sqc 2013 - exercise booklet.pdf

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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)93

4(2)

147

4(3) 30 (4) 18

3. 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≤ ≤CAT 2008

4. Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so

obtained 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≤ ≤CAT 2008

5. Ifa b c

r,b c c a a b

= = =

+ + +

then r cannot take any value except.

(1)1

2(2) – 1 (3)

1or – 1

2(4)

1 – or – 1

2

CAT 2004

6. What values of x satisfy x2/3 + x1/3 – 2 ≤ 0 ?

(1) – 8 ≤ x ≤ 1 (2) – 1 ≤ x ≤ 8 (3) 1 < x < 8 (4) 1 ≤ x ≤ 8 (5) – 8 ≤ x ≤ 8

CAT 2006

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7. If f(x) = x3 – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true:(1) – 1 < p < 2 (2) 0 < p < 3 (3) – 2 < p < 1 (4) – 3 < p < 0

CAT 2004

8. 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 Positive(3) Cannot be determined (4) Will vary according to the values of ‘b’ and ‘c’

9. 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)3

p4

< − (2) p > 6 (3) 0 < p < 6 (4)3

0 p4

< < (5)3

p4

> −

10. 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

CAT 2008

11. Find the value of 36S if2 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

12. The infinite sum2 3 4

4 9 16 251

7 7 7 7+ + + + +… equals

(1)27

14(2)

21

13(3)

49

27(4)

256

147

CAT 2003(R)

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13. If log x log 510 105 50 x ,= − then find the value of x

(1) 1 (2) 10 (3) 100 (4) 1000

14. If10 10 xlog x – log x 2log 10,= then the possible value of x is given by

(1) 10 (2)1

100(3)

1

1000(4) None of these

CAT 2003(R)

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

2 2 21 x y z

abc yz xz xy

+ +

(Given that xyz ≠ 0)

(1)1

abc(2)

27

abc(3)

3

abc(4)

9

abc(5)

1

3abc

16. Sum of the first n terms of a geometric progression is given asn

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 =

17. If a, b, c and d are integers and n is a prime number, then which of the following is always a factor of (a + b – c – d)n – (an + bn – cn – dn)?(1) n (2) a + b – c – d (3) n – 2 (4) n + 2

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18. If a, b and c are non-zero real numbers such that |a – b| =|c| and |a – c| = |b|, then what is the value of b2 + c2?(1) a2 – 2bc (2) 2bc (3) a2 (4) a2 + 2bc

19. In a regular polygon, the number of diagonals is ‘k’ times the number of sides. If the interior angle of the polygon is θ, then thevalue of k is

(1)3

2( )

θ − ππ − θ (2)

2(3 )θ − ππ − θ

(3)2( )

3

π − θθ − π

(4)( )

2(3 )

π − θθ − π

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

I.

6510 15

9

−×

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 II (3) Only I and III (4) All three

21. Find the sum of2 2 2 2 2 2

1 1 1 1 1 11 1 1

1 2 2 3 2007 2008

+ + + + + + … + +

(1)1

20082008

−

(2)1

20072007

−

(3)1

20072008

−

(4)1

20082007

−

(5)1

20082009

−

CAT 2008

Directions for questions 22 and 23: 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.

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

(1) 2n – 7 (2)21

(n 5n 6)2

− + (3)21

(n 7n 14)2

− + (4) n – 3 (5)21

(n 3n 2)2

− −

CAT 2007





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23. 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)1

n(n 3)2

− (2) n – 2 (3)21

(n 7n 16)2

− + (4)21

(n 5n 8)2

− + (5) 2n – 6

CAT 2007

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?

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

CAT 2005

25. Let n! = 1 × 2 × 3 × ... × n for integer n ≥ 1.If p = (1 × 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

CAT 2005

26. If 1 n 1 na 1and a – 3a 2 4n+= + = for every positive integer n, then a100

equals

(1) 993 – 200 (2) 993 200+ (3) 1003 – 200 (4) 1003 200+CAT 2005

27. If65 65

64 64

30 – 29R

30 29=

+, then

(1) 0 R 0.1< ≤ (2) 0.1 R 0.5< ≤ (3) 0.5 R 1.0< ≤ (4) R > 1.0

CAT 2005





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28. 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

B

CD

E

FP

(1) 30° (2) 60° (3) 40° (4) 50° (5) 20°

29. In triangle ABC, D is a point on BC such that BD = DC. BE is an altitude drawn on AC. If ∠ ADB = 45°and ∠ ACB = 30°, what

is the measure of ∠ ABE?(1) 30° (2) 60° (3) 45° (4) Cannot be determined

30. In the figure given below, ∆ABC is an isosceles right angled triangle and the “in-circle” with center O touches the side BCat D. If AB = 2 cm, find the length (in cm) of BD.

A

B CD

O

(1)1

12

− (2) 2 2− (3)1

22

− (4) 2 1−





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31. In the figure given below, a quadrant of a circle of radius 2 units is drawn. If it touches the line AB at point M, what is the areaof the shaded region (in square units)

Y

OX

(0, 0)

B

M( 2 , 2 )Ö Ö

A

(1) 12

π− (2) 4 − π (3) 2 − π (4) 2

2

π−

32. In ∆ABC; D, E, F and G are the midpoints of the sides BC, AD, BE and FD respectively. What is the ratio of the area of ∆EGDto ∆AEC?(1) 1 : 16 (2) 1 : 4 (3) 4 : 1 (4) 16 : 1

33. An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?(1) 75 (2) 90 (3) 120 (4) 135 (5) 150

CAT 2008

34. 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?

(1) 3 – 2 2 (2) 4 – 2 2 (3) 7 – 4 2 (4) 6 – 4 2CAT 2004

35. A square is inscribed in a circle and the circle is inscribed in a regular octagon. Find the ratio of the area of the square to thatof the octagon.

(1) (1 2) : 4+ (2) (2 2 1) : 4− (3) ( 2 1) : 4− (4) Cannot be determined





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36. 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.

A B

CD

(1)3

3 4

π− (2)

2 3

3 4

π− (3)

3 3

2 2

π− (4)

3

3 2

π−

37. Two circles of radii ‘r’ units and ‘2r’ units intersect each other in such a way that their common chord is of the maximumpossible length. What is the area (in square units) of the region that is common to the two circles?

(1)2r

2

π(2)

227 r

3r6

π− (3)

2 211 r 3r

6 2

π− (4)

2 27 r 3r

6 2

π−

38. 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 4

π− (2)

4 3

3 2

π+ (3)

2 3

3 2

π+ (4)

2 3

3 2

π− (5)

4 3

3 2

π−

CAT 2008

39. 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) 100

3

π (2)80

3

π(3)

120

7

π(4)

130

9

π(5)

110

7

π

CAT 2008

40. Let x 4 4 – 4 4 – ...to inf inity .= + + Then x equals

(1) 3 (2)13 – 1

2

(3)13 1

2

+

(4) 13

CAT 2005





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41. Let1

y1

21

31

23 ...

=+

++

+What is the value of y?

(1)11 3

2

+(2)

11 3

2

−(3)

15 3

2

+(4)

15 3

2

−

CAT 2004

42. 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

5

6

7

8

y

x1

1O–1–1

–5

–2

–2

–6

– 3

–3

–7

– 4

–4

2

2

3

3

4

4

5

6

7

8

y

x (2)1

1O– 1–1

–5

–2

–2

– 6

– 3

– 3

– 7– 8

–4

– 4

2

2

3

3

4

4

5

6

7

8

y

x1

1O– 1–1

–5

–2

–2

– 6

– 3

– 3

– 7– 8

–4

– 4

2

2

3

3

4

4

5

6

7

8

y

x1

1O– 1–1

–5

–2

–2

– 6

– 3

– 3

– 7– 8

–4

– 4

2

2

3

3

4

4

5

6

7

8

y

x1

1O– 1–1

–5

–2

–2

– 6

– 3

– 3

– 7– 8

–4

– 4

2

2

3

3

4

4

5

6

7

8

y

x (3)1

1O–1–1

–2

–2

–3

–3

–4

–

4

2

2

3

3

4

4

5

6

y

x1

1O–1–1

–2

–2

–3

–3

–4

–

4

2

2

3

3

4

4

5

6

y

x1

1O–1–1

–2

–2

–3

–3

–4

–

4

2

2

3

3

4

4

5

6

y

x (4)1

1O–1–1

–5

–2

–2

–6

–3

–3

–7

–4

–4

2

2

3

3

4

4

5

6

7

8

y

x1

1O–1–1

–5

–2

–2

–6

–3

–3

–7

–4

–4

2

2

3

3

4

4

5

6

7

8

y

x1

1O–1–1

–5

–2

–2

–6

–3

–3

–7

–4

–4

2

2

3

3

4

4

5

6

7

8

y

x (5) None of these

43. 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

44. 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





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45. 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 cylinderis 3/ π units and the height is 16 units.

A

B

16

3/ π

(1)5

2(2)

5

2(3)

5

2(4)

5

3

46. 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) 55CAT 2008

47. 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. Thelength of DH in cm is

(1) 2 2 – 1 (2)( )2 2 – 1

2

A B

C

D

E

H LF

N

M

O

G

(3)( )3 2 – 1

2(4)

( )2 2 – 1

3

CAT 2005





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48. 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

2

πduring next one minute,

r

4

πduring next 2 minutes,

r

8

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

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

CAT 2004

Directions for questions 49 to 51: Answer the questions on the basis of the information given below.In the adjoining figure, I and II are circles with centers P and Q respectively, The two circles touch each other and have commontangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diametersof I and II are in the ratio 4 : 3. It is also known that the length of PO is 28 cm.

I

II

P Q

RS

O

49. What is the ratio of the length of PQ to that of QO?

(1) 1 : 4 (2) 1 ; 3 (3) 3 : 8 (4) 3 : 4 CAT 2004

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

CAT 2004

51. The length of SO is

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

CAT 2004




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Directions for questions 52 and 53: 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.

52. 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

CAT 2007

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

CAT 2007

sqc 2013 - exercise booklet.pdf

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