CAT Past Year Question Paper - 2005 | Additional Study Material for CAT PDF Download

 Page 1


Sub–Section I-A : Number of questions =  10
Note: Questions 1 to 10 carry one mark each.
Directions for questions 1 to 5: Answer the questions independently of each other.
1. If ()
3333
x 16171819 =+ + +
, then x divided by 70 leaves a remainder of
(1) 0 (2) 1 (3) 69 (4) 35
2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The
chemical is being pumped from one tank to anther as follows.
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping
starts?
(1) A, 16.66 (2) C, 20 (3) D, 20 (4) D, 25
3. Two identical circles intersect  so that their centers, and the points at which they intersect, form a
square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
(1) 
4
p
(2) 
–1
2
p
(3) 
5
p
(4) 
2 – 1
4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclos-
ing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B,
start jogging simultaneously form the point where one of the circular tracks touches the smaller side
of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular

	



Instructions:
1. The Test Paper contains 90 questions. The duration of the test is 120 minutes.
2. The paper is divided into three sections. Section-I: 30 Q:, Section-II: 30 Q:, Section-III: 30 Q.
3. Wrong answers carry negative marks. There is only one correct answer for each question.
Page 2


Sub–Section I-A : Number of questions =  10
Note: Questions 1 to 10 carry one mark each.
Directions for questions 1 to 5: Answer the questions independently of each other.
1. If ()
3333
x 16171819 =+ + +
, then x divided by 70 leaves a remainder of
(1) 0 (2) 1 (3) 69 (4) 35
2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The
chemical is being pumped from one tank to anther as follows.
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping
starts?
(1) A, 16.66 (2) C, 20 (3) D, 20 (4) D, 25
3. Two identical circles intersect  so that their centers, and the points at which they intersect, form a
square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
(1) 
4
p
(2) 
–1
2
p
(3) 
5
p
(4) 
2 – 1
4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclos-
ing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B,
start jogging simultaneously form the point where one of the circular tracks touches the smaller side
of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular

	



Instructions:
1. The Test Paper contains 90 questions. The duration of the test is 120 minutes.
2. The paper is divided into three sections. Section-I: 30 Q:, Section-II: 30 Q:, Section-III: 30 Q.
3. Wrong answers carry negative marks. There is only one correct answer for each question.
tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they
take the same time to return to their starting point?
(1) 3.88% (2) 4.22% (3) 4.44% (4) 4.72%
5. In a chess competition involving some boys and girls of a school, every student had to play exactly
one game with every other student. It was found that in 45 games both the players were girls, and in
190 games both were boys. The number of games in which one player was a boy and the other was
a girl is
(1) 200 (2) 216 (3) 235 (4) 256
Directions for questions 6 and 7: Answer the questions on the basis of the information given below.
Ram and Shyam run a race between points A and B, 5 km apart, Ram starts at 9 a.m from A at a speed
of 5 km/hr, reaches B, and returns to A at the same speed, Shyam starts at 9:45 a.m. from A at a speed
of 10 km/hr, reaches B and comes back to A at the same speed.
6. At what time do Ram and Shyam first meet each other?
(1) 10 a.m (2) 10:10 a.m (3) 10:20 a.m (4) 10:30 a.m.
7. At what time does Shyam over take Ram?
(1) 10:20 a.m (2) 10:30 a.m (3) 10:40 a.m (4) 10:50 a.m
Directions for questions 8 to 10: Answer the questions independently of each other.
8. If 
65 65
64 64
30 – 29
R
30 29
=
+
, then
(1) 
0R 0.1 <=
(2) 
0.1 R 0.5 <=
(3) 
0.5 R 1.0 <=
(4) R > 1.0
9. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of
radius 20 cm?
(1) 1 or 7 (2) 2 or 14 (3) 3 or 21 (4) 4 or 28
10. For which value of k does the following pair of equations yield a unique solution of x such that the
solution is positive?
22
22
x –y0
(x – k) y 1
=
+=
(1) 2 (2) 0 (3) 
2
(4) 
2 -
Page 3


Sub–Section I-A : Number of questions =  10
Note: Questions 1 to 10 carry one mark each.
Directions for questions 1 to 5: Answer the questions independently of each other.
1. If ()
3333
x 16171819 =+ + +
, then x divided by 70 leaves a remainder of
(1) 0 (2) 1 (3) 69 (4) 35
2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The
chemical is being pumped from one tank to anther as follows.
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping
starts?
(1) A, 16.66 (2) C, 20 (3) D, 20 (4) D, 25
3. Two identical circles intersect  so that their centers, and the points at which they intersect, form a
square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
(1) 
4
p
(2) 
–1
2
p
(3) 
5
p
(4) 
2 – 1
4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclos-
ing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B,
start jogging simultaneously form the point where one of the circular tracks touches the smaller side
of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular

	



Instructions:
1. The Test Paper contains 90 questions. The duration of the test is 120 minutes.
2. The paper is divided into three sections. Section-I: 30 Q:, Section-II: 30 Q:, Section-III: 30 Q.
3. Wrong answers carry negative marks. There is only one correct answer for each question.
tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they
take the same time to return to their starting point?
(1) 3.88% (2) 4.22% (3) 4.44% (4) 4.72%
5. In a chess competition involving some boys and girls of a school, every student had to play exactly
one game with every other student. It was found that in 45 games both the players were girls, and in
190 games both were boys. The number of games in which one player was a boy and the other was
a girl is
(1) 200 (2) 216 (3) 235 (4) 256
Directions for questions 6 and 7: Answer the questions on the basis of the information given below.
Ram and Shyam run a race between points A and B, 5 km apart, Ram starts at 9 a.m from A at a speed
of 5 km/hr, reaches B, and returns to A at the same speed, Shyam starts at 9:45 a.m. from A at a speed
of 10 km/hr, reaches B and comes back to A at the same speed.
6. At what time do Ram and Shyam first meet each other?
(1) 10 a.m (2) 10:10 a.m (3) 10:20 a.m (4) 10:30 a.m.
7. At what time does Shyam over take Ram?
(1) 10:20 a.m (2) 10:30 a.m (3) 10:40 a.m (4) 10:50 a.m
Directions for questions 8 to 10: Answer the questions independently of each other.
8. If 
65 65
64 64
30 – 29
R
30 29
=
+
, then
(1) 
0R 0.1 <=
(2) 
0.1 R 0.5 <=
(3) 
0.5 R 1.0 <=
(4) R > 1.0
9. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of
radius 20 cm?
(1) 1 or 7 (2) 2 or 14 (3) 3 or 21 (4) 4 or 28
10. For which value of k does the following pair of equations yield a unique solution of x such that the
solution is positive?
22
22
x –y0
(x – k) y 1
=
+=
(1) 2 (2) 0 (3) 
2
(4) 
2 -
Sub–Section I-B : Number of questions =  20
Note: Questions 11 to 30 carry two marks each.
11. Let n! = 1 × 2 × 3 × … × n for integer 
n1. =
 If p = 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10!), then
p + 2 when divided by 11! Leaves a remainder of
(1) 10 (2) 0 (3) 7 (4) 1
12. Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each
vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates
inside the triangle (excluding all the points on the boundary) is
(1) 780 (2) 800 (3) 820 (4) 741
13. The digits of a three-digit number A are written in the reverse order to form another three-digit
number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?
(1) 100 < A < 299 (2) 106 < A < 305 (3) 112 < A < 311 (4) 118< A < 317
14. If 
1n1n
a1anda – 3a 2 4n
+
=+= for every positive integer n, then a
100
 equals
(1) 
99
3 – 200 (2) 
99
3200 + (3) 
100
3 – 200 (4) 
100
3200 +
15. Let S be the set of five-digit numbers formed by digits 1, 2, 3, 4 and 5, using each digit exactly once
such that exactly two odd position are occupied by odd digits. What is the sum of the digits in the
rightmost position of the numbers in S?
(1) 228 (2) 216 (3) 294 (4) 192
16. The rightmost non-zero digits of the number 30
2720
 is
(1) 1 (2) 3 (3) 7 (4) 9
17. Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the
length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect
repellents kept at points B and C. the ant would not go within one metre of any insect repellent. The
minimum distance in metres the ant must traverse to reach the sugar particle is
(1) 
32
(2) 1 + p (3) 
4
3
p
(4) 5
18. If x = y and y > 1, then the value of the expression 
xy
xy
log log
yx
?? ??
+
?? ??
?? ??
 can never be
(1) –1 (2) –0.5 (3) 0 (4) 1
Page 4


Sub–Section I-A : Number of questions =  10
Note: Questions 1 to 10 carry one mark each.
Directions for questions 1 to 5: Answer the questions independently of each other.
1. If ()
3333
x 16171819 =+ + +
, then x divided by 70 leaves a remainder of
(1) 0 (2) 1 (3) 69 (4) 35
2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The
chemical is being pumped from one tank to anther as follows.
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping
starts?
(1) A, 16.66 (2) C, 20 (3) D, 20 (4) D, 25
3. Two identical circles intersect  so that their centers, and the points at which they intersect, form a
square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
(1) 
4
p
(2) 
–1
2
p
(3) 
5
p
(4) 
2 – 1
4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclos-
ing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B,
start jogging simultaneously form the point where one of the circular tracks touches the smaller side
of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular

	



Instructions:
1. The Test Paper contains 90 questions. The duration of the test is 120 minutes.
2. The paper is divided into three sections. Section-I: 30 Q:, Section-II: 30 Q:, Section-III: 30 Q.
3. Wrong answers carry negative marks. There is only one correct answer for each question.
tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they
take the same time to return to their starting point?
(1) 3.88% (2) 4.22% (3) 4.44% (4) 4.72%
5. In a chess competition involving some boys and girls of a school, every student had to play exactly
one game with every other student. It was found that in 45 games both the players were girls, and in
190 games both were boys. The number of games in which one player was a boy and the other was
a girl is
(1) 200 (2) 216 (3) 235 (4) 256
Directions for questions 6 and 7: Answer the questions on the basis of the information given below.
Ram and Shyam run a race between points A and B, 5 km apart, Ram starts at 9 a.m from A at a speed
of 5 km/hr, reaches B, and returns to A at the same speed, Shyam starts at 9:45 a.m. from A at a speed
of 10 km/hr, reaches B and comes back to A at the same speed.
6. At what time do Ram and Shyam first meet each other?
(1) 10 a.m (2) 10:10 a.m (3) 10:20 a.m (4) 10:30 a.m.
7. At what time does Shyam over take Ram?
(1) 10:20 a.m (2) 10:30 a.m (3) 10:40 a.m (4) 10:50 a.m
Directions for questions 8 to 10: Answer the questions independently of each other.
8. If 
65 65
64 64
30 – 29
R
30 29
=
+
, then
(1) 
0R 0.1 <=
(2) 
0.1 R 0.5 <=
(3) 
0.5 R 1.0 <=
(4) R > 1.0
9. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of
radius 20 cm?
(1) 1 or 7 (2) 2 or 14 (3) 3 or 21 (4) 4 or 28
10. For which value of k does the following pair of equations yield a unique solution of x such that the
solution is positive?
22
22
x –y0
(x – k) y 1
=
+=
(1) 2 (2) 0 (3) 
2
(4) 
2 -
Sub–Section I-B : Number of questions =  20
Note: Questions 11 to 30 carry two marks each.
11. Let n! = 1 × 2 × 3 × … × n for integer 
n1. =
 If p = 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10!), then
p + 2 when divided by 11! Leaves a remainder of
(1) 10 (2) 0 (3) 7 (4) 1
12. Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each
vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates
inside the triangle (excluding all the points on the boundary) is
(1) 780 (2) 800 (3) 820 (4) 741
13. The digits of a three-digit number A are written in the reverse order to form another three-digit
number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?
(1) 100 < A < 299 (2) 106 < A < 305 (3) 112 < A < 311 (4) 118< A < 317
14. If 
1n1n
a1anda – 3a 2 4n
+
=+= for every positive integer n, then a
100
 equals
(1) 
99
3 – 200 (2) 
99
3200 + (3) 
100
3 – 200 (4) 
100
3200 +
15. Let S be the set of five-digit numbers formed by digits 1, 2, 3, 4 and 5, using each digit exactly once
such that exactly two odd position are occupied by odd digits. What is the sum of the digits in the
rightmost position of the numbers in S?
(1) 228 (2) 216 (3) 294 (4) 192
16. The rightmost non-zero digits of the number 30
2720
 is
(1) 1 (2) 3 (3) 7 (4) 9
17. Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the
length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect
repellents kept at points B and C. the ant would not go within one metre of any insect repellent. The
minimum distance in metres the ant must traverse to reach the sugar particle is
(1) 
32
(2) 1 + p (3) 
4
3
p
(4) 5
18. If x = y and y > 1, then the value of the expression 
xy
xy
log log
yx
?? ??
+
?? ??
?? ??
 can never be
(1) –1 (2) –0.5 (3) 0 (4) 1
19. For a positive integer n, let p
n
 denote the product of the digits of n and s
n
 denote the   sum of the
digits of n. The number of integers between 10 and 1000 for which p
n
 + s
n
 = n is
(1) 81 (2) 16 (3) 18 (4) 9
20. Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of
size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so
long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor.
The maximum number of tiles that can be accommodated on the floor is
(1) 4 (2) 5 (3) 6 (4) 7
21. In the X-Y plane, the area of the region bounded by the graph xy x y 4 ++- = is
(1) 8 (2) 12 (3) 16 (4) 20
22. In the following figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that
MN is perpendicular to AB. In addition, CG is perpendicular to AB such that AE:EB = 1:2, and DF
is perpendicular to MN such that NL:LM = 1:2. The length of DH in cm is
A
B
C
D
E
H
L
F
N
M
O
G
(1) 
22 – 1
(2) 
()
22 – 1
2
(3) 
()
32 – 1
2
(4) 
()
22 – 1
3
23. Consider the triangle ABC shown in the following figure where BC = 12 cm, DB = 9 cm, CD = 6 cm
and 
BCD BAC ?=?
Page 5


Sub–Section I-A : Number of questions =  10
Note: Questions 1 to 10 carry one mark each.
Directions for questions 1 to 5: Answer the questions independently of each other.
1. If ()
3333
x 16171819 =+ + +
, then x divided by 70 leaves a remainder of
(1) 0 (2) 1 (3) 69 (4) 35
2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The
chemical is being pumped from one tank to anther as follows.
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping
starts?
(1) A, 16.66 (2) C, 20 (3) D, 20 (4) D, 25
3. Two identical circles intersect  so that their centers, and the points at which they intersect, form a
square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
(1) 
4
p
(2) 
–1
2
p
(3) 
5
p
(4) 
2 – 1
4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclos-
ing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B,
start jogging simultaneously form the point where one of the circular tracks touches the smaller side
of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular

	



Instructions:
1. The Test Paper contains 90 questions. The duration of the test is 120 minutes.
2. The paper is divided into three sections. Section-I: 30 Q:, Section-II: 30 Q:, Section-III: 30 Q.
3. Wrong answers carry negative marks. There is only one correct answer for each question.
tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they
take the same time to return to their starting point?
(1) 3.88% (2) 4.22% (3) 4.44% (4) 4.72%
5. In a chess competition involving some boys and girls of a school, every student had to play exactly
one game with every other student. It was found that in 45 games both the players were girls, and in
190 games both were boys. The number of games in which one player was a boy and the other was
a girl is
(1) 200 (2) 216 (3) 235 (4) 256
Directions for questions 6 and 7: Answer the questions on the basis of the information given below.
Ram and Shyam run a race between points A and B, 5 km apart, Ram starts at 9 a.m from A at a speed
of 5 km/hr, reaches B, and returns to A at the same speed, Shyam starts at 9:45 a.m. from A at a speed
of 10 km/hr, reaches B and comes back to A at the same speed.
6. At what time do Ram and Shyam first meet each other?
(1) 10 a.m (2) 10:10 a.m (3) 10:20 a.m (4) 10:30 a.m.
7. At what time does Shyam over take Ram?
(1) 10:20 a.m (2) 10:30 a.m (3) 10:40 a.m (4) 10:50 a.m
Directions for questions 8 to 10: Answer the questions independently of each other.
8. If 
65 65
64 64
30 – 29
R
30 29
=
+
, then
(1) 
0R 0.1 <=
(2) 
0.1 R 0.5 <=
(3) 
0.5 R 1.0 <=
(4) R > 1.0
9. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of
radius 20 cm?
(1) 1 or 7 (2) 2 or 14 (3) 3 or 21 (4) 4 or 28
10. For which value of k does the following pair of equations yield a unique solution of x such that the
solution is positive?
22
22
x –y0
(x – k) y 1
=
+=
(1) 2 (2) 0 (3) 
2
(4) 
2 -
Sub–Section I-B : Number of questions =  20
Note: Questions 11 to 30 carry two marks each.
11. Let n! = 1 × 2 × 3 × … × n for integer 
n1. =
 If p = 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10!), then
p + 2 when divided by 11! Leaves a remainder of
(1) 10 (2) 0 (3) 7 (4) 1
12. Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each
vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates
inside the triangle (excluding all the points on the boundary) is
(1) 780 (2) 800 (3) 820 (4) 741
13. The digits of a three-digit number A are written in the reverse order to form another three-digit
number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?
(1) 100 < A < 299 (2) 106 < A < 305 (3) 112 < A < 311 (4) 118< A < 317
14. If 
1n1n
a1anda – 3a 2 4n
+
=+= for every positive integer n, then a
100
 equals
(1) 
99
3 – 200 (2) 
99
3200 + (3) 
100
3 – 200 (4) 
100
3200 +
15. Let S be the set of five-digit numbers formed by digits 1, 2, 3, 4 and 5, using each digit exactly once
such that exactly two odd position are occupied by odd digits. What is the sum of the digits in the
rightmost position of the numbers in S?
(1) 228 (2) 216 (3) 294 (4) 192
16. The rightmost non-zero digits of the number 30
2720
 is
(1) 1 (2) 3 (3) 7 (4) 9
17. Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the
length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect
repellents kept at points B and C. the ant would not go within one metre of any insect repellent. The
minimum distance in metres the ant must traverse to reach the sugar particle is
(1) 
32
(2) 1 + p (3) 
4
3
p
(4) 5
18. If x = y and y > 1, then the value of the expression 
xy
xy
log log
yx
?? ??
+
?? ??
?? ??
 can never be
(1) –1 (2) –0.5 (3) 0 (4) 1
19. For a positive integer n, let p
n
 denote the product of the digits of n and s
n
 denote the   sum of the
digits of n. The number of integers between 10 and 1000 for which p
n
 + s
n
 = n is
(1) 81 (2) 16 (3) 18 (4) 9
20. Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of
size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so
long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor.
The maximum number of tiles that can be accommodated on the floor is
(1) 4 (2) 5 (3) 6 (4) 7
21. In the X-Y plane, the area of the region bounded by the graph xy x y 4 ++- = is
(1) 8 (2) 12 (3) 16 (4) 20
22. In the following figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that
MN is perpendicular to AB. In addition, CG is perpendicular to AB such that AE:EB = 1:2, and DF
is perpendicular to MN such that NL:LM = 1:2. The length of DH in cm is
A
B
C
D
E
H
L
F
N
M
O
G
(1) 
22 – 1
(2) 
()
22 – 1
2
(3) 
()
32 – 1
2
(4) 
()
22 – 1
3
23. Consider the triangle ABC shown in the following figure where BC = 12 cm, DB = 9 cm, CD = 6 cm
and 
BCD BAC ?=?
A
B
C
D
6
9
12
What is the ratio of the perimeter of ? ADC to that of the ? BDC?
(1) 
7
9
(2) 
8
9
(3) 
6
9
(4) 
5
9
24. P , Q, S and R are points on the circumference of a circle of radius r, such that PQR is an equilateral
triangle and PS is a diameter of the circle. What is the perimeter of the quadrilateral PQSR?
(1) ()
2r 1 3 +
(2) ()
2r 2 3 +
(3) ()
r1 5 +
(4) 
2r 3 +
25. Let S be a set of positive integers such that every element n of S satisfies the conditions
I. 
1000 n 1200 ==
II. Every digit in n is odd
Then how many elements of S are divisible by 3?
(1) 9 (2) 10 (3) 11 (4) 12
26. Let 
x4 4 –44 – ...to inf inity . =+ + Then x equals
(1) 3 (2) 
13 – 1
2
??
??
??
??
(3) 
13 1
2
??
+
??
??
??
(4) 
13
27. Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is
the relation g(x+p) = g(x) necessarily true for every real x?
(1) 5 (2) 3 (3) 2 (4) 6