# CAT 2002

1) How many numbers greater than 0 and less than a million can be formed with the digits 0, 7 and 8?

a) 486

b) 1086

c) 728

d) none of these

2) If pqr = 1, the value of the expression 1/(1+p+q^-1) + 1/(1+q+r^-1) + 1/(1+r+p^-1) is equal to

a) p + q + r

b) 1/(p + q + r)

c) 1

d) (p^-1) + (q^-1) + (r^-1)

3) 7^(6n) – 6^(6n) where n is an integer >0, is divisible by

a. 13
b. 127
c. 559
d. none of these

4) In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column?

a. 56
b. 896
c. 60
d. 768

5) If u, v, w and m are natural numbers such that u^m + v^m = w^m, then one of the following is true.

a. m >= min(u, v, w)
b. m < min(u, v, w)
c. m <= min (u, v, w)
d. none of these

6) Only a single rail track exists between station A and B on a railway line. One hour after the north bound super fast train N leaves station A for Station B, a south bound passenger train S reaches station A from station B. The’ speed of the super fast train is twice that of a normal express train E, while the speed of a passenger train S is half that of E. On a particular day N leaves for station B from Station A, 20 minutes behind the normal schedule. In order to maintain the schedule both N and S increased their speed. If the super fast train doubles its speed, what should be the ratio (approximately) of the speed of passenger train to that of the super fast train so that passenger train S reaches exactly at the scheduled time at station A on that day.

a. 1:3
b. 1:4
c. 1:5
d. 1:6

7) Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is:

a. 1/2
b. 2/3
c. 1/4
d. 3/4

8) The area of the triangle whose vertices are (a, a), (a + 1, a + 1), (a + 2, a) is:

a. a^3
b. 1
c. 2a
d. 2^(1/2)

9) A train approaches a tunnel AB. Inside the tunnel is a cat located at a point that is 3/8 of the distance AB measured from the entrance A. When the train whistles the cat runs. If the cat moves to the entrance of the tunnel, A, the train catches the cat exactly at the entrance. If the cat moves to the exit, B, the train catches the cat at exactly the exit. The speed of the train is greater than the speed of the cat by what order?

a. 3:1
b. 4:1
c. 5:1
d. none of these

10) Six persons are playing a card game. Suresh is facing Raghubir who is to the left of Ajay and to the right of Pramod. Ajay is to the left of Dhiraj. Yogendra is to the left of Pramod. If Dhiraj exchanges his seat with Yogendra and Pramod exchanges with Raghubir, who will be sitting to the left of Dhiraj?

a. Yogendra
b. Raghubir
c. Suresh
d. Ajay

DIRECTIONS for questions 61 and 62: Answer these questions based on the information given below.

Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters.

11) How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)?

a. 7920
b. 330
c. 14640
d. 419430

12) How many three-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

a. 990
b. 2730
c. 12870
d. 15600

DIRECTIONS for questions 63 to 75:
Answer the questions independent of each other….

13) After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

a. 80
b. 76
c. 41
d. 53

14) Three pieces of cakes of weight 4.5 lbs, 6.75 lbs and 7.2 lbs respectively are to be divided into parts of equal weights. Further, each part must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?

a. 54
b. 7
c. 20
d. none of these

15) At a bookstore. “MODERN BOOK STORE” is flashed using neon lights. The words are individually flashed at intervals of 2.5, 4.25, 5.125 seconds respectively and each word is put off after a second. The least time after which the full name of the bookstore can be read again is:

a. 49.5 seconds
b. 73.5 seconds
c. 1744.5 seconds
d. 855 seconds

16) The number of real roots of the equation (A^2)/x + (B^2)/(x-1) = 1 where A and B are real numbers not equal to zero simultaneously is

a. None
b. 1
c. 2
d. 1 or 2

17) When 2^256 is divided by 17 the remainder would be

a. 1
b. 16
c. 14
d. none of these

18) 10 straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. The total number of regions (including finite and infinite regions) into which the plane would be divided by the lines is

a. 56
b. 255
c. 1024
d. not unique

19) Suppose, for any real number x, [x] denotes the greatest integer less than or equal to x. Let L(x, y) = [x] + [y] + [x + y] and R(x, y) = [2x] + [2y]. Then it’s impossible to find any two positive real numbers x and y for which

a. L(x, y) = R(x, y)
b. L(x, y) != R(x, y)
c. L(x, y) < R(x, y)
d. L(x, y) > R(x, y)

20) A child was asked to add first few natural numbers (that is, 1 + 2 + 3 …) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was:

a. less than 10
b. 10
c. 15
d. more than 15

21) A car rental agency has the following terms. If a car is rented for 5 hours or less the charge is 60 per hour or Rs.12 per kilometer whichever is more. On the other hand, if the car s rented for more than 5 hours, the charge is Rs. 50 per hour or Rs.7.50 per Kilometer which ever is more. Akil rented a car from this agency, drove it for 30 kilometers and ended up paying Rs.300. For how many hours did he rent the Car?

a. 4
b. 5
c. 6
d. none of these

22) Amol was asked to calculate the arithmetic mean often positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic means was 1.8 more than what it should have been. Then b – a equals

a. 1
b. 2
c. 3
d. none of these

23) If x^2 + 5(y^2) + z^2 = 2y(2x + z) then which of the following statements are necessarily true?

A. x = 2y.
B. x = 2z
C. 2x = z.
a. Only A
b. Only B and C
c. Only A and B
d. none of these

24) Let S denote the infinite sum 2 + 5x + 9(x^2) + 14(x^3) + 20(x^4) + ….., where |x| < 1 and the coefficient of x^(n-1) is n(+3)/2, (n = 1, 2, ….). Then S equals

a.  (2-x) / (1-x)^3
b.  (2-x) / (1+x)^3
c.  (2+x) / (1-x)^3

d. (2+x) / (1+x)^3

25) Shyam visited Ram on vacation. In the mornings, they both would go for yoga. In the evenings they would play tennis. To have more fun, they indulge only in one actively per day, i.e., either they went for yoga or played tennis each day. There were days when they were lazy and stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they stayed at home, and a total of 22 days when they did yoga or played tennis. For how many days Shyam stayed with Ram?

a. 32
b. 24
c. 30
d. none of these

DIRECTIONS for questions 76 and 77: Answer these questions based on the information given below.
A boy is asked to put in a basket one mango when ordered ‘One’, one orange when ordered ‘Two’, one apple when ordered ‘Three’ and is asked to take out from the basket one mango and an orange when ordered ‘Four’. A sequence of orders is given as:
12332142314223314113234

26) How many total oranges were in the basket at the end of the above sequence?

a. 1
b. 4
c. 3
d. 2

27) How many total fruits will be in the basket at the end of the above order sequence?

a. 9
b. 8
c. 11
d. 10

DIRECTIONS for questions 78 to 90: Answer the questions independent of each other….

28) A rich merchant bad collected many gold coins. He did not want anybody to know about them. One day, his wife asked, “How many gold coins do we have?” After pausing a moment, he replied, “Well! If I divide the coins into two unequal numbers, then 48 times the difference between the two numbers equals the difference between the squares of the two numbers.” The wife looked puzzled. Can you help the merchant’s wife by finding out how many coins the merchant has?

a. 96
b. 53
c. 43
d. none of these

29) On a straight road XY, 100 metres long, five heavy stones are placed two metres apart beginning at the end X. A worker, starting at X, has to transport all the stones to Y, by carrying only one stone at a time. The minimum distance he has to travel (in metres) is:

a. 472
b. 422
c. 744
d. 844

30) Four horses are tethered at four corners of a square plot of side 14 metres (m) so that the adjacent horses can reach one another. There is a small circular pond of area 20 m^2 at the centre. The area left unglazed is:

a. 22 m^2
b. 42 m^2
c. 84 m^2
d. l68 m^2

31) If f(x) = log{(l+x)/(l-x)}, then f(x) + f(y) is:

a. f(x + y)
b. f{(x + y) / (1 + xy)}
c. (x + y) f{ l/(1 + xy)}
d. f(x) + f(y)/(l + xy)

32) The length of the common chord of two circles of radii 15 cm and 20 cm, whose centers are 25 cm apart, is (cm):

a. 24
b. 25
c. 15
d. 20

33) In a triangle ABC, the internal bisector of the angle A meets BC at D. If AB = 4, AC = 3 and <A = 60°, then length of AD is:

a. 2 sqrt {3}
b. 12 sqrt {3 / 7}
c. 15 sqrt {3 / 8}
d. 6 sqrt {3/7}

34) If there are 10 positive real numbers n1 < n2 < n3 ….. <n10. How many triplets of these numbers (n1, n2, n3), (n2, n3, n4), …. can be generated such that in each that in each triplet the first number is always less than the second number, and the second number is always less than the third number?

a. 45
b. 90
c. 120
d. 180

35) 3 small pumps and a large pump are filling a tank. Each of the three small pumps works at 2/3rd the rate of the large pump. If all 4 pumps work at the same time, they should fill the tank in what fraction of the time that it would have taken the large pump alone?

a. 4/7
b. 1/3

c. 2/3
d. 3/4

36) Davji Shop sells samosas in boxes of different sizes. The samosas are priced at Rs.2 per samosa up to 200 samosas. For every additional 20. samosas, the price of the whole lot goes down by 10 paise per samosa. What should be the maximum size of the box that would maximize the revenue?

a. 240
b. 300
c. 400
d. none of these

37) It takes 6 technicians a total of 10 hours to build a new server from Direct Computer, with each working at the same rate. If six technicians start to build the server at 11.00 AM, and one technician per hour is added beginning at 5.00 PM, at what time will the server be complete?

a. 6:40pm
b. 7:00pm
c. 7:20pm
d. 8pm

38)

In the above figure, ACB is a right angled triangle. CD is the altitude. Circles are inscribed within the triangle ACD, BCD. P and Q are the centers of the circles. The distance PQ is

a. 5
b. sqrt {50}
c. 7
d. 8

39) Three travelers are sitting around a fire, and are about to eat a meal. One of them has five small loaves of bread; the second has three small loaves of bread. The third has no food, but has eight coins. He offers to pay for some bread. They agree to share the eight loaves equally among the three travelers, and the third traveler will pay eight coins for his share of the eight loaves. All loaves were the same size. The second traveler (who had three loaves) suggests that he be paid three coins and that the first traveler be paid five coins. The fir traveler says that he should get more than five coins. How much the first traveler should get?

a. 5
b. 7
c. 1
d. none of these

40) A piece of string is 40 centimeters long. It is cut into three pieces. The longest piece is 3 times as long as the middle-sized piece and the shortest piece is 23 centimeters shorter than the longest piece. Find the length of the shortest piece.

a. 27
b. 5
c. 4
d. 9

DIRECTIONS for questions 91 to 92: Answer these questions based on the following diagram. In the diagram below:
<ABC = 90°= <DCH = <DOE = <EHK = <FKL = <GLM = <LMN.
AB = BC = 2CH = 2CD = EH = FK = 2HK = 4KL = 2LM = MN

41) The magnitude of <FGO =

a. 30°
b. 45°
c. 60°
d. none of these

42) The ratio of the areas of the two quadrangles ABCD and DEFG is

a. 1 : 2
b. 2 : 1
c. 12 : 7
d. none of these

DIRECTIONS for questions 93 to 100: Answer the questions independent of each other….

43) Mayank, Mirza, Little and Jaspal bought a motorbike for \$60.00. Mayank paid one half of the sum of the amounts paid by the other boys. Mirza paid one third of the sum of the amounts paid by the other boys; and Little paid one fourth of the sum of the amounts paid by the other boys. How much did Jaspal have to pay?

a. 15
b. 13
c. 17
d. none of these

44) The owner of a local jewellery store hired 3 watchmen to guard his diamonds, but a thief still got in and stole some diamonds. On the way out, the thief met each watchman, one at a time. To each he gave 1/2 of the diamonds he had then, and 2 more besides. He escaped with one diamond. How many did he steal originally?

a. 40
b. 36
c. 25
d. none of these

45) Neeraj has agreed to mow the front lawn, which is a 20m by 40m rectangle. The mower mows a 1 m wide Strip. If Neeraj starts at one corner and mows around the lawn toward the center, about how many times would he go round before he has mowed half the lawn?

a. 25
b. 3.5
c. 3.8
d. 4.0

46) If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?

a. 5/3
b. sqrt(19)
c. 13/3
d. none of these

47) The nth element of a series is represented as Xn = (–1)^n Xn-1 If Xn = x and x > 0 then the following is always true

a. Xn is positive if n is even
b. Xn is positive if n is odd
c. Xn is negative if n is even
d. None of these

48) Number S is obtained by squaring the sum of digits of a two digit number D. If difference between S and D is 27, then the two digit number D is:

a. 24
b. 54
c. 34
d. 45

49) On a 20 km tunnel connecting two cities A and B there are three gutters. The distance between gutter 1 and 2 is half the distance between gutter 2 and 3. The distance from city A to its nearest gutter, gutter 1 is equal to the distance of city B from gutter 3. On a particular day the hospital in city A receives information that an accident has happened at the third gutter. The victim can be saved only if an operation is started within 40 minutes. An ambulance started from city A at 30 km/hr and crossed the first gutter after 5 minutes. If the driver had doubled the speed after that, what is the maximum amount of time the doctor would get to attend the patient at the hospital? Assume 1 minute is elapsed for taking the patient into and out of the ambulance.

a. 4 minutes
b. 2.5 minutes
c. 1.5 minutes

d. Patient died before reaching the hospital

50)  In the figure giver below, ABCD is a rectangle. The area of the isosceles right triangle ABE = 7cm^2. (FC) = 3(BE). The area of ABCD (in cm^2) is:

a. 21
b. 28
c. 42
d. 56

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