F) if A and B are disjoint set then n(A U B)= ____.

G) If A U B= A ∩ B then____.

H) The dual of A U (B ∩C)= (A U B) ∩ (A U C) is _____.

I) The dual of A U U= U is ___.

J) If A is a given set and ∅ is the null set then ___.

14) Let A= {a, b, c}, B={a,b}, C={a,b,d}, D{c, d} and E={d}. State which of the following statements are correct and give reasons: 

A) B ⊂ A

B) E is no subset of E

C) D⊂B

D) {a} ⊂A

15) Let A={a,b, c,d,e,f,g,h,i},

B={b,d,f,h}.

C={a,c,e,g,i}, D{c,d,e}, E={c,e}. Which set can equal X if we are given the following informations?

A) X and B are disjoint

B) X ⊂A but X is not subset of C.

C) X ⊂ D but X is not subset of B.

D) X⊂ C but X is not subset of A.

16) List the sets A U B, A ∩ B A∩ (B U C). given that,

A={p,q,r,s}, B={q,r,s,t}, C={q,r,t}

17) If P{a,b,c,d,e} and Q={a,e,i,o,u}, prove P⊂ P U Q and P ∩Q ⊂ P.

18) If A= {2,3,4,5} and B={1,2,3,4} show that B - A ≠ A- B.

19) Let S={1,2,3,4,5} be the Universal set and let A={3,4,5} and B={1,4,5} be two of its subsets. Verify (AU B)'= A' ∩B'.

20) If A{1,2,3,4}, B={2,3,4,5}, C={1,3,4,5,6,7}, find

A) A - B

B) A - C

and hence verify A - (B∩C)= (A - B) U (A - C).

21) If P={a,b, c, d, e, f} and Q={a, c,e,f}, show (P - Q) U (P ∩Q)= P.

22) If A={x : x is an integer and 1 ≤ x ≤ 10} and B={x: x is a multiple of 3 and 5 ≤ x ≤ 30}; find A U B, A∩B, A - B and B - A.

23) Let U={1,2,3,4,5,6,7,8,9,10} be the universal set. suppose, A={1,2,3,4,5,6} and B {5,6,7} are its two subsets. Write down the elements of A - B and A ∩ B'.

24) If X={x : x is an even integer and 6< x ≤ 20} and Y={x : x is a multiple of 3 and 0≤ x≤25}, find

A) X U Y 

B) X ∩ Y

C) Y - X

D) X - Y

25) Let S={1,2,3,4,5,6} be the Universal set, let A U B= {2,3,4}; find A' ∩ B' where A' , B' are complements of A and B respectively. also show that A UB, and A' ∩ B' are disjoint sets.

26) if P={p,q,r,s,t,u} and Q={q,r, v,w}; find

A) (P U Q) ∩(P U R)

B) (P - Q)U(P - R).

27) if A,B, C be three subsets of the universal set S where S={1,2,3,4,5,6,7}, A={1,3,5,6} and B ∩C ={1,2,6}, find

A) (A U B)∩(A U C)

B) B' U C'

28) Given, A={x: 0< x≤2} and B={x: 1< x < 3}; find

A) A∩B

B) A U B

C) A - B

29) Let A= {x:2 ≤ x < 5} and B={x : 3 < x < 7} be two subsets of the universal set S={x: 0 < x≤ 10}; verify that (A UB)'= A'∩B'.

30) Given X U Y={1,2,3,4}, X U Z = {2,3,4,5}, X ∩Y={2,3} and X∩Y= {2,4}; find X, Y Z.

31) If aN={ax: x ∈N}, Describe 3N∩ 7N where N is the set of natural numbers.

32) Using set operation show that the numbers 231 and 260 are prime to each other.

33) Applying set operation prove that 3 + 4 = 7.

34) Using Venn diagram or otherwise, solve the following problems:

 In a class of 70 students. Each students has taken either English or Hindi or both. 45 students have taken English and 30 students have taken Hindi. How many students have taken both English and Hindi ?

35) A market research group conducted a survey of 1000 consumers reported that 700 consumers liked product A and 480 consumers liked product B. What is the least number that must have liked both products ?

36) In a town of 60% read magazine A, 25% do not read magazine A but read magazine B. Calculate the percentage of those who do not read any magazine. Also find the highest in the lowest possible figures of those who read magazine B.

37) In a statistical investigation of 1003 families of Calcutta, it was found that 63 families had neither a radio nor a T.V., 794 families had a radio and 187 had a television. How many families in that group had both radio and TV ?

38) Three daily newspaper, E, B, H are published in a certain city. 62% of the citizens read E, 59% read B, 41% read H, 40% read both E and B, 28% read both B and H, 24% read both E and H. Find the percentage of citizens who read all the three papers.

39) In a survey of college students, it was found that 40% use their own books, 50% use library books, 30% use borrowed books, 20% use both their own books and library books, 15% use their own books and borrowed books, 10% use library books and borrowed books, and 4% use their own books, library books and borrowed books. Calculate the percentage of students who do not use a book at all.

40) In a city 3 daily news papers X, Y, Z are published; 65% of the citizens read X, 54% read Y, 45% read Z; 38% read X and Y; 32% read Y and Z; 28% read X and Z; 12% do not read any of these three papers If the total number of people in the City be 10000000, find the number of citizens who read all the three newspapers.

41) A company studies the product preferences of 300 consumers. It was found that 226 liked product A, 51 liked product B, 54 laked product C; 21 liked products A and B, 54 liked products A and C, 39 liked products B and C and 9 likwd all the three products. Prove that, the study results are not correct. [Asume that each consumers liked at least one of the three products]

42) The production manager of Sen, Sarkar and Lahiri company examined 100 items produced by the workers and furnished the following report to his boss.

Defect in measurement 50, defect in colouring 30, defect in quality 25, defect in quality and colouring 10, defect in measurement and colouring 8, defect in measurement and quality 20 and 5 are defective in all respect. The manager was penalised for the report. Using appropriate results of set theory, explain the reason for the penal measure.

43) In a survey of 150 students, it was found that 40 students studied Economics, 50 students studies Mathematics, 60 students studied Accountancy and 15 student studied all the three subjects. It was also found that 27 students studied Economics and Accountancy, 35 students studied Accountancy and mathematics. Find the number who studied only economics and the number who studied none of these subjects.

44) Out of thousand students in a college 540 played football, 465 plyed cricket and 370 played volleyball; of the total 325 played football and cricket, 260 played football and volleyball, 235 played cricket and volleyball, 125 played all the three games. How many students 

A) did not play any game

B) played only one game.

C) played just two games.

45) A group of consist of a number of students and each Student of the group can speak at least one of the languages Bengali, Hindi and English. 65 can speak Bengali, 54 Hindi and 37 English; 31 can speak both Bengali and Hindi, 17 both Hindi and English, and 18 both Bengali and English. Determine the greatest and least number of students in the group.

46) An investigator interviewed 100 students to determine their preference for the three drinks; Milk(M), coffee (C) and tea(T). He reported the following:

 10 students had all the three drinks M, C and R; 20 had M and C; 30 had C and T, 25 had M and T; 12 had M only, 5 had C only and 8 had T only.Find how many did not take any of the three drinks.

APPLICATION OF DERIVATIVES IN COMMERCE AND ECONOMICS

EXERCISE -1

1) A manufacturer finds that the production cost of each article produced by his firm is ₹25 and the other fixed costs are ₹25000. If each article is sold for ₹35, Find

A) cost function.                     C(x)= 25000+ 25x

B) Revenue function.        R(x)= 35x

C) break even point.                 2500

2) The fixed cost of a new product is ₹35000 and the variable cost per unit ₹500. If the demand function p(x)= 5000 - 10x, find the break-even values.                          10, 35

3) The fixed cost in the variable cost of x units of a product of a company are ₹ 30000 and ₹75x respectively. If each unit is sold for ₹125, find break-even point.       600

4) A company decides to set up a small production plant for manufacturing clocks. The total cost of initial set up(fixed cost) is ₹9 lakhs. The additional cost (variable cost) for producing each clock is ₹300. Each clock is sold at ₹750. During the first month 1500 clocks are produced and sold.

A) determine the cost function for C(x) for producing x clocks.       300x + 900000

B) determine the revenue function R(x) for the sale of x clocks.     750x

C) determine the profit function p(x) for the sale of x clocks.          450x - 900000

D) what profit or loss the company incurs during the first month when all 1500 clocks are sold ?        225000

E) determine the break even point.           2000

5) The printing cost of 1900 and 1300 copies of a book are 5100 and 3700 respectively. Find the equation of the cost curve of printing assuming it to be linear. If the selling price is ₹3 per copy, find the number of copies that must be printed so that

A) there is no profit or loss.      y= 7x/3 + 2000/3, 1000 copies 

B) a profit of ₹ 40.         1600 copies

6) A publishing house finds that the production cost directly attributed to each book is ₹30 and that the fixed costs are ₹15000. If each book can be sold for ₹45 then find

A) cost function.           30x+15000 

B) the revenue function.            45x

C) break-even point.                 1000

7) A firm knows that the demand function for one of its products is linear. It also knows that it can sell 1000 units when the price is ₹4 per u6, and it can sell 1500 units when the price is ₹2 per unit. Find

A) demand function.       8 - x/250

B) total revenue function.   8x -x²/250

C) marginal revenue function.      8 - x/125

8) ABC Co. Ltd. is planning to market a new model of shaving razor. Rather than set the selling price of the razor based only on production cost estimates, management pulls the retailer of the razors to see how many razors they would buy for various prizes. From this survey it is determined that the unit demand function (the relationship between the amount x each retailer would buy and the price p he is willing to pay) is 

x= 1500 p + 30000

The fixed costs to the company for production of the razors are found to be ₹28000 and the cost for material and labour to produce each razor is estimated to be ₹8.00 per unit. What price should the company charge retailer in order to obtain a maximum profit?       at 14, ₹26000, 9000

9) 

EXERCISE -2

1) The unit demand function is x= (25- 2p)/3, where x is the number of units and p is the price. Let the average cost per unit be ₹40. Find

A) the revenue function R in terms of price p.                        (25 - 2p)/3

B) the cost function C.    40(25- 2p)/3

C) the profit function P.      (-2p² + 105p - 1000)/3

D) the price per unit that maximizes the profit function.          When p= 105/4

E) the maximum profit.    

2) The demand function faced by a firm is p= 500 - 0.2x and its cost function is C= 25x + 10000 (p= price, x= output and C= cost). Find

A) the output at which the profits of the firm are maximum.      1187.50

B) the price it will charge.    262.50

3) 

7) For a manufacturer of dry cells, the daily cost of production C for x cells is given by C(x)= ₹(2.05x + 550). If the price of a cell is ₹ 3. Determine the minimum number of cells those must be produced and Sold daily to ensure no loss.       579

9) The daily cost of production C for x units of an assembly is given by C(x)= ₹(12.5 x + 6400).

A) if each units is sold for ₹25, determine the minimum number of units that should be produced and Sold to ensure no loss.            513

B) if the selling price is reduced by ₹2.50 per unit, what would be the break-even point.                    640

C) if it is known that 500 units can be sold daily, what price per unit should be charged to guarantee no loss ?                    25.30

10) A firm produces x tonnes of output per week at a total cost of ₹(x³/10 - 5x² + 60x +100). Find

a) average cost.     x²/10- 5x + 60 + 100/x

b) average variable cost.    x²/10 - 5x + 60

c) marginal cost.      3x²/10- 10x+60

11) A calculator manufacturing company introduces production bonus to the workers that increases the cost of a calculator. The daily cost of production C for y calculators is given by C(y)= ₹ 2.05y + ₹ 550.

A) If each calculator is sold for ₹3, determine the minimum number that must be produced and sold daily to ensure no loss.          579

B) if the selling price is increased by 30 paise per piece, what would be the break even point ?             440

C) if it is known that at least 500 calculator can be sold daily, what price the company should charge per piece of calculator to guarantee no loss?                3.15

12) The total cost and the total revenue of a company that produces and sales x units of a product are respectively C(x)= 10x + 400 and R(x)= 60x - x³.  Find

A) the break-even values.     10 & 40

B) the values of x that gives a profit.       10< x < 40

C) the values of x that results in a loss.                        x< 10 and x> 40

13)  The total cost and the total revenue functions of a company that produces and sales x units of a particular product are given by C(x)= 5x + 350 and R(x)= 50x - x² respectively. Find

A) the break even values of x.      10 & 35

B) the values of x that produces a profit.                          10< x < 35

C) the values of x that result in a loss.           x< 10 and x > 35

14) The total cost and the total revenue of a company that produces and sells x units of a product are respectively C(x)= 5x +350 and R(x)= 50x - x². Find

A) the break-even values.       10 or 35

B) the value of x that produces a profit.                          10< x < 35

C) the value of x that results in a loss.                x< 10 or x> 35

15) The cost function C(x) of a firm is given by C(x)= 2x² - 4x + 5. Find

A) The average cost.       

B) the marginal cost when x= 10.            16.5 units, 36 units

16) A firm produces x units of a article at a total cost of ₹(5+ 48/x + 3x²). Find the minimum value of the total cost.                At x= 2, ₹ 41

17) A firm produces x units of output at a total cost of ₹(2x/3 + 35/2).  Find the cost when the output is 4 units, the average cost of output of 10 units, and the marginal cost when output is 3 units.           ₹20.16, ₹2.42, ₹0.67

18) A firm produces x units of output per week at a total cost of ₹(x³/3 - x² + 5x + 3). Find the output levels at which the marginal cost and the average variable cost attains their respective minima.                1, 1.5

19) A firm produces x tons of a valuable metal per month at a total cost C given by C= ₹(x³/3 - 5x² + 75x +10). Find at what level of output the marginal cost attains it's minimum.                        5 tons.

20) Let the cost function of a firm be given by the equation: C(x)= 300x - 10x² + x³/3, where C(x) stands for cost function and x for output. Calculate the output at which

A) the marginal cost is minimum.   10

B)  the average cost is minimum.    15

C) average cost is equals to Marginal cost.              15

21) The efficiency E of a small manufacturing concern depends on the number of workers w and is given by 10E = - w³/40 + 39w - 392. Find the strength of the workers which gives maximum efficiency.   20

22) A company after examining its cost structure and revenue structure has determined that the following functions approximately describe its cost and revenues:

C= 100 + 0.015x² and R= 2x where C= total cost, R= total revenue and x= number of units produced and Sold. Find the output rate which will maximum profits for the firm.    66.67

23) A firm can sell x items per week at a price p= (300 - 2x) rupees. Producing items cost the firm y rupees where y= 2x + 1000. How much production will yield maximum profits ?                74

24) The total revenue function and the total cost function of a company are given by R= 21q - q² and C= q³/3 - 3q² - 7q + 16 respectively, where q is the output of the company. Find the output at which the total revenue is maximum and the output at which the total cost is minimum.      10.5, 7

25) The demand function of a firm is p= 500 - 0.2x and its cost function is c= 25x - 10000, where p is the price and x is the output. Find the output at which the profit of the firm will maximum. Also find the price it will charge.          1187.5, 262.5

26) The demand function of a producer is 3q= 98 - 4p and its average cost is 3q +2, where q is the output and p is the price. Find the maximum profit of the producer.                33.75 units

27) A radio manufacturer finds that he can sell x radio per week at ₹p each, where p= 2(100 - x/4). His cost of production of x radios per week is ₹ (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week.       1600

28) A manufacturer produces x units per month at a total cost of ₹(x²/25 + 3x + 100). There is no competition in the market and the demand follows the rule x= 75 - 3p, where p is the selling price per article. Find x such that the net revenue is maximum,. also find the monopoly price.                 30, 15

29) A firm produces x units of output at a total cost of ₹(300x - 10x² + x³/3). Find

A) output at which marginal cost is minimum.                      10

B) output at which average cost is minimum.                         15

C) output at which average cost is equal to marginal cost.             0, 15

30) The demand function of a monopolist is given by p= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue when x= 0.     1160

31) A firm produces x tons of output per week of a total cost of ₹(x³/8 - 4x² + 12x +3). Find the level of output at which average variable cost attains minimum value.     16

32) The manufacturing cost of an item consists of ₹900 as overheads, the material cost is ₹3.00 per item and the labour cost is ₹ x²/100 for x items produced. How many items must be produced to have average cost minimum.                  300

33) The total cost function of a firm is C= x³/3 - 5x² + 28x +10. Where C is total cost and x is output. A tax at the rate of ₹2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by

p= 2530 - 5x.

Where ₹ p is the price per unit of output, find the profit maximizing output and price.               50, 2280

34) Given the demand and cost functions:

p= 10 - 4x

C= 4x

Find

A) the maximum quantity, price and the profit on this level.              12

B) what will be the new equilibrium after a tax of ₹0.50 is imposed ?    12.25

*C) the tax rate that will maximizing tax revenue and determine that tax revenue.                 8

35)