CAT 2000

Q1. Let D be recurring decimal of the form, D =0.a1 a2 a1 a2 a1 a2, where digits a1 and a2 lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?
a. 18
b. 108
c. 198
d. 288

Q2.
x 1 2 3 4 5 6
y 4 8 14 22 32 44

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?
a. y = a + bx
b. y = a + bx + cx^2
c. y = e^(a + bx)
d. None of the above

Q3. If a1 = 1 and an+1 = 2an + 5, n = 1, 2, …., then
a100 is equal to:
a. (5 x 2^99 – 6)
b. (5 x 2^99 + 6)
c. (6 x 2^99 + 5)
d. (6 x 2^99 – 5)

Q4. What is the value of the following expression?
(l/(2^2-l))+(1/(4^2-1))+(l/(6^2-l))+ +(1/(20^2— 1)
a. 9/19
b. 10/19
c. 10/21
d. 11/21

Q5. A truck travelling at 70 kilometres per hour uses 30% more diesel to travel a certain distance than it does when it travels at the speed of 50 kilometres per hour. If the truck can travel 19.5 kilometres on a litre of diesel at 50 kilometres per hour, how far can the truck travel on 10 litres of diesel at a speed of 70 kilometres per hour?
a. 130
b. 140
c. 150
d. 175

Q6. Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is:
a. n
b. n+1
c. k x n, where k is a function of n
d. n+(2/7)

Q7. If x > 2 and y > -1, then which of the following statements is necessarily true?
a. xy > -2
b. – x < 2y
c. xy < -2
d. – x > 2y

Q8. One red flag, three white flags and two blue flags are arranged in a line such that,
A. no two adjacent flags are of the same colour
B. the flags at the two ends of the line are of different colours.
In how many different ways can the flags be arranged?
a. 6
b. 4
c. 10
d. 2

Q9. Let S be the set of integers x such that:
1. 100 2. x is odd
3. x is divisible by 3 but not by 7.
How many elements does S contain?
a. 16
b. 12
c. 11
d. 13

Q10. Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?
a. xyz^2 is odd
b. Z(x – y)^2 is even
c. (x + y)(x + y – z)^2 is even
d. (x- y)(y + z) (x + y – z) is odd

Q11. Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all elements of S. With how many consecutive zeros will the product end?
a. 1
b. 4
c. 5
d. 10

Q12. What is the number of distinct triangles with integral valued sides and perimeter 14?
a. 6
b. 5
c. 4
d. 3

Q13. Let N = 1421 x 1423 x 1425. What is the remainder when N is divided by 12?
a. 0
b. 9
c. 3
d. 6

Q14. The integers 34041 and 32506 when divided by a three-digit integer a leave the same remainder. What is n?
a. 289
b. 367
c. 453
d. 307

Q15. Each of the numbers x1, x2, … ,x n34, is equal to 1 or- 1. Suppose,
x1x2x3x4 + x2x3x4x5 + x3x4x5x6+ ……. +
xn-3xn-2xn-1xn+ xn-1xnx1x2+ xnx1x2x3=0, then:
a. n is even
b. n is odd
c. n is an odd multiple of 3
d. n is prime

Q16. The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.
Age-group Percentage
Below
15 years 30
15-24 17.75
25-34 17.00
35-44 14.50
45-54 12.50
55-64 7.10
65andabove 1.15

If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group?
a. 82.8
b. 90.8
c. 80.0
d. 90.0

Q17. Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number nine appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
a. 10000
b. 2430
c. 3402
d. 3006

Directions for questions 18 to 19:
A, B, C are three numbers. Let @ (A, B) = average of A and B, /(A, B) = product of A and B, and X(A, B) = the result of dividing A by B

Q18. The sum of A and B is given by:
a. /(@ (A, B),
b. X(@(A, B), 2)
c. @(/(A, B), 2
d. @ (X(A, B), 2)

Q19. Average of A, B and C is given by:
a. @ (/(@(/(B, A), 2), C), 3)
b. X(@(/(@(B, A), 3), C), 2)
c. /((X(@ (B, A), 2), C), 3)
d. /(X(@ (/(@(B, A), 2), C), 3), 2)

Directions: for Q.20 to 21: For real numbers x, y, let
f(x, y) = Positive square-root of(x + y), if(x + y)^(0.5) is real
= (x + y)^2; otherwise.
g(x, y) = (x + y)^2, if(x + y)^(0.5) is real
= – (x + y) otherwise

Q20. Which of the following expressions yields a positive value for every pair of non-zero real numbers (x, y)?
a. f(x, y) – g(x, y)
b. f(x, y) – (g(x, y))^2
c. g(x, y) – (f(x, y))^2
d. f(x, y) + g(x, y)

Q21. Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?
a. Both x and y are less than-1
b. Both x and y are positive
c. Both x and y are negative
d. y>x

Directions for Q.22 to 24: For three distinct real numbers x, y and z, let
f(x, y, z) = mm (max(x, y), max (y, z), max (z, x))
g(x, y, z) = max (min(x, y), mm (y, z), mm (z, x))
h(x, y, z) = max (max(x, y), max(y, z), max (z, x))
j(x, y, z) = mm (mm (x, y), min(y, z), mm (z, x))
m(x, y, z) = max (x, y, z)
n(x, y, z) = mm (x, y, z)

Q22. Which of the following is necessarily greater than 1?
a. (h(x, y, z) – f(x, y, z)) /j(x, y, z)
b. j(x, y, z)/h(x, y, z)
c. f(x, y, z)/g(x, y, z)
d. (f(x, y, z) + h(x, y, z)-g(x, y, z))/j(x, y, z)

Q23. Which of the following expressions is necessarily equal to 1?
a. (t(x, y, z)- m(x, y, z))/(g(x, y, z)-h(x,y, z))
b. (m(x, y, z)-f(x, y, z))/(g(x, y, z)-n(x, y, z))
c. (j(x, y, z)-g(x, y, z)) /h(x, y, z)
d. (f(x, y, z)-h(x, y, z) /f(x, y, z)

Q24. Which of the following expressions is indeterminate?
a. (f(x, y, z)-h(x, y, z))/(g(x, y, z) —j(x, y, z))
b. (f(x, y, z) + h(x, y, z) + g(x, y, z) +j(x, y, z))/(j(x, y, z) + h(x, y, z)-m(x, y, z) – n(x, y, z))
c. (g(x, y, z)-j(x, y, z))/(f(x, y, z)-h(x, y, z))
d. (h(x, y, z) fix, y, z))/(n(x, y, z) – g(x, y, z))

Directions for Q. 25 to 26:
There are five machines A, B, C, D, and E situated on a straight line at distances of 10 metres, 20 metres, 30 metres, 40 metres and 50 meters respectively from the origin of the line. A robot is stationed at the origin of the line. The robot serves the machines with raw material whenever a machine becomes idle. All the raw material is located at the origin. The robot is in an idle state at the origin at the beginning of a day. As soon as one or more machines become idle, they send messages to the robot- station and the robot starts and serves all the machines from which it received messages. If a message is received at the station while the robot is away from it, the robot takes notice of the message only when it returns to the station. While moving, it serves the machines in the sequence in which they are encountered, and then returns to the origin. If any messages are pending at the station when it returns, it repeats the process again. Otherwise, it
remains idle at the origin till the next message (s) is received.

Q25. Suppose on a certain day, machines A and D have sent the first two messages to the origin at the beginning of the first second, and C has sent a message at the beginning of the 5th second and B at the beginning of the 6th second, and E at the beginning of the 10th second. How much distance in metres has the robot travelled since the beginning of the day, when it notices the message of E? Assume that the speed of movement of the robot is 10 metres per second.
a. 140
b. 80
c. 340
d. 360

Q26. Suppose there is a second station with raw material for the robot at the other extreme of the line which is 60 metres from the origin, that. is, 10 meters from E. After finishing the services in a trip, the robot returns to the nearest station. If both stations are equidistant, it chooses the origin as the station to return to. Assuming that both stations receive the messages sent by the machines and that all the other data remains the same, what would be the answer to the above question?
a. 120
b. 140
c. 340
d. 70

Directions for questions 27 to 29:
Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
(a) if f(x)=3f(-x)
(b) if f(x)= -f(-x)
(c) if f(x) = f(-x)
(d) if 3f(x) = 6f(-x), for x^(3)0.


c. if f(x) = f(-x)
d. if 3f(x) = 6f(-x), for x^(3)0

Directions for questions 30 and 31:
There are three bottles of water, A, B, C, whose capacities are 5 litres, 3 litres, and 2 litres respectively. For transferring water from one bottle to another and to drain out the bottles, there exists a piping system. The flow through these pipes is computer controlled. The computer that controls the flow through these pipes can be fed with three types of instructions, as explained below:

Initially, A is full with water, and B and C are empty.

Q30. After executing a sequence of three instructions, bottle A contains one litre of water. The first and the third of these instructions are shown below:
First instruction: FILL (C, A)
Third instruction FILL (C, A)
Then which of the following statements about the instruction is true?
a. The second instruction is FILL (B, A)
b. The second instruction is EMPTY (C, 13)
c. The second instruction transfers water from B to C
d. The second instruction involves using the water in bottle A.

Q31. Consider the same sequence of three instructions ‘and the same initial state mentioned above. Three more instructions are added at the end of the above sequence to have A contain 4 litres of water. In this total sequence of six instructions, the fourth one is DRAIN (A). This is the only DRAIN instruction in the entire sequence. At the end of the execution of the above sequence, how
much water (in litres) is contained in C?
a. One
b. Two
c. Zero
d. None of these

Directions for questions 32 to 33:
For a real number x let
f(x) = 1/(1+x), if x is non-negative
= 1+x, if x is negative
f^(n)(x) = f(f^(n-1)(x)), n = 2, 3…..

Q32. What is the value of the product, f(2) f^2(2)f^3(2)f^4(2)f^5(2)?
a. 1/3
b. 3
c. 1/18
d. None of these

Q33. r is an integer >= 2. Then, what is the value of
f^(r-1)(-r) + f^(r)(-r) + f^(r+1)(-r)?
a. -1
b. 0
c. 1
d. None of these

Directions for questions 34 to 38:
Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated, The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup. The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q34. What is the total number of matches played in the tournament?
a. 28
b. 55
c. 63
d. 35

Q35. The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is:
a. 5
b. 6
c. 7
d. 4

Q36. What is the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of first stage?
a. 1
b. 2
c. 3
d. 4

Q37. What is the number of rounds in the second stage of the tournament?’
a. 1
b. 2
c. 3
d. 4

Q38. Which of the following statements is true?
a. The winner will have more wins than any other team in the tournament.
b. At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.
c. It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.
d. The number of teams with exactly one win in the second stage of the tournament is 4.

Directions for Questions 39 to 55: Answer each of the questions independently?

Q39. Let N = 55^3 + 17^3 – 72^3. N is divisible by:
a. both 7 and 13
b. both 3 and 13
c. both 17 and 7
d. both 3 and 17

Q40. If x^2 + y^2 = 0.1 and |x – y| = 0.2, then | x | + | y | is equal to:
a. 0.3
b. 0.4
c. 0.2
d. 0.6

Q41. ABCD is a rhombus with the diagonals AC and BD intersection at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?
a. x + y = -1
b. x – y = -1
c. x + y = 1
d. None of the above

Q42. Consider a circle with unit radius. There are 7 adjacent sectors, S1, S2, S3,….., S7 in the circle such that their total area is (1/8)th of the area of the circle. Further, the area of the jth sector is twice that of the (j-1)th sector, for j=2, …… 7.
What is the angle, in radians, subtended by the arc of SI at the centre of the circle?
a. pi/508
b. pi/2040
c. pi/1016
d. pi/1524

Q43. There is a vertical stack of books marked 1, 2 and 3 on Table-A, with 1 at the bottom and 3 on top. These are to be placed vertically on Table-B with 1 at the bottom and 2 on the top, by making a series of moves from one table to the other. During a move, the topmost book, or the topmost two books, or all the three, can be moved from one of the tables to the other. If there are any books on the other table, the stack being transferred should be placed, on top of the existing books, without changing the order of books in the stack that is being moved in that move. If there are no books on the other table, the stack is simply placed on the other table without disturbing the order of books in it. What is the minimum number of moves in which the above task can be accomplished?
a. One
b. Two
c. Three
d. Four

Q44. The area bounded by the three curves |x+y| = 1, |x| = 1, and |y| = 1, is equal to:
a. 4
b. 3
c. 2
d. 1

Q45. If the equation x^3 – ax^2 + bx – a = 0 has three real roots, then it must be the case that,
a. b =1
b. b != 1
c. a = 1
d. a != 1

Q46. If a, b, c are the sides of a triangle, and a^2 + b^2 + c^2 = bc + ca + ab, then the triangle is:
a. equilateral
b. isosceles
c. right angled
d. obtuse angled

Q47. In the figure AB = BC = CD = DE = EF = FG = GA. Then angle DAE is approximately:

a. 15°
b. 20°
c. 30°
d. 25°

Q48. A shipping clerk has five boxes of different but unknown weights each weighing less than 100 kgs. The clerk weighs the boxes in pairs. The weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, l2Oand 121 kgs. What is the weight, in kgs, of the heaviest box?
a. 60
b. 62
c. 64
d. cannot be determined

Q49. There are three cities A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the, first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly there are 23 routes from B to C (including those via A).
How many roads are there from A to C directly?
a. 6
b. 3
c. 5
d. 10

Q50. The set of all positive integers is the union of two disjoint subsets:
{t(1), f(2),…..f(n), …} and {g(1),g(2)….,g(n)…..}, where
f(1)<f(2)<…..= 1.
What is the value of g(1)?
a. Zero
b. Two
c. One
d. Cannot be determined

Q51. ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either oft he two adjacent vertices. When it reaches E, the frog stops and says there. Let a be the number of distinct paths of exactly n jumps ending in E. Then, what is the value of
a(2n-1)?
a. Zero
b. Four
c. 2n-1
d. Cannot be determined

Q52. For all non-negative integers x and y, f(x, y) is defined as below:
f(0, y) = y + l
f(x + 1, 0) = f(x, 1)
f(x+ l, y+ 1)= f(x, f(x+ l, y))
Then, what is the value of f(l,2)?
a. Two
b. Four
c. Three
d. Cannot be determined

Q53. Convert the number 1982 from base 10 to base 12. The result is:
a. 1182
b. 1912
c. 1192
d. 1292

Q54. Two full tanks, one shaped like a cylinder and the other like a cone, contain jet fuel. The cylindrical tank holds 500 litres more than the conical tank. After 200 litreS of fuel has been pumped out from each tank the cylindrical tank contains twice the amount of fuel in the conical tank. How many litres of fuel did the cylindrical tank have when it was full?
a. 700
b. 1000
c. 1100
d. 1200

Q55. A farmer has decided to build a wire fence along one straight side of this property. For this, he planned to place several fence-posts at six metre intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was five less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them eight metres apart. What is the length of the side of his property and how many posts did he buy?
a. 100 metres, 15
b. 100 metres 16
c. 120 metres, 15
d. 120 metres 16

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