This document contains 69 multiple choice questions from various topics including mathematics, probability, geometry and more. The questions range in difficulty and cover concepts such as sums, ratios, percentages, coordinate geometry and other topics. The correct answers to each question are provided as options a, b, c or d.

1. Section – II
51. ( ) ( ) ( )
What will be the remainder when 26 + 1 + 25 + 2 + 24 + 3 + ... + 1 + 26 is divided
27 27 27 27
( )
by 27?
a. 26 b. 1
c. 0 d. None of these
52. The pirce of the scenery is Rs. R1. A shopkeeper gives a discount of x% on R1 and reduces its price
to R2. He gives a further discount of x% on the reduced price R2 to reduce it further to R3, which
reduces it by Rs. 415. A customer bargains with him and takes an x% discount on R3 and buys the
scenery for Rs. 3,362.8. Find the original price R1 of the scenery.
a. Rs. 5,349 b. Rs. 4,213
c. Rs. 4,488 d. Rs. 4,613
53. If a, b and c are positive integers such that ab–c . bc –a . c a–b = 1 , then how many sets of solutions
are possible for a, b and c?
a. One b. Two
c. Infinite solutions d. All integer values
1 1 f(g(x))
54. If f(x) = x + and g(x) = , then find x for which = 1.
x 1+ x g(x)
a. –1 b. 0
c. More than one value exists d. No such value exists
55. If angles of a triangle are integers x, y and z and log (x × y × z) = 3 log x + 3 log 2 + 2 log 3, then
the angles x, y and z are respectively
a. 30°, 80° and 70° b. 10°, 80° and 90°
c. 40°, 60° and 80° d. Cannot be determined
3 2 3 3 2 2
56. If x + y + 3xy = 125 and 27x + y + 27x y + 9xy = 0 , then find (x + y).
a. –10 b. 100
c. –5 d. Data insufficient
57. How many five-digit numbers can be formed so that at even place there is an even digit and at odd
place there is an odd digit? One digit can be used twice at maximum. (Assume zero is an even
number.)
a. 1,250 b. 1,500
c. 3,000 d. 2,500
58. An equilateral triangle ABC is kept vertically with side BC on ground. The sun is in the plane
containing the line AD and perpendicular to the plane of the triangle. Thus the shaded part shown in
the diagram is the shadow of ∆ABC.
The sun ray passing through vertex A makes an angle of 30° with ground i.e. at A ′ . Find the area
of the shadow of the triangle if one of the side of ∆ABC is 4 3 .
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2. A
C
D
B
A
a. 6 cm2 b. 18 cm2
c. 36 cm2 d. 24 cm2
59. Let p × q = 300 is such that p + q = Prime number and when p is divided by q it gives the remainder
1. Find the remainder when (p – q)2 is divided by (p + q).
a. 21 b. 36
c. Data insufficient d. None of these
60. Vikram has a few gold coins. He wants to distribute these coins among his 3 sons and 3 daughters.
1
He also wants to keep some coins with himself. He keeps one coin with himself and gives of the
5
1
remaining to his first son, again keeps one and gives of the remaining to his second son, and
5
does the same with his last son. He then distributes the remaining coins among his 3 daughters in
the ratio 23 : 46 : 55. Find the minimum number of coins that Vikram would have had.
a. 246 b. 124
c. 624 d. None of these
61. If a year is of 360 days, where all months are of 30 days, what is the probability that a particular day
appears on Monday and that is an even date of an even month, if January 1 is a Monday?
1 13
a. b.
30 360
11 1
c. d.
360 28
62. If f(x) = x4 + 1, and x ≠ 0 , then which one of the following options is correct?
f(x)  1  1  1
a. = f(x) − f   b. f(x). f   = f(x) + f 
 1 x x  x
f 
x
c. Both (a) and (b) are right d. None of these
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3. Direction for questions 63 and 64: Answer the questions based on the following information.
Both Salman and Vivek want to marry Aishwarya. Salman and Vivek are standing at point A of a circular
track whereas Aishwarya is standing at point B which is diametrically opposite to point A. Salman and
Aishwarya start running towards each other along the circular track and Vivek runs along the straight line
AB and after reaching B he runs along the circular track in the direction of Aishwarya. All three start
simultaneously and eventually they meet at the same point together. (No one completes the round.)
14 0 m
A B
63. If the ratio of speed of Salman and Vivek is 1 : 1, what is the distance travelled by Aishwarya till the
meeting?
a. 40 m b. 60 m
c. 80 m d. Data insufficient
64. If the ratio of speeds of Salman and Vivek is 1 : 2 and Aishwarya runs at a speed of 20 m/sec, after
how much time since start will all three meet?
a. 2 s b. 3 s
c. 4 s d. 5 s
65. If k is any natural number such that 100 ≤ k ≤ 300 , how many values of k exist such that k! has ‘x’
zeros at the end and (k + 1)! has ‘x + 3’ zeros at the end?
a. 0 b. 2
c. 4 d. None of these
66. There are 5 questions with 4 options each. Out of 4 options one is right and 3 are wrong. A right
1
answer adds 1 mark, and one wrong answer deducts . What is the probability of getting a zero if
4
all the questions are mandatory?
1 81
a. b.
6 256
405
c. d. None of these
1024
4 9 16 25
67. The infinite sum 1 + + + + + ... equals
3 32 33 34
9 7
a. b.
2 2
5 3
c. d.
2 2
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4. 68. In the following figure, CAD and CBE are straight lines. If CA is the diameter of the circle, find
∠ADE .
D
A
C
B
E
a. 90° b. 110°
c. 45° d. Data insufficient
69. In a chessboard, if we start writing week days, starting from Monday from top-left block towards
right, then second line left to right till all the blocks of the chessboard are filled. Find the number of
Sundays in the white blocks if top most left block is white.
a. 2 b. 3
c. 7 d. 6
α 2β2 (1 + α + β) + (α + β) (αβ + 1)
70. If α and β are the roots of x2 + 3x + 4 = 0, then find the value of .
αβ
28
a. − 28 b.
3 3
14
c. d. None of these
3
Direction for questions 71 and 72: Answer the questions based on the following information.
15 0 m
X Y
A 10 m /s 15 m /s B
C 25 m /s
XY is a running track of 150 m. Three runners A, B and C start running at time t = 0 from the directions
given in the figure with the given speeds. As soon as they reach the opposite end they return and keep
running at a constant speed. A and C start from X and B starts from Y.
71. After how many seconds will A, B and C be together at any of the edges?
a. 30 s b. 60 s
c. 90 s d. Never
72. Which of the following are false?
A. If the speeds of A, B and C are doubled, then the number of meetings of A, B and C per 100 m
will also get doubled.
B. B and C will never meet at any edge.
C. The first meeting of B and C will be 2 s before the meeting of A and B for the first time.
a. A and B b. B and C
c. A and C d. A, B and C
Mock CAT – 1/2004 Page 17

5. 73.
P
B
C
Q
A E
D
Point Q lies at the centre of the square base (ABCD) of the pyramid. The pyramid’s height PQ
measures exactly one-half the length of each edge of its base, and point E lies exactly half-way
between C and D along one edge of the base. What is the ratio of the surface area of any one of the
pyramid’s four triangular faces to the area of the shaded triangle?
a. 3 : 2 b. 5 :1
c. 4 3 : 3 d. 2 2 : 1
Direction for questions 74 to 76: Answer the questions based on the following information.
Numbers from 1 to 56 are written on a chessboard as given below, from positions a1 to h7.
a b c d e f g h
1 1 2 3 4 5 6 7 8
2 9 10 11 12 13 14 15 16
3 17
4 25 32
5 33 40
6 41 48
7 49 50 51 52 53 54 55 56
8
74. If a8 = a1 + a2 + a3 + ... + a7,
b8 = b1 + b2 + b3 + ... + b7 ,
h8 = h1 + h2 + h3 + ... + h7, then what is the value of (a8 + b8 + c8 + ... + h8)?
a. 1426 b. 1596
c. 1652 d. 1540
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6. 75. If the numbers were written from 1 to 64 in the chessboard in the same fashion, the total number of
odd numbers on the white box would have been
a. 8
b. 16
c. 24
d. Depends upon the orientation of the black and white boxes
76. In the previous question, what is the sum of the numbers of the black boxes if the top-left box is
black?
a. 510 b. 520
c. 1020 d. 1040
77. If roots of x2 + ax + b = 0 are 2 and α, and roots of x2 + bx + a = 0 are 3 and β , then find the value
of (α + β).
5 8
a. − b. −
3 5
5 8
c. d.
3 5
Direction for questions 78 to 80: Answer the questions based on the following information.
Destry has five squares (A, B, C, D and E) as shown below. Each square is supposed to have a decimal
number in it, but all the squares are empty! Help, Destry find his missing numbers and put them back in
their squares. Here are some clues as to where the numbers should be put.
I. One square (the sum of 11.09, 6.21 and 5.04) is to the left of a square with the difference
between 13.27 and 1.34.
II. One of the numbers in the squares is 13.27, but not in C.
III. One square has a number larger than square B by 13.78.
IV. The square with a sum of 13.62, 3.98, 7.00 and 0.57 is between the squares B and E.
V. The smallest number is in B and the largest number is in E.
Left R ig ht
A B C D E
78. The number in square A is
a. 22.34 b. 11.98
c. 25.17 d. None of these
79. The number in square B is
a. 11.93 b. 25.17
c. 13.47 d. None of these
80. The highest number is
a. 25.17 b. 25.71
c. 26.18 d. 24.34
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7. 81. In x-y plane, O is the centre of the circle and ABCD is a rectangle. If the radius of the circle is 5 cm
and AB = 6 cm, then calculate the area of ∆ADE.
y
E B
C
F
x x
O
D A
y
a. 48 cm2 b. 24 cm2
c. 93 cm2 d. None of these
82. Ram Prasad lives in Ramnagar colony where each house has a number. If Ram Prasad’s house
number is a multiple of 7, then it falls between 200 and 299. If his house number is not a multiple of
4, then it falls between 300 to 399. If his house number is not a multiple of 9, in that case it falls
between 400 to 499. His house number can be
a. 432 b. 252
c. Neither (a) nor (b) d. Both (a) and (b)
83. A farmer grows cauliflower in his square field. Each cauliflower needs 1 sq.mt of independent area.
This year the farmer increases the area of his field but still maintains the shape of the field to be a
square. In the new field also, each cauliflower needs 1 sq. mt of independent area and the increase
in number of cauliflowers grown is 211 because of the increase of area, what is the total number of
cauliflowers produced this year?
a. 11,025 b. 11,236
c. 11,449 d. 10,816
84. If f(x) = |[x + 1] – [x]| and g(x) = Isin xI,
where [x] : greatest integer ≤ x ,
|x| : non negative value of x.
Then how many times will g(x) touch/meet f(x) between –2π and 2π?
a. 0 b. 2
c. 4 d. Infinite
85. What is the sum of all three-digit numbers, which are divisible by all prime numbers less than 10?
a. 1470 b. 1926
c. 1100 d. 2100
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8. 86. Given that
–3 ≤ x ≤ 3
−2 ≤ y ≤ 2
1≤ z ≤ 3
x2 y xy 2
p= and q =
z z
Maximum value of (p – q) is
a. 22 b. 25
c. 28 d. 30
Direction for questions 87 to 90: Each of these questions consists of a question and two statements,
I and II. Choose
a. if one of the two statements (I or II) alone is sufficient to answer the question, but cannot be
answered by using the other statement alone.
b. if each statement alone is sufficient to answer the question asked.
c. if I and II together are sufficient to answer the question but neither statement alone is sufficient.
d. if even I and II together are not sufficient to answer the question.
87. What is the LCM of a, b and c?
I. LCM of a and c is 36.
II. LCM of (a + b) and (b + c) is 270.
88. What is X in (XEN), where X, E and N are the digits of the three digit number ‘XEN’?
I. Sum of square of first X natural numbers is 55.
II. 2X2 – 31X + 15 = 0
89.
R r
A O2
O1
B
Find the area of the shaded portion.
I. R – r = 2 cm
II. AB is a chord of outer circle and tangent to the inner circle with length = 6 cm.
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9. 90.
B (0, 2 )
C A
(–2, 0) (2, 0)
E (x, y)
D (0, –2)
Area of ABCDE is 8 units square. Find the coordinates (x, y).
I. x = y
II. x and y lies on x – y = 2
91. ABCD is a rectangle. PC = 9 cm, BP = 15 cm, AB = 14 cm. Then the angles of ∆APB are such
that
A D
α
14 β P
15
γ 9
B C
a. α > β < γ b. α > γ > β
c. β > γ > α d. α > β > γ
92. Find the range for values that x can take for
|x – 1| – |x| + |2x + 3| ≥ 2x + 4
−3 −3
a. x < b. ≤x<0
2 2
−3
c. − ∞ ≤ x ≤ d. x ≤ 0
2
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10. 93.
B G A
L P K
T S
H M O F
Q R
I J
N
C D
E
In the above figure ‘a’ is the side of the square ABCD, there are four squares EFGH, IJKL, MNOP
and QRST. Area of the shaded portion is
1 2 3 2
a. a b. a
2 4
5 2 7 2
c. a d. a
16 19
94. If the graph of any function y = f(x) is symmetrical about the line x = 1, then for any real number α,
which one of the following is true?
a. f( α + 1) = f( α − 1) b. f( α ) = f ( −α )
c. f(1 + α) = f (1 − α ) d. None of these
95. If x > y > 0, which one of the following is always true?
a. x y y x > x x y y b. x x y x > x y y y
c. x x y y > x y y x d. x y y y > x x y x
96. The top and bottom radii of a frustum of a cone are respectively 6 cm and 3 cm. Its height is 8 cm.
There is a conical cavity of a height of 3 cm at the bottom. The amount of material in the solid is
a. 132 π cm3 b. 168 π cm3
c. 159 π cm3 d. Data Insufficient
1 3 5
97. Sum of n terms of the series 2 +2.4 + 2.4.6 + ... is
1 1
a. 1 − b. n
n 2 (n !)
2 (n !)
1
c. 1 + d. None of these
n
2 (n !)
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11. 98. How many numbers in the set {–4, –3, 0, 2} will satisfy the condition that
| x − 4 | ≤ 6 and | x + 1| < 5 ?
a. 3 b. 2
c. 0 d. None of these
z
x 102y
99. Given 100 = . Find the expression for z.
100
a. logx 2y b. logy (2x + 1)
c. logy (x + 1) d. logx (2y + 1)
100. There are 10 cigarette-making machines in a factory, each machine makes 3 cigarettes of 10 g
each in a batch. If in a batch one of the 10 machines starts making cigarettes of 11 g each, then
what is the minimum number of weighings required to determine which machine is faulty?
a. 1 b. 2
c. 8 d. 3
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