PROBABILITY- mixed (C)

1) A bag contains 3 green and 8 white balls. If one ball is drawn from it at random, find the chance that the ball drawn is green.  3/11

2) if a coin is tossed, what is the chance of a tail ? If three coins are tossed, find the probability that they are all tails.         1/2, 1/8

3) From a pack of 52 cards, one cards is drawn at random. Find the chance drawing a spade and also not drawing a spade.      1/4, 3/4

4) Two unbiased dice are thrown together, find the probability of obtaining 2 in both the dice.   1/36

5) if a dice is tossed, what is the chance of getting 

A) an even number.     1/2, 2/3, 1/3

B) a number greater than 2.

C) an even number greater than 2?

6) From a pack of 52 cards, one card is drawn at random. Find the chance that the card is 

A) red.                                       1/2

B) an ace.                                  1/13

C) an spade.                               1/4

C) either king or an ace.            2/13

7) What is the chance of picking a spade, or an ace not of spade, from a pack of 52 cards ?      4/13, 

8) out of 5 players of which 2 are members of a certain club, 3 are to be selected to represent the country at an international tournament, Find the probability that less than two of those selected to represent are  members of the club.      7/10

9) From a well shuffled pack of 52 cards, one card is drawn at random, Find the chance that the card is

A) neither a king or a knave.

B) neither a king nor a knave

C) neither a heart nor a diamond.

D) neither an ace, nor a king, nor a queen, nor a knave.    2/13, 11/13, 1/2 , 9/13

10) Two letters are drawn at random from the HOME. Find the probability

A) both the letters are vowels.   1/6

B) at least one is a vowel.     5/6

C) one of the letters chosen is M.   1/2

11) From a bag containing 7 white and 5 red balls, 4 balls are drawn at random. What is the probability that they are all are white ?     7/99

12) In a single cast with two dice, find the probability of throwing 7( i.e., of throwing two numbers whose sum is 7).           1/6

13) If 4 unbiased coins are tossed, find the probability that there should be two heads.        3/8

14) Find the chance of throwing at least 8 in a single cast with two dice .    

15) two balls are drawn at random from a bag containing 6 white balls and 4 Black balls. find the chance that one is white and other is black.   8/15

16) 4 balls are drawn at random from a bag containing 5 red and 7 white balls. compute the probability of getting 

A) 4 red balls.           1/99

B) two Red and two white balls.   14/34

C) three white balls and 1 red ball.   35/99

17) four balls are drawn at random from a bag containing 5 black and 8 white balls. What are the odds in favour of the balls being all white.    14: 129

18) find the chance of not throwing an ace, two or three, in the single throw with a die.       1/2

19) there are 17 balls numbered from 1 to 17 in a bag, if a person selects one at random, what is the probability that the number printed on the ball will be an even number greater than 9 .            4/17

20) what are the odds against throwing ace or 6 in a single throw with die ? and what are the odds in favour?           2:1, 1:2

21) Given are the weekly wages (in ₹) of workers in a factory: 62, 90, 78, 85, 79 and 68. If two of these workers are selected at random to serve as representatives, what is the probability that at least one will have a wage lower than the average.          3/5

22) four men in a company of 10 employees are engineers. If two men are selected random, then find the probability that atleast one of them will be an engineer.   2/3

23) five men in a company of 20 are graduate6. If 3 men are picked out of the 20 at random, What is the probability that

A) they are all graduates.     1/114

B) there is no graduate.      91/228

C) atleast one graduate ?   137/228

24) A bag contains 10 rupees coin5, 7 fifty paise coins and 4 twenty five paise coins. Find the probability of drawing

A) a rupee coin.      10/21

B) 3 rupee coins.          12/133

C) 3 coins, one of each type.   4/19

25) consider four events: A, NOT A, B, NOT B. it is given that P(B)= 0.4, P(A but NOT B)= 0.3, P(NOT A)= 0.7. find P(A UB).       0.7

26) A, B and C are three mutually exclusive events. Fi6P(B), If 1/3 P(C)= 1/2 P(A)= P(B).      1/6

27) the probability that the teacher in Statistics will take a sudden test on any day of a week is 2/5. A student abstain for 2 days in a week. what is the probability that he will miss atleast one test ?    16/25

28) An urn contains 13 balls numbering from 1 to 13. Find the probability that a ball is selected at random is a ball with number that is a multiple of three or four.   6/13

29) Find the chance of throwing at least 9 in a single case with two dice .    5/18

30) Two fair dice thrown simultaneously. The two scores are then multiplied together. Calculate the probability that the product is

A) 12.         1/9

B) even.             3/4

31) In a lot of 20 articles, 5 are defective, 4 articles the drawn at random from the lot. Find the probability that exactly two of drawn articles are defective.  70/323

32) Two cards are drawn from a well shuffle pack of 52 cards. Find the probability that 

A) both are spades.       1/17

B) both are kings.      1/221

C) one is spade and the other is a heart.       13/102

33) A Sub-committee of 6 members is be formed out of a group consisting of 7 gentlemen and 4 ladies. Calculate the probability that the sub-committee will consist of

A) exactly two ladies.   5/11

B) at least two ladies.       53/66

34) an article manufactured by a company consists of two parts, A and B, in the process of manufacture of part A, 9 out of 100 are likely to be defective. similarly, 5 out of 100 are likely to be defective in the manufacture of part B. Calculate the probability that the assembled article will not be defective.             0.8645

35) the manufacturing process of an article consists of two parts X and Y. the probabilities of defect in parts X and Y are 10% and 15% resp6. What is the probability that the assembled products will not have any defect ?         0.765

36) there are four balls in a bag-- white, red and green and blue. If a bell is drawn out at random three times in succession, what is the chance that all the three would be white? (Assume that the ball is replaced after each drawing).  1/64

37) a box contain 9 red and 6 white balls. Two balls are drawn at random one after another. Find the probability that the both the balls are white, when the first draw ball is not replaced before the second drawing.              1/7

38) a coin is tossed 3 times in succession. what is the probability of the tosses coming in the order.

A) head, tail and head.        1/8

B) head, head, head.       1/8

C) two heads and one tail.    1/8

39) From a bag containing 5 green, 6 blue and 4 red balls, 7 balls are drawn at random. find the chance that 2 green, 3 blue and 2 red balls are drawn. What are the odds against drawing 2 green, 3 blue and 2 red balls.           80/429, 349:89

40) If a die is throwing twice, what is the probability that the first throw does not show less than 4 and the second does not show more than 4   1/3

41) A pair of dice are thrown. Find the probability of getting a sum of 7, when it is known that the digit in the first die is greater than that of the second.     

42) in a single cast with two dice

A) what are the odds against throwing 8, i e. Against two numbers the sum of which is 8 ?  31:5

B) what is the chance of throwing more than 8, and what is the probability of throwing less than 8.   5/18, 7/12

C) find the probability of throwing atleast 8 in a single cast with two dice.          5/12

43) In a single cast with 3 dice, what is the probability of throwing

A) three fives.    

B) two-four-six.          

C) five-six-six ?

What are the odds against the events.     1/216, 215:1, 1/36, 35:2, 1)72, 71:1

44) Three cards are drawn from a pack of 52 cards, each card being replaced before the next one is drawn. compute the probability that all are

A) spades.      1/64

B) aces.          1/2197

C) red.          1/8

45) One bag contains 4 red and 2 black balls, another bag contains 3 red and five black balls. if one ball is drawn from each bag, determine the probability that 

A) both are red.     1/4

B) both are black.       5/24

C) one is red and one is black.  13/24

46) box A contains 4 white and 3 red balls while box B contains 5 white and 4 red balls. If one ball is drawn at random from each box, what is the probability that both the balls are of the same colour?   32/63

47) One counter foil is drawn at random from the bag containing 70 counter foils marked with the first 70 numerals. Find the chance that it is multiple of 8 ton 9.    3/14

48) From a set of 18 balls marked 1,2,3,......17, 18, one is drawn at random. What is the chance that its number is either multiple of 3 or 4 ?     1/2

49) An investment consultant predicts the odd against the price of a certain stock will go up during next week by 2:1 and the odd in favour of the price remaining same by 1 : 3. What is the probability that the price of the stock will go down during the next week?     5/12

50)  A bag contains 4 green and red balls is drawn at random and then, without replacing it, a second ball is drawn. What is the chance that a green ball is drawn each time ?  2/15

51) Two drawing, each of 4 balls, are made from a bag containing 6 red and 5 black balls, the balls not being replaced before the second trial, Find the chance of the first drawing gives 4 red balls and second gives 1 red and 3 black balls.   2/3

52) A problem in statistics is given to three students A, B, C whose chances of solving it are 1/3, 1/4, 1/5 respectively. What are the probability that the problem will be solved ?     3/5

53) the probability of solving a problem by 3 students A,B, C are 2/7, 3/8 and 1/2 respectively. if each of them tries independently, find the probability that the problem is solved. find also the probability that the problem could not be solved.     87/112,25/112

54) the independent probability that the three sections of a costing department will encounter a computer error are 0.1, 0.3, 0.3, each week respectively. Calculate the probability that there will be,

A) at least one computer error.   0.559

B) 1 only one computer error encounted by the costing department next week.      0.427

55) a bag contains 4 red and 3 blue balls. Two drawing of two balls are made. Find the probability of a drawing first 2 red balls and second 2 blue balls

A) if the balls are returned to the bag after the first draw.       2/49

B) if the balls are not returned after the first draw.          3/25

56) if 20 dates and named at random, find the chance that 5 of them will be Sundays.      29C5 x 6¹⁵/7¹⁰

57) two urns contains 3 white, 7 red, 15 black and 10 white, 6 red, 9  black balls respectively. one ball is drawn from each urn. Find the probability that both the balls are of the same colour.        207/625

58) 10 balls are distributed at random among 3 boxes. what is the probability that the first box will contain three balls ?        5120/19683

59) If P(A)= 1/4, P(B)= 2/5, find P (A U B), when find A and B are two independent events.         11/20

60) If P(A)= 1/3, P(B)= 2/2, find P (A ^ B)=1/4 find P(B/A) AND P(A' ^ B'),, where  A' and B' are the compliments of events A and B  respectively.       3/4, 5/12

61) if the two events A and B are mutually independent and P(A)= 0.3, P(B)= 0.1, find P(A U B).   0.37

62) A speaks the truth 3 times out of 5 and B speaks the truth 5 times out of 7. What is the probability that they will contradict each other in stating the same fact ?     16/35

63) If P(A)= 1/4, P(B)= 2/5, find P (A U B) = 1/2, Find P(A^ B), P(A/B) , P(B/A).           3/20,3/8,3/5

64) If A and B be events with P(A + B)= 1/4, P(A)= 2/3 and P(AB)= 1/4,  find P (A), Find P(B), P(AB).            1/3 , 2/3,3/4

65) If P(A)= 1/2, P(B)= 2/5, find P (A ^ B) = 1/3, Find P(AU B), P(A' ^/') , P(A' U/B'), P(A ^ B') , P(A' ^ B).          23/30,7/30,2/3, 1/6, 4/15

66) If P(A)= 2/3, P(B)= 1/2, find P (A U B) = 1, Find P(A/B), P(A/B') , P(B?A' ^ B').           1/3, 1, 0

67) A construction company is bidding for two contracts, A and B. the probability that the company will get contract A is 3/5, the probability that the company will get contact B is 1/3 and the probability that the company will get both the contract the 1/8, what is the probability that company will get at least one contract ?   97/120

68) The probability that a contractor will get an electric contract is 3/5, and the probably that he will not get a plumbing contract is 7/10. If the probability of getting atleast one contract is 4/7, what is the probability the company that he will get both the contracts.    23/70

69) the probability that a management account's job applicant has a postgraduate degree is 0.30, that he has had some work experience as a chief accountant is 0.70, and that he has both is 2.0. Out of 400 applicants, what number would have either a post graduate degree of some professional work experience or both ?        320

70) A bag contains 10 red and 6 green balls. Two successive drawings of 3 balls are made without replacement. Find the probability that the first drawing will get 3 red balls and second will give 3 Green balls.       15/1001

71) A bag contains 6 white and 9 black balls. 4 balls are drawn at a time. Find the probability for the first draw to give 4 white balls and the second to give 4 black balls when the balls are not replaced before the second draw.     3/715

72) a bag contains 5 red and 3 black balls, and the second bag contains 4 red and 5 black balls. one of these is selected at random and a draw of two balls is made from it. what is the probability that one of them is red and other is black.      275/504

73) There are 2 identical boxes containing 5 white and 3red balls and 4 white and 6 red balls. A box is choosen at random and a ball is drawn from it. If the ball is white, what is the probability that it is from the second box ?    16/41

74)In a bolt factory, machines X, Y, Z manufacture respectively 20, 35, and 45 percentage of the total output. If their output 8, 6 and 5% respectively are defective bolts.  One bolt is drawn randomly from the product and is found to be defective. what is the probability that it is manufactured in the machine Z ?   45/119

75)  a box containing for defaction 6 electronic calculator to calculate the don out of one by one without replacement what is the probability to calculate so Drone or good one on the two calculated the probability there are two identical earns containing respectively for white 3 red balls and what three wise and 7 balls an earn is chosen A triant Time In A ball is done from it find the probability that a ball is white if the ball Drone is white what is the probability the first ear two balls I don't want after another from a back on turning three wide 5 block balls what is the probability that the first ball is white in the second is black one ball is whiten the other is black one and contents to White two black ball a second and contents to White and four black balls if one ball is chosen from each other what is the probability that there will be of the same colour for early selected at random and one ball is drawn from it what is the probability that will be avoid Paul X and Y stands in a line to random with other persons what is the probability that there are 3% why given that independent what is the probability that a leap year selected in random will content 53 Sunday and only pure selected 53 sector in a group of equal number of man 10% man in 45% on an RM unemployed what is the probability the person selected random employed contents for white 5 black balls second hand contents 5 black balls 1 ball is transpord from passed through ball is drawn from the second ear what is the probability that it is right two girls and two boys and three boys and one girl one child selected random Pradeep group show that chance that the three selection consist of one girl into boys is 1332 speak the truth in 60% of being 75% what percent of the cases are they likely to contradict Each Other is starting the same pack a 6 phase dice is so bias that is twice as likely to Sona even number as an odd number when Throne it is thrown twice what is the probably that the sum of the two number soon is given ABC in order to Toss a Coin the first wins what are the respective chances of winning has winning the game the content indefinitely from the pack of 52 cards and even numbers of card is drawn so that the probability this consist of half rate and half black cards NB throws a pair of dice a wise if we 36 ball before be throw seven so that the chance of winning age there are three mens aged 6065 and 70 years the probability to leave 5 years more than 0.84 a 60 years old 0.6% 5 years old 0.3.70 years old person find the probably that at least two obtained persons will remain alive yourself a bag contains 5.5 block balls A ball is Donald Trump on the bag and out into the bag which contains 3 what is the what is the probability can the following represent measure the probability where sample space mutual exclusive event three fare coins are tossed construct the sample space out comes at the random experiment mention two mutual exclusively find the probability one tell apan Bihar event of connected with random experiment such that let A and B wents I find let Ab To events connected the random experiment such that three the nine digit 123456789 are arranged random order to form a 9 digit number find the probability that one two three appears in the neighbours in orders men set in a family therefore children find the probability that all of them will have different Birthday to up them will have same birthday one year 365 day the back contest from the bag and put and the bag which contain three red in 7 black balls A ball is Don randomly from the second bag what is the probability that it is rate on sort expired from each of 38 BC do not the event that is target inside by the first second and the third cons respectively that ABC and mutual Independent event that find the probability register what is the chance of getting at least one defective items are drawn randomly from the lord of content items 5% room find the probability that at least two of them will have the same birthday in the same calendar month of year Mr X is called interviews for 3 separate post at the first interview there are 5 candidates at the second four candidates at the third six candidate the selection of each candidate Polytechnic find the probability that mixture X will be selected for at least one post at least two pole it was found that a large match in football supporter the Clever what is the probability that correct result a back on 10 3 8 5 8 balls in a second Back contains 48 6.41 ball is done at random proposed Baigan second back now a ball is a Drone at random from the second find the probability that it is rate

MIXED QUESTIONS (C)

1) A shopkeeper buys a box of 10 dozen apples for ₹180. He estimates that 20% of them would be rotten. He intends to sell each apples so as to make a profit of 60% on his cost. Assuming that 20% of the apples turn out to be rotten, find:

A) the number of goods apples in the box.               96

B) the sum of money for which he should sell the good apples so as to make a profit of 60% on his cost ?   288

C) the selling price of each apple in order to earn the said profit.    3

2) A merchant bought 40 kg tea and sold it for ₹3225.60, thereby making a profit of 12%.

A) at what rate per kg was the tea bought?               72

B) if the cost price of tea raised by 8% for the merchant, and the selling price per kg remains the same, find his profit or loss on selling 50 kg tea.              144

3) A man bought 150 articles, the price of each being the same. He sold 30%of these articles at a profit of 30%, 48% of the articles at a profit of 20% and the remaining at a loss of 16%, thus making a profit of ₹4071.60 on the whole. find the cost price of each article.       180

4) One hundred and fifty articles of the same kind were bought by a supplier for ₹12750 and 12.50 per article were spent on its packing, transportation etc.

    if 60% of the articles were sold for ₹115 each and remainings for ₹120 each, find the profit percent made.     20%

5) A certain number of eggs are bought at 4 for ₹5 and an equal number at 9 for ₹10.

A) if 15% of the eggs were broken in transaction and remaining were sold at 2 for ₹3, find the profit%.   8%

B) Find the number of eggs of each kind bought, if ₹510 were gained on the whole.           2700

6) A manufacturer is to charge ₹32000 for an article manufactured by him. These charges include the cost of material and labour in the ratio 3 : 5 and a profit of 60% of the manufacturer.

   If the cost of the material doubles in the labour becomes 20% costlier but the manufacturer gets the same money (32000) for the articles; what loss of profit percent does he make  ?       1%

7) A trader gives two successive discounts of 15% and 10% on the marked priced of an article.

A) If the marked price was ₹100, what would be the selling price?

B) The actual selling price is ₹7650. Calculate the actual marked price.

C) The actual marked price is 25% more than the cost price. Calculate:

 i) the cost price ii) the profit or loss percent.         76.50, 10000, 8000, 4.375%

8) A trader gives 25% discount on his articles and still makes a profit of 25%. Find his profit percent, if he sales his articles at:

A) 10% discount.       50%

B) their marked price.          200/3%

9) the catalogue price of an article is ₹ 36000 and a manufacturer sells it to the distributor at 20% off the catalogue price. The distributor, in turn, sales it to the retailer at 10% off the catalogue price.

A) what profit percent is made by the retailer, if he sells the article to a customer at the catalogue price ?

B) what profit percent is made by the manufacturer, if the catalogue price is 50% above the cost ?     100/9%, 20%

10) A trader adds 40% of the price he pays for his goods to give the selling price. He later reduces the price of the goods, in a sale, by 25%,

A) if the trader buys an article for ₹640; find its sale price.

B) Find the price paid by the trader for an article whose sale price is ₹1575.

  In both the cases, given above, find profit percent made by the trader.     672, 1500, 5% in each case 

11) A man has 15 barrels, all of same capacity, of oil, If he sells oil at ₹ 25 per litre, he losses ₹1500 on the whole and if he sells oil at ₹27 per litre, he gains ₹300 on the whole. Find the quantity, in litres, of oil in each barrel and the cost price of oil per barrel.      60 , ₹1600

12) The population of a town increases every year at a certain fixed rate. If its population at the end of years 1994 and 1995 be 46000 and 52900 respectively; find its population at the beginning of the 1994 and at the beginning of 1997.         40000, 60835

13) the cost of a machine depreciated by ₹4752 during the second year and by ₹4181.76 during the third year. Calculate

A) its rate of depreciation.

B) its original cost.

C) its cost at the end of the 3rd year.           12%, 45000, 30666.24

14) A sum of ₹30000 is invested at 10% per year compound interest

A) what is the sum at the end of the second year?         36300

B) what is the sum due at the end of 3rd year ?      39930

C) find the difference between the B and A and find the intrest on this sum for one year.       3630, 363

D) Hence, write down the compound interest for the fourth year.               3993

15) A sum of money lent out at  12% compounded yearly earns an interest of ₹4704 in the second year. find 

A) compound interest in first year.

B) compound interest in 3rd year.

C) the sum.         4200, 5268.48, 3500 (read ₹4704 instead of 4,74)

16) Find the difference between the simple interest and the compound interest on ₹24000 for 2 years at 15% per annum.      540

17) 3 years ago, the population of a town was 9 lakh. It increased by 5% in the first year and 6% in the second year but it decreased by 10% in the third year. Find its present population.        901530.

18) A man borrowed ₹14000 for 3 years under the following terms:

15% simple interest for the 1st 2 years.

16% compound interest for the last years on the amount due after 2 years. the intrest being compounded semiannually.

Find the amount to be paid after 3 years.         21228.48

19) A sum of money lent out compound interest per year amounts to ₹ 5292 in two years and to 5556.60 in 3 years. find

A) the rate of interest 

B) The sum lent out.         5%, 4800

20) Find tha perimeter and area of the remaining, after 4 quadrants (quarter circles) of radius 4 cm are cut from the four corners of a rectangle measuring 18cm by 10cm.          129.71cm, 49.14cm

21) A piece of wire is in the form of an Arc of a circle of radius 20cm, subtending an angle of 150° at the centre of the circle. If it is bent into the form of a complete circle; Find the radius of the circle.     25/3

22) The area enclosed between two concentric circle is 2464 cm² and the difference between their perimeters is 88cm; calculate the circumference of the inner circle and the area of the outer circle.   132, 3850

23) the Radii of the three concentric circles are 14cm, x cm and 56cm;  where 14 < x< 56. If 4 times the area enclosed between the two innner circles is equals to the area enclosed between the two outer circles, Calculate the value of x.   28

24) the pulley on the jib of a crane has a diameter of 56 cm, find 

A) the circumference of the pulley,

B) the number of revolutions made by the pulley when the crane raises a load through a vertical height of 44m.      176, 25

25) A square is inscribed in a circle of radius 21√2cn. calculate the area enclosed between the circle and square.        1008

26) A circular running track, 4.2m wide, is bounded by two concentric circles. A sum of ₹20314.80 is spent in constructing this track at the rate of ₹45 per square metre. calculate the length of the diameter of the outer edge of the running track.             38.4m

27) Calculate the amount and compound interest on ₹32000 in 2 years at 10% per annum, if compounded

A)  yearly 

B) half yearly

28) Find the amount and compound interest on ₹48000 in 3/2 years at 8% per year, if compounded

A)  annually

B) semiannually

29) A town contains 64000 inhabitants, If the population increases at the rate 5/2% per annum, find the number of inhabitants at the end of the three years.

30) A man invests ₹12000 for 3 years at 25/2% p.a compound interest, compounded yearly. Income tax at 20p in a Rupee is deducted at the end of each year. Find the amount at the end of the 3rd year.

31) which is the better price (for buyer) for a property, at ₹84000 paid at once or ₹98260 in 3 years time, allowing 25/4% per annum compound interest.

32) A saving certificate costs ₹5000 and is worth ₹5832 at the end of 2 years. What rate% p.a. compound is allowed ?

How long would it take ₹2500 to increase to ₹3149.28 at the same rate ?

33) Find the compound interest on ₹ 1600 for 3/2 years at 10% per annum, compound annually.

34) A sum of ₹640 amount to ₹1250 in 3/2 years at some rate of interest compound half yearly. Find the rate of interest per annum.

35) At what rate per annum will ₹ 4800 yield a compound interest of ₹756.60 in 3 years when the interest is compounded annually.

36) In what time will ₹2000 yield a compound interest of ₹ 205 at 10% per annum, compounded semi annually ?

37) If A={0,1,2,3} and B={3,5,7} then find

A) (A U B) x (A∆B) 

B) (A - B) x B

C) B x (A - B)

38) The product of two consecutive even numbers exceeds their sum by 142. What are the numbers.

39) find the numbers which when divided by 9 gives the same quotient as 16 divided by the number.

40) Let f be such that f(x)= (x-1)² - (x+3)², x belongs to I. Find x, such that f(x)= 0.

41) solve: -7x +3/2 ≤ 1- 1px; where x belongs to R.

42) Solve: 20 - 3(x -1)< 10; given x belongs W.

43) If P={x: x belongs R and 4x - 3≥ 5} and Q ={x : x belongs to R and 5 - x > 1} then find

A) P' ∆ Q

B) P ∆ Q

44) A person bought a certain number of pens for ₹800. If he had bought 4 more pens for the same money; he would have paid ₹10 less for each pen. How many pens did he buy?

45) If a person had cycled 3 km per hour faster, he would have taken 1 hour and 20 minutes less to travel 80km. What time did he originally take to travel 80 km?

46) the sum of the two numbers is 15. If the sum of their reciprocals is 3/10; find the numbers.

47) A two digit number is such that the product of its digit is 12. When 36 is added to the number; the digits interchange their places. Determine the number.

48) the sum of the Ages of a father and his son is 45 years. 5 years ago, the product of their ages was four times the father's age at that time. Find their present ages.

49) A motor boat whose speed is 15km per hour in still water goes 30 km down stream and comes back in a total of 4 hours 30 minutes. Determine the speed of the stream.

50) A sum of ₹587 is lent for 20 years for 4% compound interest. Find (to the nearest rupees) the amount and the compound interest.

51) AB and CD are two chords of a circle intersecting at a point P inside the circle. if 

A) AB= 24cm,  AP= 4cm and PD= 8cm, determine CP.

B) AP= 3cm, PB= 2.5cm and CD= 6.5cm determine CP.

52) AB and CD are two chords of a circle intersecting at a point P outside the circle. If

A) AP= 8cm, PC= 5cm and PD= 4cm, determine AB.

B) PC =30cm, CD =14cm and PA= 24cm, determine AB.

53) If tanA= 1 and tan B= √3; evaluate:

A) cosA cosB - sinA sinB

B) sinA cosB + cosA sinB.

54) If (2cos2A -1)(tan3A -1) = 0; find all the possible value of A

55) If cosA = 9/41; find the value of 1/sin²A - 1/tan²A

56) The perimeter of a rhombus is 48cm and one angle of it is 60°. Find the lengths of its diagonals.

57) The length of a line segment AB is 10 units. If the coordinates of A are (6,9) and abscissa of B is  - 2;  find the ordinate of B.

58) A point is equidistant distance from A (-6,4) and B (2,-8). Find its coordinates, if its abscissae and Ordinates are equal.

59) Find the point on x-axis which are at a distance of 5 units from the point (2,-4).

60) find the point on y-axis which are at a distance of 13 units from the point (12,-3).

61) calculate the coordinates of the point P is divides the line joining A(-1,3) and B(5,9) so that AB: BP=  3:2.

62) A(0,4),  B(2,0) and C(6,2) are the vertices of a triangle ABC; find:

A) the length of medium through A 

B) the point of intersection of the median of the triangle ABC.

63) find the ratio in which the join the points (6,4) and (1,-7) divided by x-axis. Also, find the coordinates of the point of intersection.

64) Find the ratio in which join of points (-3, 10) and (3,5) divided by y-axis. Also, find the coordinates of the point of intersection.

65) A(3,3) , B(6,-1) and D(-5,9) are the vertices of a parallelogram ABCD. Find the coordinates of vertex C.

66) A wire is in the form of a circle of radius 42cm. It is bent into a square. Determine the side of the square.

67) From a copper plate, which is a square of side 12.5cm; a circular disc of diameter 7.0 cm is cut off. Find the weight of the remaining part, if one square centimetre of the plate 0.8 gram.

68) the inner circumference of a circular track is 440m. the track is 14m wide. Calculate the cost of

A) levelling the track at the rate of 20 paise per sq m

B) putting up fence along the outer circle at ₹2 per m.

69) A playground has the shape of a rectangle, with two semicircle on its smaller sides as diameter; added to its outside. If the sides of the rectangle are 36m and 24.5m; calculate the area of the playground.

70) a boy cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60cm, calculate the speed, in km per hour, with which the boy is cycling.

71) water flows into a tank 150m long and 100m broad through a pipe whose cross section is 2dm by 1.5dm at the speed of 15 km per hour. In what time, will the water be three metre deep ?

72) A cylindrical bucket 28cm in diameter and 72cm high is full of water. The water is emptied into a rectangular tank 66cm long and 28 cm wide. Find the height of the water level in the tank.

73) if the radius of the base of a right circular cylinder is halved, keeping the height same, what is the ratio of the volume of the reduced cylinder to that of the original.

74) the radii of two cylinders are in the ratio 2 : 3 and their height are in the ratio 5:3, calculate the ratio of the volumes.

75) find what length of Canvas, 2 m in width, is required to make a conical tent 12 m in diameter and 6.5 m in slant height. Also, find the cost of the Canvas at the rate of ₹12.5 per metre.

76) A tent is of the shape of right circular cylinder upto height of 3m and then becomes a right circular cone with a maximum height of 13.5m above the ground. Calculate the cost of painting the inner surface of the tent at 4 per square metre, if the radius of the base is 14 m.

77) a spherical ball of lead has been melted and made into smaller balls of half the radius of the original one. How many such balls can be made ? Find the ratio of the total surface of all the smaller balls to that of the original one.

78) Marbles, spherical in shape, of diameter 1.4cm are dropped into a beaker containing some water and are fully submerged. The diameter of the beaker is 7cm. Find how many Marbles have been dropped in it, if the water rises by 5.6cm.

79) 65% of children in a Sports Club play football, 70% play volleyball and 75% play basketball. what is the largest percent of children playing all the three games.    10%

80) A TV Survey gives the following data for TV viewing:

 60% see program A, 50% see programme B,  50% see programme C, 30% see programme A and B, 20% see programmes B and C, 30% see programmes A and C, 10% do not view any programme. Find the following:

A) what percentage view A, B and C.

B) what percentage view exactly 2 programme?

C) what percent view only A?   10, 50%, 10

81) In a beauty contest, half the number of set of experts voted for Mrs A and two thirds of them voted for Mrs B. How many experts were there in that set of 10 voted for both and six did not vote for either Miss A or Miss B.                24

82) A class was discussing how quickly their pocket money seemed to disappear. they noted down money on clothes, 20 had bought a present for someone and 8 of the class only had not spent money on these items. 8 people had incurred expenses in each of the three mentioned categories.12 had bought a record and nothing else and 6 had bought some clothes and a record, but not present. Twice as many had bought only clothes as had bought only presents. how many of the class had not bought on LP record during the previous week?       24

83) Two finit sets have x and y number of elements. The total number of subsets of the first set is four times the total number of subsets of the second set. Find the value of x - y .                2

84) In a group of athletics teams in a certain School, 21 are in the basketball team, 26 in the hockey team and 29 in the football team. If 14 play hockey and basketball, 12 play football and basketball, 15 play hockey and football and 8 play all 3,  find

A)  how many players are there in all ?

B) how many play only football.

85) if A={a,b, c, d}, B={b, c, d, e}, find

A) A U B

B) A' U B'

C) A - B

86) if A={2,4,6}, B={4,5,6,7} and €={x: x≤ 10, x≥ N} then find

A) (A U B)'

B) A' U B'

87) If A= {x: x²- 3x +2= 0} and B={x: x² + 4x -12= 0} find

A) A U B

B) A - B

88) If A and B are two sets such that neither A is a subset of B nor B is a subset of A then which of the following statement is true ?

A) AUB is the subset or equal to A

B) A is the subset or equal to A intersection B

C) A is subset of A UB

D) A U B is the subset or equal to B.

89) (x-8)(x+7)> 0.      -7,8

90) x²+12x +11< 0.        -11,-1

91) 8/x - 2/7 > 9/x - 1/2.       x< 0, x> 14/3

92) |3x -2|≤ |2x -5|.      -3≤ x≤ 7/5.

93) x+ 1/x < 2.5.                 

94) Find the greatest value of (x+2)/(2x² + 3x +6) for real values of x.              1/3

95) (x-6)(x +13)≥ 0, find the range of x.                -13≥ x ≤ 6

96) |2x -5|< 4x +9.       x> -2/3

97) In an election for the president, 261 valid votes are cast for the 5 contestant. Find the least number of votes a candidate requires to receive to win the election.       53

99) If x+ y= 1 and x> 0, y>0, then find the least value of (x+2/x)²+ (y+ 1/y)².            25/2

100) (x-2)/(x +2)≥ (2x-4)(4x -2).

101) Find the maximum value of the product of two numbers whose sum is 12.

102) Merchant mixes 3 kg of tea costing ₹50 per kg and 5 kg another variety costing ₹34 per kg. At what price per kg should be sell the mixture so as to make a profit of 10% on the cost price ?      44/kg

103) 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 proportion of 5:4 ?    5:2

104) A container has a capacity of 80 litres 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 ?           61.8 litres (appr)

105) A mixture of 40 litres of milk and water contains 10% of water. How much water must be added to make 20% in the new mixture.     5 

106) In what ratio should silver at ₹5 per gm be mixed with silver at ₹4.75 per gm to obtain a mixed variety of silver worth ₹4.92 per gram?             8/17

107) In what ratio must water be added to the milk to gain 25/2% by selling the diluted milk at the cost price ?              1/8

108) How many kilogram of sugar worth ₹6.50 per kg must be mixed with 80 kg of sugar worth ₹8.00 per kg so that the mixture is worth ₹ 7.50 per kg ?         40kg

109) A sum of ₹39 was divided among 45 boys and girls. Each girls 50 paise, whereas a boy gets one rupee. Find the number of boys and girls.          12, 33

110) An alloy of two metals weight 18gms is worth ₹87 but if their composition weights be interchanged, it would be ₹78.60. if the price of one metal be ₹6.70 per gram. Find the weight of the other metal in the mixture.         8gm

111) a buttler stole wine from a butt of a sherry which contain 32% spirit and then replaced what is stole by wine containing 18% spiri6. the butt was then of 24% strength only. How much butt did he steal.       4/7

112) In a Farm there are goats and Ducks. if you count the heads there are 15, if you count the legs there are 48. How many of ech kinds are there ?            6 9

113) In what ratio should milk and water mixed so that after selling the mixture at the cost price a profit of 50/3% is made.

114) A person travels 285 km in 6 hours in two stages. In the first part of the journey, he travels by bus at a speed of 40 km per hour. In the second part, he travels by train at 55 kmph. How much distance did he travel by train ?

115) A person has a chemical of ₹25/lit. In what ratio should water be mixed in that chemicals so that after selling the mixture at ₹20/ltr, he may get a profit of 25%.

116) A container contained 80 kg of milk. From this container 8 kg of milk was taken out and replaced by water. This process was further repeated two times. How much milk is now contained by the container.

117) If 2 kg of metal, of which 1/3 is zinc and rest copper, be mixed with 3 kg of metals, of which 1/4 is zinc and rest copper. What is the ratio of zinc to copper in the mixture.

118) In 3 vessels each of 10 litres capacity, mixture of milk and water is filled. The ratio of milk and water are 2:1, 3:1 and 3:2 in the 3 respective vessels. If all the three vessels are emptied into one larger vessel, final proportion and milk and water in the mixture.

119) A cask of wine when fully filled holds 10 litres, 2 litres of wine is removed and filled with water. then 4 litrd in the solution is replaced with water. Then 6 and 8 litres respectively. At the end of the fourth operation, the ratio of wine to water is:

A) 4!/5⁴ B) 4!/(5⁴-4!) C) 8!/10⁴ D) 8!/(10⁴ - 8!)

120) find the average of 30 natural numbers.        15.5

121) the average of 11 results 60. if the average of the first 6 results is 58 and that of the last six is 63. find the sixth result.             66

122) the average marks of 3 batches of students, containing 40, 50 and 60 students respectively are 50, 60 and 70.  find the average marks of all 150 students.     61.33

123) find the Geometric mean of 12, 15 ,17, 19 and 32.       17.94

124) The average age of a class was 15 years. When 5 boys whose average age was 12 years 6 months were admitted, the class average was reduced by 6 months. How many boys were there initially.   20

125) the average of 8 results is 25,  that of the first two is 20 and and the next 3 is 26. the sixth result is less than the 7th by 4 and less than the 8th by 6. Find the last result.   30

126) the average weight of 10 oarsmen in a Boat is increased by 1.8 kg when one of the crew, who weight is 53 kg is replaced by a new men. Find the weight of the new men.          71kg

127) 9 people went to a hotel for taking their meals. 8 of them spent ₹12 each on their meals and the ninth spent ₹8 more than the average expenditure of all nine. What was the total money spent by them ?               ₹117

128) There are 35 student in a hostel. Due to the admission of 7 news students, the expenses of the mess were increased by ₹42 per day, while the average expenditure per head decreased by ₹1. What was the original expenditure of the mess ?         ₹420

129) the average age of 24 boys and the teacher is 15 years. When the teacher's age is excluded. The average decrease by 1. What is the age of the teacher ?       39

130) The average of 5 consecutive numbers is n. If the next two numbers are also added. Find the new average.     n+1

131) the mean temperature of Monday to Wednesday was 37°c and of Tuesday to Thursday was 34°C. If the temperature on Thursday was 4/5th that of Monday, find the temperature on Thursday.           36

132) 3 years ago, the average age of a family of five members was 17 years. A baby having been born, the average age of the family is the same today. What is the present age of the baby ?          2 years

133) The average age of a class is 15.8 years. the average age of the boys in the class 16.4 years while that of the girls is 15.4 years. What is the ratio of the boys to girls in the class ?      2:3

134) Find the average of 3,3,3, 6,7,8, 9,9, 10 ,11 ,11,12 and 13 .    

135) the average of 11 results is 30, that of the first five is 25 and that of the last five is 28. Find the value of the 6th number.

136) A batsman in his 17th in innings makes the score of 85 and thereby increases his average by 3. What is his average after 17 innings.

137) One of 2 buses completes a journey of 300 km in 15/3 hours and the second journey of 450 kms in 10 hrs. Find the ratio of their average speeds 

138) A man spends ₹1800 per month on an average for the first four months and ₹2000 per month for the next 8 month and saves ₹5600 a year. Find his average monthly income.

139) A, B and C enter into a partnership investing ₹3800, ₹4200, ₹4000 respectively. How should a profit of ₹1800 be divided among them?                     570,630, 600

140) In a partnership deal, A contributes ₹3200 for 4 months, B contribute ₹5100 for 3 months and C contributes ₹2700 for 5 months. How should they divided a total profit of ₹1248 among themselves?        384, 459, 405

141) A, B, C enter into a partnership A contribute ₹320 for 4 months. B contributes ₹520 for 3 months, and C contributes ₹ 270 for 5 months. If the total profit ₹208, find the profit share of the partner.        64, 76.50, 67.50

142) A and B enter into a partnership for a year.  A countributes ₹ 1500 and B ₹2000. After 4 months they admit C, who contribute ₹2250. If B withdraws his contribution after 9 months, how would they share a profit of ₹900 at the end of the year ?

143) A and B enter into a speculation. A puts in ₹ 50 and B puts in ₹ 45. At the end of 4 months A withdraws half his capital and at the end of 6 months B withdraws half of his capital. C then enters with a capital of ₹70. At the end of 12 months in what ratio will the net profit be divided?

144) A began a business with ₹450 and was joined afterwards by B with ₹300. When did B join if the profit at the end of the year were divided in the ratio 2:1? 

145) A and B rent a pasture for 10 months. A put in 90 oxen for 7 months. How many can B put in for the remaining 3 months, if he pays half as much as A?

146) three partners invested ₹114000 in a business. At the end of the year one got ₹ 337.50, the second  ₹ 1125.00 and the third ₹ 675 as profit. How much did it each  invest ? what is the percentage of profit ?

147) What is 14.28% of 560.    80

148) what is 37.5% of 720.   270

149) Express 7/8 as a rate percentage.       87.5% 

150) In an examination a candidate who scores 25% of the maximum marks fails by 60 marks but another secured 42% of the maximum marks gets 8 marks more than those and necessary for passing. Find the maximum marks and the percentage necessary for passing.   400, 40%

151) in an examination 70% student passed in English and 75% in mathematics while 20% failed in both the subjects. if 195 students passed in both the subjects, find the total number of students.   300

152) find the difference between 7/4 of ₹ 100 and 7/4% of ₹100.   ₹173.25

153) if A's income be 10% more than B's, how much percentage B's income less than A's.    100/11%

154) A sample of 20 litres of gasoline is found to be adulterated to the extent of a 10%. How many litres of adulterated mixture should be replaced with pure gasoline to bring up the level of purity to 95%.   10

155) If Rajinder's salary is 20% more than Jatinder's salary, then what percentage less than Rajendr's salary is Jatinder's salary ?     50/3%

156) On account of reduction in price of Apples by 25% Narinder is able to buy 5/2 kg of Apples more for ₹120.

A) find the reduced price per kg of Apples.        12

B) find the original price per kg of Apples.       16

157) the price of a Fiat Uno costing 4 lakh is 20% less than that of a Bolero. What is the price of Bolero.  480000

158) If the length of a rectangle is decreased by 40% and the breath is increased by 30%, then find the percentage of change in the area of the rectangle.      22%

159) An ore contains 12% copper. How much ore is required to get 69 kgs of copper ?

160) Reeta loses 12.5% of her money and after spending 70% of the remainder, has 210 left. How much money did she have at first ?

161) area of square increases by 4 times its value. Find increases percentage in dimension.

162) In an examination 40% of students failed in maths, 30% failed in English and 10% failed in both. Find the percentage of students who passed in both subjects.

163) The tax on a commodity is diminished by 15% and its consumption increasing by 10%. Find the decrease percentage in the revenue derived from it and also find out with what percent increase in consumption would revenue remain the same.

164) In a village 50% are male. 20% of females are married and 10% have children. Find the percentage of females with no children.

Continue......

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GEOMETRICAL INTERPRETATION ( AREA)

AREA

STEPS: to set up a proper definite integral corresponding to a desired area:
1) Make a sketch of the graph of the given function.
2) Shade the region whose area is to be calculated.
3) In choosing the limits of integration, the smaller value of x at ordinate is drawn will be taken as lower limit and the greater as upper limit (i.e., we are to move from left to right on x-axis) and then to evaluate the definite integral.
4) Only the numericals value (and not the algebraic value) of the area will be considered, i.e., we will discard the -ve sign, if some area comes out to be -ve (after calculation).
5) If the curve is symmetrical, then we will find, area of one symmetrical portion and then multiply it by n, if there are n symmetrical portions.

SOME WELL KNOWN CURVES

1) Straight line: ax+ by+ c= 0
2) Circle: x² + y²= a²
3) Parabola: y² = 4ax
4) Ellipse: x²/a² + y²/b²= 1.
5) Hyperbola : x²/a² - y²/b²= 1.

EXERCISE -1

1) Find the area of the triangle formed by the following lines:
i) x= 0, y= 0 and x+ y= 2.            2
ii) x= 0, y= 0 and 3x+ 2y= 6.        3
iii) x= 2, y= 0 and 2x+ y=6.           1
iv) x= 2, y= 0 and y= 3x.          27/8
v) x= 0, y= 1 and 4x+ 3y= 12.      3

2) Find the area of the region bounded by y= 2x+1, the x-axis and the ordinates x=2 and x= 4.       14 sq.units

3) Find the area of the triangle bounded by the x-axis, y-axis and the line x+ y = 4.         8 sq.units

4) Find by Integration the area bounded by the curve y= x², the x-axis and the ordinates x= 1 and x= 3.            26/3 sq.units

5) Find the area region lying in the first quadrant bounded by the parabola y²= 8x, the x-axis and the ordinates x=4.        32√2/3 sq.units

6) Find the area of region bounded by y= 3x² + x , the x-axis and the ordinates x= 1 and x= 3.            30 sq.units

7) Find the area of the plane bounded by the three lines y= x², y= 0 and x= 1.          1/3 sq.units

8) Find the area of region bounded by y²= 4x, x-axis and the straight lines x= 1 and x= 4.            28/3 sq.units.

9) Find the area of the region bounded by y= 4x², y= 0, x= 1 and x= 3.        104/3 sq.units

10) Find the area bounded by the parabola y²= 16x , the x-axis and the ordinates x= 4.        64/3 sq.units

11) Find the area bounded by 4y= x² , and the line x= y.        8/3 sq.units

12) Find the area bounded by the curve y= x(4- x) and x-axis.           32/3 sq.units

13) Find the area of bounded by the parabola y= 16(x-1)(4- x)  and the x-axis.         72 sq.units

14) Find the area of bounded by the curve y= 4(x-1)(3 - x)  and the x-axis.         16/3 sq.units

15) Find the area of bounded by the curve y= 2x(x-1)(x +1) and the x-axis.         1 sq.units

16) Find the area bounded by y= 3(x -1)(5- x) and x-axis           32 sq.units

17) Find the area of region bounded by y²= 6x and x²= 6y.            12 sq.units

18) Find the area bounded by y²= 4x and x²= 4y.            16/3 sq.units

19) Find the area bounded by y= x² and x= y.            4/3 sq.units

20) Find the area of region bounded by y²= 3x ,  y-axis and lines y= 1 and y= 4.          7 sq.units

21) Find by integration the area between y²= 8x² + x and the straight line  x= 4.       64√2/3 sq.units

22) Find the area enclosed between the axes and the curve (y-2)²= 8x.            1/3  sq.units

23) Draw the graph of y= 3x²+ 2x+4. Shade the area bounded by the curve, x-axis and the lines by  x= -1 and x= 3 and hence find its area by integration.           52 sq.units

24) Find the area bounded by the curve y²= 2x² and the line y= x.              2/3 sq.units

25) Find the area included between y²= 9x and x= y.        27/2 sq.units

26) Find the area bounded by the parabola 4y= x² and its latus rectum.            8/3 sq.units

27) Find the area enclosed by the parabola y²= 4ax and its latus rectum.         8a²/3 sq.units

28) Find the area bounded by the parabola y²= 8x and its latus rectum.         32/3 sq.units

29) Find the area bounded by parabola 4y= x² and its latus rectum          8/3 sq.units

30) Find the area of the region bounded by the curve  y²= 12x, x-axis and the semi latus rectum.              12 sq.units

31) Find the area cut off from the parabola y²= 12x by its latus rectum.         24 sq.units

32) Find the area above the x-axis bounded by x - 2y +4= 0,  x= 3 and x= 6.            51/4 sq.units

33) Find the area of region bounded by the curves y= x² and y= √x.           1/3 sq.units

34) Find the area of the region bounded by y= x² and the line y= 4, (lying in 1st quadrant)      16/3 sq.units

35) Find the area enclosed between the curve y²= x , the x-axis and the ordinates x= 1 and x= 9.            52/7 sq.units

36) Find the area enclosed between the parabola y²= 8 and the straight line x= y.           32/3 sq.units

37) Find the area bounded x-axis, part of the curve by y= (1+ 8/x²) and the ordinates x= 2 and x= 4. If the ordinate at x= a divides the area into equal parts, find a.           4 sq.units, 2√2

38) Calculate the area bounded by the x-axis and the curve y= x - 3√x.          27/2 sq.units

39) Find the area enclosed between the curve by y²= x²(4- x²) , the coordinate axes and the ordinates x=2.           8/3 sq.units

40) Find the area bounded by y= 4x(x-1)(x -2) and the x-axis.           2 sq.units

41) Find the area enclosed by the curve (x-1)(5-x) and the x-axis.           32/3 sq.units

42) Find the area enclosed by the curve y= 3x - x². The x-axis and the ordinates x= 0 and x= 3.           9/2 sq.units

43) Find the area bounded by the curve 4y= x² and the straight line  x= 4y -2         27/24 sq.units

44) Find the area enclosed by the curve y= x², the y-axis and the lines y= 1 and y= 9.            52/3 sq.units

45) Find the area of the region lying in the first quadrant bounded by the parabola y²= 4x , the x-axis and the ordinates x= 4.           33/3 sq.units

46) Find the area of region common to the curves y²= 3x and x² = 2y.           4/3 sq.units

47) Shade the area of enclosed by the parabola y²= 8x and x²= y and use the method of integration to find the area so enclosed.               8/3 sq.units

48) Find the area bounded by the curve y= 4x(x -1)(x-2) and the x-axis.         1 sq.units

49) Find the area region common to the curve y= 4x²(3 - x) and the straight line passing through the point (0,0) and (2,16)         1 sq.units

50) Determine the area bounded by the straight line y= 3x+ 3 and the parabolic curve y= x²- 5x +15.                32/3 sq.units

51) Find the area of the region common to the curve y= x² and y= √x.     1/3 sq.units

52) Find the area bounded by the straight lines y= 2x - 4, x = 5 and y= 0.        9 sq.units

53) Find the area enclosed by the parabolic curve y²= 2x +2 and straight line x- y -2= 0            16/3 sq.units

54) Find the area of the circle x²+ y²= a²           πa² sq.units

55) Find the area of the ellipse x²/a²  + y²/b² = 1.        πab sq.units

Continue......

EXERCISE -2

) Showing a rough sketch of the graph y= sin 2x, 0≤ x ≤π/2 find the area enclosed by the curve and x-axis.                          1 sq.units

57) Find the area enclosed by the graph  y= sin² x, 0≤ x ≤π/2, x-axis and the line x=π/2. Also show a rough sketch of the graph.      π/4 sq.units

PROFIT, LOSS AND DISCOUNT (J)

EXERCISE-1

1) A man buys Oranges at 9 for ₹8.10 and sells them at 11 for ₹ 12:10. Find his gain or loss%

2) By selling a bicycle ₹797.50, a man loses 8%. At what price should he sell it to gain 15%.

3) A man goes from Agra to Delhi to buy an article which cost him ₹4350 in Delhi. If he spends ₹250 on his travels and transportation of the article and sells it in Agra for ₹5290; calculate his profit or loss percent in the transaction.

4) By selling tea at ₹60 per kg a grocer gains 4%. At what rate must he sell it in order to gain 17%.

5) I lose 5% by selling a piece of land for ₹46740; what would be my gain or loss percent if I sell it for ₹528290?

6) By selling 27m of clothes at a loss of 4%, I lose ₹162 altogether. What are the cost price and selling price of the cloth per meter ?

7) 15% was gained by selling a dining table for ₹1610 and 12% was lost by selling a black and white TV for ₹3080. Find, the total gain or loss percent.

8) A man sold two articles for ₹2970 each, gaining 10% on one and losing 10% on the other. Find his total loss or gain percent.

9) Smith sells 2 bicycles for ₹450 each, losing 20% on one and gaining 25% on the other. Find total loss or gain percentage.

10) By selling wheat at ₹4.50 per kg, a shopkeeper gains 25/2%. Find the cost of find the cost of 50 kg wheat.

 Also, find his profit percent if he sells the wheat at ₹460 per quintal. 

11) A shopkeeper buys 40kg ghee at ₹45 per kg and another 60kg at ₹40 per kg. If he mixes them and sell the mixture at ₹46.30 per kg, find his gain%.

12) A Steel almirah, costing ₹ 900 to the manufacturer, is sold at 15%. It is estimated that 5% of the selling price must be set aside to allow for overhead charges. Find the net profit, correct to one rupee.

13) John buys three identical watches and sells one watch at 20% profit, other at 12% profit and third at 15% loss. If, on the whole, he earns Rs 85; find the cost price of five such watches.

14) By selling 35m of cloth at a profit of 5%. I altogether, gain 70. calculate the cost price and selling price per metre.

15) 75 unfinished articles of the same kind were bought for ₹1275 and ₹3 per article was spent on their finishing. Find the profit percent made if, one third of these articles were sold at ₹26 each and the remaining at ₹22 each.

EXERCISE -2

1) A bought a radio and spent ₹ 110 on its repairs. He then sold it B at 20% profit. B sold it to C at a loss of 10%. What is the amount for which A bought the radio, if C paid ₹ 1080 for it ?

2) The cost of 19 Apples is equal to the selling price of 16 apples. Find the gain%.

3)  A reduction of 20% in the price of sugar enables Mrs. Jain to buy 4 kg sugar more for ₹80. Find

A) the reduced price per kg

B) the original price per kg

4) A dishonest dealer professes to  sell his goods are the cost price, but uses of weight 900 gram per 1 kg weight. Calculate his gain%.

5) A bicycle passes through the hands of three dealers each of whom gains 25%. if the third dealer sales it for ₹750. What did the first pay for it.

6) John sold an article to Sumit at a gain of 10%. Smith spend to ₹260 on the repairing of the article and sold it to Peter at a gain of 15%. If Peter sold the article to some unknown person for ₹1966.50 and lost 5%, calculate its cost price for John.

7) the selling price of 10 note books is same as the cost of 12 notebooks. find the profit percent.

8) By selling 20 pens, a shopkeeper gains equal to the cost price of 4 pens. Calculate his gain%.

9) By selling 20 pens, a shopkeeper gains equal to the selling price of 4 pens. calculate his gain%.

10) By selling 20 pens, a shopkeeper losses equal to the cost price of 4 pens. calculate his loss percent.

11) A rise of 25% in the price of sugar, compels a man to buy 5 kg sugar less for ₹200.

Calculate

A) the increased price per kg.

B) the original price per kg.

12) A dishonest fruit vendor professes to sale his goods at cost price, but he uses a weight of 460gm for per half- kg. find his gain%.

13) When a certain quantity of tea is sold at ₹53.76 per kg, the gain percent is 12% and the total gain is ₹51.84, calculate the quantity of tea sold.

14) A fruit seller bought some oranges at 4 for ₹3 and an equal number at 5 for₹3. He mixed the two lots and sold these at 90 paise each. find the gain or loss percent.

15) A woman bought two parcels of mangoes with equal number mangoes in each. She bought the first parcel at one rupee per mango and the second at 3 for ₹2. She mixed them all together and sold at 13 per dozen. find her gain%.

16) A fruit seller bought a certain number of oranges for ₹640 and sold half of them at 20% profit. half of the remainder wre sold at 10% profit. If on the whole he makes a profit of 15%;  find the profit or loss percent made on the remaining oranges.

17) A fruit seller loses  8% by selling oranges at 5  for ₹2. How many oranges should be sell for ₹5 to gain 15% ?

18)  A bicycle dealers sells 9 bicycles at a gain of 8% and 5 more at a gain of 10%. Had he sold all the bicycles at a gain of 9%; he would have received ₹48 more. find the CP of one bicycle if, all the bicycles cost the same.

19) a man has 15 barrels, all of same capacity, of oil. If he sales oil at ₹25 per litre, he loses ₹1500 on the whole and if he sales oil at ₹27 per litre, he gains ₹300 on the whole. find the quantity, in litres, of oil in each barrel and the cost price of oil per barrel.

20) Sonal sold a book at a gain of 12%. had she sold it for ₹3.60 more, the gain would have been 18%. Find the cost price of the book.

21) a man bought an article and sold it at a gain of 25%. if he had bought it at 20% more and sold it for ₹150 more, the gain would have been 7.5%. find the cost price of the article.

22) A certain number of same kind of articles are purchased for ₹13500 and 1/3 are sold at a loss of 10%. At what gain% should the remainder be sold, so as to gain 20% on the whole transaction ?

23) A man bought a certain number of goods of which he sold 1/3 at a profit of 14%, 3/5 at a profit of 35/2% and the remainder at a gain of 20%. What was his profit percentage on the whole.

EXERCISE -3

1) A trader sales his goods at a discount of 20% and still makes a profit of 25%. Find the profit percent made, if he sells the goods at the marked price only.

2) the list price of an article is ₹900 and is available at two successive discounts of 20% and 10%. Find  the selling price of the article.

3) Find the single discount which is equivalent to successive discount of 20%, 15% and 10% .

4) A shopkeeper marks the price of an article at ₹800. find in each case of the selling price, if he allows:

A) two successive discount of 10% and 8% .

B) three successive discount of 10%, 5% and 3%,

5) the list price of a watch is ₹400. A customer gets two successive discounts, the first being 10%. Find as the percentage, the second discount if the customer pays ₹ 306 for it.

6) find which of the following discount series is better:

A) 30%, 20% and 10%.

B) 25%, 20% and 15%.

7) find the single discount which is equivalent to three successive discount of 20%, 10% and 5%.

Hence, find the selling price of an article marked at ₹ 250.

8) A shopkeeper allows two successive discounts of 10% and 15% on his articles. if he gets ₹459 for an article, find its marked price.

9) A tradesman marks his goods 30% above the cost price, but allows a discount of 15% on the marked price for cash pay5. Find his gain percentage.

10) A bicycle agent allows 20% discount on his advertised prices and still makes a profit of 20%. what is the advertised price of a bicycle that cost him ₹ 900?

11) A shopkeeper marks his goods 20% above the cost price but allows 5% cash discount. Find his gain on an article for which he charges Rs 190.

12) A dealer marks his good 35% above the cost price and then allows 15% discount on it. what is the cost price of an article on which he gains ₹88.50?

13) A shopkeeper allows 20% discount on the advertised price of his goods and still makes a profit of 12% on his cost price. calculate the advertised price of an article on which he gains ₹135.

14) A manufacturer gives a  discount of 15% on the market price of the goods manufactured by him and yet makes a profit of 20%. had he sold his goods at the marked price, what profit percent would have been made by him ?

15) A tradesman marks his goods 60% above the cost price. He sells one-third of the goods at this price, one-third at a discount of 20% and the remaining at the discount of 30%. calculate his gain percentage altogether.

16) A publisher gives his distributor a discount of 30% on the printed price of the books. The distributor sells those books to a bookseller at 20% discount on the printed price and the bookseller sells these books at their painted price. calculate the profit percent made by:

A) the distributor

B) the bookseller.

MISCELLANEOUS-1 

1) A publisher sells books to a retailer at ₹15 per copy, but gives 60 copies for the price of 50. if the retailer sells each of the 60 copies at ₹16.50, what profit percent does he make ?

2) A profit or 20% is made on goods when a discount of 20% is given on the marked price. what profit percent will be made when a discount of 30% is given on the marked price.

3) A manufacturer marks his goods 80% above the cost price, but gives five articles at the price of 4 and also 10% discount for cash payment. Calculate the profit percent made by the manufacturer, when the customer pays in cash.

4) The actual cost of an article is 70% of the price at which the article is marked. it is, however, sold at a discount of 7.5%. how much percent does the seller gain.

5) A sells his car to B at gain of 30%, but B becoming bankrupt, pays only 80 paise per for Rupee. calculate A's profit or loss percent.

6) what profit percent is made by selling an article at certain price, if by selling it at two-thirds of that price, there would be a loss of 20%.

7) Geeta bought two old TV sets for ₹8100 and sold one at a gain of 7.5% and the other at a loss of 6%. find the cost price of each set, if on the whole she neither lost nor gained.

8) by selling horse and a carriage for ₹31200, a man gains 10% on the horse and 25% on the carriage, but if sales them for ₹30800, he gains 25% on the horse and 10% on the carriage. calculate the cost price of each.

9)  Manoj travels from Meerut to Delhi to buy goods which he gets 10% cheaper in Delhi than in Meerut. if his journey expenses are ₹160 and he earns ₹240 by selling  goods at meerut, calculate

A) the price of the goods, he paid at Delhi.

B) his profit percent.

10) A manufacturer makes 750 articles at a cost of 60 paise per article. He fixes the selling price such that if only 600 articles are sold, he would make a profit of 40% on his outlay, find his actual profit or loss as percent, if he could sell

A) 630 art6

B) 560 articles

11) on every purchase of ₹1800 at 10% discount, a dealer gives goods wort5 ₹120 free as a gift. If he still makes a profit of 20%, find the profit percent made by the dealer, had he sold the goods without any discount and without any gift.

12) A poultry keeper sells 18 dozen eggs of which 6 dozen are large grade eggs selling at ₹10.50 per doz6, 4 dozen are medium grade eggs selling at ₹9.75 per dozen and the rest are small. find the selling price per dozen of the small eggs if  the poultry keeper receives an average price of 75 paise per egg on the whole.

13) a bookseller gets certain number of copies of textbook for ₹ 3600. Had the price per copy been 20% higher, he would have got 50 copies less. find

A) the price of each copy.

B) the number of copies he got.

14) The catalogue price of a T. V is ₹ 70000. If it is sold at a discount of 50/3% of the catalogue price, a gain of 25% is made  find the gain or loss percent, if the TV is sold for ₹1600 below the catalogue price.

15) A dealer buys a certain number of goods at a uniform price and sells 80% of them at a profit of 15% and the remainder at loss of 5%. what money did he pay for goods, if his total profit is ₹1870?

16) A business man sells an old machine for ₹3450 after making some profit. Had he sold it for ₹2730 his loss would have been 7/5 of the profit. find the cost price of the machine.

STRAIGHT LINE IN SPACE

λ γ μ ₁₂₃₄ ₙ ₐ ₓ ˣⁱⁿ⁺ ⁻ ₊ ₋ 

1) Vector equation of a line:

The vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b is 

r= a+ λb, where λ is scalar.

Note :

a) In the above equation r is the position vector of any point (x,y,z) on the line. Therefore, r= xi + yj+ zk.

b) The position vector of any point on the line is taken a+ λb.

2) The vector equation of a line passing through two points with position vector a and b is:

r= a + λ(b - a).

3) The Cartesian equation of a straight line passing through a fixed point (x₁ , y₁ , z₁) and having direction ratios proportional to a, b, c is:

(x - x₁)/a = (y - y₁)/b = (z - z₁)/c.

Note:

a) The above form of a line is known as the symmetrical form of a line.

b) The parametric equations of the line (x - x₁)/a = (y - y₁)/b = (z - z₁)/c. are x= x₁ + aλ, y= y₁ + b λ ,.z=  z₁ + cλ, where λ is the parameter.

c) The coordinates of any point on the line (x - x₁)/a = (y - y₁)/b = (z - z₁)/c are 

(x₁ + aλ,  y₁ + b λ , z₁ + cλ) where λ belongs R.

d) Since the direction cosines of a line are are its direction ratios. Therefore, equation of a line passing through (x₁ , y₁ , z₁) and having direction cosines l,.m n are

(x - x₁)/l = (y - y₁)/m = (z - z₁)/n

e) Since x, y and z-axes pass through the origin and have direction cosines 1,0,0; 0,1,0; and 0,0,1 respectively. Therefore, their equation are.