EduCAT: June 2012

Jun 10, 2012

Quant Mock Test CAT Series: 001

3. Find all natural numbers n that satisfy the equation:
(n − 1) + (n − 1) = n!

4. Find the ordered 4-tuple of positive integers (a, b, c, d) so that:

5. Each vertex of a given square is the center of one of four circles. The circles are all congruent and each
one is tangent to two others. What is the probability that a randomly chosen point in the figure is
inside both a circle and the square?

6. There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people
and the remaining rooms are empty. Each minute, the following happens: for each room containing
more than one person, someone in that room decides it is too crowded and moves to the next room.
All these movements are simultaneous (so nobody moves more than once within a minute). After one
hour, how many different rooms will have people in them?

7. What is the largest whole number that is equal to the product of its digits?

8. Suppose f is a function that assigns to each real number x a value f(x), and suppose the equation
f(x1 + x2 + x3 + x4 + x5) = f(x1) + f(x2) + f(x3) + f(x4) + f(x5) − 8 holds for all real numbers x1, x2, x3, x4, x5. What is f(0)?

9. How many ways can you mark 8 squares of an 8 × 8 chessboard so that no two marked squares are in
the same row or column, and none of the four corner squares is marked? (Rotations and reflections
are considered different.)

10. A rectangle has perimeter 10 and diagonal sqrt 15. What is its area?

11. Find the ordered quadruple of digits (A,B,C,D), with A > B > C > D, such that
ABCD
− DCBA
= BDAC.

12. Let ACE be a triangle with a point B on segment AC and a point D on segment CE such that BD
is parallel to AE. A point Y is chosen on segment AE, and segment CY is drawn. Let X be the
intersection of CY and BD. If CX = 5, XY = 3, what is the ratio of the area of trapezoid ABDE to
the area of triangle BCD?

13. You have a 10×10 grid of squares. You write a number in each square as follows: you write 1, 2, 3, . . . , 10 from left to right across the top row, then 11, 12, . . . , 20 across the second row, and so on, ending with a 100 in the bottom right square. You then write a second number in each square, writing 1, 2, . . . , 10
in the first column (from top to bottom), then 11, 12, . . . , 20 in the second column, and so forth.
When this process is finished, how many squares will have the property that their two numbers sum
to 101?

14. Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn
is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of
selecting an urn and drawing out a ball, without returning the first ball. What is the probability that
the first ball drawn was red, given that the second ball drawn was black?
15. A floor is tiled with equilateral triangles of side length 1, as shown. If you drop a needle of length 2
somewhere on the floor, what is the largest number of triangles it could end up intersecting? (Only
count the triangles whose interiors are met by the needle — touching along edges or at corners doesn’t
qualify.)

16. Find the largest number n such that (2004!)! is divisible by ((n!)!)!.

17. Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence
HH) or flips tails followed by heads (the sequence TH). What is the probability that she will stop
after flipping HH?

18. How many ordered pairs of integers (a, b) satisfy all of the following inequalities?
a^2 + b^2 < 16
a^2 + b^2 < 8a
a^2 + b^2 < 8b.

19. A horse stands at the corner of a chessboard (above), a white square. With each jump, the horse can
move either two squares horizontally and one vertically or two vertically and one horizontally (like a
knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a
carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the
square on which it began, it can leave. What is the maximum number of carrots the horse can earn
without touching any square more than twice?

20. Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge
games — that is, into unordered pairs of unordered pairs of people?

21. a and b are positive integers. When written in binary, a has 2004 1’s, and b has 2005 1’s (not necessarily consecutive). What is the smallest number of 1’s a + b could possibly have?

22. Farmer John is grazing his cows at the origin. There is a river that runs east to west 50 feet north of
the origin. The barn is 100 feet to the south and 80 feet to the east of the origin. Farmer John leads
his cows to the river to take a swim, then the cows leave the river from the same place they entered
and Farmer John leads them to the barn. He does this using the shortest path possible, and the total
distance he travels is d feet. Find the value of d.

23. A freight train leaves the town of Jenkinsville at 1:00 PM traveling due east at constant speed. Jim, a
hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle,
traveling along a straight road in a northeasterly direction (but not due northeast) at 10 miles per
hour. At 1:12 PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He
wakes up and immediately begins walking at 3.5 miles per hour directly towards the road on which
Julie is riding. Jim reaches the road at 2:12 PM, just as Julie is riding by. What is the speed of the
train in miles per hour?

24. Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the
given one through its center. What is the volume of their intersection?

25. A lattice point is a point whose coordinates are both integers. Suppose Johann walks in a line from
the point (0, 2004) to a random lattice point in the interior (not on the boundary) of the square with
vertices (0, 0), (0, 99), (99, 99), (99, 0). What is the probability that his path, including the endpoints,
contains an even number of lattice points?

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Soln :

3. Answer: 2

Quant Test : 006

DIRECTIONS for questions 51 and 52: These questions are based on the following data.

Rama went to the market and bought some apples, mangoes and bananas. He bought 42 fruits in all. The number of bananas is less than half the number of apples; the number of mangoes is more than one-third the number of apples and the number of mangoes is less than three-fourths the number of bananas.
51. How many apples did Rama buy?
(1) 20 (2) 23 (3) 26 (4) 28
52. How many bananas did Rama buy?
(1) 8 (2) 9 (3) 10 (4) 11

DIRECTIONS for questions 53 and 54: These questions are based on the information given below.

Car C1 starts at town T1 at 5 a.m. and reaches town T2 at 10 a.m. Car C2 starts at town T2 at 7 a.m. and reaches town T1- at 11 a.m.

53. If the distance between towns T1 and T2 is 320 km, what is the distance between the two cars 15 min after they meet each other?
(1) 36 km (2) 40 km (3) 48 km (4) Cannot be determined

54. One hour after C1 starts, another car C3 – whose speed is 25% more than that of C1 – starts from T1 towards T2. How many of the following statements is/are true?
I. Cars C1 and C3 reach T2 at the same time.
II. C3 meets C2 20 min after C1 meets C2.
III. When C3 meets C2, C2 has to still travel for 1 hr 30 min to reach T1.
(1) Exactly one of the three statements is true. (2) Exactly two of the three statements are true.
(3) All the three statements are true. (4) None of the statements are true.

DIRECTIONS for question 55 to 58: Select the correct alternative from the given choices.

55. Amit found that he needs to multiply a natural number N with at least p to make it a perfect square and with at least q to make it a perfect cube. He also found that he needs to multiply N with at least r to make it a perfect cube as well as a perfect square. If p, q and r are natural numbers, then which of the following expresses the relationship between p, q and r?
(1) p x q = r (2) p x q > r (3) p x q < r (4) Depends on N

56.In the given rectangle ABCD, E and F are points on BC such that AB : BE : EF : FC = 1 : 2 : 1 : 4. Which of the following is true of the values of ΔEAF and ΔACF?
(1) ΔEAF > ΔACF (2) ΔEAF = ΔACF
(3) ΔEAF < ΔACF (4) Cannot be determined

57. There are three equal circles C1, C2 and C3 each of radius 6 cm, where C1 and C3 pass through
the centre of C2. What is the area of the shaded region? (in sq.cm) C1
(1) 36 3 −12π (2) 48 3 −16π
(3) 36π − 54 3 (4) None of these

58. Malini and Shalini play a game in which they first write down the first n natural numbers and
then take turns in inserting plus or minus signs between the numbers. When all such signs
have been placed the resulting expression is evaluated (i.e., the additions and subtractions are
performed) Malini wins if the sum is even and Shalini wins if the sum is odd. Assuming that the concept of even and odd (i.e., even and odd parities) is defined for all integers, which of the following statements is true?
(1) Malini wins if n is a multiple of 4 (2) Shalini wins if n is even
(3) Shalini wins if n is odd (4) Malini loses if n is a multiple of 4

DIRECTIONS for questions 59 and 60: These questions are based on the following data.
A teacher found that the performance of her students in the mid-term exams, comprising 6 subjects – A, B, C, D, E and F, is as follows:

59. If the number of students who passed in all the six subjects is 10, then find the number of students who passed in exactly five subjects.
(1) 10 (2) 15 (3) 20 (4) Cannot be determined.

DIRECTIONS for questions 61 to 63: These questions are based on the data given below.
Everyday, Saddam, the office attender fetches water for the office in container A which has certain rated capacity. However, because of a dent at the bottom of the container, only 80% of the rated capacity of the container can be used to fill water. This water is transferred periodically into a smaller container B - for people in the office to use this water for drinking. There is an outlet (a faucet) in B from which water is let out. Since the faucet is fixed at a level above the base of B, water upto 10% of the rated capacity of B cannot be let out through the faucet. Everyday in the morning, after Saddam fetches water in container A, he cleans B and fills B to the brim by pouring water from A into B. Whenever the water level falls to the faucet level in B, he again fills B to the brim by pouring water from A into B. The questions in this set are independent of each other.

61. On a particular day, Saddam finds that he filled B five times (including the first time) and at the end of the day, A was empty. The water level in B reached the faucet level. What is the ratio of the rated capacities of A and B?
(1) 4.6 : 1 (2) 5 : 1 (3) 5.75 : 1 (4) 6.25 : 1

62. If Saddam gets the dent in container A removed (so that water can be fetched in this container to its rated capacity) how many times can he fill container B (including the first time in the morning) given that the rated capacities of the two containers are in the ratio 10 : 1?
(1) 9 times (2) 10 times (3) 12 times (4) 11 times

63. Saddam gets the dent in container A removed. He also gets the faucet in container B refixed so that all the water filled into B can be used. He keeps filling B from A everytime B gets emptied. After he pours out water from A into B the last time (i.e., A gets emptied), what percentage of B is empty? The ratio of the rated capacities of A and B is 7.5 : 1?
(1) 0% (2) 331/3% (3) 25% (4) 50%

DIRECTIONS for questions 64 and 65: These questions are based on the following data.

Amar, Akbar and Anthony sold their three cycles manufactured in different years to Mr.Kishanlal. Mr.Kishanlal gave a total of Rs.1700 to the three and said that Amar should get about one-half of the total amount as his cycle was used less. Akbar’s cycle being used more than Amar’s, he should get about one-third of the total amount and the last one gets about one-ninth. Each individual gets his amount only in denominations of Rs.100.

64. What is the difference between the amounts received by Amar and Anthony?
(1) Rs.900 (2) Rs.700 (3) Rs.800 (4) Rs.600

65. The amount that Amar has is how much more than what Akbar and Anthony together have?
(1) Rs.200 (2) Rs.300 (3) Rs.100 (4) Rs.400

Directions for questions 66 to70: Select the correct alternative from the given choices.

66. A, B and C start running simultaneously from the points P, Q and R respectively on a circular track. The distance (when measured along the track) between any two of the three points P, Q and R is L and the ratio of the speeds of A, B and C is 1 : 2 : 3. If A and B run in opposite directions while B and C run in the same direction, what is the distance run by C before A , B and C meet for the first time?
(1) 310L (2) 311L (3) All three of them will never meet. (4) Cannot be determined

67. A circle of radius 1cm circumscribes a square. A dart is thrown such that it falls within the circle. What is the probability that it falls outside the square?
(1) 1/2π (2) (2π - 1) /2π (3) (π - 1) /π (4) (π - 2) /π

68. Fifteen boys went to collect berries and returned with a total of 80 berries among themselves. What is the minimum number of pairs of boys that must have collected the same number of berries?
(1) 0 (2) 1 (3) 2 (4) 3

69. A cube of edge 12 ft is placed on the floor with one of its faces touching a wall. A ladder of length 35 ft is resting against that wall and is touching an edge of the cube. Find the height at which the top end of the ladder touches the wall, given that it is more than the distance of the foot of the ladder from the wall?
(1) 11 ft (2) 23 ft (3) 21 ft (4) 28 ft

70. Two circles touch each other externally. One of the circles is 300% more in area than the other. If A is the centre of the larger circle and BC is the diameter of the smaller circle and either AB or AC is a tangent to the smaller circle, then find the ratio of the area of the triangle ABC to that of the smaller circle?
(1) 2 : π (2) 3 : π (3) 2 2 : π (4) π : 4 2

DIRECTIONS for questions 71 and 72: Select the correct alternative from the given choices.

71. a1, a2, a3, a4 and a5 are five natural numbers. Find the number of ordered sets (a1, a2, a3, a4, a5) possible such that a1 + a2 + a3 + a4 + a5 = 64.
(1) 64C5 (2) 63C4 (3) 65C4 (4) None of these

72. In the above question if a1, a2, a3, a4 and a5 are non-negative integers then find the number of ordered sets (a1, a2, a3,a4 and a5) that are possible.
(1) 64C5 (2) 63C4 (3) 68C4 (4) None of these

DIRECTIONS for questions 73 to 75: Each question gives certain information followed by two quantities A and B.Compare A and B, and then
Mark 1 if A > B
Mark 2 if B > A
Mark 3 if A = B
Mark 4 if the relationship cannot be determined from the given data.

73. A baker had a certain number of boxes and a certain number of cakes with him. Initially he distributed all the cakes equally among all the boxes and found that there was no cake left without a box. He later found that he had one more box with him and so he redistributed all the cakes equally among all the boxes and found that there was one cake less per box than initially and one cake was left without a box with the baker.
A. The number of cakes per box in the first case.
B. The total number of boxes with the baker.

74. A trader gives a discount of r% and still makes a profit of r%. A second trader marks up his goods by r% and gives a discount of r%.

A. The cost price of the first trader.
B. The cost price of the second trader.

75. A piece of work is carried out by a group of men, all of equal capacity, in such a way that on the first day one man works and on every subsequent day one additional man joins the work. A group of women, all of equal capacity is engaged to carry out a second piece of work with ten women starting the work on the first day and one woman leaving the work at the end of everyday. The second piece of work is thrice as time consuming as the first piece of work while each man is thrice as efficient as each woman. It is known that one man working alone can complete the first piece of work in 6 days.
A. Number of days in which the first piece of work is completed.
B. Number of days in which the second piece of work is completed.

DIRECTIONS for questions 76 and 77: Select the correct alternative from the given choices.

76. A number when divided by a certain divisor, left a remainder of 8. When the same number was multiplied by 12 and then divided by the same divisor, the remainder is 12. How many such divisors are possible?
(1) 1 (2) 2 (3) 4 (4) 5

77. Consider the equation x² + y² + z² = 1. Let (x1, y1, z1) and (x2, y2, z2) be two sets of values of (x, y, z) satisfying the given equation and let A = (x1 – x2)² + (y1 – y2) ² + (z1 – z2)². What is the maximum possible value that A can assume? (assume that all the quantities involved are real numbers)
(1) 1 (2) 2 (3) 4 (4) 6

Quant Test : 005

26. The inhabitants of Planet Rahu measure time in hours and minutes which are different from the hours and minutesof our earth. Their day consists of 36 hours with each hour having 120 minutes. The dials of their clocks show 36 hours. What is the angle (in Rahuian degrees) between the hour and minute hands of a Rahuian clock when it shows a time of 9:48? Rahuians measure angles in degrees (°) the way we do on earth. But for them, the angle around a point is 720 Rahuian degrees [instead of 360° that we have on earth].
(1) 112° (2) 100° (3) 24° (4) None of these

27. A ladder is placed against a wall at an angle. Let the area enclosed by the ladder be A1. The ladder slides on the floor by a few feet and makes a new angle and let the area enclosed be A2. Which of the following is true?
(1) A2 > A1 (2) A2 < A1 (3) A2 = A1 (4) Data insufficient

28. The set Y consists of the following numbers. Y = {1, 31/2, 3, 33/2, ……, 39, 319/2, 310}. In how many ways can a pair of distinct numbers be selected from the set Y such that their product is greater than or equal to 310? Assume that a x b is the same as b x a.
(1) 110 (2) 210 (3) 105 (4) 100

29. A stone weighing 121 kg fell from a height of 10 m and broke into exactly 5 pieces - all of different weights. Find the sum (in kg) of the weights of the smallest piece and the largest piece, if it is known that it is possible to weigh any weight (using a common balance) in kg from 1 to 121 kg using the 5 pieces?

(1) 118 (2) 82 (3) 65 (4) Cannot be determined

30. Six friends share a circular pizza equally by cutting it into six equal sectors. If three of them cut out and eat only the largest possible circle from their respective slices and leave the rest while the others eat their whole slice, then the approximate
percentage of pizza wasted is
(1) 11% (2) 15% (3) 17% (4) 22%

32. In a regular hexagon of side 4 cm, the midpoints of three alternate sides are joined in order to form a triangle. What is the area of this triangle?
(1) 8 3 sq.cm. (2) 12 3 sq.cm. (3) 9 3 sq.cm. (4) 18 3 sq.cm.

33. In a number system to the base 20, letters A, B, C, …. to K of the English alphabet are sequentially used to digitallyrepresent the values 10, 11, 12, …. to 20 (to the base 10). Calculate the decimal equivalent of the value (in base 10)
of [CAKE](20)- [BAKE](20).
(1) 1483 (2) 1488 (3) 1000 (4) 8000

34. R and S are the centres of two unequal circles touching externally at the point T. P and Q are the points of contact ofa direct common tangent with the larger and smaller circles respectively and the common tangent at T intersects PQ at U. What is the measure of the angle RUS?
(1) 45° (2) 90° (3) 135° (4) None of these

35. How many small squares are crossed by the diagonal in a rectangular table formed by 16 x 17 small squares?
(1) 32 (2) 33 (3) 34 (4) None of these

36. Let S = 141414 …. Upto 202 digits. What is the remainder when S is divided by 909?
(1) 115 (2) 216 (3) 418 (4) 721

38. In the following figure, find the ratio of the areas of the triangles ABE and DCE given that TA
: TD : TF = 5 : 4 : 10.
(1) 16 : 25 (2) 25 : 36 (3) 36 : 49 (4) 25 : 49

39. How many three-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 and 8 without any repetition of thedigits and wherein the tens digit is greater than the hundreds digit but less than the units digit?
(1) 48 (2) 56 (3) 64 (4) 72

40. The set of all positive integers is divided into two subsets {a1, a2, a3, …an, …} and
{b1, b2, b3, …bn, …} where ai < ai + 1, bi < bi + 1 and ai = bj for any i, j. Also, bi =2ai−1 + ai
for all i except i = 1. What is the value of b1?
(1) 1 (2) 2 (3) 4 (4) Cannot be determined

41. If three playing squares are chosen at random from the 64 playing squares of a 8 x 8 chessboard, then find theprobability that exactly two of them are of the same colour?
(1) 9/21 (2) 16/21 (3) 14/21 (4) 18/21

42. In the above figure, ABCD is a parallelogram. P and Q are the points of trisection of AB and R is
the midpoint of DC. What is the ratio of the area of the parallelogram ABCD to that of the
quadrilateral PBCR?
(1) 16 : 7 (2) 15 : 7 (3) 2 : 1 (4) 12 : 7

43. Four people need to cross a stream. At a time only two people can cross the stream using a certain boat which isavailable. The times taken by the four people to cross the stream individually are 3, 7, 11, 17 minutes respectively.If the faster person on the boat drives it and no person drives the boat more than two trips in total, what is the least time required for all the four to cross the stream? (Reaching from one bank to the other bank is one trip).
(1) 23 minutes (2) 59 minutes (3) 31 minutes (4) 37 minutes

DIRECTIONS for questions 44 and 45: These questions are based on the following data.
There are 100 players numbered 1 to 100 and 100 baskets numbered 1 to 100. The first players puts one ball each in every basket starting from the first basket (i.e., in the baskets numbered 1, 2, 3, …and so on upto 100), the second player then puts two balls each in every second basket starting from the second (i.e., in the baskets numbered 2, 4, 6, … and so on upto 100), the third player puts three balls each in every third basket starting from the third (i.e., in the baskets numbered 3, 6, 9, ... and so on upto 100), and this is comtinued so on till the hundredth player puts 100 balls in the 100th basket.

44. Which basket will finally have the maximum number of balls?
(1) 96 (2) 98 (3) 100 (4) None of these

45. How many baskets will finally have exactly twice the number of balls as the number on the basket itself?
(1) 8 (2) 6 (3) 4 (4) 2

DIRECTIONS for questions 46 to 50: Select the correct alternative from the given choices.

46. A cylindrical vessel has its radius and height in the ratio 1 : 12 and it can hold the same quantity of water as another conical vessel whose height is one-third of its height. What is the ratio of the lateral surface area of the cylinder and that of thecone? (Ignore the thickness of the vessel in both cases)
(1) 3 : 2 (2) 8 : 5 (3) 1 : 1 (4) None of these

47. If (21)n x (36)n = (776)n and (12)n x (63)n = (x)n then find x.
(1) 510 (2) 540 (3) 756 (4) 776

48. What is the area enclosed by x = 0, x = 3, y = 0 and y = | x - 1 | + | x - 2 | ?
(1) 4 sq.units (2) 4.5 sq.units (3) 5 sq.units (4) 6 sq.units

49. There are two parallel lines and a circle in a plane dividing the plane into distinct non-overlapping regions. What is the maximum number of regions into which the plane can be divided?
(1) 8 (2) 5 (3) 6 (4) 7

50. The area of a triangle which is inside a semicircle is equal to the area outside the triangle but within the semicircle. What is the ratio of the area of the complete circle to that of a parallelogram formed with its base as the diameter of the circle and height equal to the height of the triangle, if the base of the triangle is the diameter of the circle and the third vertex of the triangle lies on the circle?
(1) 1 : 2 (2) 4 : 1 (3) 2 : 1 (4) 1 : 4

Quant Test : 004

DIRECTIONS for questions 1 to 4: Select the correct alternative from the given choices.

1. There are certain number of cookies with each of Payal, Richa and Sapna. If we add the number of cookies with anytwo girls at a time, the ratio is 3 : 4 : 5 (Payal+Richa : Richa+Sapna : Sapna+Payal respectively). Which of theollowing statements is/are true?
I. The number of cookies with Sapna is 50% of those with all three of them together.
II. Payal has twice as many cookies as Richa.
III. Richa and Sapna together have twice as many cookies as Payal.
(1) Only I and II (2) Only II and III (3) Only III and I (4) All three statements

2. Today, Susheel made a total of exactly one hundred calls from his cell phone to ten different people. He could only remember the fact that he had called each person at least four times. In how many ways could Susheel have distributed his calls among the ten different people today? Ignore the order in which the calls were made.
(1) 100C10 (2) 99C6 (3) 60C6 (4) 69C9

3. A boy starts to paint a fence on one day. On the second day two more boys join him and on the third day three more boys join the group and so on. If the fence is completely painted this way in exactly 20 days, then find the number of days in which 10 men painting together can paint the fence completely where every man can paint twice as fast as a boy can.
(1) 20 days (2) 40 days (3) 45 days (4) 77 days

4. On a certain sum, the difference between the compound interest and the simple interest for the second year is Rs.3,600 and the same for the third year is Rs.11,340. What is the sum?
(1) Rs.1,60,000 (2) Rs.1,20,000 (3) Rs.1,80,000 (4) Cannot be determined

DIRECTIONS for questions 5 and 6: These questions are based on the following data.

A box contains a certain number of red, green and blue balls. The number of balls of each colour is more than one. The ratio of the number of red balls to the number of green balls is the same as the ratio of the number of green balls to the number of blue balls.

5. If the total number of balls in the box is 61, how many green balls are there in the box?
(1) 16 (2) 20 (3) 25 (4) Cannot be determined
6. If the number of green balls in the box is 21, then the total number of balls in the box can be
(1) 63 (2) 89 (3) 101 (4) 117

DIRECTIONS for questions 7 to 43:
Select the correct alternative from the given choices.

7. Consider z = 22225555 + 55552222. Which of the following statements is/are true?
(1) z is a multiple of 7 but not 11. (2) z is a multiple of 11 but not 7.
(3) z is a multiple of both 7 and 11. (4) None of these

8. Two candles of equal length are lighted simultaneously. After 15 minutes of burning, the length of the first candlebears a ratio of 4 : 5 to that of the second candle. If the first candle burns out completely in 45 minutes how much more time does the second candle take to burn out completely?
(1) 30 minutes (2) 60 minutes (3) 45 minutes (4) None of these

9. The number of two-digit numbers [where neither digit is a zero] whose product of the digits is a square are
(1) 16 (2) 17 (3) 18 (4) None of these

10. Bakul and Manohar start from two points P and Q respectively on a river and head towards each other. Had theybeen travelling in still water, they would have met at a point R, which is twice as distant from P as it is from Q. If Bakul had been travelling along the current and Manohar against it, then they would have met in 24 minutes. Find the time they would take to meet, if Bakul were to travel against the current and Manohar along the current.
(1) 12 minutes (2) 24 minutes (3) 36 minutes (4) 48 minutes

11. If p + 7 > 0 and (25 - p2) < 0, how many integer solutions are possible for p given that it lies between -151/6 and 471/7?
(1) 41 (2) 42 (3) 43 (4) 45

12. If the sum of the first 10 terms of an arithmetic progression is 100 and the sum of the first 100 terms of
the sameprogression is 10, then the sum t101 + t102 + …..t110 is
(1) –90 (2) –100 (3) –110 (4) –120

13. A cuboidal box of dimensions 10 cm x 8 cm x 12 cm is partitioned completely into cubicles of dimensions 1 cm x1 cm x 1 cm. Amy the ant is in the top left corner cubicle towards the front of the box. If Amy can move only between any two cubicles that have a common face, then find the number of ways in which Amy can reach the bottom right corner cubicle that is towards the back of the box. Assume that Amy visits no cubicle more than once and that it is allowed to move only in downward, rightward and backward directions.
(1) 960 (2) 10!8!2!30! (3)9!7!11 !27 !(4)20C8 * 30C10

14. ABCD is a rhombus of side 12 cm. The diagonals of the rhombus meet at the point P. Line segments PX and PY are joined, where X and Y are the midpoints of the sides AD and CD respectively. If the length of the line segment PD is 10cm, find the length of the line segment XY.
(1) 2 11 (2) 3 11 (3) 4 11 (4) Cannot be determined

15. A group of new students whose total age is 221 years joins a class, because of which the strength of the class goes up by 50% but the average age of the class comes down by one year. What is the new average of the class if it is known to be a natural number after the new group of students have joined, given that the original strength of the class was a two digit number greater than 30?
(1) 15 yrs (2) 17 yrs (3) 19 yrs (4) 16 yrs

16. If x is an integer, then which of the following statements is true of z = (x + 1) (x + 2) (x + 3) (x + 4).
(1) z - 1 is a prime number. (2) z2 - 1 is a prime number.
(3) z + 1 is a perfect square (4) None of these

18. All Analysts are Engineers. One-third of all Engineers are Analysts. Half of all Technicians are Engineers. One Technician is an Analyst. Eight Technicians are Engineers. If the number of Engineers is 90, how many Engineers are neither Analysts nor Technicians?
(1) 65 (2) 79 (3) 82 (4) 53

19. If a is an integer greater than –7 but less than 5, b is an integer less than 7 but greater than –5 while c is an integer that is not greater than 6 and not less than –2, which of the following statements is/are always true?
I. –36 < (ab + bc + ca) < 84
II. –384 < a(b2 + c2) + b(c2 + a2) + c(a2 + b2) < 912
(1) I only (2) II only (3) Both I and II (4) None of these

20. Let S = 7A68G023535928. If S is divisible by 792, what is the value of A?
(1) 5 (2) 9 (3) 7 (4) 8

21. Amar and Ajeeth start simultaneously from the same point on a circular track, of length 5 km, and run in opposite directions. Their speeds are doubled every time they cross each other. Find the number of times that they will meet within the first hour, given that they started the race with respective speeds of 6 kmph and 4 kmph.
(1) 4 times (2) 6 times (3) 7 times (4) None of these

22. How many ordered pairs of integers (a, b), are there such that their product is a positive integer less than 100?
(1) 99 (2) 545 (3) 635 (4) 1090

23. Among four persons A, B, C and D, C works half as fast as A while D works a third as fast as B. If C and D, when working individually, complete the work in 24 and 54 days respectively more than the time in which they complete the work when working together, then find the time in which A and B, working together, will complete the work.
(1) 15 (2) 18 (3) 24 (4) 30

24. There are 51 coins in a bag. The coins are first divided into two separate bags after which the coins in one of the two bags are taken and again divided into two separate bags and so on until we are left with 51 bags containing one coin each. If after every division of the coins in a bag into two bags the product of the number of coins in the two bags is written down, what is the sum of all the numbers written down?
(1) 1020 (2) 1275 (3) 1551 (4) 1525

25. Train A leaves station X for station Z at 0800 hrs and travels at a constant speed of 36 kmph. Train B leaves station Z for station X at 0830 hrs and runs at a constant speed of 27 kmph. Both trains have a stop at station Y but train A stops for 10 minutes while train B stops for 15 min. If the distance between the stations X and Y is 300 km and that between Y and Z is 405 km, where do the two trains meet?
(1) 450 km from station X (2) 450 km from station Z
(3) 408 km from station X (4) 408 km from station Z

Jun 8, 2012

DILR Test: 001

DILR WORKSHOP – 1

Questions 1-5:

The fitness school marathon attracted 16 entrants (including sports wiz) this year. Each of the five houses (IIM-A, IIM-B, IIM-C, IIM-L, and IIM-K) was represented by a team of 3 runners and the field was made up by the sports wiz, Aditya. The school houses were competing for the trophy. The number of points by each entrant would be equal to his finishing position.

The five houses were tied for the cup, their totals being equal, although no two entrants tied for the same position .In order to determine the order in which the houses would hold the cup (they agreed to hold it for 73 days each), they multiplied the finishers’ positions together in each house. The house with the smallest product, IIM-K, would hold the cup first and so on to the house with the largest product, IIM-A, which held it last. Unfortunately IIM-B and IIM-C houses were still tied, and had to be separated by the toss of coin.

Aditya later noted that no house had two finishers in consecutive positions, although IIM-B would have achieved this had he not managed to get in between two of their runners.

1. The possible position of Aditya is

a) 1 only b) 6 only c) 11 only d) 6 or 11 e) 16 only

2. The possible positions of IIM-C are

a) 4,6,15 b) 3, 10, 12 c) 3,5,12 d) 6, 11, 16 e) none of these

3. The product of IIM-K is

a) 136 b) 360 c) 120 d) 220 e) 128

4. The product of IIM-A is

a) 180 b) 320 c) 214 d) 455 e) 198

5. IIM-L’s positions are

a) 2,8,14 b) 2,9,14 c) 3,8,15 d) 1,8,13 e) 2,8,13

Questions 6-9:

A game of card is played with a normal pack of 52 cards. Each plyer is dealt 13 cards and values his suits separately, each card counting its face value (Ace – 1, Jack, Queen, king – 11, 12 and 13 respectively).

Then the total value of the spade suit is multiplied by 4, the heart suit by 3, the diamond suit by 2, the club suit is counted by its face value.

The total of these 4 suits is then aggregated and the player whose hand has the highest aggregate is the winner.

I must admit, I am rather fond of playing cards. I was recently dealt a hand with more spades than hearts, in which no two cards had the same face value. If the spade had been clubs , the heart spades, the diamond hearts and the club diamonds, my score would have been 27 more than what it was. If the spades had been diamonds, the heart clubs, the diamond spades and the club hearts, my score would have been 30 less than what it was.

If the spade had been hearts, the hearts diamonds, the diamonds clubs and the clubs spades, my score would have been 11 less than what it was.

And you know what, if I told you the color of the king in my hand, you could easily deduce all my cards.

the total face value of the 13 card was:

1.91 2.66 3. 52 4.39

7. Which of the following is not a combination of the clubs?

1. Jack, 9 2.queen, 8 3.king, 7 4. Ten, 9

8. Which of the following is the likely king?

1. Clubs 2.hearts 3. Spades 4. 1 or 3

Which of the following is not a card of the hearts?

1. Queen 2. Jack 3. Five 4. Four

Questions 10-14:

Eight dogs in an obedience class are learning to follow two commands—"heel" and "stay." Each dog is either a shepherd, a retriever, or a terrier, and each of these three breeds is represented at least once among the group. All female dogs in the group are retrievers. The results of the first lesson are as follows:

At least two of the dogs have learned to follow the "heel" command, but not the "stay" command.
At least two of the dogs have learned to follow the "stay" command, but not the "heel" command.
At least one of the dogs has learned to follow both commands.
Among the eight dogs, only terriers have learned to follow the "stay" command.

Which of the following statements CANNOT be true?
.
(A) The group includes more females than males.
(B) The group includes fewer terriers than.shepherds.
(C) The group includes more shepherds than retrievers.
(D) More of the dogs have learned to stay than to heel.
(E) More of the dogs have learned to heel than to stay.
If each dog has learned to follow at least one of the two commands, all of the following must true EXCEPT:
.
(A) All retrievers have learned to heel.
(B) All shepherds have learned to heel.
(C) All terriers have learned to stay.
(D) No retriever has learned to stay.
(E) No shepherd has learned to stay.
If four of the dogs are male and four of the dogs are female, all of the following must be true EXCEPT:
.
(A) One of the dogs is a shepherd.
(B) Four of the dogs are retrievers.
(C) Three of the dogs are terriers.
(D) Three of the dogs have learned to stay.
(E) Four of the dogs have learned to heel.
If the group includes more shepherds than terriers, the minimum number of male dogs among the group that have learned to heel is
.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
If each dog has learned to follow at least one of the two commands, and if two of the dogs have learned to heel but not stay, it could be true that
.
(A) two of the dogs are female
(B) all of the dogs are male
(C) only one male dog has learned to heel
(D) one female dog has learned to stay
(E) two of the dogs are retrievers

Questions 15-18:

A particular seafood restaurant serves dinner Tuesday through Sunday. The restaurant is closed on Monday. Five entrees—snapper, halibut, lobster, mahi mahi, and tuna—are served each week according to the following restrictions:

Halibut is served on three days each week, but never on Friday.
Lobster is served on one day each week.
Mahi mahi is served on three days each week, but never on consecutive days.
Halibut and snapper are both served on Saturday and Sunday.
Tuna is served five days each week.
No more than three different entrees are served on any given day.

15 . On which of the following pairs of days could the restaurant's menu of entrees be identical?
.
(A) Friday and Sunday
(B) Tuesday and Wednesday
(C) Saturday and Sunday
(D) Wednesday and Friday
(E) Thursday and Friday

16 . Which of the following is a complete and accurate list of the days on which halibut and lobster may both be served?
.
(A) Tuesday, Thursday
(B) Tuesday, Wednesday, Thursday
(C) Monday, Tuesday, Wednesday
(D) Tuesday, Wednesday, Thursday, Friday
(E) Tuesday, Wednesday, Thursday, Saturday

17. If mahi mahi is served on Saturday, it could be true that
.
(A) snapper and mahi mahi are both served on.Sunday
(B) snapper and halibut are both served on Tuesday
(C) lobster and halibut are both served on Thursday
(D) tuna and snapper are both served on Saturday
(E) lobster and snapper are both served on Friday

18. Which of the following statements provides sufficient information to determine on which three days halibut is served?
.
(A) Mahi mahi and lobster are served on the same..day.
(B) Lobster and snapper are both served on Tuesday.
(C) Tuna is served on Saturday, and lobster is served on Tuesday.
(D) Mahi mahi is served on Saturday, and snapper is served on all but one of the six days.
(E) Tuna is served on Sunday, and snapper is served on Tuesday and Thursday.

Questions 19-23
Two or more essences out of a stock of five essences-- L, M, N, O, and P are used in making all perfumes by a manufacturer. He has learned that for a blend of essences to be agreeable it should comply with all the rules listed below.
A perfume containing L, should also contain the essence N, and the quantity of N should be twice as that of L.
A perfume containing M, must also have O as one of its components and they should be in equal proportion.
A single perfume should never contain N as well as O.
O and P should not be used together.
A perfume containing the essence P should contain P in such a proportion that the total amount of P present should be greater than the total amount of the other essence or essences used.

19. Among the following which is an agreeable formula for a perfume?

One part L, one part P
Two parts M, two parts L
Three parts N, three parts L
Four parts O, four parts M
Five parts P, five parts M

20.Adding more amount of essence N will make which of the following perfumes agreeable?

One part L, one part N, five parts P

B. Two parts M, two parts N, two parts P

C. One part M, one part N, one part P

D. Two parts M, one part N, four parts P

E. Two parts N, one part O, three parts P

21.Among the following, the addition of which combination would make an unagreeable perfume containing two parts N and one part P agreeable?

(A) One part L (B) One part M (C) Two parts N (D) One part O (E) Two parts P

22.Among the following which combination cannot be used together in an agreeable perfume containing two or more essences?

A.L and M

B.L and N

C.L and P

D.M and O

E.P and N

23.Among the below mentioned formulas, which can be made agreeable by the eliminating some or all of one essence ?

A. One part L, one part M, one part N, four parts P

B. One part L, two parts N, one part O, four parts P

C. One part L, one part M, one part O, one part P

D. Two parts L, two parts N, one part O, two parts P

E. Two parts M, one part N, two parts O, three parts P

ANSWER KEY

1. C

2. A

3. E

4. D

5. B

6. A

7. D

8. D

9. D

10. A

11. C

12. E

13. C

14. B

15. D

16. B

17. E

18. E

19. D

20. A

21. E

22. A

23. B

24.

Quant Test: 003

Marks: --/1

In the nineteenth century a person was X years old in the year X². How old was he in 1884?

Choose one answer.

	a. 43
	b. 68
	c. 78
	d. 58

Marks: --/1

A three-digit number in base 10 is written in base 9 and base 11 to give two numbers N₁ and N₂,respectively. What is the probability that N₁and N₂ are also three-digit numbers?

Choose one answer.

	a. 0.67
	b. 0.33
	c. 0.88
	d. 0.55
	e. 0.42

Marks: --/2

What is the remainder when (17)³⁶ + (19)³⁶ is divided by 111?

Choose one answer.

	a. 2
	b. 0
	c. 1
	d. 109

Marks: --/2

, then x is equal to

Choose one answer.

	a. 2
	b. 1/2
	c. -1
	d. 1

Marks: --/1

The smallest positive integer N such that sqrt(N) - sqrt(N - 1) is less than 0.01 is

Choose one answer.

	a. 2499
	b. 2501
	c. 2498
	d. 2502
	e. 2500

Marks: --/1

The numbers 123 456 789 and 999 999 999 are multiplied. How many times does digit â€˜9â€™ come in the product?

Choose one answer.

	a. 2
	b. 3
	c. 0
	d. 9

Marks: --/1

All the divisors of 360, including 1 and the number itself, are summed up. The sum is 1170. What is the sum of the reciprocals of all the divisors of 360?

Choose one answer.

	a. 2.75
	b. 1.75
	c. 3.25
	d. 2.5

Marks: --/1

The sum of the digits of Kalluâ€™s nightclub number is

Choose one answer.

	a. an even number
	b. a perfect square
	c. a perfect number
	d. a prime number

Marks: --/1

(x, y are integers)

Choose one answer.

	a. 7
	b. 9
	c. 3
	d. 5

Marks: --/1

To number the pages of a book, exactly 300 digits were used. How many pages did the book have?

Choose one answer.

	a. 136
	b. 135
	c. 138
	d. 137

Marks: --/1

The product of three consecutive odd numbers is 531117. What is the sum of the three numbers?

Choose one answer.

	a. 273
	b. 213
	c. 183
	d. 243

31
Marks: --/1
In a national hockey single elimination tournament, 303 teams are participating. How many games will be played before a team becomes the national champion?
Choose one answer.
a. 152

b. 77

c. 303

d. 302
32
Marks: --/1
The value of A + B that satisfies (630 + 6-30)(630 - 6-30) = 3A8B - 3-A8-B is
Choose one answer.
a. 40

b. 20

c. 60

d. 80

33
Marks: --/1
The digits 1, 2, 3, 4, and 5 are each used once to compose a five-digit number abcde such that the three-digit number abc is divisible by 4, bcd is divisble by 5, and cde is divisble by 3. Find the digit a.
Answer:
34
Marks: --/1
The number A4531B, where A and B are single-digit numbers, is divisible by 72. Then A + B is equal to
Choose one answer.
a. 8
  b. 4
  c. 7
  d. 5

35
Marks: --/1
What is the value of n such that n! = 3! Ã— 5! Ã— 7!
Answer:

36
Marks: --/2
What is the remainder when  1 + (11)¹¹+ (111)¹¹¹+....
+ (111...111)^111...111 is divided by 100? The last term contains ten 1's within the bracket as well as the power.
Choose one answer.
a. 30
  b. 10
  c. 40
  d. 0

37
Marks: --/1
What is the sum of the real values of x satisfying the equation
4 Ã— 3^2x+2 â€“ 9^2x = 243?
Choose one answer.
a. 5/2

b. 3/2

c. 3

d. 1
38
Marks: --/1
The last two digits of 4¹⁹⁹⁷ are
Choose one answer.
a. 36

b. 84

c. 64

d. 24

39
Marks: --/1
Let S = p2 + q2 + r2, where p and q are consecutive positive integers and
r = p Ã— q. Then (S)1/2 is
Choose one answer.
a. always irrational

b. an even integer

c. an odd integer

d. sometimes irrational

40
Marks: --/1
If the remainder when x¹⁰⁰
^{is divided by x^2 - 3x + 2 is ax+b, then value of a and b are :--}Choose one answer.
a. 2^100 and 1

b. 2^100 and 2 - 2^100

c. 2^100 and 1- 2^100

d. 2^100 - 1 and 2 - 2^100

41
Marks: --/2
What is the remainder when the number is divided by 99?
Choose one answer.
a. 18

b. 36

c. 33

d. 27

Marks: --/1

53⁵³ â€“ 27²⁷ is certainly divisible by

Choose one answer.

	a. 7
	b. 9
	c. 10
	d. 11

Marks: --/1

What is the largest prime whose cube divides 1!2!â€¦1001!?

Choose one answer.

	a. 977
	b. 997
	c. 973
	d. 991

Marks: --/1

The difference between the cubes of two consecutive positive integers is 1027. Then the product of these integers is

Choose one answer.

	a. 132
	b. 306
	c. 342
	d. 552

Marks: --/2

Let

, where a, b, c, and d are not equal to zero. Then the set of intersection of all values of S and T is

Choose one answer.

	a. {1}
	b. {0}
	c. {phi}
	d. {4}

Marks: --/1

Let q and r be the quotient and remainder when M, a five digit number, is divided by 100. For how many values of M is q + r divisible by 99?

Choose one answer.

	a. 909
	b. 908
	c. 989
	d. 900

Marks: --/1

A two-digit number is divided by the sum of its digits. The answer is 6. What is the product of the digits?

Choose one answer.

	a. 54
	b. 24
	c. 18
	d. 20

Marks: --/1

(CAT 2006)
When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?

Choose one answer.

	a. 7
	b. 8
	c. 5
	d. 6
	e. 10

Marks: --/2

Let n be the smallest positive number such that the number S = (8ⁿ)(5⁶⁰⁰) has 604 digits. Then the sum of the digits of S is

Choose one answer.

	a. 8
	b. 11
	c. 10
	d. 19

Marks: --/2

If n is a natural number such that 10¹² < n < 10¹³ and the sum of the digits of n is 2, then the number of values n can take is

Choose one answer.

	a. 12
	b. 11
	c. 13
	d. 10

Marks: --/1

The number (2n)! is divisible by
I. (n!)²
II. ((n âˆ’ 1)!)²
III. n! Ã— (n+1)!

Choose one answer.

	a. II and III only
	b. I and II only
	c. I and III only
	d. I, II and III

Marks: --/1

The number (2n)! is divisible by
I. (n!)²
II. ((n âˆ’ 1)!)²
III. n! Ã— (n+1)!

Choose one answer.

	a. II and III only
	b. I and II only
	c. I and III only
	d. I, II and III

Marks: --/2

Vinay has 128 boxes with him. He has to put least 120 oranges in one box and 144 oranges at the most. Then the least number of boxes containing the same number of oranges is

Choose one answer.

	a. 24
	b. 6
	c. 103
	d. 5

Marks: --/1

In a village of 2029 inhabitants, at least x villagers have the same English initials for their first name and their surname. The least possible value of x is

Choose one answer.

	a. 4
	b. 3
	c. 6
	d. 5
	e. 2

Marks: --/1

, then
(CAT 2005)

Choose one answer.

	a. R > 1.0
	b. 0.1 < R â‰¤ 0.5
	c. 0 < R â‰¤ 0.1
	d. 0.5 < R â‰¤ 1.0

Marks: --/2

The unit digit of

Choose one answer.

	a. 3
	b. 1
	c. 9
	d. 7

Marks: --/1

In how many ways can the number 105 be written as a sum of two or more consecutive positive integers?

Choose one answer.

	a. 4
	b. 6
	c. 5
	d. 7

Marks: --/1

For how many values of k is 12¹² the least common multiple of 6⁶, 8⁸, and k?

Choose one answer.

	a. 1
	b. 25
	c. 12
	d. 24

Marks: --/1

(CAT 2006)
The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

Choose one answer.

	a. 25
	b. 67
	c. 21
	d. 41
	e. 73

Marks: --/1

The single digits a and b are neither both nine nor both zero. The repeating decimal 0.abababab... = V />= O />is expressed as a fraction in lowest terms. How many different denominators are possible?

Choose one answer.

	a. 5
	b. 4
	c. 6
	d. 3

Marks: --/1

The sum of 20 distinct numbers is 801. What is their minimum LCM possible?

Choose one answer.

	a. 360
	b. 42
	c. 840
	d. 480

Marks: --/1

What is the remainder when

is divided by 13?

Choose one answer.

	a. 6
	b. 7
	c. 1
	d. 10