Updated SQC

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Principles

1. If ab × cd = 1073 and ba × cd = 2117, find the value of (ab + cd) given that ab, ba and cd are all two digit positive integers.1. 66 2. 65 3. 63 4. 95

2. Find the maximum value of |30 + 9x – 3x2|, where –1 x 4.≤ ≤

1. 934

2. 147

43. 30 4. 18

3. In the given figure, ABCD is a rectangle which is divided into four equal rectangles by PS, QT and RU. If BC = 3 cm andAB = 8 cm then MN is

UTS

O

M

N

B

C

P Q RA

D

1. 0.33 cm 2. 0.66 cm 3. 1 cm 4. 1.25 cm

4. In the given figure, P is a point on the circumcircle of ∆ABC. From P, perpendiculars PD, PE & PF are drawn on the sides

AC, BC and AB respectively. If BPF 30 ,∠ = ° then find the measure of <DPC

A C

F

B P

D

E

1. 30° 2. 60° 3. 40° 4. 50° 5. 20°

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5. An ant starts from point A and crawls along the surface of the cylinder to reach the point B, vertically above A. The pathfollowed by the ant is equal to that of 4 identical spirals as shown in the figure. Find the radius of the circle that circumscribesa square such that the perimeter of the square is equal to the distance traversed by the ant. The diameter of the cylinder is3/π units and the height is 16 units.

A

B

16

3/π

1. 52

2. 5

23.

52

4. 5

3

6. If both the roots of the quadratic equation (– 2x2 + bx + c) = 0, are negative, then what is the sign of ( )( )b c

bc

+?

1. Always Negative 2. Always Positive3. Cannot be determined 4. Will vary according to the values of ‘b’ and ‘c’

7. Find the value of 36S if 2 3

6 1 7 1 8 1S

(2) (3) (4) 2 (3) (4) (5) (4) (5) (6)2 2= × + × + × +

× × × × × × ... till infinite terms.

1. 4 2. 4.5 3. 5 4. 6

8. If 10 10 xlog x – log x 2log 10,= then the possible value of x is given by

1. 10 2. 1

1003.

11000

4. None of these

9. The natural number 555...5 consisting of 65 fives, is equal to

I. 6510 1

59

−×

II. ( )( )13 26 65 2 125 1 10 10 ... 10 1 10 10 .... 10× + + + + + + + +

III. ( ) ( )5 10 15 60 45 1 10 10 10 ... 10 1 10 ... 10× + + + + + × + + +

1. Only I 2. Only I and II3. Only I and III 4. All three

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10. If x = a(b – c), y = b(c – a) and z = c(a – b), then find the value of

+ + ≠

2 2 21 x y zxyz 0.

abc yz xz xy

1. 1

abc2.

27

abc3.

3

abc4.

9

abc5.

1

3abc

11. Find all the values of p, such that 6 lies somewhere between the roots of the equation ( )+ + =2x 2 p – 3 x 9 0 (‘x’ is a

real number)

1. < − 3p

42. p > 6 3. 0 < p < 6 4. < < 3

0 p4

5. > − 3p

4

12. Find the range of values of x, where + ≥ +| 3x 7 | | 5x 6 | . (‘x’ is a real number)

1. − ≤ ≤13 1x

8 22. − < ≤6 1

x5 2

3. ≥ 1x

24. ≤ −13

x8

5. − ≤ < −13 6x

8 5

13. Find the sum of the series + + + +

3 9 27 n3

1 1 1 1.......

log 9 log 9 log 9 log 9 .

1. +n(n 1)

22. +

2

n(n 1) 3. + +n(n 1)(2n 1)

124.

+n(n 1)

45. +

4

n(n 1)

14. Sum of the first n terms of a geometric progression is given as

n

n1

S .3

= α + β . If the sum of infinite terms of this series is

unity, then

1. 1α + β = 2. 3 1α −β = − 3. 2 1α + β = − 4. 2 3 1α + β =

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15. Two dice are thrown together n times in succession. Find the probability of obtaining ‘1’ on the top face of both the dice in atleast one of the throws.

1. 1

362.

n351

36 −

3. n1

36

4. n35

36

16. Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls. A white ball is picked frombag A and put into bag B. Then, three balls are picked from bag B and put into bag A. Find the probability that a ball picked nowfrom bag A is black.1. 1/4 2. 1/3 3. 7/12 4. 5/12 5. 11/24

17. Which of the following graphs, represents the curve y = 2|x| + 3[x]? Here, [x] represents the greatest integer function.[Figures not drawn to the scale]

1. 1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7

– 4

– 4

2

2

3

3

4

4

5678

y

x 2. 1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7– 8

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7– 8

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7– 8

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7– 8

– 4

– 4

2

2

3

3

4

4

5678

y

x 3. 1

1O–1–1

–2

–2

–3

–3

–4

–4

2

2

3

3

4

4

56

y

x1

1O–1–1

–2

–2

–3

–3

–4

–4

2

2

3

3

4

4

56

y

x1

1O–1–1

–2

–2

–3

–3

–4

–4

2

2

3

3

4

4

56

y

x 4. 1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7

– 4

– 4

2

2

3

3

4

4

5678

y

x1

1O– 1– 1

– 5

– 2

– 2

– 6

– 3

– 3

– 7

– 4

– 4

2

2

3

3

4

4

5678

y

x 5. None of these

18. At what time between 6pm and 7pm will the hands of the clock coincide?1. 6:32:44 pm 2. 6:33:46 pm 3. 6:32:30 pm 4. 6:34:42 pm 5. 6:33:24 pm

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19. A function f is defined for all whole numbers n by the following relation f(n + 2) + f(n) – 2f(n + 1) = 0 If f(16) = 4 and f(24) = 7,what is the value of f(16 + 24)?1. 10 2. 13 3. 170 4. 3340

20. Two identical Re. 1 coins are kept on a table touching each other as shown in the figure below. One of the coins is fixed onthe table whereas the other coin rolls (without sliding) along the periphery of the fixed coin, touching it at all times. How manycomplete rotations has the rolling coin made, when it reaches its initial position again for the first time?1. 1 2. Between 1 and 2 3. 2 4. Between 2 and 3 5. 3

21. ABCD is a square. Arc AC and BD are drawn on the square ABCD with centres at D and C respectively. Find the ratio of areaof the shaded region to the area of the square ABCD.

� �

��

1. π − 3

3 42.

π −2 3

3 43.

π −3 3

2 24.

π − 3

3 2

CAT 2008

1. The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, anytwo numbers, say a and b, currently on the blackboard are erased and a new number a + b – 1 is written. What will be thenumber left on the board at the end?(1) 820 (2) 821 (3) 781 (4) 819 (5) 780

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2. What are the last two digits of 72008?(1) 21 (2) 61 (3) 01 (4) 41 (5) 81

3. If the roots of the equation x3 – ax2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

(1) 1

3− (2) –1 (3) 0 (4) 1 (5)

1

3

4. A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half theremaining amount plus half a kg of rice. Then the third customer also buys’ half the remaining amount plus half a kg of rice.Thereafter, no rice is left in the shop. Which of the following best describes the value of x?(1) 2 x 6≤ ≤ (2) 5 x 8≤ ≤ (3) 9 x 12≤ ≤ (4) 11 x 14≤ ≤ (5) 13 x 18≤ ≤

12. Suppose, the seed of any positive integer n is defined as follows: seed(n) = n, if n < 10

= seed(s(n)), otherwise,where s(n) indicates the sum of digits of n. For example,seed(7) = 7, seed(248) = seed(2 + 4 + 8) = seed(14) = seed (1 + 4) = seed (5) = 5 etc. How many positive integers n, suchthat n < 500, will have seed (n) = 9?(1) 39 (2) 72 (3) 81 (4) 108 (5) 55

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13. Two circles, both of radii 1 cm, intersect such that the circumference of each one passes through the centre of the other. Whatis the area (in sq. cm.) of the intersecting region?

(1) 3

3 4π − (2)

2 33 2π + (3)

4 33 2π − (4)

4 33 2π + (5)

2 33 2π −

23. Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum soobtained is perfect square whose square root equals the total of the three original integers. Which of the following bestdescribes the minimum, say m, of these three integers?

(1) 1 m 3≤ ≤ (2) 4 m 6≤ ≤ (3) 7 m 9≤ ≤ (4) 10 m 12≤ ≤ (5) 13 m 15≤ ≤

25. Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of theflat surfaces resting on the base of the cone. Find the largest possible total surface area (in sq. cm) of the cylinder.

(1) 1003

π (2) 803

π (3) 120

7π

(4) 130

9π

(5) 110

7π

CAT 2007

1. How many pairs of positive integers m, n satisfy 1 4 1m n 12

+ = , where, ‘n’ is an odd integer less than 60?

(1) 7 (2) 5 (3) 3 (4) 6 (5) 4

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3. Suppose you have a currency, named Miso, in three denominations: 1 Miso, 10 Misos and 50 Misos. In how many ways canyou pay a bill of 107 Misos?(1) 18 (2) 15 (3) 19 (4) 17 (5) 16

Directions for question 6: Answer the following questions based on the information given below:Let a1 = p and b1 = q, where p and q are positive quantities. Define

n n–1a pb= , n n–1b qb= , for even n > 1,

and n n–1a pa= , n n–1b qa= , for odd n > 1.

6. Which of the following best describes n na b+ for even ‘n’?

(1) ( ) ( )1

n–12q pq p q+ (2) ( )

1n–1

2qp p q+ (3) ( )1

n2q p q+

(4) ( )1

n n2 2q p q1

+ (5) ( ) ( )1 1

n–1 n2 2q pq p q+

8. In a tournament, there are n teams 1 2 nT , T ,..., T , with n > 5. Each team consists of ‘k’ players,

k > 3. The following pairs of teams have one player in common:

1 2 2 3 n–1 n n 1T & T ,T & T , ...,T & T , and T & T

No other pair of teams has any player in common. How many players are participating in the tournament, considering all the‘n’ teams together?(1) n(k –2) (2) k(n –2) (3) (n – 1)(k – 1) (4) n(k – 1) (5) k(n – 1)

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9. Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many suchnumbers are perfect squares?(1) 4 (2) 0 (3) 1 (4) 3 (5) 2

Directions for questions 12 and 13: Answer the following questions based on the information given below:

Let S be the set of all pairs (i, j) where, 1 i j n≤ < ≤ and n 4≥ . Any two distinct members of S are called “friends” if they have one

constituent of the pairs in common and “enemies” otherwise. For example, if n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}.Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.

12. For general ‘n’, how many enemies will each member of S have?

(1) 2n – 7 (2) ( )��

�(3) ( )��

� ��

(4) n – 3 (5) ( )��

�

13. For General n, consider any two members of S that are friends. How many other members of S will be common friends ofboth these members?

(1) ( )��

�(2) n – 2 (3) ( )��

� ��

(4) ( )��

�(5) 2n – 6

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14. Two circles with centres P and Q cut each other at two distinct points A and B. The circles have the same radii and neither Pnor Q falls within the intersection of the circles. What is the smallest range that includes all possible values of the angle AQPin degrees?(1) Between 0 and 45 (2) Between 0 and 90 (3) Between 0 and 30 (4) Between 0 and 60 (5) Between 0 and 75

Directions for questions 17 and 18: Answer the following questions based on the information given below:Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airlineoperating non-stop flights between A and B. All the times indicated are local and on the same day.

Departure Arrival

City Time City Time

B 8:00 am A 3:00 pm

A 4:00 pm B 8:00 pm

Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowingfrom east to west at 50 km per hour.

17. What is the time difference between A and B?(1) 1 hour (2) 1 hour and 30 mins (3) 2 hours(4) 2 hours and 30 mins (5) Cannot be determined

18. What is the plane’s cruising speed in km per hour?(1) 500 (2) 700 (3) 550(4) 600 (5) Cannot be determined

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20. Consider the set S = {2, 3, 4, ……, 2n + 1}, where n is a positive integer larger than 2007. Define X as the average of the oddintegers in S and Y as the average of the even integers in S. What is the value of X – Y?

(1) ��

� (2)

��

��(3) 2008 (4) 0 (5) 1

CAT 2005

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 (2 Marks)

A B

C

D

E

H LF

N

M

O

G

(1) 2 2 – 1 (2) ( )2 2 – 1

2(3)

( )3 2 – 1

2(4)

( )2 2 – 1

3

26. Let x 4 4 – 4 4 – ...to inf inity .= + + Then x equals (2 Marks)

(1) 3 (2) 13 – 1

2

(3) 13 1

2

+

(4) 13

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8. If 65 65

64 64

30 – 29R

30 29=

+, then (1 Mark)

(1) 0 R 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 Mark)(1) 1 or 7 (2) 2 or 14 (3) 3 or 21 (4) 4 or 28

14. If 1 n 1 na 1and a – 3a 2 4n+= + = for every positive integer n, then a100 equals (2 Marks)

(1) 993 – 200 (2) 993 200+ (3) 1003 – 200 (4) 1003 200+

21. In the X-Y plane, the area of the region bounded by the graph x y x y 4+ + − = is (2 Marks)

(1) 8 (2) 12 (3) 16 (4) 20

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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 adiameter of the circle. What is the perimeter of the quadrilateral PQSR? (2 Marks)

(1) ( )2r 1 3+ (2) ( )2r 2 3+ (3) ( )r 1 5+ (4) 2r 3+

CAT 2004

43. On January 1, 2004 two new societies s1 and s2 are formed, each n numbers. On the first day of each subsequent month, s1

adds b members while s2 multiples its current numbers by a constant factor r. Both the societies have the same number of

members on July 2, 2004. If b = 10.5n, what is the value of r? (1 Mark)(1) 2.0 (2) 1.9 (3) 1.8 (4) 1.7

44. If f(x) = 3x – 4x p+ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true (1 Mark)(1) –1 < p < 2 (2) 0 < p < 3 (3) –2 < p < 1 (4) –3 < p < 0

47. If a b c

rb c c a a b

= = =+ + +

then r cannot take any value except. (1 Mark)

(1) 12

(2) –1 (3) 1

or – 12

(4) 1

– or – 12

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48. Let1

y1

21

31

23 ...

=+

++

+

(1 Mark)

What is the value of y?

(1) 11 3

2+

(2) 11 3

2−

(3) 15 3

2+

(4) 15 3

2−

Directions for questions 53 to 55: Answer the questions on the basis of the information given below.In the adjoining figure I and II, are circles with P and Q respectively, The two circles touch each other and have common tangent thattouches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II arein the ratio 4 : 3. It is also known that the length of PO is 28 cm.

I

II

P Q

R

S

O

53. What is the ratio of the length of PQ to that of QO? (1 Mark)(1) 1 : 4 (2) 1 ; 3 (3) 3 : 8 (4) 3 : 4

54. What is the radius of the circle II? (1 Mark)(1) 2 cm (2) 3 cm (3) 4 cm (4) 5 cm

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55. The length of SO is (1 Mark)

(1) 8 3 cm (2) 10 3 cm (3) 12 3 cm (4) 14 3 cm

57. Each family in a locality has at most two adults, and no family has fewer than 3 children. Considering all the families together,there are more adults than boys, more boys than girls, and more girls than families, Then the minimum possible number offamilies in the locality is (1 Mark)(1) 4 (2) 5 (3) 2 (4) 3

58. The total number of integers pairs (x, y) satisfying the equation x + y = xy is (1 Mark)(1) 0 (2) 1 (3) 2 (4) None of these

61. A sprinter starts running on a circular path of radius r metres. Her average speed (in metres/minute) is πr during the first 30

seconds, r

2π

during next one minute, r

4π

during next 2 minutes, r

8π

during next 4 minutes, and so on. What is the ratio of the

time taken for the nth round to that for the previous round? (2 Marks)(1) 4 (2) 8 (3) 16 (4) 32

65. If the lengths of diagonals DF, AG and CE of the cube shown in the adjoining figure are equal to the three sides of a triangle,then the radius of the circle circumscribing that triangle will be (2 Marks)

G F

E

B

AD

C

(1) equal to the side of cube (2) 3 times the side of the cube

(3) 1

3 times the side of the cube (4) impossible to find from the given information.

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68. A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure.What is the radius of the smaller circle? (2 Marks)

(1) 3 – 2 2 (2) 4 – 2 2 (3) 7 – 4 2 (4) 6 – 4 2

72. The remainder, when (1523 + 2323) is divided by 19, is (2 Marks)(1) 4 (2) 15 (3) 0 (4) 18

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