From one of the walls of a square room of side length 1 unit, Maradona kicks a frictionless, perfectly elastic ball which bounces of the three other walls once each and returns to him. What is the distance traveled by the ball?
The parallel sides of a trapezium are 10 units and 15 units in length. Find the length of the segment extending from one leg to the other, passing through the intersection of the diagonals and parallel to the bases.
How many distinct right triangles with integer side lengths are there one of whose sides is of length 60?
Which is the smallest number which can form the side of 10 distinct right triangles with integer side lengths?
A natural number is written on each face of a cube. On each vertex a number, which is the product of all the three numbers on the faces meeting at that vertex, is assigned. The sum of these numbers assigned to the vertices is 2008. What are the possible sums of the numbers written on the faces?
The hour and minute hands of a clock are exactly similar. How many times during the day would it be impossible to tell the correct time?
I am thinking of a polynomial P(x) with whole number coefficients. Although I will not tell you the degree of the polynomial, I will tell you the value of P(x) for every x that you tell me. What is the least number of queries should you ask to determine P(x)?
The quadratic polynomial f(x) = ax2 + bx + c has integer coefficients such that f(1), f(2), f(3), and f(4) are all perfect squares of integers but f(5) is not. What is the value of a, b and c?
You are in a dark room and face a large pile of coins on a table. You are told that 2008 of them are showing heads. Can you divide the coins into two groups so that each group is showing the same number of heads?
Given that f is a continuous function such that f(1000) = 999 and f(x) × f(f(x)) = 1 for all real x. What is the value of f(891)?
An isosceles right triangle ABC has three squares inscribed in it, and three circles inscribed between the squares and the triangle, as shown in the figure. If the largest circle in the following figure has radius 7569 and the smallest has radius 100, what is the radius of the middle circle?
What are the last two digits of (1 + 52n+1)/6?
Kumar claims that cigarette smoking hasn't dimmed his stamina and he can still run twice as fast as TG. TG is anyhow lazy and anyone can perform the feat but still TG accepts the challenge. Kumar challenges TG to a race on Barakhamba road (between two red lights) and back. They both start from the same red light but Kumar starts when TG is already halfway across. Kumar catches up with TG at a point 50 m from the other red light. Kumar continues to the end and turns back and meets TG only 20 m from the same red light. He continues running to the starting point and starts running back again to the other end while TG reaches the other end and starts his return trip. Assuming they were running at constant speeds, how far from the first red light do they meet for the third time?
Rajat, Kumar and TG are playing a game to determine how will they bear the kharcha of the next 'daroo party' they are planning (Manoj doesn't drink so he's safe). Three distinct natural numbers a, b, and c are written on three cards, one number on each card. In each round the three cards are dealt among the three and each of them deposits as many 100 rupee notes on the table as the number on his card. They leave the money on the table but redeal the cards. After a few rounds Rajat, Kumar, and TG have deposited respectively 3100, 2600, and 2800 rupees. How many rounds were played?
Rajat finally decided on january 1st 2009 that he will come to the CP center on time on every month whose name has an 'r' in it and every day whose name also has an 'r' it. TG decided to come early on all the other days. As TG doesn't like to wake up early, on how many days will he wake up with a sullen face?
The two altitudes of triangle ABC are shown with AD = 8 units. What is the length of ED?
The equation x3 + kx2 -1329x = 2007 has integral roots which may or may not be distinct. What is the value of k?
Find the sum of all values of x that satisfies the equation
If a < b < c, then the minimum value of
is
Ten thousand sailors are arranged around the edge of their ship. Starting with the first one, every other sailor is pushed overboard until they are all gone.
1) Where would you be standing to be the last survivor?
2) How many times is the last survivor skipped over before he is finally pushed overboard?
3) Find the location of the
a) second to last survivor
b) third to last survivor
c) 100th to last survivor.
2) How many times is the last survivor skipped over before he is finally pushed overboard?
3) Find the location of the
a) second to last survivor
b) third to last survivor
c) 100th to last survivor.
If ABC × CBA = 65125, then A + B + C = ?
On a straight line, five points are located. You choose any two points and note down the distance between them. The ten distances so obtained are written down in an increasing order as follows: 2, 4, 5, 7, 8, n, 13, 15, 17 19. What is the value of n?
All three-digit permutations (ABC, CAB, etc.) of the digits of a three-digit number ABC are written and added. The sum is divisible by 23. Then A + B + C = ?
In the figure, CD is perpendicular to chord AB. Find the radius of the circle.
In the figure, the square and the circle are intersecting each other such that AB = BC. If the radius of the circle, with the centre as O, is 1 unit and OB = 1/2, then find the length of AB.
6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?
6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?
If 9 < x < 13, then how many solutions to the equation (x - 1)(x - 2)...(x - 19)(x - 20) + 1 = 0 exist?
Two cars leave simultaneously from points A and B on the same road in opposite directions. Their speeds are constant, and in the ratio 5 to 4, the car leaving at A being faster. The cars travel to and fro between A and B. They meet for the second time at the 145th milestone and for the third time at the 201st. What milestones are A and B?
On a very busy day, the four senior partners of the law firm Smith, Smith, Smith, and Smith had some sandwiches delivered from the corner delicatessen. One of the partners had ordered three salami Sandwiches; one had ordered two salami and one bologna; one had ordered one salami and two bologna; and the fourth had ordered three bologna. Each knew that the four orders were different.
Shortly after they started eating, the owner of the delicatessen came running in to apologize for his new waitress, who had accidentally mixed up the orders so that each order was placed in the wrong bag.
Ann Smith said, "Oh. I have already eaten two salami sandwiches, so I know what the third sandwich in my bag must be."
"In that case," Bob Smith declared, "since I know what Ann ordered and I have already eaten one salami sandwich, I know what the other two sandwiches in my bag must be."
Carla Smith said, "I have not yet opened my bag but, based on what I have just heard, I must have received three bologna sandwiches."
Assuming that the delicatessen owner's statement implies that each of the four received the order intended for one of the others, what did John Smith (who at that moment was out of the room and had not yet started eating) order, and what did he receive?
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Adam, Robert, Clifton, Stephen, and Brent are the five starters on the Doylestown Dribblers basketball team. Two are left handed and three right handed. Two are over 6 feet tall and three are under 6 feet. Adam and Clifton are of the same handedness, whereas Stephen and Brent use different hands. Robert and Brent are of the same height range, while Clifton and Stephen are in different height ranges. The man who plays center is over 6 feet and is left handed. Who is he?
Circles A and B are concentric, and the area of circle A is exactly 20% of the area of circle B. The circumference of circle B is 10. A square is inscribed in circle A. What is the area of that square?
Let a function f satisfy the following: f (x) + 2f (27 - x) = x. Find f (11).
For a deck containing an even number of cards, define a 'perfect shuffle' as follows: divide the deck into two equal halves, the top half and the bottom half, then interweave the cards one-by-one between the two halves starting with the top card of the bottom half, then the top card of the top half, etc. For example, if the deck has 6 cards, labeled '123456' from top to bottom, after a perfect shuffle the order of the cards will be '415263'. Determine the minimum (positive) number of perfect shuffles needed to restore a 94-card deck to its original order.
1, 3 and 22 are the first three natural numbers, the sum of whose divisors is a perfect square, i.e. 1 + 3 = 4, and 1 + 2 + 11 + 22 = 36. What is the fourth such natural number?
Everyday at noon a ship leaves Le Havre for New York and another ship leaves New York for Le Havre. The trip lasts 7 days and 7 nights. How many New York-Le Havre ship will the ship leaving Le Havre today meet during the journey to New York?
Kumar starts rowing a boat upstream from Point A. His hat falls in the water at point B, which is 1 km from point A. He takes 5 min to realize the same and starts rowing back immediately. He catches the hat exactly at point A. What is the speed of the stream?
Two cars leave simultaneously from points A and B on the same road in opposite directions. Their speeds are constant, and in the ratio 5 to 4, the car leaving at A being faster. The cars travel to and fro between A and B. They meet for the second time at the 145th milestone and for the third time at the 201st. What milestones are A and B?
A train leaves a station precisely on the minute, and after having traveled 8 miles, the driver consults his watch and sees the hour-hand is directly over the minute-hand. The average speed over the 8 miles is 33 miles per hour. At what time did the train leave the station?
One day a young man and an older man left the village for the city, one on horse, one in car. Soon it was apparent that if the older man had ridden three times as far as he had, he would have half as far to ride as he had, and if the young man had ridden half as far as he had, he would have three times as far to ride as he had. Who rode the horse?
If f(x) = x2 + 12x + 30, then the value of x satisfying f(f(f(f(f(x))))) = 0 is
The largest factor of 1001001001 that is less than 10000 is
In triangle ABC, AB = 7 and the other two lengths are natural numbers. D is a point on AC such that AD = 3 and BD = 5. What is the largest possible value for BC?
Let S be a set of points in the plane containing at least 2 points and satisfying the following properties.
- For any two distinct points A and B in S, the distance between A and B is 2nwhere n is a positive integer depending on A and B
- For each positive integer n there is at most one pair of points in S having distance 2n between them.
Find the exact number of points in S.
Let's see the property of two square numbers:
9801 = (98 + 01)2
3025 = (30 + 25)2
Find the third number which has the same property. Also, find the shortest way to reach the number through properties of numbers.
For which real values of p and q are the roots of the polynomial x3 - px2 + 11x - q three successive integers? Give the roots.
9801 = (98 + 01)2
3025 = (30 + 25)2
Find the third number which has the same property. Also, find the shortest way to reach the number through properties of numbers.
For which real values of p and q are the roots of the polynomial x3 - px2 + 11x - q three successive integers? Give the roots.
On a sheet of paper statements numbered 1 to 20 are written.
- If statement numbered n says "exactly n of the written statements are false," how many statement(s) written on the sheet are true? What is/are their number(s)?
- If statement numbered n says "at least n of the written statements are false," how many statement(s) written on the sheet are true? What is/are their number(s)?
2. A railway track runs parallel to a road and a cyclist, whose speed is 12 km/h, meets a train at the crossing everyday at the same time. One day, the cyclist started 25 min late and met the train 6 km ahead of the railway crossing. What is the speed of the train?
3. One day I started from A to B at exactly 12 noon. My friend started from B to A at exactly 2:00 pm. We met on the way at five past four and reached our destinations at exactly the same time. What time was it?
The equation 5(x - 2) = 27(x + 2)/ 7 is written throughout in base 9. Find the value of x in base 10.
For the same set of integral values of x an y, 3x + y and 5x + 6y are divisible by
(a) 7
(b) 11
(c) 13
(d) 17
Tortoise travels uniformly 20 km a day. The hare, starting from the same point three days later to overtake the tortoise, travels at the uniform rate of 15 km the first day, at a uniform rate of 19 km the second day, and so forth in arithmetic progression. After how many days does the hare catches up with the tortoise?
Find the total 5-digit numbers made by arranging the digits 1, 2, 3, 4, and 5 which are divisible by 11.
What is the remainder when 1993 - 1399 is divided by 162?
The numbers 479, 698 and 907 are in arithmetic progression.
A. yes
B. no
C. DG is trunk
D. I can't think
The square root of 3B58261 (B = 11) in base 12 is...
The number of ordered pairs (A, B) such that 2143A62701B4632 is divisible by 99 is...
The values of x which satisfy the equation (x2 + 6x + 8)(x2 -8x + 15) = 72 are
Let S(n) denote the sum of the digits of a natural number n. Find all numbers n for which n = 7S(n)
Five professors, residing at five different cities, teach five different subjects. From the clues, determine the subjects that each professor teaches and the place where he/she is residing
- Mr Antony does not stay at either Bangalore or Lucknow. He teaches Philosophy.
- Mr Bhagat does not stay at either Hyderabad or Lucknow. He teaches Math.
- Mr Diwakar stays at Jaipur.
- Miss Elena does not stay either at Bangalore or at Delhi. She teaches Geography.
- Mr Chand does not take History and he stays at Delhi.
Let a = (log2b)2 - 6 log2b + 11. Then, the number of solutions of the equation ba = 64 is
On a Collision Course 93 identical balls move along a line, 59 of them from left to right with speed v; the remaining 34 balls move from right to left toward the first group of balls with speed w. When two balls collide, they exchange their speeds and direction of motion. What is the total number of collisions that will occur?
In a certain Olympic, the number of races on each day varied jointly as the number of days from the beginning and end of the Olympic up to and including the day in question. On three successive days, there were respectively 6, 5, and 3 races. Then, the number of days for which the Olympic last was
A number N written in base b is represented as a two-digit number A2, where A = b - 2. What would N be represented as when written in base b - 1?
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The remainders when F(x) is divided by x - 99 and x - 19 are 19 and 99, respectively. What is the remainder when F(x) is divided by (x - 19) (x - 99)?
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If polynomials P1(x) = x2 + (k - 29)x - k and P2(x) = 2x2 + (2k - 43)x + k are both factors of a cubic polynomial P(x), what is the highest value of k?
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My ABN AMRO ATM Pin is a four-digit number. My HDFC BANK ATM Pin is also a four-digit number using the same digits, in a different order, as those in my ABN AMRO Pin. When I subtract the two numbers, I get a four-digit number whose first three digits are 2, 3 and 9. What is the unit digit of the difference?
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Ann and Bill take turns writing digits of 0 or 1 from left to right next to each other. Ann starts. The game ends when each of them has written down exactly 2007 digits. The resulting sequence of zeros and ones is read as a number written in binary notation. Ann wins if the result is the sum of two perfect squares and Bill wins if it is not. Which of them has a winning strategy?
What is the highest two-digit prime number which will divide the number 200C100?
If a + 13b is divisible by 11 and a + 11b is divisible by 13, what is the least value of a + b?
What is the remainder when (22008 + 2042)3 is divided by 511?
A right triangle has perpendicular sides of 8 units and 15 units. A square is inscribed in the triangle with two of the vertices of the square lying on the hypotenuse and the other two vertices of the square lying on the legs of the triangle. What is the length of a side of the square?
Find the remainder when 192001 is divided by 23.
Let N be a natural number consisting of only digits 0 and 1. If M is an integer such that M = N/12, what is the least possible value of M?
Let x and y be two four-digit palindromes (numbers that read the same forwards and backwards) and z be a five-digit palindrome. If x + y = z, how many values of z are possible?
Suppose you write the numbers 0, 1, 2, 3,..., 20 on the board. You perform following operation 20 times- You pick up two numbers at random (call them a and b), erase them, and write down mod(a - b). Then, the final number left on the board is
A. odd
B. even
C. more than 10
D. less than 10
Find the last two digits of 1110 - 9.
If S = 1 × 2 + 2 × 22 + 3 × 23 + 4 × 24 + ... + 20 × 220, find the remainder when S is divided by 19.
A group of men working at the same rate can finish a job in 45 hours. However, the men report to work, one at a time, at equal intervals over a period of time. Once on the job, each man stays until the job is finished. If the first man works five times as many hours as the last man, find
If S = 1 × 2 + 2 × 22 + 3 × 23 + 4 × 24 + ... + 20 × 220, find the remainder when S is divided by 19.
A group of men working at the same rate can finish a job in 45 hours. However, the men report to work, one at a time, at equal intervals over a period of time. Once on the job, each man stays until the job is finished. If the first man works five times as many hours as the last man, find
- the number of hours the first man works
- the total number of men in the group
Previously, the autorickshaw metres in Delhi showed 8 Rs for the first 900 m and then the reading increased by 35 paisa per 100 m. Recently, after the increase in fares, the metres show 10 Rs for the first 900 m and then the readings increase by 45 paisa per 100 m. What is the second whole number value of the fare that will be common to both the old and the new metres?
You and two other people have numbers written on your foreheads. You are told that the three numbers are primes and that they form the sides of a triangle with prime perimeter. You see 5 and 7 on the other two people, both of whom state that they cannot deduce the number on their own foreheads. What number is written on your forehead?
A snail starts crawling from one end along a uniformly stretched elastic band. It crawls at a rate of 1 foot per minute. The band is initially 100 feet long and is instantaneously and uniformly stretched an additional 100 feet at the end of each minute. The snail maintains his grip on the band during the instant of each stretch. At what points in time is the snail (a) closest to the far end of the band, and (b) farthest from the far end of the band?
Music on the planet Alpha Lyra IV consists of only the notes A and B. Also, it never includes three repetitions of any sequence nor does the repetition BB ever occur. What is the longest Lyran musical composition?
The letters of the word 'ANNIVERSARY' are arranged in all possible orders to form words with or without meaning. The words are arranged in alphabetical order. What is the 10th letter of the 50th word?
Find the smallest positive natural number that ends with 56, is divisible by 56, and has sum of its digits is equal to 56.
Find a 5-digit number that is 45 times the product of its digits.
Find the smallest positive natural number that ends with 56, is divisible by 56, and has sum of its digits is equal to 56.
Find a 5-digit number that is 45 times the product of its digits.
Dagny's rectangular image in surrounded with a two pixel wide frame composed of 504 pixels. If the width of her image is increased by 10% then the frame needs 524 pixels. How many pixels does Dagny’s image have?
What is the difference between the largest and smallest root of x4 -6x3 -2x2 + 6x + 1 = 0?
You have Rs16 with you currently. A fair coin is flipped, if it comes up heads your money increases by Rs8, otherwise half the money is lost. What is the probability that you will have less than Rs16 after the fourth flip?
Find the highest natural number N, less than 400, such that N can be written as sum of consecutive natural numbers in 11 ways but cannot be written as sum of 11 consecutive natural numbers.
According to the rule of reflection, the angle of incidence (angle that the incident ray makes with the normal) is equal to the angle of reflection (angle that the reflected ray makes with the normal). A ray of light, originating perpendicularly at point P on the diameter AB of a semicircular mirror, is reflected at two points, D and E, on the mirror before passing through point B. What is the measure of angle POD?
A number lock has a four-digit combination made by turning four wheels, each of which has the digits 0 through 9 in order. After 9 comes 0 again. Unfortunately, the lock is partially broken; each time you turn one of the four wheels, one of the adjacent wheels also turns. The correct combination to open the lock is 2000. From which of the following starting numbers is it possible to open the lock? More than one answer may be correct.
0000 1999 2001 3456 6543 7777 8161 8181
If x and y are positive numbers and the average of 4, 20, and x is equal to average of y and 16, then the ratio x:y is
On each day of its life a doxy squares its number of legs. For example, if a doxy had two legs on the first day of its life, it would have 4 legs on the second day of its life, 16 legs on the third day of its life, and so on. Harry Potter bought some newborn doxies from the pet store. Some had 2 legs to start with, some had 3 legs, and some had 5 legs. Before they have a chance to increase their number of legs, the total leg count came out to 58 legs. The next day the total leg count came out as 164. The day after that the total number of legs increased to 1976. How many doxies did Harry buy altogether?
The display on a digital clock reads 6:38. What will the clock display twenty-seven digit changes later?
Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by Rs2.00 from that of each adjacent book. For the price of the book at the extreme right, a customer can buy the middle book and an adjacent one. Then
a. The cheapest book sells for Rs4.00.
b. The middle book sells for Rs36.00.
c. The most expensive book sells for Rs64.00.
d. The adjacent book referred to is the one to the right of the middle book.
e. None of the statements above is true.
b. The middle book sells for Rs36.00.
c. The most expensive book sells for Rs64.00.
d. The adjacent book referred to is the one to the right of the middle book.
e. None of the statements above is true.
During the end of the year, many software programmers have to work overtime, so a supervisor at one software company planned a late-night snack for the employees. He ordered 1 extra large pizza for every two programmers, 1 large bag of potato chips for every three programmers and 1 two-liter bottle of cola for every four programmers. When the order arrived, 26 items were delivered. How many employees were working that evening?
A right triangle is partitioned into five congruent triangles as shown in the figure above. If the hypotenuse has length 5 find the length of the shorter leg.
Find the remainder when 1 × 2 + 2 × 3 + 3 × 4 + ... + 98 × 99 + 99 × 100 is divided by 101.
Which is larger 9911 + 10011 or 10111?
The number 523abc is divisible by 7, 8 and 9. Then a × b × c is equal to
How many numbers between 1 and 1000 are there such that n2 + 3n + 5 is divisible by 121?
How many ordered pairs (a, b) satisfy a2 = b3 + 1, where a and b are integers?
A pentagon is made by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon, not in any particular order around the pentagon have lengths 13, 19, 20, 25, and 31. What is the area of the pentagon?
For how many natural numbers m is m3 - 8m2 + 20m - 13 is a prime number?
A square paper ABCD of side length 1 unit is folded so that corner A touches the midpoint of BC as shown in the figure. Find the length of CG.
Find the remainder when 139 + 239 + 339 + 439 + ... + 1239 is divided by 39.
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At 10:00 am, Jinny starts writing consecutive natural numbers, starting with 1, in a row from left to right at a rate of 60 digits per minute. At 10:15 am, Johnny starts rubbing out digits from left to right, starting from the first digit, at a rate of 90 digits per minute.
(a) What is the difference between the rightmost and the leftmost digit one minute before Johnny catches up with Jinny?
(b)If Jinny stops the moment Johnny catches up with her, what is the last digit to be erased?
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The number N = 123456789(10)(11)(12)(13)(14) is written in base 15. What is the remainder when N is written in base 10 and divided by 7?
Five numbers A, B, C, D and E are to be arranged in an array in such a manner that they have a common prime factor between two consecutive numbers. These integers are such that: (XAT 2005)
A has a prime factor P
B has two prime factors Q and R
C has two prime factors Q and S
D has two prime factors P and S
E has two prime factors P and R
1. Which of the following is an acceptable order, from left to right, in which the numbers can be arranged?
(A) D, E, B, C, A
(B) B, A, E, D, C
(C) B, C, D, E, A
(D) B, C, E, D, A
2. If the number E is arranged in the middle with two numbers on either side of it, all of the following must be true, EXCEPT:
(A) A and D are arranged consecutively
(B) B and C are arranged consecutively
(C) B and E are arranged consecutively
(D) A is arranged at one end in the array
3. If number E is not in the list and the other four numbers are arranged properly, which of the following must be true?
(A) A and D can not be the consecutive numbers.
(B) A and B are to be placed at the two ends in the array.
(C) A and C are to be placed at the two ends in the array.
(D) C and D can not be the consecutive numbers.
4. If number B is not in the list and other four numbers are arranged properly, which of the following must be true?
(A) A is arranged at one end in the array.
(B) C is arranged at one end in the array.
(C) D is arranged at one end in the array.
(D) E is arranged at one end in the array.
5. If B must be arranged at one end in the array, in how many ways the other four numbers can be arranged?
(A) 1
(B) 2
(C) 3
(D) 4
Two cars A and B drive straight towards the same point P with speeds a and b respectively. At the start, A, B and P form an equilateral triangle. After some time A and B move to the new positions, B covers 80 miles, and ABP becomes a right triangle. At the moment when A arrives at P, B is 120 miles away from P. What is the distance between A and B in the beginning?
What is the smallest number which has divisors ending in every digit from 0 to 9?
In triangle ABC, points D and E lie on BC and AC, respectively. If AD and BE intersect at T so that AT/DT=3 and BT/ET=4, what is CD/BD?The sum of the reciprocals of four different natural numbers is 1.85. Which of the following could be the sum of the four numbers?
What is the largest factor of 11! that is of the form 6k + 1?
How many values of k are there such that the system of equations x2 = y2, (x - k)2 + y2 = 1 has exactly 3 solutions (x, y)?
The length of the side of the square is 2. Find the radius of the smaller circle.
If the equation x2 - y2 = a has only one solution then a is
I. a prime number
II. an even number
III. a composite number
IV. a square of a prime number
The true statement (s) among the above statements is (are)
a) I only b) III only c) I and II d) I and IV e) III and IV
The value(s) of n for which n(n + 16) is a perfect square is (are)
TG and Dagny run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after TG has run 100 meters. They next meet after Dagny has run 150 meters past their first meeting point. Each person runs at a constant speed. What is the length of the track in meters?
TG has five brothers. When ages of any two brothers are added, 10 different sums are given. Here are the results: 637, 699, 794, 915, 919, 951, 1040, 1072, 1197. What is the age of the oldest brother?
Grass in lawn grows equally thick and in a uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole of the grass.
How many cows are needed to eat the grass in 96 days?
What is the maximum number of cows that can eat in the lawn for infinite amount of time?
A contractor had employed 100 laborers for a flyover construction task. He did not allow any woman to work without her husband. Also, at least half the men working came with their wives. He paid five rupees per day to each man, four rupees to each woman and one rupee to each child. He gave out 200 rupees every evening. How many men, women and children were working with the constructor?
Find the remainder when 139 + 239 + 339 + 439 + ... + 1239 is divided by 39.
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At 10:00 am, Jinny starts writing consecutive natural numbers, starting with 1, in a row from left to right at a rate of 60 digits per minute. At 10:15 am, Johnny starts rubbing out digits from left to right, starting from the first digit, at a rate of 90 digits per minute.
(a) What is the difference between the rightmost and the leftmost digit one minute before Johnny catches up with Jinny?
(b)If Jinny stops the moment Johnny catches up with her, what is the last digit to be erased?
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The number N = 123456789(10)(11)(12)(13)(14) is written in base 15. What is the remainder when N is written in base 10 and divided by 7?
Five numbers A, B, C, D and E are to be arranged in an array in such a manner that they have a common prime factor between two consecutive numbers. These integers are such that: (XAT 2005)
A has a prime factor P
B has two prime factors Q and R
C has two prime factors Q and S
D has two prime factors P and S
E has two prime factors P and R
1. Which of the following is an acceptable order, from left to right, in which the numbers can be arranged?
(A) D, E, B, C, A
(B) B, A, E, D, C
(C) B, C, D, E, A
(D) B, C, E, D, A
2. If the number E is arranged in the middle with two numbers on either side of it, all of the following must be true, EXCEPT:
(A) A and D are arranged consecutively
(B) B and C are arranged consecutively
(C) B and E are arranged consecutively
(D) A is arranged at one end in the array
3. If number E is not in the list and the other four numbers are arranged properly, which of the following must be true?
(A) A and D can not be the consecutive numbers.
(B) A and B are to be placed at the two ends in the array.
(C) A and C are to be placed at the two ends in the array.
(D) C and D can not be the consecutive numbers.
4. If number B is not in the list and other four numbers are arranged properly, which of the following must be true?
(A) A is arranged at one end in the array.
(B) C is arranged at one end in the array.
(C) D is arranged at one end in the array.
(D) E is arranged at one end in the array.
5. If B must be arranged at one end in the array, in how many ways the other four numbers can be arranged?
(A) 1
(B) 2
(C) 3
(D) 4
Two cars A and B drive straight towards the same point P with speeds a and b respectively. At the start, A, B and P form an equilateral triangle. After some time A and B move to the new positions, B covers 80 miles, and ABP becomes a right triangle. At the moment when A arrives at P, B is 120 miles away from P. What is the distance between A and B in the beginning?
What is the smallest number which has divisors ending in every digit from 0 to 9?
Two squares, each of side lengths 1 unit and having their centres at O, are rotated with respect to each other to generate octagon ABCDEFGH, as shown in the figure above. If AB = 43/99, find the area of the octagon.
In rectangle ABCD, points E and F are on BC and AB, respectively, such that areas of triangles AFD, FBE, and ECD are all equal to 1. What is the area of triangle FDE?
In triangle ABC, points D and E lie on BC and AC, respectively. If AD and BE intersect at T so that AT/DT=3 and BT/ET=4, what is CD/BD?The sum of the reciprocals of four different natural numbers is 1.85. Which of the following could be the sum of the four numbers?
What is the largest factor of 11! that is of the form 6k + 1?
How many values of k are there such that the system of equations x2 = y2, (x - k)2 + y2 = 1 has exactly 3 solutions (x, y)?
The length of the side of the square is 2. Find the radius of the smaller circle.
If the equation x2 - y2 = a has only one solution then a is
I. a prime number
II. an even number
III. a composite number
IV. a square of a prime number
The true statement (s) among the above statements is (are)
a) I only b) III only c) I and II d) I and IV e) III and IV
The value(s) of n for which n(n + 16) is a perfect square is (are)
- In the figure, ABCD is a square with side length 17 cm. Triangles AGB, BFC, CED and DHA are congruent right triangles. If EC = 8, find the area of the shaded figure.
- ABC is a triangle with area 1. AF = AB/3, BE = BC/3 and ED = FD. Find the area of the shaded figure.
- What is the length of the side of the largest cube that can be inscribed inside a hemisphere of radius r?
TG and Dagny run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after TG has run 100 meters. They next meet after Dagny has run 150 meters past their first meeting point. Each person runs at a constant speed. What is the length of the track in meters?
TG has five brothers. When ages of any two brothers are added, 10 different sums are given. Here are the results: 637, 699, 794, 915, 919, 951, 1040, 1072, 1197. What is the age of the oldest brother?
Grass in lawn grows equally thick and in a uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole of the grass.
How many cows are needed to eat the grass in 96 days?
What is the maximum number of cows that can eat in the lawn for infinite amount of time?
A contractor had employed 100 laborers for a flyover construction task. He did not allow any woman to work without her husband. Also, at least half the men working came with their wives. He paid five rupees per day to each man, four rupees to each woman and one rupee to each child. He gave out 200 rupees every evening. How many men, women and children were working with the constructor?
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