Funde Again

8/20/2019 Funde Again

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1)When a four digit number is multiplied by N,the four digit number repeats itself to give an 8 digitnumber .If four digit number has all distinct digits then N is a multiple of ?a)11 b) 37 c)73 d) 27 Ans 73

2)Find the greatest and least values of f(x) = x^4 +x^2+5 / (x^2+1)(x^2+1)for real xa)5,0.95 b ) 5,-1 c)4,0 d)3,2 Ans a

3)Out of m persons sitting at a round table in how many ways no two of the persons A,B, C are sitting together?a)m(m-1)(m-2) b)m(m-2)(m-3) c)m(m-4)(m-5) d)m(m-1) Ans c

4)A bookworm starts eating pages of a book from a certain page number in ascending order ofpage numbers without missing a single page between the first and last pages eaten by it .When itfinished with its job it counts the sum of the page numbers of all the pages eaten and finds that the sum is 198.Find the page number where it stopped

eating?The cover page of the book is page number 1 and the worm always eats both sides of asheet.a)11 b) 22 c) 33 d ) 44 Ans c

5)The number from 1 to 33 are written side by side as follows : 123456...........33.If this number isdivided by 9 what is the remainder?a)0 b)1 c)3 d) 6 Ans c

6) A positive three digit number X is such,when divided in two unequal three digit number,thelarger part is the aritmetic mean of X and the smaller part.How many values can X take?a)300 b)234 c)198 d)None

Ans b

7)A 101 digit number 222.........2X3...333 is formed by repeating the digit 2,50 times followed bythe digit X and then repeating the digit 3,50 times.The number is a multiple of 7.Find the value ofX.?a)2 b)3 c ) 4 d ) 5 Ans d

8) There are ten coin making machine.Nine of them produces coins of 10 gm each while the tenthmachine produces coins with 11 gm weight . If one has a weight measuring instrument tomeasure weight in grams,how many minimum number of readings are required to determine which machine produces heavier coins?a)1 b)3 c)4 d)None of these

Ans a

9)A student is solving a maths problem.He attempts to solve a particular problem where he has todetermine the sum of the squares of the roots of the given polynomial.The print having faded over time,he can see only 3 terms of the fifth degree polynomialx^5-11x^4+....-13=0 is all that is visible.He still manages to find the answers What is it given that -1 is the root and all other roots are integers?1)171 2)173 3)169 4)None Ans 2


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10) The number 444444....(999 times) is definitely divisible by:a)22 b)44 c)222 d)444 Ans c

11)If a three digit number is divided into three two digit numbers and if all these three two digitnumbers from an A.P. with a common difference of 20.How many three digit numbers satisfy thiscondition?a)54 b)45 c)46 d) 55 Ans c

12) If first 23 terms of the series 1,11,111 are added together ,what digit would occupy thethousands place?a)3 b)0 c)9 d)2 Ans d

13)A chord is drawn arbitrarily in a given circle.What is the probability that the length of the chordis less than or equal to the radius of the circle?a)0.5 b)0.25 c)0 d)0.33 Ans d

14) Ravi,Deepak and Amod started out on a 100 mile journey.Ravi and Deepak went by car atthe rate of 25mph,while Amod walked at the rate of 5 mph.After a certain distance,Deepak got offand started walking at 5 mph,while Ravi drove back for Amod and got him to the destination in the car at the same time that Deepak arrived.Q)What is the time taken by Amod to finish the journey?a)6 b)7 c)8 d)None Ans cQ)What was the total distance(in miles) travelled by Ravi?a)100 b)200 c)250 d)150 Ans b

15)How many natural numbers are factors of 7560 and multiples of 14?a)20 b)22 c)24 d)28

Ans c

16)Definition:If a,k and n are positive integers with k>1,and n=k*a ,then a is calledproper divisor of n.How many positive integers less than 54 are equal to the product of theirproper divisors?a)1 b)8 c)10 d)14 Ans d

17) Given that A is a six digit number with unit digit of 1,and B is a natural number which is thefourth root of A,what is the largest possible value of B?a)23 b)29 c)31 d)37 Ans 31

18)How many integers greater than 40,00,000 and less than 90,00,000 are perfect squares?a)100 b)999 c)1000 d)1999

19)Six integers are selected from 1 to 100 in such a way that the smallest positive differencebetween any two of them is as large as possible.What is this difference?a)16 b)17 c)19 d)20

20)How many different points in the xy plane are at a distance of 5 from the origin and havecoordinates (a,b) ,where a and b are integers?a)4 b)6 c)8 d) 12


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A survey was conducted in a city to determine the choice of channel (DD, BBCand CNN) among viewers in viewing the news. The viewership for these threechannels is 80, 22 and 15 percent respectively. Five percent of therespondents do not view news at all.

Ques: What is the maximum percent of viewers who watch all the threechannels?

1)22 2)15 3)11

ANS: We know thatn(A)+n(B)+n(C)-n(AintersectionB)-n(BintersectionC)-n(AintersectionC)- (2 x All three) = 95To have a max. intersection, the intersecting pairs must be 0Or, 80+22+15 -2X = 95Or, x=11

A fruit vendor bought 100 pounds of berries for > > $2.00 per pound and> > expected to double his investment by selling the> > berries for $4.00> > per pound at an open-air market. The vendor only> > managed to sell 50> > pounds of berries the first day and he sold the> > remainder on the> > second day. The fresh berries had a content of 99%> > water, but because> > of the hot weather, the berries dehydrated and> > contained only 98%> > water on the second day. How much profit did the> > vendor make?

Rohit Arora is absolutely right about the first day .. so whatever hesold the second day is the profit. But explanation for second day iswrong as the weight of berry can't increase .. instead water has todry out..

Initially vendor had 100 pounds in all at 99% which means he had 99pounds of water and 1 pound of actual berry.

Now at the end of the first day the vendor is left with 50 pound(half

of 100) in all i.e. he had 49.5 pounds of water(half of initial) and0.5 pound of berry(half of initial). [Remeber this is still the firstday so %age of water in berry should be 99% which is actually correctas 49.5/50 * 100 = 99% ]

Now let the water in berry be x pounds (after water dries out a bit)also weight of actual berry is 0.5 pound (as this will not dry out) Also it is given that percentage of water on second day is 98%

=> Water/(Water+Berry) = 98/100


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=> x/(x+0.5) = 98/100=> 100x = 98(x+0.5)=> 100x = 98x + 49=> 2x = 49=> x = 24.5 pounds

So on second day ... water = 24.5 pounds and actual berry = 0.5 pounds(as this will not dry out) => total berry = 25 pounds

Now this berry is sold at $4 per pound which gives a total profit of 25*4 = $100 which is the answer.=>

1. A club with x members is organised into 4 committees according to the following rules :(i) Each member belongs to exactly 2 committees(ii) Each pair of committees has exactly 1 member in common.Then find the value of x….?

2.The no. of integer lying between 3000 and 8000(both included) that have at least 2 digits equal.

3.The coefficient of x^4 in the expansion of (1+x-2x^2)^7 is .

4. A closet has pairs of shoes. The no ways in which 4 shoes can be chosen from it so that therebe no complete pairs.

5. Distance passed over by a pendulum bob in successive swing are 16, 12, 9, 6.75..cm. Thenthe total distanced traversed b the bob before it come to rest is (in cm) A. 60 B. 64 C.65 D.67.

6 The point (2,1), (8,5) and (x,7) lie on a straight line , find the value of x..?.

7.Eight singers participate in an art festival where m songs are performed. Each song is

performed by 4 singers, and each pair of singers performs together in the same number of songs.Find the smallest m for which this is possible. A. 12 B.14 C.16 D.18

When two fat fish of identical volume were placed inside the tank, the water level rose to a heightof 25 cm.(a) What is the volume of one fat fish?(b) How many small turtles of volume 250 would be needed to replace the fish?

10. In D ABC, ÐC = 90, ÐA = 30 and BC = 1. Find the minimum of the length of the longest sideof a triangle inscribed in ABC (that is, one such that each side of ABC contains a different vertexof the triangle).

11. Starting at (1, 1), a stone is moved in the coordinate plane according to the following rules:

(i) From any point (a, b), the stone can move to (2a, b) or (a, 2b).(ii) From any point (a, b), the stone can move to (a-b, b) if a > b, or to (a, b-a) if a < b.For which positive integers x, y can the stone be moved to (x,y)?

12. The numbers from 1 to 100 are written in an unknown order. One may ask about any 50numbers and find out their relative order. What are the fewest questions needed to find the orderof all 100 numbers? A. 5 B.6 C.7 D. none of these.


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13. Let k be a fixed odd positive no. find minimun value of x^2 +^2, where x, y are nonnegativeinterger and x+y =k.?

14. How many natural number less than 10^8 are there , whose sum of digit = 7.?

15. All permutations of the letter a, b, c, d, e were written down and arranged in alphabetical orderas in dictionary. Thus in arrangement abcde is in the first place and edcba in last , find theposition of debac.?

16. If the area of circumcircle of a regular polygon with n sides is A then the area of circleinscribed in polygon is..(in term of A) ?

17. Find the no. of rational number m/n where m ,n are relatively prime positive no. satisfyingm 5) whose centers form an equilateral triangle. Prove that m = n = 6,and find the angles of ABC.

25. Find all real numbers x such that X [x [x [x]]] = 88:

26. Two players take turns drawing a card at random from a deck of four cards labeled 1,2,3,4.The game stops as soon as the sum of the numbers that have appeared since the start of the

game is divisible by 3, and the player who drew the last card is the winner. What is the probability that the player who goes first wins?

27. Find all triples (x, y, n) of positive integers with gcd(x, n + 1) = 1 such that x^n + 1 = y^n+1.

28. The point P lies inside an equilateral triangle and its distances to the three vertices are 3, 4, 5.Find the area of the triangle.

29. A triangle ABC has positive integer sides, angle A = 2 angle B and angle C > 90. Find theminimum length of the perimeter of ABC.


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30. A country has 1998 airports connected by some direct flights. For any three airports, sometwo are not connected by a direct flight. What is the maximum number of direct flights that can beoffered?

31. The side lengths of a triangle and the diameter of its incircle are 4 consecutive integers in anarithmetic progression. Find all such triangles.

32. We have a 102 * 102 sheet of graph paper and a connected figure of unknown shapeconsisting of 101 squares. What is the smallest number of copies of the figure, which can be cutout of the square?

33. Remainder when 3^37 is divided by 79. ? A.78 b.1 c.2 d.35

34. Number of ordered pair of integer x^2+6x+y^2 = 4 is. A.2 b.4 c.6 d.8.

35. Remainder when 3^2002 + 7^2002 + 2002 is divided by 29.

36. a1, ... , a8 are reals, not all zero. Let cn = a1n + a2n + ... + a8n for n = 1, 2, 3, ... . Given thatan infinite number of cn are zero, find all n for which cn is zero.

37. In how many ways can Rs. 18.75 be paid in exactly 85 coins , consists of 50 paise, 25 p and10p coins.

38. In a certain community consisting of p persons , a % can read and write : of the males alone b% , and of the females alone c % can write : find the no. of males and females in the community(in a, b, c and p terms) .

39. A no. of 3 digit in base 7 when expressed in base 9 has it’s digits reversed in order : find theno.

40 A man has to take a plot of land fronting a street : the plot is to be rectangular , and 3 times it’sfrontage added to 2 it’s depth is to be 26 metres. What is the greatest no of square metres he can

take.

41. A farmer sold 10 sheep at a certain price and 5 others at Rs. 50 less per head : the sum hereceived for each lot was expressed in rupee by the same 2 digits: find the prices per sheep.

42. If A tell B , that he has thought of a 3 digit no. , in such a way that the sum of all of its digithappens to be a prime no. If based on this information only B tries to find the no. , what is theprobabilities that she will find the prime no. in the first guess .

43. The area of intersection of 2 circular discs each of radius r and with the boundary of each discpassing through the center of the other is ..

44. Find 3 consecutive integer each divisible by a square greater than 1.

45.Find the 3 consecutive no , the first of which is divisible by a square , the second by a cubeand third by a fourth power .

46. A gang of 17 pirates steal a sack of gold coins. When the start dividing the loot among themthere was 3 coin left. They start fighting for the 3 left over coins and a pirates was killed. Theydecided to re-divided the coins , but again there was 10 coin left over. Again the fight and another pirates was killed. However they were third time luckyas this time loot was equally distributed. Now tell the least no. of coin satisfying above condition.


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47. Find all prime for which the quotient (2^p-1 - 1)/p is a square.

48. find all a, b ,c such that the roots of x^3+ax^2+bx-8=0

49. Find all pairs of integer (x,) such that (x-1)^2=(x+1)^2+(y+1)^2.

50. ABC is an isosceles triangle with AC = BC . The median AD and BE are perp. To each otherand intersect at G . If GD = a unit , find the area of the quadrilateral CDGE.

51 A regular octagon with sides 1 unit long is inscribed in a circle . Find the radius of the circle.

52 A regular octagon with sides 1 unit long is circumscribed in a circle . Find the radius of thecircle.

53. In a recent exam, your teacher asked 2 difficult questions : "Eugenia ,which is further north,Venice or Vladivoski?" and "What is the latitude and longitude of the north pole?".Of the class, 331\3% of the students were wrong on the latitude question , but only 25% missed the other one.20% answered both questions incorrectly. 37 students answered both questions correctly . Howmany students took the examination?

54. Two people agree to meet at a given place between noon and 1 p.m. By agreement, the firstone to arrive will wait 15 min for the second, after which he will leave. What is the probability thatthe meeting actually take place if each of them selects his moment of arrival at random during 12 noon to 1 p.m?

55. A positive no n, not exceeding 100 is chosen in such a way that if n is less than or equal to50, then probabilities of choosing n is 10 and if n > 50 then it is 3p. What is the probability ofchoosing a square?

56. What is the greatest area of a rectangle th esum of whose 3 sides is equal = 100. Ans. 1250 .

57. A plane figure consists of a rectangle and an equaliteral triangle constructed on one side ofthe rectangle . How large is the area of that figure in the term of perimeter. Ans. 6+ sqrt(3)/132 p^2.

58. A reservoir is shaped like a regular triangle . A ferry boat goes from point A to point B bothlocated on the bank . A cyclist who has to get form A to B can use the ferry boat or to ride alongthe bank . hat is the least ratio between the speed of cyclist and the speed of the boat at which the use of the boat does not give any gain in time, thelocation of the point A and B being arbitrary ? Ans. 2.

59. Find the greatest volume of the cylinder inscribred in the given cone. Ans. V(R/3) = 4/27 * pie H*R*R

60. In a party with 1982 persons, among any group of four there is at least one person whoknows each of the other three. What is the minimum no. of people in the party who knoweveryone else. Ans. ??(not have)

61. In a party with 1982 persons, among any group of four there is at least one person whoknows each of the other three. What is the minimum no. of people in the party who knoweveryone else.


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62 A chord PQ is drawn on a circle. If the length of PQ is equal to the radius of the circle, thenwhat is the probability that any line drawn at random from point P through the circle is smallerthan PQ A. 1/2 B. 1/3 C. 2/3 D. 3/5.

63 Between 2 station the first, second and the third class fares were fixed in the ratio 8:6:3, butafterwards the first class were reduced by 1/6 and second class by 1/12. In a year no. of firstsecond and third class passenger were in ratio 9:12:26 and the money at the booking counter was 1326. How much was paid by the first class passenger? A. 320 B. 390 C.420 D. None. Of these.

64 . A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a redone, a white one, and a blue one, so that each box contains at least one card. A member of theaudience selects two of the three boxes, chooses one card from each, and announces the sum of the numbers on the chosen cards. Given this sum, themagician identifies the box from which no card has been chosen. How many ways are there toput all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

65. On a circle are marked 999 points. How many ways are there to assign to each point one ofthe letters A, B, or C, so that on the arc between any two points marked with the same letter,

there are an even number of letters differing from these two

66. On a train are riding 175 passengers and 2 conductors. Each passenger buys a ticket onlyafter the third time she is asked to do so. The conductors take turns asking a passenger whodoes not already have a ticket to buy one, doing so until all passengers have bought tickets. Howmany tickets can the conductor who goes first be sure to sell?

67. Around a table are seated representatives of n countries (n greater than or equal to 2), suchthat if two representatives are from the same country, their neighbors on the right are from twodifferent countries. Determine, for each n, the maximum number of representatives.

A person join a job at 20 yrs.First 3 years sal =10,000 p.a. Afterwards every year inc of 2,000 peryear for 10 year. Then sal become const till retirement. at retirement avg sal is 25,000. ( thro’outcareer) what age he retires.?

Diamond\’s value is proportional to its weight2 .When the diamond broke wts of pieces in ratio1:2:3;4:5.Total loss in value is 85,000.What is the value of the diamond twice the wt of the originaldiamond.

Two trains are traveling at 18kmph and are 60 km apart. There is fly in the train. It flies at80kmph. It flies and hits the second train and then it starts to oscillatebetween the two trains. At one instance when the two trains collide it dies.Distance traveled by the fly when both trains collide is?

Two cars are 500 cm apart. each is moving forward for 100 cm at a velocity of 50 cm/s andreceding back for 50 cm at 25 cm/s at what time they will collide with each other?

Answers to the questions

1) he got 10K for 13 years , 2k for 1 year . 4 k for 1 year.....20 k for 1 year and then 30K for , say,n years

so, (10X13 + ( 2+4+6+......+20) + 30Xn)/(n+13) = 25


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=> n = 17 . hence, RETIREMENT AGE IS 20 + 13 + n = 50 YEARS

2 ) 450,000

let weight of the original diamond be 15k ( This number is the sum of the numbers in the ratiogiven ) ,so, it's value would be 225k^2. Now, the broken diamond's value adds upto 55k^2. So,loss of value is 170k^2 which is 85,000. This gives k^2 = 500.

We need (30k)^2 = 900k^2 = 900* 500 = 450,000

3) Dunno....I am getting some 400/3 kms as the answer..if that is the correct answer, i can explainhow i got that..otherwise , it 's too boring to see that humongous solution

4 ) 14 seconds

SIMPLE -the trains will meet at 60/(18+1 or 5/3 hrdistance trAvelled by fly = 80 x 5/3 or 400/3 km

each car moves forward (then backward) for 2 sec; in 4 sec, they cover 50 cm each;they will together cover 300 cm in 12 sec. then, a 100 cm thrust from both ewill ensure that theycollide! this will happen after another 2 secthus, 14 secs.

The ratio of Boys to Girls is 6:4. 60% of the boys and 40% of the girls take lunch in the canteen.What % of class takes lunch in canteen?

I think the answer for this should be 52%, Total Students 10x Students taking food in canteen =3.6x + 1.6 x = 5.2 x

Hence 52%

==

>>>>>>>>>>>>>>>>ANSWERS>>>>>>>>>>>>>>>>>>>>>>>>>>>

Q1- X=6let the committeees be A,B,C,Dconsider each com as a square where exactly 1/3 of each square overlap with each other andeah 1/3 has 1 member for eg let C overlap with 1/3 A , 1/3 B and 1/2 D= 3 members

II'ly D with 1/3 A and 1/3 B =2 mem

and A with 1/3 B = 1mem

hence X=6

Q2--- 2481 Sol Total num of num. between 3000 to 8000 are 5001. Of them number of num. withno digit in common == 5*9*8*7 = 2620Thus total no. of no with atlest 2 digit in common = 5001 – 2620=2481.

Q3--- -91(again want sol from U people)

Q4 – 80 sorry Q was in complete it say’s A closet has 5 pairs. Now solve. 5p4


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Q5 – This time haripi is right 64 is the ans.

Q 6 –Haripi U are quite close to ans. , must have done some minor mistake well ans. is –11.

2) Answer 1400:

3000-8000using P & C2 digits repeating: The first digit filled in 10 ways ..other digit can be chosen in 9 ways.. 3rd onecan be chosen in only one way..as it is repaeting...so 90 waysthe repeating digits can take 3 combinations ..3c2... therefore total number of ways is 90*3 =270

all 3 repeating...10 ways..

therfore between 3000 -4000 there are 280 numbers..3000-8000 there will be 1400 numbers..

4) answer 16 ways...divide shoe pairs into left and right. left pairs into 4 and right pairs into four

L1 L2 L3 L4 --------- R1 R2 R3 R4

YOU CAN CHOOSE 4 c 4 + 4c 4 +4c1 + 4c 2 +4c 3 = 16i.e all 4 left + all four right + 1 left 3 right + 2 left 2 right + 3 left 1 right

5) its an infinite gp using a/(1-r) where r = 3/4 we get answer as 64. it is infinite GP hence sum =a/1-r where r=.75(16*.75=12 and so on)= 16/.25=64

6)find the eqn: of teh line using first two coordinates.. then substitute values of x and 7 in thatequation... answer x = 22. use the formulae x-x1/(x2-x1) = y-y1/(y2-y1) to form the line eq andsub y= 7

Ans : X= 11

7— m = 14 , using the arrangement(1; 2; 3; 4) (5; 6; 7; (1; 2; 5; 6) (3; 4; 7;(3; 4; 5; 6) (1; 3; 5; 7) (2; 4; 6; (1; 3; 6;(2; 4; 5; 7) (1; 4; 5; (2; 3; 6; 7) (1; 4; 6; 7)(1; 2; 7; (2; 3; 5; :

8— If n+1 is composite, then each prime divisor of (n+1)! is a prime less than n, which alsodivides n! and so does not divide n! + 1. Hence f(n) = 1. If n + 1 is prime, the same argumentshows that f(n) is a power of n + 1, and in fact n + 1|n! + 1 by Wilson's theorem. However, (n + 1)^2 does not divide (n + 1)!, and thus f(n) = n + 1.

9 A 4000 cm^3B 16 Assumption 30 cm is the heigth of the tank

9--- If 32 discs are placed in an 8 *4 rectangle, they can all move up, left, down, right, up, etc.Game with 33 discs would stop. (Help me in the sol.).

10--- Q is wrong, sorry.

11---It is necessary and sufficient that gcd (x, y) = 2^s for some


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Non negative integer s. We show necessity by noting that gcd (p, q) = gcd (p, q - p), so an oddcommon divisor can never be introduced, and noting that initially gcd (1, 1) = 1. As for sufficiency, suppose gcd (x, y) = 2^s. Of those pairs (p, q) from which (x, y) can bereached, choose one to minimize p+q. Neither p nor q can be even, else one of (p/2, q) or (p, q/2)is an admissible pair.If p > q, then (p; q) is reachable from ((p + q)/2, q), a contradiction, similarly p < q is impossible.Hence p = q, but gcd (p; q) is a power of 2 and neither p nor q is even. We conclude p = q = 1,and so (x; y) is indeed reachable.

12--- 5 (again help me in the sol.).

Distance passed over by a pendulum bob in successive swing are 16, 12, 9, 6.75..cm. Then thetotal distanced traversed b the bob before it come to rest is (in cm) A. 60 B. 64 C.65 D.67.

ans = 64.S = 16 + 12 + 9 + ....= 16 (1 +(3/4) + (3/4)^2 + ..... )= 16 * 1 / (1 - (3/4) )= 16 * 4 = 64

luckynawabs wrote:

6 The point (2,1), (8,5) and (x,7) lie on a straight line , find the value of x..?.

slope will be same coz these points r in straight line...

(7-5)/(x- = (5-1)/(8-2)2/(x- = 4/3x = 11

Q-14: Ans . 36

Q-15total no of permutations = 5! =120

no of elements starting in a,b,or c = 72

no of elements starting in da,db or dc = 18

no fo elements in dea = 2hence tot no before debac = 92hence Ans =93

16. If the area of circumcircle of a regular polygon with n sides is A then the area of circleinscribed in polygon is..(in term of A) ?

Let the radius of the outer circle be 'r'.

So, pi * r*r = A

Let the radius of the inner circle be h. So, pi * h * h = A', where A' is the area of the inner circle.The relation between r and h is pretty simple.

r * cos (180/n) = h


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so, A' = A * (cos (180/n))^2

18. Let ABC be an isosceles triangle with AB = BC = 1 cm and angle A = 30 . Find the volume ofsolid obtained by revolving the triangle about the line AB. ?

check the above (or below) diagram .... I ahve added 2 extra points C' and D.

Now, when the triangle rotates, it will form a cone (ACC'). But the inner part of this cone (BCC')will be hollow.

the following deductions can be made by elementary geometry and trigonometry ->

A = 30, B = 120, C = 30BDC = 90, DBC = 60, BCD = 30 degrees.now, AB = BC = 1

so, DC = 1 * cos (30) = sqrt(3)/2 = 1.73 / 2BD = 1 * sin (30) = 1/2

Now height of cone ACC' = AD = AB + BD = 1 + 0.5 = 3/2

Radius of cone ACC' = DC = sqrt(3)/2 (sorry, i couldn't find the underoot sign on my keyboard )

Volume of cone ACC' = 1/3 * pi * DC^2 * AD= 1/3 * pi * (3/4) * (3/2)

Now let's consider the inner cone.

Height of cone BCC' = BD = 1/2Radius of cone BCC' = DC = sqrt(3)/2

Volume of BCC' = 1/3 * pi * DC^2 * BD

So, volume formed by solid on revolution =

Volume of ACC' - Volume of BCC'= 1/3 * pi * DC^2 * AD - 1/3 * pi * DC^2 * BD= 1/3 * pi * DC^2 * (AD - BD)= 1/3 * pi * 3/4 * (3/2 - 1/2)= 1/3 * pi * 3/4 * 1= pi/4

Correct me if I am wrong.

13. k^2/2 +1/2OR Ans 5 let x & y be 1,2 or am i missing some thing?

min value of x^2+y^2=(k^2)/2sol: (x+y)^2=k^2x^2 + y^2 + 2*x*y = k^2for (x^2 + y^2) to be min, 2*x*y should be max


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this can happen only when x=y => 2*x*y = (k^2)/2so (x^2 + y^2)=(k^2)/2

14. find it yaar, require some time15. 9316. A*(cos^2(pi/n))17. 51218. Use the formula19. Trigo problem, requires a pen. The radius of the circle is 2.5 if the length is 4.20. 120 degree21. Hmmm...22. 4.46 or 4.9823. 4

27. put n=1 to generate (x,y, n) at will...

25. let D. ans x ques / min then the equn is10x + 9(x-3)=201soivn we get x=12hence D.= 120 & S. = 81

28. 6again requires pen/paper, so may be wrong...It's 19.5 for the side is square root of (25 + 12*(3^(1/2)))

28. Got it, after 30 minutes. The area is 19.5 square units(approx.)How is this question going to be asked in CAT, and attempted?Nawab, tell us the 2 minute solution...

30. 1998C2 -1998though not sure

31. One triangle

or its generation 3a, 4a, 5a...

Q-33: 3^37=3*3^36=3*9^18=3*81^9=3*2^9=6*2^8=6*16^2=6*256=6*19

=114=>reminder is 35

Q-37: Ans. 20 of 50p , 15 of 25p and 50 of 10p coin's

Q-38: (a-c)p/b-c males (b-a)p/b-c females

40: Ans:since 3x+2y=26i am assuming that x must be divisible by 3(correct me if i am wrong) and even

hence it can take vals 6,12,18,24 thus corres ponding x is 2,4,6,8 here we find that x=4 & corresy=7yeilds the max area of 28

43. Answer = 2 * r * r * (pi / 3 - sqrt(3) / 4)

53: Ans:of stu ans atleast 1 ques wrong= 20+5+13 1/3=38 1/3% =>% of stu ans rite = 61 2/3% =37 ; tot no stu = 37*100 / 185/3 = 60


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Q-63:initial no of passengers in first class = 8x*5/6initial no of passengers in 2nd class = 6x*11/12initial no of passengers in 3rd class = 3x

cost of a ticket in 1st class = 9ycost of a ticket in 2nd class = 12ycost of a ticket in 3rd class = 26y

revenue 20x/3*9y = 60xyrevenue 11x/2*12y = 66xyrevenue 3x*26y = 78xy

tot revenue = 1326 = 204xy=> revenue from 1st class = 60/204*1326 = Rs 390

hence choice B is the ans

Pagalguy:

f 7 planes are flying between 2 points and the slowest reaches the destination in 15 minutes andthe fastest in 5 minutes, then which one of the following cannot be the average of the time takenby all 7 planes to fly the entire distance.1. 9 2. 11 3. 12 4. 14

Without calculations we can say the answer is 14 as that is closest to the defining range 5 and15.

If questions give none of these as an answer option, then we may have to calculate..

Shoot your shortcuts here...

Ambi

A has got 10 breads and B has got 6breads C has got none but he has some money with him.Now A, B and C divide the breads equally among themselves. C will divide the money he haswith him among A and B in the ratio of the amount of bread given to C by them.find the ratio

well my approach to the question would befirst of all calculate the amount of bread each of them will get after dividing equallythat would come out to be 13/3

that means A would give (10 - 16/3) to Cand B would give (6- 16/3) to Cso the ratio would be 14:2

Let's say we want to find 14.25% of 3267.What will 10% of 3267 be… 326.7 And 1%… obviously 32.67. Then what would 4% be… 128 for 32 and 2.68 for 0.67..., i.e. 130.6


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Thus, 14% will be 326.7 + 130.6, i.e. 457.3If I want an even more accurate answer, if 1% is 32.6, then 0.25% will be 1/4 of 32.6, i.e 8.1514.25% of 3267 will be 465.4

Knowing reciprocal percentage equivalent, I should have thought of an even better factorisationas 14.28% – 0.03% and since 14.28% is nothing but 1/7, the answer can directly be found bydividing 3267 by 7, i.e. 466.7

Let's see another example where I can save a lot of calculations.

If I have to find 4835/7280 is of what percentage and the alternatives to choose from, area. 59.6%b. 63.8%c. 66.4%d. 71.4%

Just focus on denominator. 10% of the denominator would be 728. Thus, the answer will bedefinitely less than 70% (since 7* 10% i.e. 70% of the denominator will be more than 4900 i.e.more than the numerator) and also answer will be more than 60% (since 6*10% will be around

4360). 2/3 of 7280 will be 4860. Since the numerator is less than 4860, the answer choice has tobe less than 2/3 or 66.66% and hence (c) is the obvious choice.

Take the chord as AB with centre C. Let O be the centre of the circle. COD is the perpendicularbisector of AB to the major arc. By the problem, AB = CD = 36.

Let the circle be of radius r. So, the lengths OD = OA = OB = r.OC = CD - OD = 36-r.In triangle COA (or COB for that matter) applying pythogoras theorem,(36-r)^2 + 18^2 = r^2

Solving for r, we get r = 22.5.

oops... I didn't look into that... sorry. I meant the bisector but missed the term somehow.

Hope that helps.

Regards,Bharathi.

Let ABC be an equilateral triangle and D be a point outside it. The distances AD, BD and CD are10, 10 and 18 units respectively. Find the side of the equi. triangle.

Is this the answer?

Consider a triangle ABC

let D be a point such that AD=BD=10 and CD=18.Let CD cut AB at O.Now focus on Triangles AOC and BOC.. It shouldnt be difficult to prove that they both arecongruent triangles....Thus angle AOC=angle BOC=90 degrees

Let CO=x and OD=18-x. Let AO=y As angle BAC=60 deg hence CO=AO tan60 ie x=y.sqrt(3)


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In traingle AOD, AD^2= OD^2+ AO^2

or 100 = (18-x)^2 + y^2Replace x by y.sqrt(3)

Thus y =9.96 cm and hence AC = AO cos 60 = 19.92 or 20 cm

Also y can b 5.6 cm then AC = 11.2 cm...

Regards

A group of friends went on a holiday to a hill station. It rained for 13 days. But when itrained in the morning, the afternoon was lovely. And when it rained in the afternoon, theday was preceded by clear morning. Altogether there were 11 very nice mornings and 12 very nice afternoons. How many days didtheir holiday last?

Let the number of total mornings be M and the number of total afternoons be EThe wet mornings must be M-11 and the horrid afternoons are E-12So, M-11+E-12=13 or M+E = 36but M+E is one day so the number of days must be 18!

Which of the following day(s) can't be the last day of a century? Sunday, Monday, Tuesday,Wednesday, Thursday, Friday, Saturday.

Ans:easiest one.

The last day of a century can not be Tuesday, Thursday or Saturday.( hope everybody aware of calendar probs. 5317 - fri,wed,mon,sun alone be the last days ofcenturies)

Here it goes.

A normal year has 365 days whereas a leap year has 366 days. Every year which is divisible by 4is called a leap year. Also, every 4th century is a leap year but no other century is a leap year.1 normal year = 365 days = 52 weeks + 1 day1 leap year = 366 days = 52 weeks + 2 day

Thus, a normal year has 1 odd day whereas a leap year has 2 odd days.

100 years = 76 normal years + 24 leap years = 76*[52 weeks + 1 day] + 24*[52 weeks + 2 day] =(76*52) weeks + 76 days + (24*52) weeks + 48 days = 5200 weeks + 124 days = 5217 weeks + 5days

i.e. 100 years contain 5 odd daysSimilarly, 200 years contain 10 odd daysi.e. 3 odd days. 300 years contain 15 odd daysi.e. 1 odd days.

400 years contain (20+1) odd days i.e. 0 odd days.


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Note that 400 years contain one more leap year.

Also, we have Sunday for 0 odd day, Monday for 1 odd day, Tuesday for 2 odd days, and so on...Thus, last day of first century is Friday. (5 odd days) Last day of second century is Wednesday. (3odd days) Last day of third century is Monday. (1 odd days)

Last day of forth century is Sunday. (0 odd days)

Since the order is repeating in successive cycles, the last day of a century can not be Tuesday,Thursday or Saturday.

A ship went on a voyage. After it had travelled 180 miles a plane statrted with 10 times the speed of the ship.Find the distance when they meet from starting point.

200 miles

Let's say the ship is travelling v mi/hr

and plane took t hours to meet the ship

Than, vt is the distance ship travelled after plane started(10v)t = 10vt is the distance plane travelled.

So we have, vt + 180 = 10vtor 9vt = 180or vt = 20

Hence distance ==> 180 + 20 = 200 miles

But.... one more soln. with me.

By the time the plane flies 180 miles the ship would have travelled 18 miles.

180 + 18 = 198 miles

By the time the plane flies 18 miles the ship would have travelled 1.8 miles.

198 + 1.8 = 199.8 miles

By the time the plane flies 1.8 miles the ship would have travelled .18 miles.

199.8 + .18 = 199.98 miles

By the time the plane flies .18 miles the ship would have travelled .018 miles.

199.98 + .018 = 199.998 miles

By the time the plane flies .018 miles the ship would have travelled .0018 miles.

199.998 + .0018 = 199.9988

And so on, assuming the plane never tries to overtake the ship


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speeds x and 10 xrel dis = 180rel speed = 10x - x = 9xtime = 180/9x = 20/x

Ans = Dis Travelled by plane = 10x * 20/x = 200 miles

Quote:There are some chicken in a poultry. They are fed with corn. One sack of corn will come for 9days. The farmer decides to sell some chicken and wanted to hold 12 chicken with him. He cutsthe feed by 10% and sack of corn comes for 30 days. So initially how many chicken are there?

.9x - 30 days

1 x - 30*.9 days = 27 days for 12 chicken= 9 days for t chicken= 27 * 12 days for 1 chicken= 27 * 12 / 9 days for t chickent = 27*12/9 = 36 chicken initially.Quote:There is an element which triplicates in every hour. Each of these 3items inturn reproduce exactly 3 other items. If a single compund is kept in acontainer at noon and the container is full by midnight. After how many hours is the container 1/3full.

9 pm.or 9 hours (1/3 becomes 3/3 in 3 hours - therefore 12-3 = 9) - not very sure - have to check.Quote:

100 Kg. Potatoes with 98% water content are dried in an oven. The percentage of waterdecreases to 50 %. What is the weight of potatoes now.

98 kg + 2kg of solid matter of water becomes 2 kg of water + 2 kg of solid matter (2 = 50% of 4)(assuming 50% of original weight - if it is 50% of the weight of water - then it will be 49kg water +2kg solid = 51kg - but then the wording should have been "The percentage of water decreases by50 %" )Quote:There are n Railway stations. Ticket facility is 10available between everytwo stations. Recently, m new stations are bulit so that 42 new ticketsare to be printed. How many stations were there? How many are newlyconstructed.

For the m new stations you will need to print 42 new tickets = 21 new tickets one way.Now, 21 = trains between 1 each of n stations and 1 of m stations = m*n + trains between each ofthe m new stations = mC221 = m*n+mC2 = m(2n + m-1)/23*7 = m(2n+m-1)/2Taking m = 37 = (2n+3-1)/214 = 2n + 2n = 6


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If we take m = 7, (2n+6) = 6 => n = 0Therefore, either there were no stations initially and 7 new stations were added or 6 originally, 3new were added.

36 chickens11 pm51 kg

1(part)*9*x= 0.9(part)*12*30therfore x=36 (chicks)

1/3 becomes 3/3 in 1 hr) - therefore 12-1 = 11 ==> 11pm.

n(n-1) is the no. of tickets available between n no. of stations.

therefor n1(n1-1) - n(n-1) =429*8 - 6*5 =42.n=6, n1 =9. 3 new stations.

1. The street of a city are arranged like the lines of a chessboard , there are m street runningnorth and south and n east and west . Find the no. ways in which a man can travel from N.W tothe S.E corner, going by the shortest possible distance.

2. How many different arrangement can be made out of the letters in the ex-pression a^3b^2c^4when written at full length? Ans. 1260.

3. How many 7-digit numbers exist which are divisible by 9 and whose last but one digit is 5?

4. You continue flipping a coin until the number of heads equals the number of tails. I then awardyou prize money equal to the number of flips you conducted .How much are you willing to pay me

to play this game?

5. Consider those points in 3-space whose three coordinates are all nonnegative integers, notgreater than n. Determine the number of straight lines that pass through n of these points.

6 Four figures are to be inserted into a six-page essay, in a given order. One page may contain atmost two figures. How many different ways are there to assign page numbers to the figures underthese restrictions?

7. How many (unordered) pairs can be formed from positive integers such that, in each pair, thetwo numbers are coprime and add up to 285?

8. ( 1+ x) ^ n – nx is divisible by

A. x B. x.x C. x.x.x. D. None of these

9. The root of the equation (3-x)^4 + (2-x)^4 = (5-2x)^4 area. ALL REAL B. all imaginary C. two real and 2 imaginary D. None of these

10. The greatest integer less than or equal to ( root(2)+1)^6 is A. 196 B.197 C.198 D.199.

11. How many hundred-digit natural numbers can be formed such that only even digits are usedand any two consecutive digits differ by 2?


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12. If x1 , x2, x3 are the root of x^3-1=0 , then A. x1+x2+x3 not equal to 0 B. x1.x2.x3 not equal to 1 C.(x1+x2+x3)^2 = 0 D. None of these

13. How many different ways are there to arrange the numbers 1,2,...,n in a single row such thatevery number, except the number which occurs first, is preceded by at least one of its originalneighbours?

14. There are N people in one room. How big does N have to be until the probability that at leasttwo people in the room have the same birthday is greater than 50 percent? (Same birthdaymeans same month and day, but not necessarily same year.)

15. A 25 meter long wound cable is cut into 2 and 3 meter long pieces. How many different wayscan this be done if also the order of pieces of different lengths is taken into account?

16. You are presented with a ladder. At each stage, you may choose to advance either one rungor two rungs. How many different paths are there to climb to any particular rung; i.e. how manyunique ways can you climb to rung "n"? After you've solved that, generalize. At each stage, youcan advance any number of rungs from 1 to K. How many ways are there to climb to rung "n"?

17. Find the no. of rational number m/n where m ,n are relatively prime positive no. satisfyingm 5) whose centers form an equilateral triangle. Prove that m = n = 6,and find the angles of ABC.


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25. The area of triangle formed by the point (p,2-2p) , (1-p,2p) and (-4-p,6-2p) is 70 units. Howmany integral values of p are possible.? A. 2 B. 3 C. 4 D. None of these

26. In a park, 10000 trees have been placed in a square lattice. Determine the maximum numberof trees that can be cut down so that from any stump, you cannot see any other stump. (Assumethe trees have negligible radius compared to the distance between adjacent trees.) A. 2500 B.4900 C.6400 D. none of these.

27. A certain city has a circular wall around it, and this wall has four gates pointing north, south,east and west. A house stands outside the city, three miles north of the north gate, and it can justbe seen from a point nine miles east of the south gate. The problem is to find the diameter of thewall that surrounds the city.

28. Income of A,B and C are in the ratio 7:10:12 and their expenses in the ratio 8:10:15 . If Asaves 20% of his income then B’s saving is what % more/less than that of C’s A.100% more B. 120% more C. 80% less D. 40% less

29. A triangle ABC has positive integer sides, angle A = 2 angle B and angle C > 90. Find theminimum length of the perimeter of ABC.

30. We have k identical mugs. In an n-storey building, we have to determine the highest floorfrom which, when a mug is dropped, it still does not break. The experiment we are allowed to dois to drop a mug from a floor of our choice. How many experiments are necessary to solve theproblem in any case, for sure? [/glow][/color][/font]

31. The side lengths of a triangle and the diameter of its incircle are 4 consecutive integers in anarithmetic progression. Find all such triangles.

32. We have a 102 * 102 sheet of graph paper and a connected figure of unknown shapeconsisting of 101 squares. What is the smallest number of copies of the figure, which can be cutout of the square?

33. Between points A and B there are two railroad tracks. One of them is straight and is 4 mileslong. The other one is the arc of a circle and is 5 miles long. What is the radius of curvature of thecurved track?

34. Number of ordered pair of integer x^2+6x+y^2 = 4 is. A.2 b.4 c.6 d.8.

35. 100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least onecard in each box). How many ways can this be done so that if two boxes are selected and a cardis taken from each, then the knowledge of their sum alone is always sufficient to identify the thirdbox?

36. A cube is divided into 27 pairwise congruent smaller cubes. Find the maximum number ofsmall cubes that can be stabbed by a straight line.

37. Suppose that you have a table with 8 rows and 8 columns and that the numbers from 1 to 64are placed in the table in such a way that all the rows and columns add up to the same number.What is this number?

38 A certain shade of gray paint is obtained by mixing 3 parts of white paint with 5 parts of blackpaint. If 2 gallons of the mixture is needed and the individual colors can be purchased only in one-gallon or half- gallon cans, what is the least amount of paint, in gallons, that must be purchased inorder to measure out the portions needed for the mixture?(A) 2 (B) 2 .5 (C) 3 (D) 3.5 (E) 4


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39. What is the least number of digits (including repetitions) needed to express 10^100 in decimalnotation?(A) 4 (B) 100 (C) 101 (D) 1,000 (E) 1,001

40 You are at a track day at your local racecourse in your new Porsche. Because it's a crowdedday at the track, you are only allowed to do two laps. You haven't driven your car at the track yet,so you took the first lap easy, at 30 miles per hour. But you do want to see what your ridiculoussports car can do. How fast do you have to go on the second lap to end the day with an averagespeed of 60 miles per hour?

41. A farmer sold 10 sheep at a certain price and 5 others at Rs. 50 less per head : the sum hereceived for each lot was expressed in rupee by the same 2 digits: find the prices per sheep.

42. If A tell B , that he has thought of a 3 digit no. , in such a way that the sum of all of its digithappens to be a prime no. If based on this information only B tries to find the no. , what is theprobabilities that she will find the prime no. in the first guess .

43. The area of intersection of 2 circular discs each of radius r and with the boundary of each discpassing through the center of the other is.

44. An acrobat thief enters an ancient temple, and finds the following scenario:1. The roof of the temple is 100 meters high.2. In the roof there are two holes, separated by 1 meter.3. Through each hole passes a single gold rope, each going all the way to the floor.4. There is nothing else in the room.The thief would like to cut and steal as much of the ropes as he can. However, he knows that ifhe falls from height that is greater than 10 meters, he will die. The only thing in his possession isa knife.How much length of rope can the acrobat thief get?

45. A 25 meter long wound cable is cut into 2 and 3 meter long pieces. How many different wayscan this be done if also the order of pieces of different lengths is taken into account?.

46. A gang of 17 pirates steal a sack of gold coins. When the start dividing the loot among themthere was 3 coin left. They start fighting for the 3 left over coins and a pirates was killed. Theydecided to re-divided the coins, but again there was 10 coin left over. Again they fight andanother pirates was killed. However they were third time lucky as this time loot was equallydistributed. Now tell the least no. of coin satisfying above condition.

47. The coefficient of x^43 in the product (x-2)(x-5)(x- …….(x-131) is A. 3087 B.4462 C.5084 D.2926

48. Find all a, b ,c such that the roots of x^3+ax^2+bx-8=0

49. A can hit a target 4 times in 5 shots; B 3 times in 4 shots ; and C 2 times in 3 shots. They all

fire shot , find the probabilitiy that 2 shots at least hit.? Ans. 6/13

50. ABC is an isosceles triangle with AC = BC . The median AD and BE are perp. To each otherand intersect at G . If GD = a unit , find the area of the quadrilateral CDGE.

51. A regular octagon with sides 1 unit long is inscribed in a circle . Find the radius of the circle.


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52 The pages of a report are numbered consecutively from 1 to 10. If the sum of the pagenumbers up to and including page number x of the report is equal to one more than the sum ofthe page numbers following page number x, then x =(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

53. Two urns contain the same total numbers of balls, some blacks and some whites in each.From each urn are drawn n balls with replacement, where n >= 3. Find the number of drawingsand the composition of the two urns so that the probability that all white balls are drawn from thefirst urn is equal to the probability that the drawing from the second is either all whites or allblacks.

54. Two people agree to meet at a given place between noon and 1 p.m. By agreement, the firstone to arrive will wait 15 min for the second, after which he will leave. What is the probability thatthe meeting actually take place if each of them selects his moment of arrival at random during 12noon to 1 p.m?

55. A positive no n, not exceeding 100 is chosen in such a way that if n is less than or equal to50, then probabilities of choosing n is 10 and if n > 50 then it is 3p. What is the probability ofchoosing a square?

56. What is the greatest area of a rectangle the sum of whose 3 sides is equal = 100. Ans. 1250 .

57. A plane figure consists of a rectangle and an equilateral triangle constructed on one side ofthe rectangle . How large is the area of that figure in the term of perimeter. Ans. 6+ sqrt(3)/132 p^2.

58. A reservoir is shaped like a regular triangle . A ferry boat goes from point A to point B bothlocated on the bank . A cyclist who has to get form A to B can use the ferry boat or to ride alongthe bank . hat is the least ratio between the speed of cyclist and the speed of the boat at whichthe use of the boat does not give any gain in time, the location of the point A and B being arbitrary? Ans. 2.

59. Find the greatest volume of the cylinder inscribed in the given cone. Ans. V(R/3) = 4/27 * pie H*R*R



62 A chord PQ is drawn on a circle. If the length of PQ is equal to the radius of the circle, thenwhat is the probability that any line drawn at random from point P through the circle is smallerthan PQ A. 1/2 B. 1/3 C. 2/3 D. 3/5.

63 Between 2 station the first, second and the third class fares were fixed in the ratio 8:6:3, butafterwards the first class were reduced by 1/6 and second class by 1/12. In a year no. of firstsecond and third class passenger were in ratio 9:12:26 and the money at the booking counterwas 1326. How much was paid by the first class passenger? A. 320 B. 390 C.420 D. None. Ofthese.


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64 . A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a redone, a white one, and a blue one, so that each box contains at least one card. A member of theaudience selects two of the three boxes, chooses one card from each, and announces the sum ofthe numbers on the chosen cards. Given this sum, the magician identifies the box from which nocard has been chosen. How many ways are there to put all the cards into the boxes so that thistrick always works? (Two ways are considered different if at least one card is put into a differentbox.)

65. On a circle are marked 999 points. How many ways are there to assign to each point one ofthe letters A, B, or C, so that on the arc between any two points marked with the same letter,there are an even number of letters differing from these two?

66. On a train are riding 175 passengers and 2 conductors. Each passenger buys a ticket onlyafter the third time she is asked to do so. The conductors take turns asking a passenger whodoes not already have a ticket to buy one, doing so until all passengers have bought tickets. Howmany tickets can the conductor who goes first be sure to sell?

67. Around a table are seated representatives of n countries (n greater than or equal to 2), suchthat if two representatives are from the same country, their neighbors on the right are from twodifferent countries. Determine, for each n, the maximum number of representatives.

68. 98 points are given on a circle. Maria and Joe take turns drawing a segment between two ofthe points which have not yet been joined by a segment. The game ends when each point hasbeen used as the endpoint of a segment at least once. The winner is the player who draws thelast segment. If Joe goes first, who has a winning strategy?

69. A computer screen shows a 98 * 98 chessboard, colored in the usual way. One can selectwith a mouse any rectangle with sides on the lines of the chessboard and click the mouse button:therefore, the colors in the selected rectangle switch (black becomes white, white becomesblack). Find, the minimum number of mouse clicks needed to make the chessboard all one color.

70. A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in length. What is themaximum area he can enclose by connecting the pieces?

71. Somewhere in Northern Asia, groups of 20 man were planning a special group suicide thisyear. Each of the them will be placed in a random position along a thin, 100 meter long plank ofwood which is floating in the sea. Each man is equally likely to be facing either end of the plank. At time t=0, all of them walk forward at a slow speed of 1 meter per minute. If a man bumps intoanother man, the two both reverse directions. If a man falls off the plank, he drowns. What is thelongest time that must elapse till all the man have drowned?

72. I have chosen a number from 1 to 144, inclusive. You may pick a subset (1, 2, .. 144), i.e. Ucan frame any subset based on AP, GP, or any other thing and then ask me whether my numberis in the subset. An answer of “yes" will cost you 2 dollars, an answer of “no" only 1-dollar. Whatis the smallest amount of money you will need to be sure to find my number?

73. The side lengths of a triangle and the diameter of its incircle are four consecutive integers inan arithmetic progression. Find all such triangles.

74. A 10-digit number is said to be interesting if its digits are all distinct and it is a multiple of11111. How many interesting integers are there?

75. Two matching decks have 36 cards each; one is shuffled and put on top of the second. Foreach card of the top deck, we count the number of cards between it and the corresponding cardof the second deck. What is the sum of these numbers?


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76. A cube of side length n is divided into unit cubes by partitions (each partition separates a pairof adjacent unit cubes). What is the smallest number of partitions that can be removed so thatfrom each cube, one can reach the surface of the cube without passing through a partition?

77. A room has its dimension as 4 ,5 and 6 mt. An ant wants to go from one corner of the room tothe diagonally opposite corner. The minimum distance it has to travel is A.10.1 B. 10.8 C. 10.9 D. 11.2

78. A rumor began to spread one day in a town with a population of 100,000. Within a week,10,000 people had heard this rumor. Assuming that the rate of increase of the number of peoplewho have heard the rumor is proportional to the number who have NOT yet heard it, how long willit take until half of the population of the town has heard the rumor? A. 44 B.46 C.45 D. none ofthese

79. A camel must travel 15 miles across a desert to the nearest city. It has 45 bananas but canonly carry 15at a time. For every mile camel walks, it needs to eat a banana. What is themaximum number of bananas that can transport to the city? A. 10 B.8 C. 15 D. none of these

80. So far this basketball season, all of Siddhart points have come from two- point and three pointfield goals. He had scored 43 points. He has made one more than twice as many three pointers

as two pointers. How many of each kind of field goal has Siddhart made?

81. Let g (t)=9t^4+6t^2+2. For which value of t is this function a minimum?

82. Consider any two-digit number whose digits are not zero and are not the same. What is thegreatest integer that divides evenly the difference between the square of the number and thesquare of the reverse? ANS Largest difference = 91^2 - 19^2 = 7920 Hence, the greatest integer is 7920.

83. What is the maximum number of Friday the thirteenths that can occur in any given year?

84. What letter is in the 150th entry of the pattern ABBCCCDDDD......?

85. Tanisha and Richa had lunch at the mall. Tanisha ordered three slices of pizza and two colas.Richa ordered two slices of pizza and three colas. Tanisha's bill was Rs60.0, and Rachel's billwas Rs52.5. What was the price of one slice of pizza? What was the price of one colon?

86. An architect plans a room to be 14m x 30m. He wishes to increase both dimensions by thesame amount to obtain a room with 141 sq. metres more area. By how much must he increaseeach dimension? Ans 3/2, -47/2

87. A geometry teacher drew some quadrilaterals on the chalkboard. There were 5 trapezoids, 12rectangles, 5 squares, and 8 rhombuses. What is the least number of figures the teacher couldhave drawn? Ans 20

88. Given an n x n square board, with n even. Two distinct squares of the board are said to beadjacent if they share a common side, but a square is not adjacent to itself. Find the minimumnumber of squares that can be marked so that every square (marked or not) is adjacent to atleast one marked square.

89. A painted wooden cube, such as a child's block, is cut into twenty-seven equal pieces. Firstthe saw takes two parallel and vertical cuts through the cube, dividing it into equal thirds; then ittakes two additional vertical cuts at 90 degrees to the first ones, dividing the cube into equalninths. Finally, it takes two parallels and horizontal cuts through the cube, dividing it into twenty-


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seven cubes. How many of these small cubes are painted on three sides? On two sides? On oneside? How many cubes are unpainted?

90. It was claimed that the shepherd was the shepherd of 2000 sheep. The shepherd exclaimed,"I am not the shepherd of two thousand sheep!" Pointing to his flock, he added, "If I had thatmany sheep plus another flock as large as that, then again half as many as I have out there, Iwould be the shepherd of two thousand sheep." How many sheep were in the shepherd's flock?

91. Leon, who is always in a hurry, walked up an escalator, while it was moving, at the rate of onestep per second and reached the top in 20 steps. The next day he climbed two steps per second(skipping none), also while it was moving, and reached the top in thirty-two steps. If the escalatorhad been stopped, how many steps did the escalator have from the bottom to the top?

92. A, B and C were asked to sell fun-fair tickets priced at $10 each. A sold 1/4 of them. B and Csold the remaining tickets in the ratio 5:7. A received $90 less than C.What fraction of tickets were sold by C? b) How many tickets did the 3 girls sell altogether?

93.If 500ml of popcorn and 1 box of potato chips cost $1.64, and 250ml of popcorn and 2 boxesof potato chips cost $1.99, and 250ml of popcorn is half the price of 500ml of popcorn, then how

much is one box of popcorn?

94.There are three circles, they share a similar area. The similar area is 1/10 of the biggest circle,1/6 of the second biggest circle, and half of the smallest circle. What is the ratio of the circles?(biggest, second biggest, smallest)

95. Sally and Jeff took 37 hours to complete a whole project. If Sally had worked 5 hours less andJeff had worked 6 hours more, Jeff would still have put in 2 more hours than Sally. How manyhours did Sally put in for the project? (Assume that Sally and Jeff worked separately on their ownproject.)

96. Mrs Lee purchased 240 rings and some watches. The number of rings was more than thenumber of watches by 20% of the total number of both items bought. She sold all the watches at

$36 each. From the total sale of the rings and the watches, she earned $2240 from her total sale.If each ring was sold at 25% less than the selling price of each watch,a) find the no. of watches purchased by Mrs Lee, andb) find the total cost price of all the rings and watches

97. We play the following game with an equilateral triangle of n(n+1)=2 pennies (with n pennieson each side). Initially, all of the pennies are turned heads up. On each turn, we may turn overthree pennies which are mutually adjacent; the goal is to make all of the pennies show tails. Forwhich values of n can this be achieved? I. 17 II. 7 III. 8 IV. 6 A. all of the above B. none of the aboveC. for more than 2 of the above including IV D. Exactly 2 of them.

98. In a certain town, the block are rectangular with stress running East-West, the avenues North-

South. A man wishes to go from one corner to another m block East, n block North. In how manyways the shortest path can be achieved.

99. Two players take turns drawing a card at random from a deck of four cards labeled 1,2,3,4.The game stops as soon as the sum of the numbers that have appeared since the start of thegame is divisible by 3, and the player who drew the last card is the winner. What is the probabilitythat the player who goes first wins?

100. We are given 1997 distinct positive integers, any 10 of which have the same least commonmultiple. Find the maximum possible number of pairwise coprime numbers among them.


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Ans 9.

101. A rectangular strip of paper 3 centimeters wide and of infinite length is folded exactly once.What is the least possible area of the region where the paper covers itself? A. 4.5 B. 4 C. 3.45 D. 5.4

102. A company has 50000 employees. For each employee, the sum of the numbers of hisimmediate superiors and of his immediate inferiors is 7. On Monday, each worker issues an orderand gives copies of it to each of his immediate inferiors (if he has any). Each day thereafter, eachworker takes all of the orders he received on the previous day and either gives copies of them toall of his immediate inferiors if he has any, or otherwise carries them out himself. It turns out thaton Friday, no orders are given. The least employees who have no immediate superiors are. A. 97B. 518 C.216 D. none of these.

103.The numbers from 1 to 37 are written in a line so that each number divides the sum of theprevious numbers. If the first number is 37 and the second number is 1, what is the third number?

104. The centre of the circumcircle of triangle ABC with angle C = 60 is O and its radius is 2 .Find the radius of the circle that touches AO , BO and the minor arc AB.

105 . ABC is an equilateral triangle in which B is extended to K such that CK = ½ BC. Find AK^2 *1/ CK^2 A. 3 B. 7 C. 5 D. 6

106. Let S be the set of prime numbers bigger than 100 but smaller than 150. Let n be theproduct of all the numbers in the set S . Which of the following statement is FALSE? A. 5 does not divide n B. 17 divide n. C. n is an odd number D. n is a product of distinct primes.

107. A cube of side ‘a’ is converted into a sphere by adding clay on all the faces of the cube. Theminimum volume of clay required is Ans. a^3( pie * sqrt(3)/2 - 1) .

108. The product of all integer from -129……….+130, ends inclusive , then which of the following

values is closet to the result? A. 65^65 B. 9^2 C. -65^65 20^12.

109. Four people, A, B, C, and D, are on one side of a bridge, and they all want to cross thebridge. However, it's late at night, so you can't cross without a flashlight. They only have oneflashlight. Also, the bridge is only strong enough to support the weight of two people at once. Thefour people all walk at different speeds: A takes 1 minute to cross the bridge, B takes 2 minutes,C takes 5 minutes, and D takes 10 minutes. When two people cross together, sharing theflashlight, they walk at the slower person's rate. How quickly can the four cross the bridge?

110. A cube is divided into 27 pairwise congruent smaller cubes. Find the maximum number ofsmall cubes that can be stabbed by a straight line.

ANS:

.Four figures are to be inserted into a six-page essay, in agiven order. One page may contain at most two figures. How manydifferent ways are there to assign page numbers to the figures underthese restrictions?

Ans:consider the case that one page may contain 1 fig the no of ways is 6c4consider the case wher 1 page only can contain 2 fig the 6c3


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consider 2 pages can contain 2 figures then 6c2 Ans:6c4+6c3+6c2

8. ( 1+ x) ^ n – nx is divisible by A. x B. x.x C. x.x.x. D. None of these

Ans:guess it is none of these as (1+x)^n is never div by x or am i wrong

12. If x1 , x2, x3 are the root of x^3-1=0 , then A. x1+x2+x3 not equal to 0 B. x1.x2.x3 not equal to 1 C.(x1+x2+x3)^2 = 0 D. None of these

Ans: A as x1=x2=x3=1

15. A 25 meter long wound cable is cut into 2 and 3 meter long pieces.How many different ways can this be done if also the order of piecesof different lengths is taken into account?

Ans: no of 3 meter pieces 2meter pieces no of arrangments1 11 12c1

3 8 11c35 5 10c57 2 9c228. Income of A,B and C are in the ratio 7:10:12 and their expenses in the ratio 8:10:15 . If Asaves 20% of his income then B’s saving is what % more/less than that of C’s A.100% more B. 120% more C. 80% less D. 40% less

Ans: A

7x-8y=1/5(7x)=> 8y= 4/5(7x) => 10y=7x=> b saves 3x & c saves 1.5x hence the ans

30. We have k identical mugs. In an n-storey building, we have to

determine the highest floor from which, when a mug is dropped, it stilldoes not break. The experiment we are allowed to do is to drop a mugfrom a floor of our choice. How many experiments are necessary tosolve the problem in any case, for sure?

Ans: log(n)worst case senario of binary search or is thre any other faster search

happy solving

37. Suppose that you have a table with 8 rows and 8 columns and thatthe numbers from 1 to 64 are placed in the table in such a way that allthe rows and columns add up to the same number. What is this number?

Ans:260

as sum of 1 to 64= 2080 there r 8 rows & the sum of each row is the sameand the sum of all the rows is 2080 hence the sum of each row is 2080/8=260

38 A certain shade of gray paint is obtained by mixing 3 parts of whitepaint with 5 parts of black paint. If 2 gallons of the mixture is neededand the individual colors can be purchased only in one-gallon or half-gallon cans, what is the least amount of paint, in gallons, that must


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be purchased in order to measure out the portions needed for the mixture?(A) 2 (B) 2 .5 (C) 3 (D) 3.5 (E) 4

Ans: E 4 gallons

for 2 gallons of grey shade we need 3/4 of white and 5/4 of black since the minunit thet can be purchased is 1/2 we need to by 3/2 of W and 5/2 of B hence4 units of paint totally

39. What is the least number of digits (including repetitions) neededto express 10^100 in decimal notation?(A) 4 (B) 100 (C) 101 (D) 1,000 (E) 1,001

Ans: C 101as 10^n has n+1 digits

40 You are at a track day at your local racecourse in your new Porsche.Because it's a crowded day at the track, you are only allowed to do twolaps. You haven't driven your car at the track yet, so you took thefirst lap easy, at 30 miles per hour. But you do want to see what your

ridiculous sports car can do. How fast do you have to go on the secondlap to end the day with an average speed of 60 miles per hour?

Ans: it is impossible to do so.

42. If A tell B , that he has thought of a 3 digit no. , in such a waythat the sum of all of its digit happens to be a prime no. If based onthis information only B tries to find the no. , what is the probabilitiesthat she will find the prime no. in the first guess .

Ans:1/58

sum of the nos is 2 using 1 1 0= 2 ways, 2 0 0=1way =3 ways

3 using 3 0 0= 1, 1 2 0 =4, 1 1 1 =1 =6ways5 using 1 0 4= 4, 2 0 3 =4, 1 1 3=3 ,1 2 3=6, 2 2 1=3,5 0 0=1 21ways7 using 1 0 6=4,2 0 5=4,3 0 4=4,7 0 0=1,1 1 5=3,1 2 4=6,1 3 3=3,2 2 3 =3 =28 wayshence totally 58 ways hence the ansdid i miss any combinationthis method is time consuming r there any shortcut

.A B C have certain mangoes with them. B has 10% less mangoes than A & C 20% less than

A.By what % is the no. of mangoes with B more than those with C

2.the price of sugar decreased by 20% & then increased by 50%.by what percentage should ahousewife now increse or reduce her consumption so that the expenditure on sugar remains thesame.

3.the indian cricket team played 25 ODI in a particular season in a year & won 40% of theirmatches.A new coach was brought in who wanted a successs rate of 75% of the matches played


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by the end of the year.how many minimum matches more should be played so as to achieve athe target

4.the price of an item went up by 25% & the family decided to reduce its consumption so that thetotal expenditure would increse by only 8% as compared to the previous expenditure if 25 kg ofthe item was consumed previously then how many kg of the item would be consumed now.

5. a salesman used to get 8%on the total sales as commission under a new policy he now gets afixed amount Rs.2000 & 5% commission on any amount of salesabove Rs.10000.the salesmanthus got Rs 540 more this month.what were his total sales for this month

6.the price of an item increased by by the same percentage over the last 4 weeks.If at thebeginning it was 512 /gm & in the end it became 1250/gm find the percentage increaseeveryweek.

7.the rate of inflation of a country is 600% what would be the cost of an item 4 years from now ifits present cost is Rs. 125 & the rate of inflation remained unchanged throughout.

ANS:

1)Let A has 100 mangoesthen B will have 90 and C will have 80 mangoestherefore % of mangoes with B more than those with C = (90-80)*100/80=12.5%

2) Net change in prize of sugar = -20+50+(-20)(50)/100=20%in order to keep the expenditure same she has to reduce her consumption by100-(100/1.2)= 17.66%

3) 40% of 25 = 1075% of 60 = 45.therefore 35 matches has to b played .

4) previous expenditure = 25 kglet the prize per kg = 4therefore total cost = 100prize went up by 25%.......prize per kg = 4.25if the total expenditure increases by 8% only instead of 25%therefore total cost =108consumption = 108/4.25 = 25.4 Kg

Quote:2.the price of sugar decreased by 20% & then increased by 50%.by what percentage should ahousewife now increse or reduce her consumption so that the expenditure on sugar remains thesame.

my ans isLet price of sugar be 100 then current pice = first 80 & then 120by % table in pagal neha %tables cons must decrease by 16.66%Quote:3.the indian cricket team played 25 ODI in a particular season in a year & won 40% of theirmatches.A new coach was brought in who wanted a successs rate of 75% of the matches playedby the end of the year.how many minimum matches more should be played so as to achieve athe target


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easy way

its a simple alligation problem 4 those who dont knw this is how u approach it

40% u have won till now & have to will 100% matches frm now on till u have won 75% overall

40 1007525 35( u cross subtract there fore the ratio is 25:35=> u have to win next 35 matches.)Quote:4.the price of an item went up by 25% & the family decided to reduce its consumption so that thetotal expenditure would increse by only 8% as compared to the previous expenditure if 25 kg ofthe item was consumed previously then how many kg of the item would be consumed now

let it go up by 25% then cons has to dec by 20% to maintain same pricecons = 20 kg but rise of 8% can be absorbed by family so 8% of 20 =1.6=> consumption = 21.6kg

Quote:5. a salesman used to get 8%on the total sales as commission under a new policy he now gets afixed amount Rs.2000 & 5% commission on any amount of salesabove Rs.10000.the salesmanthus got Rs 540 more this month.what were his total sales for this month

.08x+540=.05(x-10000)+2000

.03x+540=2500

.03x=1960=>arnd 65KQuote:6.the price of an item increased by by the same percentage over the last 4 weeks.If at the

beginning it was 512 /gm & in the end it became 1250/gm find the percentage increaseeveryweek.

x^4 * 512= 1250find xps: is there ne easy way to solve these kinda probs i personally wund not solve it until i dunno neother probs & that happens all the timeQuote:7.the rate of inflation of a country is 600% what would be the cost of an item 4 years from now ifits present cost is Rs. 125 & the rate of inflation remained unchanged throughout.

ans is 6^4*125=1296*125

1.11.11%

2.16.66%

3.35

4.21.6kg

5.20800


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6.25%

7.125*7*7*7*7=====

Let ABC be an equilateral triangle and D be a point outside it. The distances AD, BD and CD are10, 10 and 18 units respectively. Find the side of the equi. triangle.

Consider a triangle ABC

let D be a point such that AD=BD=10 and CD=18.Let CD cut AB at O.Now focus on Triangles AOC and BOC.. It shouldnt be difficult to prove that they both arecongruent triangles....Thus angle AOC=angle BOC=90 degrees

Let CO=x and OD=18-x. Let AO=y As angle BAC=60 deg hence CO=AO tan60 ie x=y.sqrt(3)

In traingle AOD, AD^2= OD^2+ AO^2

or 100 = (18-x)^2 + y^2Replace x by y.sqrt(3)

Thus y =9.96 cm and hence AC = AO cos 60 = 19.92 or 20 cm

Also y can b 5.6 cm then AC = 11.2 cm...

Regards

Arcade

Precisely... that was it!

Now, CO is the altitude of triangle ABC and OD is the median ( and altitude ) of the triangle ABD.

If a is the side of the equilateral triangle, thenCO = alt = [sqrt(3)/2] * aOD = 18 - altIn triangle, ABD,

AD^2 + BD^2 = 2 (OD^2 + OB^2)

OB=a/2 (since equilateral)

This gives the solution a=9*sqrt(3) (+/-) sqrt(19) which comes to 19.9 / 11.2

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