`quant replica questions that have appeared in cat in the last 4 years

8/10/2019 `QUANT REPLICA QUESTIONS THAT HAVE APPEARED IN CAT IN THE LAST 4 YEARS

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QUANT REPLICA QUESTIONS THAT HAVE APPEARED IN

CATIN THE LAST 4 YEARS

Permutations and Combinations

Directions for question 1: Select the correct

alternative from the given choices.

1. X = {0, 1, 2, 3, 5}I is the set of integers greater than 999 and notexceeding 5000, formed by using one or moreelements of X as its digits. Find the number ofelements in I.(1) 376 (2) 375 (3) 500(4) 499 (5) 501

Directions for questions 2 and 3: Answer thesequestions based on the following information.

The figure below is the plan of a town T. The streets

are at right angles to each other. Shoba is a residentof T whose house is at H and whose office is at O.R is a rectangular park in which there is a road alongthe diagonal AB and P is a prohibited region in T.

2. Shoba wants to go from H to O taking the shortestpath. How many paths can she choose from?(1) 60 (2) 108 (3) 90(4) 126 (5) 72

3. Shoba wants to go from H to the club C, shown inthe figure through O, taking the shortest path.How many paths can she choose from?(1) 1638 (2) 936 (3) 1170(4) 1404 (5) 1560

Directions for question 4: Select the correctalternative from the given choices.

4. K = (p +q +r)15. Find the number of distinct termsin the expansion of the bracket.(1) 136 (2) 153 (3) 145(4) 120 (5) 128

Directions for questions 5 and 6: Answer thesequestions based on the information below.

X is the set of all pairs (q, p) where p and q satisfy

N p >q 1, where N 4. If any two distinct membersin X have one number in common, they are calledmates. Otherwise they are called non-mates. Forexample if N = 4, then X = {(3, 4), (2, 3), (2, 4), (1, 2),(1, 3), (1, 4)}. In X, (1, 3) and (1, 4) are mates (2, 3)

and (3, 4) are also mates but (3, 4) and (1, 2) arenon-mates.

5. For any N, the number of non-mates of eachmember of X is

(1) 2

1

(N

2

3N +2) (2) 2N 5

(3) 3N 10 (4)2

1(N

25N +6)

(5)2

1(N

27N +16)

6. For any N, the number of common mates oftwo mates in X is

(1) N 2 (2) 3N 8

(3)2

1(N

26N +12) (4) N 1

(5)2

1(N

23N)

Directions for questions 7 to 19:Select the correctalternative from the given choices.

7. How many five-digit numbers divisible by 9 canbe formed using the digits 0, 1, 2, 4, 6, 8, 9 suchthat each digit occurs atmost once in any suchnumber?(A) 96 (B) 192 (C) 288 (D) 144

8. How many five-digit numbers divisible by 4 canbe formed using the digits 0, 7, 5, 2, 4, 6 withoutrepeating any digit?(A) 126 (B) 186 (C) 204 (D) 180

9. Six foot ball players, seven hockey players, eighttennis players and nine cricket players are to bearranged in a row. In how many can they bearranged such that all the players who play thesame game sit together?(A) 6!7!8!9! (B) 4!(C) 30! (D) 4!6!7!8!9!

10. In how many ways can seven students be sentinto five different sections?(A) 7

5 (B) 5

7

(C) 35 (D) None of these

11. The sum of all the five-digit numbers that can be

formed by using the digits 2, 4, 6, 7, 8 withoutrepetition is(A) 7919982 (B) 7199928(C) 7999182 (D) 7919928

12. There are 40 lines in a plane of which a set of12 lines are concurrent at A, another set of15 lines are concurrent at B and the set ofremaining lines are parallel. What is the numberof points of intersection of these 40 lines, giventhat the three sets are disjoint?(A) 611 (B) 531 (C) 533 (D) 638

13. The coefficients of how many terms in theexpansion of (x + y z)

100are negative?

(A) 2450 (B) 2500 (C) 2550 (D) 2600

H

O

B

R

AP

C


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14. All possible words are formed using all the lettersin the word EQUATION exactly once and arearranged such that no two vowels and no twoconsonants are in alphabetical order. How maywords are there?(A) 56 (B) 140 (C) 720 (D) 4914

15. How many 5 - digit numbers, comprising only thedigits 4 and 5 are divisible by 3?(A) 10 (B) 11(C) 12 (D) More than 12

16.

There is a rectangular grid consisting of unit cellsas shown in the figure given below. One needs totravel from point A to point C. One can travel onlyto the right and upwards. In how many ways canone go from point A to point C without travelingvia B?(A) 3915 (B) 3150(C) 2520 (D) 6434

17. How many three-digit odd number can be formedsuch that if 2 is one of the digits, the followingdigit is 4?(A) 5 (B) 365 (C) 300 (D) 372

18. A coaching institute was to send 9 parcels for itspostal students, four of whom were from Biharand the remaining five were from Delhi. Thedispatch clerk made a mistake in addressing theparcels. He addressed them in such a way thatno student received the correct parcel althoughall the parcels for Bihar were sent to Bihar and allthe parcels for Delhi were sent to Delhi. Find thetotal number of ways in which he could haveaddressed those 9 parcels.(A) 396 (B) 2737 (C) 475 (D) 89496

19. Three, four and five points are selected on the

sides AB, BC and CD respectively of ABC.None of the 12 points coincide with A, B and C.

How many triangles can be formed by usingthese 12 points as vertices?(A) 220 (B) 217(C) 210 (D) 205

Geometry & Mensuration

Directions for questions 1 to 40: Select the correctalternative from the given choices.

1. PQR is a triangle. PQ = 19.5 cm and QR = 11 cm.

PS is the altitude of PQR of length 5 cm. C is acircle circumscribing the triangle PQR. Find theradius (in cm) of C.

(1) 18.05 (2) 28.85 (3) 27.25(4) 21.45 (5) 31.25

2. ABC is an obtuse angled triangle whose sidesare 7 cm, 16 cm and y cm where y is an integer.Find the number of possible values of y.(1) 9 (2) 10 (3) 6 (4) 16 (5) 17

3. PQRS is a square. M and N are the midpoints ofPS and QR respectively. X and Y are points lying

on the line joining M and N inside PQRS, suchthat PXS = QYR = 60. Find the ratio of thearea of the hexagon PQXRSY and the remainingarea inside PQRS.

(1) 4 3 (2) 2 3+3 (3) 2 31

(4) 1 + 3 (5) 3+1

4. Each of two circles passes through the centre ofthe other. The radius of either circle is 2 cm.Find the area of the intersecting region (in sq.cm).

(1)3

4 3 (2)

3

8+2 3

(3) 3

82 3 (4) 3

8+ 3

(5)2

3

3

2

5. The radius of the base of a right circular cone is6 cm and the height is 21 cm. The cylinder havingthe maximum possible total surface area isplaced inside the cone such that one of its flatsurfaces rests on the base of the cone. Find thetotal surface area of the cylinder, in sq.cm.

(1)8

1353 (2)

8

1321 (3)

8

1383

(4) 8

1473

(5) 5

441

6. C1 and C2 are two equal circles whose centersare M and N respectively. They intersect at X andY. Neither of the centers lies inside the other

circle. If XMN = , what is the range of possible

value of ?(1) 0 30 (2) 0 45

(3) 0 60 (4) 0 75(5) 0 90

7. Two identical circles intersect each other.The radii of the circles and the distance joiningthe centres of both the circles are in the ratio of

1 : 3 . Find the ratio of the area of the regioncommon to both the circles, to that of a circle.

(A)

4

3 (B)

12

323

(C)

6

3 (D)

6

332

8. A large number (greater than 1200) of equilateraltriangles of side 2 units are available to formequilateral triangles. Equilateral triangles of sides16 units, 18 units, 30 units are formed usingthe available triangles. Find the total number oftriangles used.(A) 1050 (B) 1100 (C) 1150 (D) 1200

B

C

A


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9. Find the side of the smallest equilateral triangle

that can contain a semicircle of radius 8 3 cm.

(in cm)

(A) 16 3 (B) 32 (C) 32 3 (D) 40

10. In the figure below, ABC is

equilateral. Three identicalcircles are shown in it. Eachcircle touches the other twocircles and has two of thesides of the triangle beingtangents to it. Find the ratio of the side of thetriangle and the radius of each circle.

(A) ( 322 + (B) ( )3212 + (C) ( )312 + (D) ( )3312 +

11. N points are marked on the circumference of acircle. The number of triangles which can bedrawn by using these points as vertices is 210

more than the number of triangles which can bedrawn by using all but one of these points asvertices. Find N.(A) 21 (B) 22 (C) 23 (D) 24

12. Let X denote the product of the sides of a triangle.Let Y denote the product of its semi-perimeter,

circumradius and its inradius.X

Y=

(A)4

1 (B)

6

1

(C)3

1 (D) Cannot be determined

13. In the figure above, PQRSis a square. A sector of acircle with centre S isshown. TQVU is arectangle such that TU = 9and UV = 32. Find PS.

(A) 60(B) 65(C) 70(D) Cannot be determined

14. Two intersecting circles are said to be orthogonal,if a pair of tangents drawn to them at any point ofintersection are perpendicular to each other.

C1 and C2 are orthogonal circles. Each of themhas a radius of 8 cm. Find the area of the regioncommon to both the circles. (in sq cm)

(A) 16(21) (B) 16(23)

(C) 32(1) (D) 32(2)

15. On the periphery of a square grass field ABCD,two poles are fixed. One of them is fixed at themidpoint of side AB while the other one is fixed atthe midpoint of side BC. Side of ABCD is 16 m.A cow is tied to one of the poles and another istied to the other pole. If each cow is tied with a8 m long rope, then find the ungrazable area inthe field. (in sq. m)

(A) 256 32 (B) 216 32(C) 192 32 (D) 240 32

16. PQR is an isosceles triangle with PQ = QR. QA isthe median to PR. B is a point on QR (or QR

extended) such that AQ = AB. PQA = 30.

Find ABQ.(A) 20 (B) 30(C) 40 (D) Cannot be determined

17. A triangle has an area of 960 sq m. Two of itssides are 68 m and 32 m. Find its third side if it isgiven that the triangle is right angled (in m).(A) 52 (B) 58(C) 62 (D) None of these

18. A right circular cone has a radius of 18 cm and aheight of 54 cm. From it, a right circular cylinderof radius 15 cm was cut. Find the volume of theconical part of the remaining solid (in cubic cm).(A) 4540 (B) 4375 (C) 4780 (D) 3375

19. Two plastic cubes each of which have an integralside (in cm), have the sum of their lateral surface

areas equal to 468 sq cm. The sum of theirvolumes is 945 cubic cm. Find the volume of thehemisphere whose radius equals the sum of theedges of the two cubes (in cubic cm)(A) 2150 (B) 2250 (C) 2325 (D) 2400

20. In ABC, AB = 32, BC = 24 and AC = 40. UsingB as a centre, a circle is drawn. This circle cutsAB and BC at D and E respectively. The radius ofthis circle equals the circumradius of ABC.Find AD : EC.(A) 3 : 1 (B) 2 : 1 (C) 3 : 2 (D) 5 : 2

21. There are 13 spherical balls, each of radius

3 2 . Nine of the balls are arranged in layer in a3 x 3 square formation. Four balls are placed in asecond layer over the four depressions which areformed. Find the height of the smallest cuboidwhich can enclose these balls.

(A) 6 + 6 2 (B) 6+6 3

(C) 2 3 +2 6 (D) 3 2 +6

22. Four points are chosen at random in a regioncomprising an equilateral triangle of side

4 3 and its interior. Which of the following is

true?(A) There are at least two points P, Q such that

PQ 4.(B) There are at least two points P, Q, such that

PQ 4.(C) There are at least two points PQ, such that

PQ > 4.(D) There are at least two points PQ, such that

PQ < 4.23. In the figure below, PQRS is a rectangle with

PQ = 9 and PS = 6. MNOP is a smaller rectanglewith MN parallel to PQ. Also the lengths of MNand MP are integers and A, B, C, D are themidpoints of the sides of the rectangle MNOPshown in the figure. The area of the shadedregion is 16

2/3% of the area of the rectangle

PQRS. Find the length of AB.

QTP

S R

UV


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M A N

B

OCP

D

QP

S R

PB

CF

G

J

RIHEDAQ

A

CB

D

F

E

G(A) 5

(B) 5)2/3(

(C) 52

(D) Cannot be uniquely determined

24. There are two concentric circles with their centreat O. Chord AB of the outer circle is tangent tothe inner circle. If the measure of the line AB is40 cm and the radii of both the inner and theouter circle are integral number of centimeters,then which of the following cannot be the lengthof the diameter of the outer circle?(A) 50 (B) 58 (C) 96 (D) 104

25. In the triangular field PQR, PQ = 54 m, QR = 240 m

and Q =900. [PQ, AB, CD, EF, GH, and IJ are

all equispaced and all the lines are perpendicularto QR. Also HI = IR]. If one moves alongPQABCDEFGHIJR, then what is the totaldistance covered (in meters)?

(A) 270m (B) 594m(C) 351m (D) None of these

26. The maximum number of acute angles in aconvex octagon is(A) 1 (B) 2(C) 3 (D) None of these

27. There are 6 boys B1, B2, B3, .B6 standingat the vertices A to F respectively of a regular

hexagon. They start walking simultaneouslyalong the perimeter in the clockwise direction withspeeds in the ratio 1 :2 :3 :4 :5 :6. WhenB6completes 3 full rounds, which are the verticesat which there are no boys?(A) A, C (B) B, E(C) A, C, E (D) B, D

28. On square ABCD, points P, Q, R and S are onsides DA, AB, BC, CD such that AP = AQ = CR =CS X and Y are points on AB and CDrespectively, which are equidistant from P and R.

FindCRAP

YDXB

+

+

(A) 0.5 (B) 0.75(C) 1 (D) Cannot be determined

29. In the figure below, circles with centres A, B andC have equal radii and similarly circles withcentres D, E and F have equal radii. Find the ratioof the radii of circles with centres at D and G.

(A)11

31013+ (B)

11

31017+

(C)11

31019 (D)

11

)22(10 +

30. Which of the following statements is/are true?() A triangle exists with altitudes measuring 6,

9 and 15() A triangle exists with altitudes measuring 4,

9 and 12 which of the following is true.(A) Only (B) Only II

(C) Both and (D) Neither I or

31. A, B, C and D are four friends who are standingat the four corners of a rectangular field in theanticlockwise order. They decide to meet at apoint P inside the field. If the shortest distance ofthe point P from the initial position of A, B, C, andD are 20 m, 40 m, 60 m and x m respectively,then find the value of x.

(A) 30 2 (B) 20 6 (C) 40 3 (D) 50

32. When a rectangle ABCD (with AB < BC) is foldedsuch that the vertex C touches vertex A, thelength of the crease XY formed is equal to thelength of the rectangle. How many times thelength of the rectangle is its breadth?

(A)2

1

2

12

+ (B)

21

2

12

(C)2

1

2

15

+ (D)

21

2

15

33. Two perpendicular chords PQ and RS intersect atT PT= 4 and QT=18. If TS is 2 times TR, thenfind the radius of the circle

(A) 110 (B) 2 30 (C) 130 (D) 5 6

34. On side AB of rectangle ABCD, P is a point such

that APD : DPC: CPB = 2 : 5 : 5. Find AB :BC.

(A) 3 (B) 232

(C) 2 (D) Cannot be determined

35. A cuboid has a square base. The length of its

longest diagonal is 2502 . All dimensions of the


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cuboid are integral. How many such cuboids arepossible?(A) 1 (B) 2 (C) 3 (D) 4

36. In a cyclic quadrilateral, the diagonals AC and BDintersect at P at right angles. If PA = 8, PC = 9and the radius of the circumcircle of ABCD is

25.81 , then find PD given that where PD < PB.

(A) 12 (B) 9 (C) 8 (D) 6

37. In trapezium ABCD, the diagonals AC and BDintersect at P. Three times the area of thepentagon APBCD is equal to 7 times the area ofPCD. Find AP : PC.

(A)3

2 (B)

4

3 (C)

3

4 (D)

2

3

38. The semiperimeter of a right angled triangle is60 and the inradius is 8. Find the length of thesmaller leg.

(A) 30 (B) 20 (C) 24 (D) 18

39. There are two vertical poles AB and CD whereB and D are on the ground. AB = 4, CD = 7.5,BD = 11.5. On line BD, P is a point where thepoints A and C subtend a right angle and Q is apoint that is equidistant from A and C. Find PQ.(A) 3 (B) 3.5(C) 4 (D) Cannot be determined

40. In triangle PQR, Q =900

and PQ = QR. D is a

point on the same side of PR as Q such that PR

= 2 QD. Find the measure of angle PDR.

(A) 221/2

(B) 30(C) 45(D) Cannot be uniquely determined

Special Equations


1. In a certain country, the currency used was calledfemto. Raju had his lunch in one of the hotels in it.His bill came to 47 femtos. He had notes in only3 denominations 1-femto, 2-femto and 20-femto.In how many ways can he settle the bill?

(1) 24 (2) 18 (3) 48 (4) 16 (5) 42

2. A bank teller was making the payment for acheque presented by Mohan. As he was in aconfused state of mind, he transposed the rupeesand paise and hence gave more than what heshould have. Mohan left the bank and bought abiscuit from a nearby store for `1.50.The amount remaining with him was 4 times theamount on the cheque. The amount remainingwith him must have been between(1) `70 and `71 (2) `79 and `80(3) `85 and `94 (4) ``93 and `94(5) `98 and `99

3. Nagu bought a new bike and went for a drive,liding at a uniform speed on a highway. At 9 amhe passed a milestone. He turned his head backand read the number on the milestone.He continued driving and at 10 am, he passedanother milestone. Again he turned his head backand saw that the number was the reverse of the

number on the first milestone. He continueddriving and at 11 am he passed a third milestone.Again, he turned his head back to note thenumber. The sum of the digits of the number onthis milestone was equal to the sum of the digitsof the number on the first milestone. Which of thefollowing is the speed at which he travelled?(Assume that the numbers he saw wereincreasing).(A) 9 km/hr(B) 45 km/hr(C) 18 km/hr(D) Cannot be determined.

Equations, Ratio, Proportion, Variation


1. Amar, Bhuvan, Chetan and Dinesh arefour friends. Amar has m marbles with him.He gives Bhuvan 1 less than half the number ofmarbles he has. Then he gives Chetan 1 lessthan half the remaining number of marbles hehas. Finally, Amar gives Dinesh 1 less than halfthe remaining number of marbles he has and isleft with 4 marbles. Which of the following bestdescribes the value of m?

(1) 1 m 4 (2) 5 m 9(3) 9 m 13 (4) 10 m 14

(5) m 14

2. If a b

1 and b c

2, then find the value of

9a + 5c, when b = 27. Given that when b = 3 thena = 4 and c = 6.(A) 94 (B) 49 (C) 76 (D) 38

3. If 3a + 5b + 7c = 1.25 k and 2a + b + 3c = 0.75 k,then 7b + 5c is what percentage of k?(A) 25% (B) 50% (C) 35% (D) 75%

4. There are some two rupee coins and five rupeecoins in a bag. If the number of five rupee coins istripled, then the amount in the bag is increasedby 75%. Which of the following can be thenumber of five rupees coins in the bag?(A) 13 (B) 20 (C) 18 (D) 32

5. There are 100 questions in a test paper. Four marksare awarded for each right answer and two marksare deducted for each wrong answer. If Abhilashattempts more than 85 questions and get70 marks, What is the minimum number ofquestions that he could have answered correctly?(A) 40 (B) 38 (C) 39 (D) 41


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6. Arun, Varun and Kiran have a total of `8,000 withthem. They spend `50, `100 and `200respectively, Now they have a money is the ratioof 14 : 22 : 15. What is the amount with the Varuninitially?(A) 2150 (B) 3400(C) 3300 (D) 2100

Time and Distance


1. P and Q are 2 stations. Raju plans to drive fromcity R, located 840 km directly to the north of Q,

to Q, at a speed of3

210 kmph so that he can

catch a train arriving there from P. The trainleaves P at 9 a.m. and travels at a speed of

84 3 kmph. P is between east and south-east

of R with RQ at 60 to RP. Also P is between thenorth and the north-east of Q with PQ at 30 toRQ. If Raju has to reach P at least 25 minutesbefore the train, then find the approximate latesttime at which he should start from R.(1) 6:50 a.m. (2) 6:40 a.m. (3) 6:20 a.m.(4) 6:30 a.m. (5) 7:00 a.m.


Cities P and Q are 4800 km apart. P is to the west ofQ. Both cities are in different time zones BestAirlines was an airline which operates non-stopflights between the cities. All its planes cruise at the

same speed in both directions. However, the effectivespeed of any plane is influenced by a steady windblowing from west to east at 100 kmph. The tablebelow shows the departure time of the planes fromeach city and their arrival time at the other city.

(Given below are local times of the respective cities)

Departure ArrivalCity Time City Time

P 7:00 a.m. Q 5:00 p.m.Q 6:00 p.m. P 4.00 a.m.

2. Find the time in P when the plane landed in Q.

(1) 4:00 p.m. (2) 3:00 p.m.(3) 3:30 p.m. (4) 2:30 p.m.(5) Cannot be determined

3. Find the cruising speed of the plane (in kmph).(1) 600 (2) 550 (3) 500(4) 450 (5) Cannot be determined


4. A starts from point P at 9:00 a.m. and travels eastat 45 km/hr. B starts a bit later from P and travelssouth at 30 km/hr for 48 minutes. At that instant,he is 74 km from A. When does B start from P?

(A) 9:42 a.m. (B) 9:45:20 a.m.(C) 9:48 a.m. (D) 9:51:40 a.m.

5. In an x-metre race, A beats B by 180 m andC beats by 351 m. In the same race, B beatsC by 198 m. Find x?(A) 1200 (B) 1000 (C) 1122 (D) 1320

6. Towns A and D are 36 km apart. Three friends,Tarun, Varun and Arun start together from

A towards D. While Arun sets off on foot, Varuntakes Tarun along on his bike and travels at45 km/hr. He drops Tarun at a point C and turnsback for Arun. He meets Arun at a point B, andthen turns back towards D. All the three friendsreached D together. If both Arun and Tarunwalked at 5 km/hr, how long do the 3 take tocover AD?

(A) 3 (B) 33

2 (C) 3

9

1 (D)

15

131

7. Two friends P and Q start simultaneously fromthe opposite ends of a race track AB of length150 m at speeds 60 m/s and 40 m/s respectively.

P starts from A and Q starts from B. Once eachreaches an end, he immediately turns back andmoves towards the other end. They keep movingto and from between the two ends. Find thedifference between the distance covered by Pand Q by the 8

thmeeting.

(A) 300 m (B) 450 m (C) 600 m (D) 1200 m

8. Ajay and Sanjay start simultaneously from thesame point on a circular track. If they travel inopposite directions, they meet at 7 distinct pointson the track whereas if they travel in the samedirection, then they meet at n distinct points on itwhere n is a prime number. If Ajay is faster thanSanjay and Sanjay's speed is P% less than thatof Ajay, then which of the following can be avalue of P?(A) 25 (B) 50 (C) 75 (D) 83

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Quadratic Equations

Directions for questions 1 and 2: Select the correctalternative from the given choices.

1. Three consecutive positive integers are taken in

descending order. The first, second and the third

are raised to the first, second and third powers

respectively. The powers are added and the

result is multiplied by 3. The square root of the

result is found to be the sum of the originalintegers. The least of the integers is denoted by

L. Which of the following holds true?

(1) 1 L 4 (2) 5 L 8(3) 9 L 12 (4) 13 L 15(5) L > 15

2. The roots of the equation x3px

2+qx r = 0 are

a, b and c, which are consecutive integers.

Find the least possible value of q.

(1) 0 (2) 1 (3) 2

(4) 1 (5) 2


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Directions for questions 3 and 4: Answer thesequestions using the information below.

Let g(x) = px2+qx +r, where p, q and r are constants

and r 0. One root of g(x) = 0 is 4 and 8g(3) = 3g(6).

3. Find the other root of g(x) = 0.

(1) 1 (2) 3 (3) 6(4) 2 (5) Cannot be determined

4. Find the sum of p, q and r.(1) 21 (2) 20 (3) 19(4) 18 (5) Cannot be determined

Directions for questions 5 and 6: Answer thesequestions based on the given information.

Raju makes and sells an item in a market every day.He sells each unit of that item at `106. The cost of

producing x units per day is 200 +px +qx2, where p

and q are constant. If Raju increased his dailyproduction from 10 units to 15 units, his daily

production cost would increase by 888

/9%. If heincreased his daily production from 15 units to20 units, his daily production cost would increase by64

12/17%. Assume that there is a high demand for the

item and that Raju can sell whatever he produces.He wishes to maximize his profit.

5. Rajus daily production must be (in units).(1) 10 (2) 15 (3) 8 (4) 5 (5) 12

6. Find the maximum daily profit that Raju canobtain (in Rs).(1) 400 (2) 425 (3) 350(4) 375 (5) 184


7. E(y) is a quadratic expression. It has the

minimum value of 1 when y = 3 and E(2) = 2.Find E(4).(1) 40 (2) 45 (3) 50 (4) 55 (5) 60

8. If the equations ax2+ bx + c = 0; bx

2+ cx + a = 0

have one root in common, then which of thefollowing is definitely true?(A) a

3+ b

3= 3abc

(B) a3+ b

3+ c

3= 3abc

(C) a3 b

3 c

3= 3abc

(D) a

2

+ b

2

+ c

2

= 2ab + 2bc + 2ca

9. If , and are the roots of the equation

x3+ 2x

2 5x 6 = 0 and , , are the roots of

the equation x3+ px

2+ qx + r = 0; then the value of

p is(A) 6 (B) 5 (C) 5 (D) 6

10. Let and be the roots of a quadratic equation

and = 9 and || || = 5 then find theproduct of roots.(A) 14 (B) 21 (C) 16 (D) 7

11. What is the minimum value of the square of thedifference of the roots of the quadratic equation

x

2

(k + 7) x (3k 15) = 0?(A) 160 (B) 180 (C) 160 (D) 180

12.A certain number of cups of tea are available for`90. If the price of each cup increases by `1.50,the number that can be bought for the sameamount decreases by 10. Find the actual cost ofeach cup of tea (in rupees).(A) 3 (B) 3.50 (C) 4 (D) 4.50

13. The expression x4 + y4 2x2 y2 32x2 32y2 +256 is completely factorized into real factors.Which of the following statements about thefactors is true?(A) There are 2 irreducible quadratic factors.(B) There is a cubic factor and a linear factor.(C) There are 4 linear factors.(D) None of these

14. The expression ax2 + bx + c takes a maximum

value of 5 at x = 1 and takes the value of -1 atx = 0. Find the value of the expression at x = 5.(A) 89 (B) 91(C) 210 (D) 211

15. The roots of the equation x3

+ ax2+ bx + c = 0,

where c > 0, are k, kr, kr2 where k and r are

integers. If the sum of the squares of the roots is364, then find the value of c(A) 180 (B) 72 (C) 108 (D) 216

16. If the sum of the roots (not necessarily real) of a

quadratic equation is 6 and the sum of the

squares of the roots is 16, then find the product of

the roots.

(A) 8

(B) 10

(C) 9

(D) Such an equation doesn't exist

17. The roots of x3 21x

2 + px 280 = 0 are in

arithmetic progression. Find the value of p.

(A) 138

(B) 128(C) 118(D) Cannot be determined

18. If cubic equation ax3+ bx

2+ cx + d = 0 has two

positive roots where a, b, c, d are real and d0,

then which of the following is true?

(A) c and b are of opposite signs.

(B) a and d are of the same signs.

(C) b and c are of the same sign.(D) a and d are of opposite signs.

Progressions / Series


1. Find the number of common terms of thesequences 24, 29, 34, .. 474 and 25, 29, 33,, 485(1) 20 (2) 21 (3) 25 (4) 24 (5) 23

2. Nilgiris coffee cost `(110 + 0.2N) per kg on the

Nth day of 2006 where 1 N 200. Its price


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8

remained constant that year from the 200th day.

Coorg coffee cost `(97 + 0.25N) per kg on the

Nth day of 2006 where 1 N 365. Find the datein 2006 on which the prices of the two varieties ofcoffee were equal.(1) June 25 (2) August 10(3) July 31 (4) August 20

(5) July 21

Directions for questions 3 and 4: Answer thesequestions based on the given information

M and N are positive quantities. Let g1= M and h1= N.When x is even, gx= Mhx 1and hx= Mgx 1When x is odd, gx= Nhx 1and hx= Ngx 1

3. Which of the following equals gx+hxwhen x is even?

(1) M(MN)x

2

1

(M +N)

(2) MN1x

2

1

(M +N)

(3) M(MN)1x

2

1

(M +N)

(4) MM 21

(M +N)

(5) M(MN)2x

2

1

(M +N)

4. If M =4

1and N =

4

3, find the least odd x for which

gx+hx< 0.02?(1) 3 (2) 5 (3) 7 (4) 9 (5) 11


5. If k =13.12.11.10

1 +14.13.12.11

1 +

+102.101.100.99

1, then the value

of k is

(A)5616600

2149 (B)

5666100

1249

(C)5666100

1429 (D)

5616600

4219

6. Find the sum of all the five digit numbers whichleave a remainder of either 4 or 6 when dividedby 8.

(A) 659100000 (B) 561900000(C) 1318200000 (D) 1237522500

7. If the sum of the first n terms of an arithmeticprogression in 2400 and the sum of next n termsas 7200, then find the ratio of first term andcommon difference.(A) 3 : 2 (B) 2 : 1 (C) 1 : 2 (D) 2 : 3

8. If the nthterm of a series is given by tn =

2

tt 1n1n + + ,

then27

31

t

tis

(A) 0 (B) 1(C) 1 (D) None of these

9. Find the sum of the first 40 terms of the series2 + 3 + 4 + 6 + 8 + 9 + 16 + 12 + . . .(A) 6

21 (B) 2

21+ 628

(C) 220

+ 628 (D) 620

+ 629

10. If S =

124

126

63

65

26

28

7

9ad infinitum, S =

(A)2

3 (B)

2

1 (C)

3

5 (D) 3

11. If S1=210

27...

30

9

20

7

12

5

6

3

2

1++++++ and

S2 =14

1...

4

1

3

1

2

1++++ , then find the value of

2S2 S1

(A)5

4 (B)

4

3 (C)

2

1 (D)

2

3

12. If a, b, c, are 3 positive numbers in geometric

progression and a12= b15= cx, then x =(A) 18 (B) 20 (C) 21 (D) 24

13.3.2.1

1+

4.3.2

1+

5.4.3

1+ . . . +

15.14.13

1 is equal

to

(A)105

26 (B)

105

31 (C)

7

1 (D)

210

59

14. If S=1

1

2

1+

3

1

4

1+

5

1

6

1+. . . +

99

1-

100

1, the value of S is also equal to

(A)1

1+

2

1+

3

1+

4

1+... +

25

1

(B)51

1+

52

1+

53

1+ ... +

100

1

(C)50

1+

51

1+

52

1+ ...

100

1

(D) None of these

15. The nthterm tnof a sequence is defined by tn= tn-1

tn-2. If t1= 3, t2= 4, and tntmis a multiple of 10,which of the following is not a possible value of(m, n)?(A) (10, 16) (B) (16, 10)

(C) (7, 1) (D) (8, 2)

16. The sum of n terms of an arithmetic progression,starting with the 11th term is directly proportionalto n

2. The sum of 2 of the terms of the arithmetic

progression is 0. Which of the following cant beeither of these two terms?(A) 5

th (B) 11

th (C) 21

st (D) 16

th

17.)10(12...)3(5)2(4)1(3

)12(10....)5(3)4(2)3(12222

2222

++++

++++=

(A)3795

4784 (B)

11

15 (C)

23

29 (D)

5

6


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9

Functions


1. G(x) is a function satisfying

=

b

aG

)b(G

)a(G for all

real a, b If G(3) =9

1, find G(9).

(1)81

1 (2)

9

1 (3) 81

(4) 9 (5) Cannot be determined

2. G(y) is a function satisfying the following conditions.

G(1) = 5400 and =

x

1y

G(y) = X2G(X). where X is a

natural number greater than 1. Find G(6).

(1)7

2100 (2)

7

2400 (3)

7

1800

(4) 7

1500 (5) 240

3. If f(xy) = f(x) + f(y) + f(x) f(y) 9

10and f(6) =

3

2,

then the value of f(1/6) is

(A)3

2 (B)

8

9 (C)

9

17 (D)

8

17

4. If f(x) =x1

x

+, then the value of f(f(f(f(f(f( 3))))) =

(A)26

5 (B)

25

4 (C)

19

3 (D)

37

6

5. If f(x +2) = f(x) + 7 and f(1) = 2; f(2) = 5 then theratio of f(150) to f(75) is(A) 523 : 261 (B) 253 : 621(C) 427 : 673 (D) None of these

6. If f (x + y) = f(x) + f(y) , where x 1, y 1 and f(7) =24.5, then find the value of f(1) + f(2) + f(3) + f(4) +f(5) + f(6) + f(7).(A) 63 (B) 84(C) 98 (D) None of these

7. If f(x+1) f(x) = ax + b, then which of thefollowing is true about the graph of y = f(x)?

(A) A line with a slope of a(B) A line with a slope of b(C) A parabola(D) None of these

8. Consider the functionf(x) = (x + 3)

p(x + 2)

q(x - 1)

r(x 3)

s, where p. Q.

R. s are nonnegative integers.f(3.5) = a, f(2.5) = b, f(0.5) = cf(1.5) = d, f(3.5) = e

Also, ab > 0,c

b 0 and eded =+ .

Which of the following can be the value of(p, q, r, s)?

(A) (2, 3, 5, 1) (B) (4, 3, 2, 0)(C) (3, 2, 5, 1) (D) (0, 1, 2, 3)

9. The continuous function f(x) satisfies thefollowing conditions x

f(x + 2) f(x) = 4 for x 2f(x + 2) + f(x) = 6 for 0 < x < 2

and f(x + 2) f(x) = 4 for x 0also f(0) = 5Which of the following is f(x)?

(A) 14x2 + (B) 5x2 +

(C) 32x4 + (D) 14x2 +

10. If f(x + 1) f(x) = x and f(x) is an nth degree

expression in x, then n =(A) 2 (B) 3(C) 4 (D) Cannot be determined

11. If f1(x) =

2x7

1x

+

, f

n(x) = f

1[f

n-1(x)], find f

8(x)

(A)1x7

1x2

+

(B)

1x7

1x2

+

(C)1x7

1x2

+ (D)

1x7

1x2

Averages


1. Twelve years ago, the average of the ages of themembers of a joint family having ten memberswas 25 years. Four years later a member aged50 years died and a child was born in the familythat year. Four years after that, another memberaged 50 years died and another child was born.Find the present average age of the members ofthe family (in years).(1) 26 (2) 27 (3) 28 (4) 29 (5) 30

2. There are two groups (A & B) of children in a jointfamily. There are 3 more children in B ascompared to A. While the average age of thechildren in A is 6 years more than that of B, thecombined age of the children in A is less thanthat of B. After 4 years, the difference in thecombined ages of group A and group B willdouble. If there are less than 10 children in thefamily, which of the following can be the averageage (in years) of the children in B?

(A) 9 (B) 8 (C) 7 (D) 5

3. The number and the average weights of differentgroups of children are given below:Number Average Weight (Kg.)M 40M + 2 45M + 2 507 60If the average weight of the entire class is 50 kg.Find the total number of students.(A) 20 (B) 21 (C) 22 (D) 23

4. There are two vessels P and Q, P containing 120L of milk and Q containing 120 L of water. In the

first operation 30 L is removed from P and poured


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10

into Q and then 30 L from Q is poured back intoP. Like this, one more operation takes places i.e.transferring 30 L from P to Q and thentransferring 30 L from Q to P. What is the ratio ofmilk and water in P, finally?(A) 2 : 1 (B) 13 : 6(C) 15 : 7 (D) 17 :8

5. Dinku Beora was a chronic alcoholic andbecause of persistent, health problems hedecided to quit drinking. He devised aningenuous way of doing so. He bought a 750 mlgoodbye bottle of Old Monk. On the first day, hedrank 5% of the contents in it and replaced thatquantity with water. Next day he drank 10% of thecontents in the bottle and replaced it with water.Like this he continued on the 19

th day he drank

95% of the contents in the bottle and replaced itwith water and on the 20

thand last day he drank

the entire contents in the bottle. Find the ratio ofthe total quantity of alcohol and water that he

drank in the entire process.(A) 2 : 21 (B) 2 : 19(C) 3 : 28 (D) None of these

Quant Based Reasoning

Directions for questions 1 and 2: Answer thesequestions based on the following information. P, Q, R,S and T are 5 horses. They participated in a race.The following are the rules of the race.

(1) A person who bets on the winning horse gets4 times the bet amount.

(2) A person who bets on the horse coming secondgets 3 times of the bet amount.

(3) A person who bets on the horse coming in thirdgets back his amount.

(4) Other persons lose their amount.

Mohan had placed his bets on Q, R and T.The amounts he bet on Q, R and T were `2000,`4000 and `6000 respectively. He ended up with nogain and no loss.

1. Which of the following is not possible?(1) At least two horses finished before P.

(2) There were three horses between Q and P.

(3) T finished last.

(4) S came in second.

(5) There were three horses between T and R.

2. Suppose S finished in the fourth position. Then

which of the following is not possible?

(1) P finished first.

(2) One horse finished between Q and R.

(3) Q came in second

(4) T finished last.

(5) R came second.

Data Sufficiency

Directions for questions 1 and 2: Answer thesequestions based on the instruction below.

In the questions below, each question has twostatements A and B following it. Mark your answer as

(1) if the question can be answered from A alone butnot from B alone.

(2) If the question can be answered from B alone butnot from A alone.

(3) if the question can be answered from A alone aswell as from B alone.

(4) if the question can be answered from A and B

together but not from any of them alone.(5) if the question cannot be answered even from A

and B together.A certain number of players participated in atournament, played according to the followingrules. The number of players at any stage isdenoted as N.

(i) if N is even, the players are grouped into2

Npairs.

The players in every pair play against each other.The resulting winners move on to the next round.

(ii) If N is odd one player is allowed to move on tothe next round. He is said to be given a bye. The

remaining N 1 players are grouped in to 2

1N

pairs who play against each other. The resultingwinners move on to the next round. The playerswho lose are eliminated from the tournament.From the rules above, it follows that if there are

N players in a round, then2

Nplayers move on to

the next round if N is even and2

1N+ players

move on to the next round if N is odd.This process continues until the final round,which is played between two players. The winnerin this round is the champion.

1. Find the number of matches played by the champion.

A. In the first round, there were 169 players.

B. The champion was given a bye only once.

2. The number of players in the first round was Mwhere 129 < M < 256. Find M.A. One player received a bye while moving from

the third to the fourth round.B. Only one player received a bye in the entire

tournament.

Directions for questions 3 to 6: Each question isfollowed by two statements A and B. Indicate yourresponses based on the following directives:

Mark (1) if the question can be answered using Aalone but not using B alone.

Mark (2) if the question can be answered using Balone but not using A alone.

Mark (3) if the question can be answered using Aand B together, but not using either A orB alone.

Mark (4) if the question cannot be answered evenby using A and B together.

3. Class X has 80 students. The average height ofthe students in it is 140 cm. It has two sections,A and B, with equal number of students in each

section. The average height of A exceeds that ofB. Mohan is the tallest in A and Sohan is the


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11

shortest in B. If each of these students istransferred to the other section, the averageheights of the sections would get interchanged.Find the height of Mohan.(A) The average heights of A and B differ by 2 cm.(B) If Sohan shifted from B to A, the average

heights of the sections would become equal.

4. A company has to store at least 270kilolitres ofwater at all times to meet safety and regulatoryrequirements. It is considering having a sphericaltank whose wall thickness is uniform and whoseouter radius is 6 meters for this purpose. Will thetank meet the company requirements?

(A) When empty, the tank weighs 36000kg. It ismade of a material whose density is 4 gm/cc.

(B) The tanks inner radius is at least 4.5 metres.

5. P, Q and R are three integers. Find the maximum

value of PQ +QR +PR.(A) P = Q R

(B) P +Q +R = 84

6. P is a point on AB. Rohit wanted to draw asquare ABCD but failed to do so. Why did he fail?(A) PC = 3 cm

(B) PC =3

PD

Percentage, Profit & Loss


Mohan was considering three alternatives for investinga certain amount. He wanted to get the maximum

possible assured return on his investment. The threealternative are given below. He could make use ofeach completely or partially along with the others.Alternative 1: Invest in the mutual funds of PQR Ltd.

A rise in the stock market will result in a return of 8%and a fall will result in a return of 10%.

Alternative 2: Invest in the mutual funds of RQP Ltd.

A rise in the stock market will result in a return of 5%

and a fall will result in return of 4%.

Alternative 3: Invest in a public sector bank which

promises a 0.4% return.

1. Find the greatest assured return for Mohan.

(1) 0.4% (2) 0.5% (3)3

2%

(4) 0.8% (5)6

5%

2. Find the strategy which will maximize theguaranteed return to Mohan.(1) 100% in alternative 3.(2) Equal investment in each alternative.(3) Investments in alternatives 1 and 2 in the

ratio 1 : 2(4) Investments in alternatives 1 and 2 in the

ratio 2 : 1

(5) Investments in alternatives 1, 2 and 3 in theratio 2 : 3 : 4


3. The price of coffee was increased by 40% but Rajwas willing to increase his expenditure by 12%.Find by what percentage should he decrease hisconsumption.

(A) 10% (B) 15% (C) 20% (D) 25%

4. The ratio of the populations of cities X, Y and Z in2008 was 3 : 5 : 6. The percentage increases inthe populations of X, Y and Z from 2008 to 2009were 10%, 12.5% and 15% respectively. Find thepercentage increase in their total population from2008 to 2009.(A) 12% (B) 14%(C) 13% (D) None of these

5. If the cost of a ball pen reduces by 20%, Raj can

buy 90 more ball pens for `3600. Find the cost (in

`) of a ball pen.

(A) 12.5 (B) 10 (C) 15 (D) 20

6. Ashok made a loss of 15% by selling 96 apples

for `2040. How many apples must he sell for

`2600 to make a 30% gain?(A) 80 (B) 100 (C) 65 (D) 104

7. Three persons A, B and C have their monthly

incomes in the ratio 6 : 7 : 8. Their monthly

expenditures are in the ratio 5 : 6 : 10. The monthly

savings of C is 37.5% of his monthly income. Find

what percent of Bs savings was As savings?(A) 50% (B) 66

2/3% (C) 83

1/5% (D) 87.5%

8. Pradeep bought a puppy for a certain price.He sold it to his neighbour at a profit percentwhose magnitude was equal to the profit realisedby him in the transaction. But after a couple ofdays the neighbour sold it back to Pradeep at20% loss. Effectively, 27.5% of the cost price wasrefunded to Pradeep. At what profit percentagedid he sell the puppy to his neighbour?(A) 30% (B) 37.5% (C) 27.5% (D) 32.50%

Simple Interest Compound Interest


1. The difference between the simple interest andthe compound interest for two years on a suminvested at 16% p.a. is `384. Find the sum(in `)(A) 13500 (B) 15000 (C) 14250 (D) 12750

2. Ramu took a certain loan at Simple Interest in2000 for a period of 4 years. The rate of interestwas constant throughout the loan periodwhereas had he cleared the loan after 9 years,he would have paid `90,000. Where as had hecleared the loan after 12 years, he would havepaid `105,000. Find the amount (in `) he paid toclear the loan.

(A) 45000 (B) 65000 (C) 70000 (D) 75000


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12

Time and Work


1. Pipe X can fill a tank in a certain time. It wasopened at 12 pm. Due to a leak at the bottom of

the tank, the tank was filled only at 12:50 pm.If the leak can empty the tank in 200 minutes,then find the time (in minutes) in which X can fillthe tank.(A) 30 (B) 45 (C) 40 (D) 50

2. 8 men can build a wall, 8 m long in 8 days,working 8 hours a day. Find the number ofhours per day for which 16 men are required towork, to build a wall 16 m long in 16 days.(A) 2 (B) 8 (C) 16 (D) 4

3. 50 men can complete a job in 20 days working9 hours a day. They started the job. They

worked 9 hours each day for the first x days.At the end of x days, 5 men left. The remainingjob was completed by the remaining menworking 8 hours a day in 12.5 days. Find x.(A) 12.5 (B) 8 (C) 9 (D) 10

4. If A, B, C, D work independently, the amount thatA is paid for 3 days is equal to that paid to B for4 days. The amount paid to C for 3 days is equalto that paid to D for 2 days. B and D are paidequal amounts for equal duration. A and Btogether complete a piece of work for which theyare paid `1680. If all 4 had completed the samework together, what would A's share have been(in rupees)?

Assume that equal amounts are paid for equalwork.(A) 960 (B) 800 (C) 560 (D) 630

5. For workers working on a construction site, therate of doing work for men and women increasesin winter with respect to that in summer by

333

1% and 50% respectively. 4 men and 12

women can complete a certain piece of work in120 days in summer. The time taken triples if thewomen do not turn up for the work. If a men andb women can complete two times the given workin 180 days in winter then find the possible

number of ordered pairs of (a, b).(A) 2 (B) 20 (C) 10 (D) 5

6. A and B are two daily labourers who work on amaintenance site. The daily wage of A is 40%less than the daily wage of B. A and B togetherworked on a certain project and completed it in72 days. As a result, at the end of the project theytogether received a certain amount. If that entireamount were to be earned by A alone, then forhow many days would he need to work ?(A) 144 (B) 216 (C) 288 (D) 192

7. A and B can together do a piece of work in

12 days. If A completed half the work and theother half is completed by B with only one of

them working each day, the total time takenwould be 24

1/2days. Find the number of days A

alone would take to complete the work if B isfaster than A.(A) 28 (B) 21 (C) 24 (D) 25

Venn diagrams


1. In a class, 30% of the students like tea and 40%of the students like coffee. 20% of the studentswho like tea also like coffee. Find thepercentage of the studies who like neither teanor coffee.(A) 32% (B) 34% (C) 38% (D) 36%

2. In a locality, 180 residents watch only Sony TV,210 residents watch only Star Plus and 150residents watch only Zee TV. 540 residents

watch atleast one of Sony TV and Star Plus. Atmost 340 residents watch Star Plus. 90residents watch all the three channels. Find theminimum possible number of students whowatch Sony TV and Zee TV but not Star Plus.(A) 40 (B) 50 (C) 20 (D) 30

3. In a group, 60% of the boys and 50% of the girlslike cricket. 45% of the boys and 55% of thegirls like volleyball. Number of students who likecricket is 7 more than that of students who likevolleyball. The difference of the number of boysand girls who like cricket and that of the numberof boys and girls who like volleyball are in the

ratio 16 : 5. Find the strength of the group.(A) 90 (B) 100 (C) 110 (D) 120

Trigonometry


1. T1and T2are two towers and Raju was on the topof the tower T1. He realized that there were twopoints on the ground such that the angle ofelevation of T1s top from each of those points

was . The distances from, T2s bottom to the topof T1 as well as to each of the points was30 feet. The area of the triangle formed by the top

of T1 and each of the points can be (in feet).(T2 's bottom and the two points on the groundare collinear)(A) 225 (B) 960(C) Both (A) and (B) (D) Neither (A) nor (B)

2. A ladder has a length of 10 m. It makes an angleof 45

with a wall. It touches the wall at a point P.

There are two points on the ground. The angle ofelevation of P from each of these points is 60.Find the distance between these points (in m).(The two points on the ground and the bottom ofthe wall are collinear)

(A) 3

67

(B) 3

610

(C) 3

68

(D) 3

64


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13

3. Given that ,rqp

rpqrpqcos

222 ++

++= where p, q, r are

real numbers such that the sum of any twoexceeds the third, which of the following is not apossible value of ?

(A) 20 (B) 40 (C) 50 (D) 80

Indices, Logarithms, Surds


1. 33x + 2

92x 1

= 118098, find the value of 6x+ x

6.


2. Find the value of

121120120121

1...

18171718

1

17161617

1

+++

++

+

(A)15

20

(B)

44

7 (C)

44

37 (D)

15

13

3. If ,3x

1x =+ find the value of x

6+ .

x

16

(A) 729 (B) 326 (C) 322 (D) 324

4. If a = 5 2 6 and b = 5 + 2 6 , evaluate

33

44

ba

ba

+

+.

(A)49

4901 (B)

485

4799 (C) 10 (D)

485

4801

5. If log2log4x = log4log2x, find x.

(A) 16 (B) 8 (C) 8 2 (D) 24

6. If 1, log7(4x+ 5), log7(4

x + 1 1) are in arithmetic

progression, which of the following is a possiblevalue of x?(A) 1 (B) 1.5 (C) 2 (D) 2.5

7. Let t = 3(log3a)2 9log3a + 9. Which of the

following is true about the equation at=27?

(A) It has exactly one distinct solution for t.(B) It has exactly two distinct solutions for t.(C) It has exactly three distinct solutions for t.

(D) It has no real solutions for t.

Modulus / Inequalities


1. How many integer values of x exist that do not satisfythe inequality (x

2+ 4x 32) (x

2+ 2x 8) > 0?

(A) 7 (B) 8 (C) 6 (D) 4

2. Under which of the following conditions the

inequation baa

b

b

a 22 is true?

(A) ab and ab < 0 (B) a b and ab > 0(C) Either (a) or (b) (D) None of these

3. What is the set of values of x for which the

inequation6x5x

8x9x42

2

+> 3 is satisfied

(A) (6, 1) (1,) (B) (, 6) (1, )

(C) (, ) (D) (, 1) (6, )

4. If x1, x2, x3, x4, x5and x6are positive and x1x2x3x4 x5 x6= 1; then the minimum value of (x1+ 4)(x2+ 4) (x3+4) (x4+4) (x5+4) (x6+4) is(A) 15625 (B) 1425(C) 13225 (D) 11025

5. If S =512

1...

4

1

3

1

2

1

1

1+++++ , then S lies in

which of the following ranges?(A) (4, 5.5) (B) (5.5, 9)(C) (9, 10.5) (D) (10.5, 12)

6. P is any positive number such that it is possibleto find 4 positive numbers whose product is P

and whose sum S P. Among the numbers 4, 5,6, 7, how many are possible values of P?(A) 0 (B) 1 (C) 2 (D) 3

7. If x3+ y

3+ z

3= 125, a

2+ b

2+ c

2= 16, ax + by + cz =

20 and x, y, z, a, b, c are integers, then which ofthe following cannot be the value of

( )222 zyx4cba

++

++?

(A)25

1 (B)

27

1 (C)

33

1 (D)

49

1

8. How many integral values of x satisfy thefollowing inequality

7x3x5x14 ++++< < 25?

(A) 9 (B) 11 (C) 13 (D) 16

9. Find the area described by the inequality

2yxyx ++

(A) 3 (B) 2 (C) 2.5 (D) 4

10. In triangle PQR, p, q and r are the lengths of thesides opposite P, Q, and R respectively. If (p + q+ r)

2= 3(pq + qr + rp), what can be said about the

triangle PQR?(A) It is equilateral.(B) It is isosceles.(C) It is scalene.(D) Cannot be uniquely determined

Operator Based Questions


1. Given that a b = 4a + 3b + 7ab and a a < b b, then which of the following is true?(A) (a b) (a + b + 1) < 0(B) (a + b) (a b + 1) > 0(C) (a b) (a + b+ 1) > 0(D) (a+ b) (a b + 1) > 0


14/48

14

2. If x @ y =

++

+

y

3y4x

x

3x4y , then the value

of 6 @6

1is

(A)

36

3891 (B) 1

(C)36

4179 (D)

36

1297

3. The operations * and are defined asa * b = a + b + ab and a b = ab (a + b).

Find the value of (3 * 4) (4 5).(A) 19 (29) (113) (B) (85) (30)(C) 19 (180) (199) (D) 84 (29) (113)

Numbers


1. Raju wrote the first 50 natural numbers one afteranother on a black board. He then carried out thefollowing procedure 49 times. In each instance,he erased two numbers, say p and q andreplaced them by a single number p + q 1.Find the final number left on the board.(1) 1224 (2) 1275 (3) 1276(4) 1274 (5) 1226

2. Find the last two digits of 73024

.(1) 41 (2) 81 (3) 21(4) 01 (5) 61

3. N is a natural number. Div(N), a function of N, isdefined as follows:

Div(N) = N if N 9= Div(S(N)), otherwise,

where S(N) is the sum of the digits of N.For instance, Div(8) = 8,

Div(625) = Div(6 +2 +5) = Div(13) = Div(1 +3)= Div (4) = 4 etc.

Find the number of positive integer values of Nless than 600, for which Div (N) = 9.(1) 44 (2) 55 (3) 66(4) 77 (5) 67

4. Find the value of 2222 31

211

21

111 +++++ +

upto 2005 terms

(1) 2005 2006

1 (2) 2006

2006

1

(3) 2006 2005

1 (4) 2005

2005

1

(5) 2006 2007

1

5. Set A = {3, 4, .. 2N +1, 2N +2}, where N is anatural number. Each odd element in it wasincreased by 3 and each even element in it was

increased by 1. P denotes the average of theresulting odd elements and Q denotes the average

of the resulting even elements, thenP Q =(1) 2 (2) 1 (3) 1

(4) N (5) N1

6. X and Y are natural numbers. X is odd and lessthan 100. Find the number of solutions of

Y

3

18

1

X

1= .

(1) 4 (2) 3 (3) 2 (4) 1 (5) 5

7. A tournament had 2N +1 teams t1, t2, .. t2N+1where N > 6. Each team had x players wherex > 4. The following pairs of teams have a

common player : t1and t2N+1, t2and t2N,.. tNand

tN+1. These are the only pairs of teams who havea common player. Find the total number ofplayers in the 2N +1 teams.

(1) N(x 1) +x (2) N(2x 1) +x(3) N(2x 2) +x (4) N(x +1) +x

(5) N(x +2) +x8. A four digit number has the form AABB. It is also

a perfect square. How many (A, B) are possible?(1) 0 (2) 3 (3) 4 (4) 1 (5) 2

9. Let M = 3(3!) + 4(4!) + .. + 15(15!)What is the remainder when M 15 is divided by14! 2?(A) 14! 443 (B) 14! 459(C) 459 (D) 443

10. A number when divided by 5, 6, 7 and 8 leavesremainders of 3, 4, 5 and 6 respectively.How many such 4-digit numbers are there?(A) 11 (B) 10 (C) 9 (D) 12

11. If x = ....4

1

3

1

2

1222

+++ , what is the value of

....7

1

5

1

3

1222

+++ in terms of x?

(A)4

1x3 (B)

4

x3 1 (C)

2

1x (D)

2

x 1

12. If k =7979

8080

4249

4249

+

, then

(A) 0 < k 1 (B) 1 < k 4(C) 4 < k 7 (D) k > 7

13. If A = 71421 . 98 105 112 ...189 196,what is the remainder when A is divided by 9?(A) 1 (B) 3 (C) 7 (D) 5

14. If 2a + 5b = 7(a2b

5)1/7

and x =2

ba +, what is

the value of 2(x + a)2+ 5(x b)

5?


15. Which of the following is a rational number?

(A) (7 + 4 3 )50

+ (7 4 3 )50

(B) (7 4 3 )50

+ (7 + 4 3 )50

(C)

734

log

34 +

347log

7(D) None of these


15/48

15

16. Which of the following is prime?(A) 2

70+ 1 (B) 2

96+ 1

(C) 2160

+ 1 (D) None of these

17. A 200 page book is formed by using 50 sheets,folded in the middle and stapled along the fold,with each sheet providing 4 pages. The pages

are then numbered from 1 to 200. The n thsheet isremoved from the book. What is the sum of thepages on this sheet?(A) 400 2n (B) 400 n(C) 200 + 4n (D) None of these

18. If u + v + w + x + y = 15, what is the maximumvalue of uvx + uvy + uwx + uwy?(A) 125 (B) 144 (C) 3125 (D) 243

19. The remainder when 130110

1301 is divided by21 is(A) 1 (B) 2 (C) 19 (D) 20

20. If a, b c, d are natural numbers such that23 < a < b < 40 < c < d < 50, how many valuesare possible for the quadruplet (a, b, c, d)?(A) 4284 (B) 4320 (C) 4200 (D) 4165

21. A number N leaves a remainder of 7, 10 and17 respectively when divided by 15, 21, and 35.What is the remainder when N is divided by 105?(A) 73 (B) 59 (C) 52 (D) 55

22. How many five digit numbers with digits that arenot necessarily distinct are divisible by 6 but notby 12?(A) 7500 (B) 7499 (C) 14999 (D) 15000

23. Which of the following numbers does not divide(4

12 1)?

(A) 5 (B) 63 (C) 255 (D) 127

24. Find the number of factors of 243243 which aremultiples of 21.(A) 20 (B) 21 (C) 22 (D) 24

25. What is the remainder when 4911

+ 5011

+ 5111

+52

11is divided by 202?

(A) 0 (B) 101 (C) 201 (D) 1

26. Q is the number formed by knocking off all theterminal zeros from 256!. What is the index of thehighest power of 12 that divides Q?


27. Let N = (31)(32)(33)(98)(99)(100)If N is divisible by 12

n where n is a natural

number, find the maximum value that 'n' can take.(A) 30 (B) 34 (C) 35 (D) 40

28. 'a' and 'b' are prime numbers and n is an integer

such that 1 n ab. What is the sum of all thepossible values of n such that n and ab arecoprime to each other?

(A) (a 1) (b 1) (B)2

1baab +

(C) ab (D) 2

)1b)(1a(ab

29. Set A = {1, 4, 7, 10, 13, , 100}. B is a nonempty proper subset of A such that all theelements of B are divisible by 7. Find the numberof such subsets.(A) 5 (B) 31 (C) 32 (D) 63

30. There are 4 distinct integers p, q, r and s, such

that the equation (n - p) (n - q) (n r) (n s) = 9is satisfied for some integral values of n. Howmany integral values of n satisfy the givenequation for a particular set of values of p, q, r,and s?(A) 0 (B) 1(C) 2 (D) More than 2

31. Find the remainder when (3333344444 +

44444

33333) is divided by 7

(A) 0 (B) 2 (C) 3 (D) 6

32. Find the remainder when 4 + 44 + 444+ 4444+ ...+ 444 . . . (50 digits) is divided by 9. (A) 0 (B) 3 (C) 6 (D) 8

33. Consider the equation6

1

b

1

a

1=+ , where a and b

are integers. How many ordered pairs (a, b) existwhich satisfy the given relation?(A) 9 (B) 10 (C) 17 (D) 18

34. The difference between the squares of twointegers is 420. How many such pairs of integersexist?(A) 4 (B) 8 (C) 12 (D) 16

35. How many 9 digit multiples of 6 can be formedusing only the digits 8 or 9?(A) 56 (B) 84(C) 86 (D) None of these.

36. (a + b + c + d)5 (a

5+ b

5 + c

5 + d

5) is always

divisible by(A) 24 (B) a + b + c + d(C) 9 (D) 5

37. If A = 1!+ 2!+ 3!+ 4!+ .......+ 49!andB = 1!+ 2!+ 3!+ 4!+......+168!, then which of thefollowing is true?(A) A is a perfect square.(B) (B) is a perfect square.(C) Both A and B are perfect square.(D) Neither A nor B is a perfect square.

38. What is the sum of the even factors of 3600?(A) 4030 (B) 12896(C) 6046 (D) 12090

39. Consider all 5-digit numbers for which the sum ofthe digits is 41. How many of these are divisible by11?(A) 10 (B) 12(C) 16 (D) More than 16

40. In how many ways is it possible to express 36 asa product of 3 positive integers?(A) 8 (B) 6

(C) 12 (D) None of these


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16

41. W is a whole number. The function root(W) isdefined as follows

root (W) = W if W 2

= root (W 3) if 3 W 9= root (sum (W)) otherwise, where sum (W)denotes the sum of the digits of WFor example

Root (7) = root (7 3) = root (4) = root (4 3)= root (1) = 1Root (245) = root (2 + 4 + 5) = root (11)= root (1 + 1)= root (2) = 2How many values of W less than 500 satisfy thecondition root (w) = 0?(A) 165 (B) 166 (C) 167 (D) 168

42. The 18-digit number X has a 7 in the units place.If this 7 is transposed to the front of the number,the resulting 18-digit number is 5 X. Find X.(A) 124587124587124587(B) 124857124857124857(C) 142857142857142857

(D) 142587142587142587

43. How many numbers less than or equal to 1108are coprime to 252?(A) 311 (B) 312 (C) 316 (D) 318

Directions for questions 44 to 46: These questionsare based on the following data.

Professor Calculus devised a game to pass his time.The game was played in the following manner. Hekept 500 coins numbered from 1 to 500, on a table insequence all showing heads.

In round 1, he flipped all the coins such that eachshowed a tail. In round 2, he flipped only those coinswhose position was a number divisible by 2.In round 3, he flipped only those coins whose positionwas a number divisible by 3 and so on.He continued flipping the coins for 500 rounds.

44. What was the greatest number of consecutivecoins all showing heads at the end of the game?(A) 16 (B) 41 (C) 42 (D) 43

45. Among the following coins, which coin wasflipped the greatest number of times?(A) 210 (B) 324 (C) 288 (D) 240

46. How many coins did Professor Calculus flip inround 13, which were flipped only once in theprevious rounds?(A) 7 (B) 8 (C) 11 (D) 38

Directions for questions 47 to 49: Select thecorrect alternative from the given choices.

47. If the sum of all integers between 3n and

3n+3

(both exclusive) is divisible by 70 where n is apositive integer, then which of the following isnecessarily true about n?(A) It is divisible by 4 (B) It is a multiple of 6(C) It is divisible by 7 (D) It is divisible by 5.

48. Find the units digit in 13+ 2

3+ 3

3+ +90

3.

(A) 0 (B) 5 (C) 6 (D) 1

49. If S(N) denotes the sum of the digits of N, thenwhat is the remainder when S(1) + S(2) + S(3) +..+ S(99) is divided by 99?(A) 0 (B) 9 (C) 18 (D) 36

Probability


1. There are 50 tokens numbered 1 to 50 kept on atable. You are asked to pick two tokens atrandom from the table one after the other. Whatis the probability that the difference between thenumbers on the tokens picked by you liesbetween 1 and 6 (both excluded)?

(A)245

36 (B)

1225

186

(C)1225

372 (D)

2450

186

2. A number is selected at random from 1 to 105.Consider the following 3 events.P: X is coprime to 3`Q: X is coprime to 5R: X is coprime to 7Which of the following is true about the eventsP, Q, R?(A) They are pairwise independent and mutually

exclusive.(B) The are pairwise independent but not

mutually exclusive.(C) They are not pairwise independent but are

mutually exclusive.(D) They are neither pairwise independent nor

mutually exclusive.

3. Alsi Arora worked in a private company where the

working hours were from 9:00 am to 6 pm. In a

year on 25% of the days he arrived late to office

whereas on 35% of the days he left early from

office. If p is the probability of the number of days

that he worked for the entire working day, then

which of the following best describes the possible

values of p?

(A) P = 0.4875(B) 0.25 P 0.65(C) 0.25 P 0.65 .

(D) 0.40 P 0.65

4. Ramesh doesn't always have time to read the

daily newspaper and whenever he reads it he

does it either in the morning or in the evening.

The probability that he reads it in the morning is

10% and the probability that he reads it in the

evening is 30%. Find the expected number of

days in a month of 30 days on which he doesn't

read the paper at all.(A) 16 (B) 18.9 (C) 21 (D) 20.5


17/48

17

Miscellaneous

Directions for question 1: Select the correct

alternative from the given choices.

1. There is a sequence of 7 sets A1, A2, A4,each

consisting of 6 elements and another sequence ofn sets B1, B2, ,,,, Bneach consisting of 3 elements.

The union of all the A's is equal to the union of all

the B's. If each element of this union occurs in

exactly 3 of the A sets and 6 of the B sets, then

find n.

(A) 28 (B) 21

(C) 14 (D) Cannot be determined

Coordinate Geometry


1. If the equation 9x2

y2

+ 6y = A where A is aconstant, represents a pair of straight lines, thenwhich of the following gives the point ofintersection of those two lines?(A) (1, 2) (B) (0, 3)(C) (0, 3) (D) cannot be determined

Inequations


1. A triangle with sides x, y and z is such thatx

3+ y

3+z

3= 3xyz. Which of the following is true

regarding the triangle?(A) It is an isosceles triangle(B) It is an equilateral triangle(C) It is an obtuse angled triangle(D) It is a right angle triangle

Statistics


1. The median of 11 numbers is x. If the greatestnumber is removed the median of the remainingnumbers is 10.5. If the smallest is removed the

median of the remaining numbers is 13. If x (orone of the numbers which is equal to x) isremoved the median would be(A) 11 (B) 11.5(C) 12 (D) Cannot be determined


18/48

1

SOLUTIONS FOR QUANT REPLICA QUESTIONS THAT HAVE

APPEARED IN CATIN THE LAST 4 YEARS

Permutations and Combinations

Solutions for question 1:

1. Least element of I = 1000.Greatest element of I = 5000.Let the required number be N.N = Number of integers from 1000 to 5000 which canbe formed with 0 or 1 or 2 or 3 or 5.If the integers are less than 5000, the thousands digit hasthree possibilities (1 or 2 or 3). Each of the remainingdigits has five possibilities 0, 1, 2, 3 or 5

A total of (3) (5) (5) (5) or 375 integers can be formedless than 5000.The only integer not less than 5000 is 5000 itself.

N = 375 +1 = 376 Choice (1)

Solutions for questions 2 and 3:

Any shortest path from H to O must include AB.

The number of shortest paths from H to O is mn, where mis the number of shortest paths from H to A and n is thenumber of shortest paths from B to Om =

4C2= 6 and n =

7C2= 21.

mn = 126.

6 6

5

4

3

2

1

In the figure above, the number at a node indicates thenumber of shortest paths to reach that node from O.

2. mn = 126 Choice (4)

3. There are 126 shortest paths from H to O and13 shortest paths from O to C.

There are a total of (126) (13) or 1638 shortest pathsfrom H to C via O. Choice (1)

Solutions for question 4:

4. The terms in the expansion of (p + q + r)15

have theform Kp

xq

yrzwhere x + y + z = 15. The number of terms

is the number of non-negative integral solutions ofx + y + z = 15, which is equal to the number of positiveintegral solutions of x + y + z = 18, which is

17C2or 136.

Choice (1)

Solutions for questions 5 and 6:

5. Let (a, b) be a member of X.Now non mates of this member will have any of theremaining elements other than a and b.Remaining elements in N other than a and be is N 2

elements = N 2 C2=1.2

)3N()2N( =2

6N5N2

+=

Now all these2

6N5N2 + will be non-mates of the

member (a, b) Choice (4)

6. Let (a, b) and (b, c) be two mates in XThey have element b in common.Now all the members which have b as one of theelements will be a common mate to both (a, b) and(b, c). Other than a, b and c, there are. N 3 elements.Now with each of these n 3 elements with element bcan form a common mate.In addition the member (a, c) will also be a common mateto both the members (a, b) and (b, c).So, total number of common mates = N 3 + 1= N 2. Choice (1)

Solutions for questions 7 to 19:

7. If the sum of the digits of a number is divisible by 9,

then the number is divisible by 9.Sum of the given digits is 0 + 1 + 2 + 4 + 6 + 8 + 9 = 30.We leave two digits such that the sum of the other fivedigits is multiple of 9.0, 4, 6, 8, 9 or 0, 1, 2, 6, 9 the number of arrangementsin each case 4! 41 hence 2.4.4! = 192. Choice (B)

8. If the last two digits are divisible by 4, then the numberis divisible by 4.Given digits is 0, 2, 4, 5, 6, 7.The possible two digits are 04, 20, 40, 60, 24, 52, 56,64, 72, 76

The number of five digit number formed which areend with 04 is _ _ _ 04 =

4P3.

The number of five digit numbers formed which areend with 04 or 20, 40 or 60 = 4

4P3= 96.

The number of five digit number formed which are endwith 24 is _ _ _ 24 = 3 3 2 = 18

The number of five digit number formed which are

end with 24, 52, 56, 64, 72 or 76 = 6 18 = 108. Hence the total number of five digit number whichare divisible by 4 can be formed using the given digits= 96 + 108 = 204 Choice (C)

9. All players who play the same game are treated as oneunit. Then four units can be arranged in a row in 4!ways.Now the players in each game can be arranged amongthemselves in 6! 7! 8!and 9! ways respectively.The required number of arrangements = 4!6!7!8!9!

Choice (D)

10. Each student can be sent into any of the five sections.Seven students can be sent 5 5 5 5 5 5 5i.e 5

7ways. Choice (B)

11. The sum of the all the five digit numbers formed by thedigits a, b, c, d, e without repetition in 4! (a + b + c + d+ e) (11111).The sum of the all the five digit number formed by thedigits 2, 4, 6, 7, 8 is 4! (2 + 4 + 6 + 7 + 8) (11111)= 7199928. Choice (B)

12. Since 40 lines are given, The number of points ofintersection of these 40 lines is

40C2. But 2 lines are

concurrent they are intersected at only one point.Similarly, 15 lines are concurrent they intersect at onlyone point. 13 lines are parallel.They do not intersect.

Hence required number of points of intersections=

40C2 66 105 78 + 2 = 533. Choice (C)

H C

O

B

R

A

P

c 13 7 1

1

1

1

O

P1

1


19/48

2

19.511

5

S RQ

P

RS

YM

PT Q

NX

30

30

13. Let E = (x + y z)100

The general term in the expansion is

nC1x

ay

bz

c, where

a + b + c = 100.For the terms which are negative c is odd, i.e., c = 1, 3,5, 99.

a + b = 99, 97, --- 1 -------(A)The number of negative terms is equal to the sum of thenumber of non-negative integral solutions of the fifty

equations denoted by (A) which is 100C1+ 98C1+ . 2C1= 100 + 98 + .. + 2

= .25502

)51()50(2 = Choice (C)

14. In the word EQUATION the vowels are A, E, I, O and Vwhereas the consonants are N, Q, and T. Since no twovowels or no two consonants are to be in thealphabetical order, the vowels and the consonants mustoccur in the reverse alphabetical order.Vowels U, O, I, E, A consonants T, Q, NThe total number of arrangements possible with the 8letters is 8! Among these the vowels can be arrangedamong themselves in 5! ways, of which only onesequence is permissible.

Similarly, the consonants can be arranged amongthemselves in 3! ways, out of which only one sequenceis permissible.

Therefore, the number of such words =!3!5

!8= 56.

Choice (A)

15. We can have four 4 s and one 5 or one 4 and four 5 s.So, the number of 5-digit numbers that can be formed =

!1!4

!5+

!1!4

!5 Choice (A)

16. The total number of ways in which one can go from A to

C = 715 C .

The no. of ways in which one can go from A to C

via B = 49 C x 3

6 C .

Therefore the number of ways in which one can go from

A to C but not via B = 715 C .- 4

9 C x 36 C .

= 6435 2520 = 3915 Choice (A)

17. Case If 2 is a digit of the number, it must be of the form 24 .Here the units digit can be filled with any of the five odddigits i.e. by 1, 3, 5, 7, 9.

5 such numbers can be formedCase If 2 is not a digit of the number, then it can be any three-digit odd number not containing 2.

8 9 5 i.e. 360 such numbers can be formed.

A total of 365 such numbers can be formed.Choice (B)

18. The total number of ways in which the parcels for Biharcan be addressed

= 4!(!2

1

!3

1+

!4

1)

= 24 (2

1

6

1+

20

1)

= 24 (24

1412 +) = 9

The total number of ways in which the parcels for Delhican be addressed

= 5! (!2

1

!3

1+

!4

1

5

1)

= 120 (2

1

6

1+

24

1

120

1)

= 120 (120

152060 +) = 44

Therefore the total number of ways in which all theparcels can be addressed = 9 (44) = 396.

Choice (A)

19. The number of triangles that can be formed is12

C33C3

4C3

5C3

= 220 (1 + 4 + 10) = 205 Choice (D)

Geometry / Mensuration

Solutions for questions 1 to 40:

1. Area of triangle PQR =2

1(PS) (QR)

Let the circumradius of PQR be R1for a triangle with

sides a, b, c. Then area of the triangle =1R4

abc

21 (PS (QR)

=( ) ( ) ( )

1R4

QRPRPQ

2

1(5) (QR)

=( ) ( ) ( )

1R4

QR115.19

R = 21.45 Choice (4)

2. Let AB = 7, BC = 16 and CA = y (where y is an integer)As ABC is an obtuse triangle, the square of its longestside must be greater than the sum of the squares onthe other two sides.

Max(7, 16, y) = 16 or y.If Max = 16, 16

2> 7

2+y

2.

y2 < 207

i.e., y 14.

By triangle inequality y +7 > 16 i.e., y > 9.y = 10, 11, 12, 13 or 14.If Max = y, y

2> 16

2+7

2.

y2> 305 ie., y 18.

By triangle inequality, 16 +7 > y i.e., y < 23.y = 18, 19, 20, 21 or 22.

Total number of possibilities of y is 10. Choice (2)

3.

Let QN = 1.

NY = 3 (Similarly MX = 3 ) and PQ = 2

MY = PQ NY = 2 3 (Similarly NX = 2 3 )

Ar (XQR) + Ar (YPS) = 2(2 3 )

= 4 2 3 Ar(PQRS) = 4.


20/48

3

P

TS

Q R

U

v

A

C2C1

B

120

Y

N

X

M

M NX

Y

Bx

G

yyx

CA

E

D

F

areamainingRe

RSYXPQofAr=

)32(2

)324(4

=

)32(2

32

=32

3

= 3 (2 + 3 ) = 2 3 + 3

Alternative Method:

isoscelestwotheofAr

trapeziumdoubletheofAr

=QNXArPMYAr

XYPQAr

+

=TYMPAr

QNYTAr=

YM

NY=

32

3

= 2 3 + 3

Choice (2)

4.

Let C1and C2be the centers of the circles and A and Bbe the intersection points common radius for bothcircles

C1C2A is equilateral.Similarly C1C2B is equilateral.AC1C2= BC1C2= 60

AC1B = 120Required area = 2(area of the segment AC2B)Area of segment AC2B = Area of the sector C1AC2B Area of triangle C1C2A

=360120 (2)

2

21 (C1A) (CB) Sin ACB

=3

4

2

1(2)(2) sin120 =

3

3

4sq.cm

Required area =

32

3

8sq.cm.

Choice (3)

5. Triangles PUT andPVR are similar.

VR

PV

UT

PU=

=

2

7

6

21=

Let UT = x cm

PU =2

7x cm

UV = (21 2

x7) cm

Total surface area of the cylinder

= (2x) (x + 21 2

x7) cm

2= x(42 5x) cm

2

=5

(5x)(42 5x) cm

2

As the sum of the two factors 5x and 42 5x isconstant, the product is maximum when 5x = 42 5x or

x =.5

21

The corresponding area is

2

5

21

+

10

14721042cm

2

=5

441cm

2Choice (5)

6. The two extreme cases are shown below.

XMN = 0

XMN = 60

0 60 Choice (3)

7.

Let A and B be the centres of the circles. Let D and Ebe the points of intersection of the two circles. Let C bethe intersection of AB and CD.Let AF = x, FC = CG = y and GB = x.Radius of each circle = x + 2y.

3

1

xy2x

y2x=

++

+i.e.

3

2

yx

y2x=

+

+.

cosCBD = .2

3

y2x

yx

BD

CB=

+

+=

CBD = .6

Common region area = 2 (Area of sector BED Area of

BED) = 2 ( ) ( )

EBDsinBD2

1BD.

2

6

2

22

= 2

4

3

6

BD2 . [EBD =

3

]

Required ratio =

( )

=

6

332

BD

4

3

6BD2

2

2

.

Choice (D)

8. Number of equilateral triangles which can be formed ofside a cm using equilateral triangles of side b cm

=

2

b

a

Total number of triangles used

=

222

2

30......

2

18

2

16

+

= 8

2+ 9

2+ 15

2


21/48

4

F

H

CB

A

I

G

E

D

S

9 32

QTP

R

UV

(PQ 9)

(PQ 32)

W

CB

D

A

88

E B

D

F

P

A

C

P

A

RQ

30

B

A

D CB

608 3

8 3

= 12+ 2

2+ .. 15

2 (1

2+ 2

2+ .. 7

2)

=( )( )( ) ( )( )( )

6

1587

6

311615 = 1100. Choice (B)

9.

ABC is equilateral such that radius = 8 3 cm

BD =60sin

38=16.

side of equilateral triangle = 32 cm. Choice (B)

10. Let the radius of eachcircle be r.AB = AG + GI + IB.

A = 60

GAD = 2

A

= 30.

AG = 3rGADtan

GD=

.

Similarly IB = r 3 .

GI = DE = sum of theradii of the circles whose centres are D and E = 2r.

AB = r 3 + 2r + r 3 = 2(1 + 3 )r.

( )312r

AB+= Choice (C)

11. Number of triangles which can be drawn by using theN points as vertices =

NC3.

Number of triangles which can be drawn by using all but

one of the N points as vertices =

N1

C3.NC3N1

C3= 210.

( )( ) ( )( )( )6

3N2N1N

6

2N1NN

= 210

(N 1) (N 2) = 420N

2 3N 418 = 0

(N 22) (N + 19) = 0

As N > 0, so N 19.N = 22. Choice (B)

12. Let the sides of the triangle be a, b and c. Let its areabe A.X = abc

Y =4

abc

2

cba

A.

A4

abc.

2

cba=

++

++

4

1

X

Y= Choice (A)

13.

PQRS is a square.

S = 90 and PQ = QR

= RS = PSSU2= UW

2+ SW

2

PQ2= (PQ 9)

2+ (PQ 32)

2 [PQ = SU]

Considering PQ = x, we get

x2 82x + 1105 = 0

(x 17) (x 65) = 0

x = 65 or 17As x > 32, x = 65. Choice (B)

14.

Let B and C be the centres of the circles.Let A and D be the points of intersection.AB = AC = 8 cm.

B = C =2

A180 = 45

ABD = 90.Area of the shaded region = 2(Area of the sector BAD

Area of BAD) = 2 ( ) ( )( )

BDBA.

2

1BA.

360

90 2

= ( ) ( )22 82

2AB1

2

=

= 32( 2) sq cm.

Choice (D)

15.

Area of region P = 2(Area of sector EFB Area of

EFB)

=2 ( )

EB.EF.

2

1EB.

360

90 2

= ( ) ( )22 82

2EB1

2

=

Therefore the area of field accessible to the cows

=( )

2

82

2

Area of region P

Required area = AB2 ( ) ( )

22 82

2

2

82

= 162 (64 ( 2) 32)

= 256 64+ 32 64= (192 32) sq m. Choice (C)

16.

The figure above is not to scale.


22/48

5

A

CB

G

ED

F

C4

40

24B

32

20

12

20

A

E

P Q

RS

D

A

E

B

C

E

A B

(2r+r 2 )

E

B

AD

O

C

P C O

A NM

D B

In PQA and QAR, PQ = QR.

PA = AR ( QA is the median to PR)

QA is a common side

PAQ QAR.

PQA = AQB i.e., AQB = 30.

ABQ = AQB = 30 ( AQ = AB). Choice (B)

17. The sides given are 68 and 32.

68 = 4 17; 32 = 4 8As 8, 15, 17 from an right angled triangle, the third side

= 4 15 = 60 Choice (D)

18. Let F and G be the midpoints on the diameters of thecylinder.Required volume = Volume of the cone ABC

=3

1(FC)

2(AF)

GE

FC

AG

AF= .

18

15

54

AF= .

AF = 45.

Required volume =3

1

(15)2(45) = 3375cubic cm. Choice (D)

19. Let the sides of the cubes be a cm and b cm respectively.4a

2+ 4b

2= 468 and a

3+ b

3= 945

a2+ b

2= 117 and a

3+ b

3= 945 (1)

Let a be b.Then a

3b

3.

(1) 2a3945 a

3472.5. Also a

3< 945. (1)

a is an integer. a3is a perfect cube

(1)a3= 512 or 729.

a3= 512b

3= 433. a

3= 729b

3= 216.

b is an integer. b3is a perfect cube.

b3= 216 and a3= 729 i.e., b = 6 and a = 9.

Volume of the hemisphere =3

2(9 + 6)

3=

( )

3

23375

= 2250cubic cm. Choice (B)

20.

322= 1024, 24

2= 576 and 40

2= 1600.

322+ 24

2= 40

2.

The triangle is right angled at B.

Circumradius of ABC =2

AC= 20.

AD : EC = 12 : 4 = 3 : 1. Choice (A)

21. Let PQRS be the base of the cuboid. PQ = QR = 6r =

18 2 The bottom layer is shown in the Figure 1 and thedotted lines represent the second layer. The relativeposition of the centers of 4 balls in one layer and thecenter of the fifth ball which fits into the depression thatis formed is shown in Figure2. Figure 3 gives the front

view from which we can get the height of the cuboid.

Figure 1

Figure 3

Figure 2

In figure 2, AB = AD = 2r

BD = 2(BO) = 2r2

BO = 2r

Now, in EOB, EO OBEB = 2r (The distance between the centers of twospheres touching each other)

EO = ( ) 22 )OB(EB

=22 )2r()r2( = 2r

Figure 3 shows that the height of the structure must be

greater than or equal 2rr2 + = (2+ 2 ) 3 2 =

6 2 + 6 Choice (A)

22. We are free to choose the points as close to each otheras possible. There is no lower limit to the distancebetween any two points. (We can reject choices Band C) we have to try to spread the points out as far aspossible, i.e, choose the 3 vertices and the centroid.This particular choice of 4 points satisfies the conditionsin choice A (while any other choice of points wouldsatisfy both A and D). Choice (A)

23. Let the length and breadth of the rectangle MNOP be land b respectively where l and b are integers.

Length of MO = Length of PN =22 bl +

Joining the midpoints of a rectangle we will give a

rhombus whose area is half the area of the rectangle.

Area of the rhombus ABCD is6

1Area of PQRS

= )54(6

1= 9

The area of the rectangle MNOP has to be 18.18 can be written as the product of two integers in 3ways

18(1), 9 (2), 6(3).

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