Quantitative Aptitude section in CATOver the past four years, the CAT paper has been stuctured into three sections.Quantitative Aptitude forms one of these three sections.Tables I and II give an analysis of the total number of questions and the topic-wise break-up of thequestions in this section in CAT over the last 4 years, from 2001 through 2004.

Different ways of attempting a questionSpeed and accuracy play a key role in success in CAT. Proper selection of questions is significant. So, one should be

prepared with different tactics and techniques by which one can select and solve a question effectively and efficiently.The most important thing is to be aware of different ways of successfully answering questions.

(i) Direct method(ii) Elimination method(iii) A combination of direct and elimination method.(iv) Taking numerical valuesLet us take an example from CAT 2003 (Feb)

TABLE INUMBER OF QUESTIONS

TABLE IITOPIC WISE BREAK-UP OF QUESTIONS

INTRODUCTION TO QUANT

2004 (Nov) 2004 (Feb) 2002 (Nov) 2001(Dec)

Subsection - A

Subsection - B

(Each question = 15of 2 marks)

Total 35

(Each question = 20of 1 mark)

(Each questionof 1 mark)

50 5050

(Each questionof 1 mark)

(Each questionof 1 mark)

1. Equations

S.No. Topic 2004(Nov)

2004(Feb)

2002(Dec)

2. Ratio, Proportion, Variation

– – – 5

1 –

3. Percentages, Profit and Loss

4. Numbers 12 7

5. Simple Interest / Compound Interest – – – –

6. Averages and Mixtures 1 – – 1

7. Time and Work 1 – – 2

8. Time, Speed and Distance 3 1 4 3

9. Quadratic Equations / Progressions 2 3 4 –

10. Permutations and Combinations 2 2 3 3

11.

12.

Special Equations – – 1 8

13.

Inequalities – – 1 –

14.

Indices, Logs and Surds – – 3 –

15.

Functions and Graphs 2 2 – 1

16.

Geometry / Mensuration 5 4 14 9

Others – – 7 11

Total Number of Questions 20 15 50 50

2001(Dec)

2

5

–

3

2

1

10

2

50

1 – 1

– – – –

3 3

PartA

PartB

3

6

5

1

9

–

–

Quantitative Aptitude

Triumphant Institute of Management Education Pvt. Ltd.

The ���� coaching institute in India

Q. What is the sum of the ‘n’ terms in the series:

log m + log ⎟⎟⎠

⎞⎜⎜⎝

⎛

n

m2

+ log ⎟⎟⎠

⎞⎜⎜⎝

⎛2

3

n

m + log ⎟⎟⎠

⎞⎜⎜⎝

⎛3

4

n

m + ............

(1)log 2/n

)1n(1n

mn

⎥⎦⎤

⎢⎣⎡

+−

(2) log2/n

nm

nm

⎥⎦⎤

⎢⎣⎡

(3)log2/n

)m1()n1(

nm

⎥⎦⎤

⎢⎣⎡

−−

(4) ⎥⎦⎤

⎢⎣⎡

−+

)1n()1n(

nmlog

Sol.1st method: (Direct method)

log m + log ⎟⎟⎠

⎞⎜⎜⎝

⎛

n

m 2

+ log ⎟⎟⎠

⎞⎜⎜⎝

⎛2

3

n

m + log ⎟

⎟⎠

⎞⎜⎜⎝

⎛3

4

n

m+ ............

= (logm + logm2 + logm3 + ......... logmn) – (logn + log n2 + .......... logn(n–1))= log [ ] [ ])1n.......(21)n.........21( nlogm −++++ −

= log ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+

2

)1n(n

2

)1n(n

n

m = log

)2/n(

1n

1n

n

m

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

+

Choice (4)Thus, we see that a candidate must be acquainted with the knowledge of indices, logarithms and progressions.

2nd method: (Taking numerical values)

This is the fastest approach. It requires only a little knowledge of what logarithms are.In the given example, whatever be the answer, it must be valid for all the possible values of n.If n = 1Required sum = log m.

Choice (1) log2/1

2

0

m)1(

⎥⎦

⎤⎢⎣

⎡ = log ⎥⎦⎤

⎢⎣⎡m

1 ≠ log m Choice (2) log [ ] 2/1mm ≠ log m

Choice (3) log2/1

)m1(0

1m

⎥⎦⎤

⎢⎣⎡

− = 0 ≠ log m Choice (4) log2/1

02

1m

⎥⎦⎤

⎢⎣⎡ = log m

By elimination, choice (4) is the answer.

Let us take another example from CAT2004 paper.

Q. If ,rba

cac

bcb

a =+

=+

=+ then r cannot take any value except

(1)1/2 (2) −1 (3)2

1 or −1 (4)2

1− or −1

Sol. 1st method: (Direct method)

2

1

)cba(2

cba

ba

c

ac

b

cb

a =++++=

+=

+=

+⇒ r =

2

1

1)ac()cb(

ba

ac

b

cb

a −=+−+

−=+

=+

Also, 1)ba()ca(

cb

ba

c

ca

b −=+−+

−=+

=+

⇒ r = –1

Thus, r can be (–1) or 2

1

Choice (3)2nd method: (Taking numerical values)

If a = b = c = 1

⇒ rba

c

ac

b

2

1

cb

a =+

=+

==+

⇒ r = 2

1

If a = –2, b = 1, c = 1

111

2

cb

a −=+−=

+

121

1

ac

b −=−

=+

112

1

ba

c −=+−

=+

⇒ r = –1

Thus r = –1 or .2

1Choice (3)

Pitfalls and Surprises:Since both speed and accuracy are tested in CAT, sometimes in the process of speeding up to arrive at an answer faster,

certain points may get overlooked. One needs to be aware of such pitfalls.

Example:Q. Makhanchand was supposed to sell his sweets for Rs.100. But because of the competitive scenario, he was forced to

sell his sweets for Rs.70 to Ramesh. Find the loss incurred by Makhanchand.(1)Rs.30 (2) 0(3)Cannot be determined (4) None of these

Sol: This is a concept based question on profit/lossLoss or profit is not calculated on the planned selling price or on marked price. It is calculated onthe cost price (the only exceptions being the cases wherein it is explicitly mentioned otherwise), which is notgiven. Hence, loss for Makhanchand can’t be determined.Similarly, if it were asked to find the profit made by Ramesh, then again the similar logic prevails.Therefore, profit or loss made by Ramesh can’t be determined. Choice (3)

A pitfall in CAT paper can be compared to a typical manhole or a bump on an otherwise smoothhighway. The best way to tackle them is to prepare thoroughly in advance by being aware that such areas doexist and, one must drive past these areas a bit carefully.

SURPRISES:The biggest surprise which CAT may produce is to bring in no surprises.

Just imagine a scenario, when CAT2005 paper comes out to be exactly identical to CAT2004 paper. Would we say thatCAT2005 is a big surprise (since never before have any two consecutive papers been identical) or would we say thatCAT2005 did not bring in any surprise (as the questions were identical)? Whatever may be the case, the outcomewould not have been affected for the majority.

The level of competition may go up, or remain the same or, go down, but that would be the same for the ajority ofcandidates. It is the sincerity in preparation which finally pays off, and that is reflected through the extensive practicethat zone goes through.

Thus, surprise is the most overvalued element in CAT paper. However, there has been a bit of deviation in the pattern ofCAT paper. This change in the pattern is not the surprise element. Instead it is treated as a tool by the IIMs to check theflexibility and “coolness” quotient of the candidates. If the fundamentals are clear, a smart and sincere candidate caneasily adapt himself/herself to differing conditions or circumstances.

Some of the areas where changes in pattern can happen1. Number of choices: Till CAT 2004, IIMs have been providing four options. This may change to three or five

options. Exams like IIFT and IRMA have five options (or choices) for each question. CAT2005 may bring insome changes (or no change) in this segment.

2. Structure/Model of the paper: CAT may give only one consolidated section where questionson different areas may get organised, not area wise but on other factors. OR, The structuring of paper may be doneon the basis of marks. i.e., all 1/2 mark questions could be brought into the first section, 1 mark questions(whether on quant or on other areas) into the second section, and the 2 mark questions into the third section.

3. Type and the number of questions: The kind of questions which have been asked over the past four years havebeen mentioned earlier in TABLE II whereas TABLE I provides an illustration of the total number of questions.A change in the type of questions asked could be brought about by introducing a few topics that have so far notappeared in CAT but are a regular feature in some other exam. For example co-ordinate geometry and trignometryare relatively new topics on the CAT examiners’ list. Topics like probability, stocks and shares, calculus, might beintroduced in CAT over the comimg years. However, the weightage of the new topics will be, at least initially, soless that one need not worry about them. Students should instead focus on their strong areas and the topics thatcontribute a majority of the questions.

How to prepare for Quant?1. Start early: Unlike engineering or degree exams, success in CAT can’t be guaranteed by last minute mugging up

of formulae. An aspiring candidate should give himself a time period of at least 6 months. The first 4 monthsshould be devoted to brushing up fundamentals and concepts in different areas. This can be achieved by practisinga good variety of questions in different areas. The last two months should be devoted to analysis and consolidation.

2. Preparation of topics in Quant: Topics like Numbers, Geometry, Mensuration, Time and Work, Time, Speedand Distance contribute heavily to this section. One must go through all the different concepts in each of thesetopics and solve a variety of questions related to each concept.

3. Improving speed of calculation: 80% of the time is spent on cracking the concept in any question, 20% is spenton calculation. To ensure speedier calculation, everyday in the morning (for at least five days a week), one mustpractice calculations like addition, subtraction, multiplication, and division for about10 minutes. This practicecan save lot of precious time in the actual exam.

4. Practise, Practise, Practise: How does one learn the art of driving, or swimming or any other activity? Knowledgeof traffic rules or symbols (at least for driving) does help. Knowledge about one’s bike also helps and knowledgeof how to swim also helps. But all this contributes just 1%. The remaining 99% success is achieved by actuallygetting on to the road or by jumping into the water. So, get on the CAT track, gear yourself up, and itsjust the perfect time for you to get, set and go!

DIRECTIONS for questions 1 to 4: Select the correct alternative from the given choices.

1. There are certain number of cookies with each of Payal, Richa and Sapna. If we add the number of cookies with anytwo girls at a time, the ratio is 3 : 4 : 5 (Payal+Richa : Richa+Sapna : Sapna+Payal respectively). Which of theollowing statements is/are true?I. The number of cookies with Sapna is 50% of those with all three of them together.II. Payal has twice as many cookies as Richa.III. Richa and Sapna together have twice as many cookies as Payal.(1) Only I and II (2) Only II and III (3) Only III and I (4) All three statements

2. Today, Susheel made a total of exactly one hundred calls from his cell phone to ten different people. He could onlyremember the fact that he had called each person at least four times. In how many ways could Susheel havedistributed his calls among the ten different people today? Ignore the order in which the calls were made.(1) 100C

10(2) 99C

6(3) 60C

6(4) 69C

9

3. A boy starts to paint a fence on one day. On the second day two more boys join him and on the third day three more boysjoin the group and so on. If the fence is completely painted this way in exactly 20 days, then find the number of days inwhich 10 men painting together can paint the fence completely where every man can paint twice as fast as a boy can.(1) 20 days (2) 40 days (3) 45 days (4) 77 days

4. On a certain sum, the difference between the compound interest and the simple interest for the second year isRs.3,600 and the same for the third year is Rs.11,340. What is the sum?(1) Rs.1,60,000 (2) Rs.1,20,000 (3) Rs.1,80,000 (4) Cannot be determined

DIRECTIONS for questions 5 and 6: These questions are based on the following data.

A box contains a certain number of red, green and blue balls. The number of balls of each colour is more than one. Theratio of the number of red balls to the number of green balls is the same as the ratio of the number of green balls tothe number of blue balls.

5. If the total number of balls in the box is 61, how many green balls are there in the box?(1) 16 (2) 20 (3) 25 (4) Cannot be determined

6. If the number of green balls in the box is 21, then the total number of balls in the box can be(1) 63 (2) 89 (3) 101 (4) 117

DIRECTIONS for questions 7 to 43: Select the correct alternative from the given choices.

7. Consider z = 22225555 + 55552222. Which of the following statements is/are true?(1) z is a multiple of 7 but not 11. (2) z is a multiple of 11 but not 7.(3) z is a multiple of both 7 and 11. (4) None of these

8. Two candles of equal length are lighted simultaneously. After 15 minutes of burning, the length of the first candlebears a ratio of 4 : 5 to that of the second candle. If the first candle burns out completely in 45 minutes how muchmore time does the second candle take to burn out completely?(1) 30 minutes (2) 60 minutes (3) 45 minutes (4) None of these

9. The number of two-digit numbers [where neither digit is a zero] whose product of the digits is a square are(1) 16 (2) 17 (3) 18 (4) None of these

10. Bakul and Manohar start from two points P and Q respectively on a river and head towards each other. Had theybeen travelling in still water, they would have met at a point R, which is twice as distant from P as it is from Q. IfBakul had been travelling along the current and Manohar against it, then they would have met in 24 minutes. Findthe time they would take to meet, if Bakul were to travel against the current and Manohar along the current.(1) 12 minutes (2) 24 minutes (3) 36 minutes (4) 48 minutes

Test Ref: Tep0501 Time: 60 minutes

11. If p + 7 > 0 and (25 - p2) < 0, how many integer solutions are possible for p given that it lies between -151/6 and 471/

7?

(1) 41 (2) 42 (3) 43 (4) 45

12. If the sum of the first 10 terms of an arithmetic progression is 100 and the sum of the first 100 terms of the sameprogression is 10, then the sum t

101 + t

102 + …..t

110 is

(1) –90 (2) –100 (3) –110 (4) –120

13. A cuboidal box of dimensions 10 cm x 8 cm x 12 cm is partitioned completely into cubicles of dimensions 1 cm x1 cm x 1 cm. Amy the ant is in the top left corner cubicle towards the front of the box. If Amy can move onlybetween any two cubicles that have a common face, then find the number of ways in which Amy can reach thebottom right corner cubicle that is towards the back of the box. Assume that Amy visits no cubicle more than onceand that it is allowed to move only in downward, rightward and backward directions.

(1) 960 (2) 2!8!10!30!

(3)!11!7!9

!27(4) 8

2010

30 C.C

14. ABCD is a rhombus of side 12 cm. The diagonals of the rhombus meet at the point P. Line segments PX and PY arejoined, where X and Y are the midpoints of the sides AD and CD respectively. If the length of the line segment PD is 10cm, find the length of the line segment XY.

(1) 112 (2) 113 (3) 114 (4) Cannot be determined

15. A group of new students whose total age is 221 years joins a class, because of which the strength of the class goes upby 50% but the average age of the class comes down by one year. What is the new average of the class if it is knownto be a natural number after the new group of students have joined, given that the original strength of the class wasa two digit number greater than 30?(1) 15 yrs (2) 17 yrs (3) 19 yrs (4) 16 yrs

16. If x is an integer, then which of the following statements is true of z = (x + 1) (x + 2) (x + 3) (x + 4).(1) z - 1 is a prime number. (2) z2 - 1 is a prime number.(3) z + 1 is a perfect square (4) None of these

17. Three positive numbers a, b and c are such that .cabcabcba

abc ++=++

What is the value of

?c

bac

b

acb

a

cba ++++++++

(1) 0 (2) (a + b + c)(3) (a + b + c + abc) (4) None of these

18. All Analysts are Engineers. One-third of all Engineers are Analysts. Half of all Technicians are Engineers. OneTechnician is an Analyst. Eight Technicians are Engineers. If the number of Engineers is 90, how many Engineersare neither Analysts nor Technicians?(1) 65 (2) 79 (3) 82 (4) 53

19. If a is an integer greater than –7 but less than 5, b is an integer less than 7 but greater than –5 while c is an integer thatis not greater than 6 and not less than –2, which of the following statements is/are always true?I. –36 < (ab + bc + ca) < 84II. –384 < a(b2 + c2) + b(c2 + a2) + c(a2 + b2) < 912(1) I only (2) II only (3) Both I and II (4) None of these

20. Let S = 7A68G023535928. If S is divisible by 792, what is the value of A?(1) 5 (2) 9 (3) 7 (4) 8

21. Amar and Ajeeth start simultaneously from the same point on a circular track, of length 5 km, and run in oppositedirections. Their speeds are doubled every time they cross each other. Find the number of times that they will meetwithin the first hour, given that they started the race with respective speeds of 6 kmph and 4 kmph.(1) 4 times (2) 6 times (3) 7 times (4) None of these

Income Group

I < 1 lac

1 lac < I < 2 lac

2 lac < I < 3 lac

3 lac < I < 4 lac

4 lac < I < 5 lac

5 lac < I < 6 lac

Percentage

20

11.25

15.75

22.5

10

10.5

22. How many ordered pairs of integers (a, b), are there such that their product is a positive integer less than 100?(1) 99 (2) 545 (3) 635 (4) 1090

23. Among four persons A, B, C and D, C works half as fast as A while D works a third as fast as B. If C and D, whenworking individually, complete the work in 24 and 54 days respectively more than the time in which they completethe work when working together, then find the time in which A and B, working together, will complete the work.(1) 15 (2) 18 (3) 24 (4) 30

24. There are 51 coins in a bag. The coins are first divided into two separate bags after which the coins in one of the twobags are taken and again divided into two separate bags and so on until we are left with 51 bags containing one coineach. If after every division of the coins in a bag into two bags the product of the number of coins in the two bags iswritten down, what is the sum of all the numbers written down?(1) 1020 (2) 1275 (3) 1551 (4) 1525

25. Train A leaves station X for station Z at 0800 hrs and travels at a constant speed of 36 kmph. Train B leaves stationZ for station X at 0830 hrs and runs at a constant speed of 27 kmph. Both trains have a stop at station Y but train Astops for 10 minutes while train B stops for 15 min. If the distance between the stations X and Y is 300 km and thatbetween Y and Z is 405 km, where do the two trains meet?(1) 450 km from station X (2) 450 km from station Z(3) 408 km from station X (4) 408 km from station Z

26. The inhabitants of Planet Rahu measure time in hours and minutes which are different from the hours and minutesof our earth. Their day consists of 36 hours with each hour having 120 minutes. The dials of their clocks show 36hours. What is the angle (in Rahuian degrees) between the hour and minute hands of a Rahuian clock when it showsa time of 9:48? Rahuians measure angles in degrees (°) the way we do on earth. But for them, the angle around apoint is 720 Rahuian degrees [instead of 360° that we have on earth].(1) 112° (2) 100° (3) 24° (4) None of these

27. A ladder is placed against a wall at an angle. Let the area enclosed by the ladder be A1. The ladder slides on the floor

by a few feet and makes a new angle and let the area enclosed be A2. Which of the following is true?

(1) A2 > A

1(2) A

2 < A

1(3) A

2 = A

1(4) Data insufficient

28. The set Y consists of the following numbers. Y = {1, 31/2, 3, 33/2, ……, 39, 319/2, 310}. In how many ways can a pair ofdistinct numbers be selected from the set Y such that their product is greater than or equal to 310? Assume that a x bis the same as b x a.(1) 110 (2) 210 (3) 105 (4) 100

29. A stone weighing 121 kg fell from a height of 10 m and broke into exactly 5 pieces - all of different weights. Find thesum (in kg) of the weights of the smallest piece and the largest piece, if it is known that it is possible to weigh anyweight (using a common balance) in kg from 1 to 121 kg using the 5 pieces?(1) 118 (2) 82 (3) 65 (4) Cannot be determined

30. Six friends share a circular pizza equally by cutting it into six equal sectors. If three of them cut out and eat only the largestpossible circle from their respective slices and leave the rest while the others eat their whole slice, then the approximatepercentage of pizza wasted is(1) 11% (2) 15% (3) 17% (4) 22%

31. The table below shows the income-wise distribution of the population of Bangalore.

A

B

C D

E

T

F

A P Q B

D R C

If in Bangalore the population whose income is below 3 lacs is 3000 thousand and the ratio of the number of women to thenumber of men in the 4-5 lacs income group is 0.94, then what is the approximate number of women in that income group?(1) 474 thousands (2) 408 thousands (3) 393 thousands (4) 309 thousands

32. In a regular hexagon of side 4 cm, the midpoints of three alternate sides are joined in order to form a triangle. Whatis the area of this triangle?

(1) .cm.sq38 (2) 12 3 sq.cm. (3) 9 3 sq.cm. (4) 18 3 sq.cm.

33. In a number system to the base 20, letters A, B, C, …. to K of the English alphabet are sequentially used to digitallyrepresent the values 10, 11, 12, …. to 20 (to the base 10). Calculate the decimal equivalent of the value (in base 10)of [CAKE]

(20)- [BAKE]

(20).

(1) 1483 (2) 1488 (3) 1000 (4) 8000

34. R and S are the centres of two unequal circles touching externally at the point T. P and Q are the points of contact ofa direct common tangent with the larger and smaller circles respectively and the common tangent at T intersects PQat U. What is the measure of the angle RUS?(1) 45° (2) 90° (3) 135° (4) None of these

35. How many small squares are crossed by the diagonal in a rectangular table formed by 16 x 17 small squares?(1) 32 (2) 33 (3) 34 (4) None of these

36. Let S = 141414 …. Upto 202 digits. What is the remainder when S is divided by 909?(1) 115 (2) 216 (3) 418 (4) 721

37. Given that a and b are two prime numbers while ‘n’ is a natural number such that n

1

ab

1

b

1

a

1 =++ , find the value of

| | a – b | – n |.(1) 0 (2) 1 (3) 2 (4) 3

38. In the following figure, find the ratio of the areas of the triangles ABE and DCE given that TA: TD : TF = 5 : 4 : 10.(1) 16 : 25 (2) 25 : 36 (3) 36 : 49 (4) 25 : 49

39. How many three-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 and 8 without any repetition of thedigits and wherein the tens digit is greater than the hundreds digit but less than the units digit?(1) 48 (2) 56 (3) 64 (4) 72

40. The set of all positive integers is divided into two subsets {a1, a

2, a

3, …a

n, …} and

{b1, b

2, b

3, …b

n, …} where a

i < a

i + 1, b

i < b

i + 1 and a

i = b

j for any i, j. Also, b

i =

2

aa i1i +−

for all i except i = 1. What is the value of b1?

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

41. If three playing squares are chosen at random from the 64 playing squares of a 8 x 8 chessboard, then find theprobability that exactly two of them are of the same colour?(1) 9/21 (2) 16/21 (3) 14/21 (4) 18/21

42. In the above figure, ABCD is a parallelogram. P and Q are the points of trisection of AB and R isthe midpoint of DC. What is the ratio of the area of the parallelogram ABCD to that of thequadrilateral PBCR?(1) 16 : 7 (2) 15 : 7 (3) 2 : 1 (4) 12 : 7

43. Four people need to cross a stream. At a time only two people can cross the stream using a certain boat which isavailable. The times taken by the four people to cross the stream individually are 3, 7, 11, 17 minutes respectively.If the faster person on the boat drives it and no person drives the boat more than two trips in total, what is the leasttime required for all the four to cross the stream? (Reaching from one bank to the other bank is one trip).(1) 23 minutes (2) 59 minutes (3) 31 minutes (4) 37 minutes

DIRECTIONS for questions 44 and 45: These questions are based on the following data.

There are 100 players numbered 1 to 100 and 100 baskets numbered 1 to 100. The first players puts one ball each in everybasket starting from the first basket (i.e., in the baskets numbered 1, 2, 3, …and so on upto 100), the second player then putstwo balls each in every second basket starting from the second (i.e., in the baskets numbered 2, 4, 6, … and so on upto 100),the third player puts three balls each in every third basket starting from the third (i.e., in the baskets numbered 3, 6, 9, ... andso on upto 100), and this is comtinued so on till the hundredth player puts 100 balls in the 100th basket.

44. Which basket will finally have the maximum number of balls?(1) 96 (2) 98 (3) 100 (4) None of these

45. How many baskets will finally have exactly twice the number of balls as the number on the basket itself?(1) 8 (2) 6 (3) 4 (4) 2

DIRECTIONS for questions 46 to 50: Select the correct alternative from the given choices.

46. A cylindrical vessel has its radius and height in the ratio 1 : 12 and it can hold the same quantity of water as another conicalvessel whose height is one-third of its height. What is the ratio of the lateral surface area of the cylinder and that of thecone? (Ignore the thickness of the vessel in both cases)(1) 3 : 2 (2) 8 : 5 (3) 1 : 1 (4) None of these

47. If (21)n x (36)

n = (776)

n and (12)

n x (63)

n = (x)

n then find x.

(1) 510 (2) 540 (3) 756 (4) 776

48. What is the area enclosed by x = 0, x = 3, y = 0 and y = | x - 1 | + | x - 2 | ?(1) 4 sq.units (2) 4.5 sq.units (3) 5 sq.units (4) 6 sq.units

49. There are two parallel lines and a circle in a plane dividing the plane into distinct non-overlapping regions. What isthe maximum number of regions into which the plane can be divided?(1) 8 (2) 5 (3) 6 (4) 7

50. The area of a triangle which is inside a semicircle is equal to the area outside the triangle but within the semicircle.What is the ratio of the area of the complete circle to that of a parallelogram formed with its base as the diameter of thecircle and height equal to the height of the triangle, if the base of the triangle is the diameter of the circle and the thirdvertex of the triangle lies on the circle?(1) 1 : 2 (2) 4 : 1 (3) 2 : 1 (4) 1 : 4

DIRECTIONS for questions 51 and 52: These questions are based on the following data.

Rama went to the market and bought some apples, mangoes and bananas. He bought 42 fruits in all. The number of bananasis less than half the number of apples; the number of mangoes is more than one-third the number of apples and the numberof mangoes is less than three-fourths the number of bananas.

51. How many apples did Rama buy?(1) 20 (2) 23 (3) 26 (4) 28

52. How many bananas did Rama buy?(1) 8 (2) 9 (3) 10 (4) 11

DIRECTIONS for questions 53 and 54: These questions are based on the information given below.

Car C1 starts at town T

1 at 5 a.m. and reaches town T

2 at 10 a.m. Car C

2 starts at town T

2 at 7 a.m. and reaches town T

1- at 11 a.m.

53. If the distance between towns T1 and T

2 is 320 km, what is the distance between the two cars 15 min after they meet each other?

(1) 36 km (2) 40 km (3) 48 km (4) Cannot be determined

54. One hour after C1 starts, another car C

3 – whose speed is 25% more than that of C

1 – starts from T

1 towards T

2. How

many of the following statements is/are true?I. Cars C

1 and C

3 reach T

2 at the same time.

II. C3 meets C

2 20 min after C

1 meets C

2.

III. When C3 meets C

2, C

2 has to still travel for 1 hr 30 min to reach T

1.

(1) Exactly one of the three statements is true. (2) Exactly two of the three statements are true.(3) All the three statements are true. (4) None of the statements are true.

DIRECTIONS for question 55 to 58: Select the correct alternative from the given choices.

55. Amit found that he needs to multiply a natural number N with at least p to make it a perfect square and with at least q tomake it a perfect cube. He also found that he needs to multiply N with at least r to make it a perfect cube as well as a perfectsquare. If p, q and r are natural numbers, then which of the following expresses the relationship between p, q and r?(1) p x q = r (2) p x q > r (3) p x q < r (4) Depends on N

56.In the given rectangle ABCD, E and F are points on BC such that AB : BE : EF : FC = 1 : 2 : 1 : 4. Which of thefollowing is true of the values of ∆EAF and ∆ACF?(1) ∆EAF > ∆ACF (2) ∆EAF = ∆ACF(3) ∆EAF < ∆ACF (4) Cannot be determined

57. There are three equal circles C1, C

2 and C

3 each of radius 6 cm, where C

1 and C

3 pass through

the centre of C2. What is the area of the shaded region? (in sq.cm) C

1

(1) π−12336 (2) π−16348

(3) 35436 −π (4) None of these

58. Malini and Shalini play a game in which they first write down the first n natural numbers andthen take turns in inserting plus or minus signs between the numbers. When all such signshave been placed the resulting expression is evaluated (i.e., the additions and subtractions areperformed) Malini wins if the sum is even and Shalini wins if the sum is odd. Assuming that the concept of even andodd (i.e., even and odd parities) is defined for all integers, which of the following statements is true?(1) Malini wins if n is a multiple of 4 (2) Shalini wins if n is even(3) Shalini wins if n is odd (4) Malini loses if n is a multiple of 4

DIRECTIONS for questions 59 and 60: These questions are based on the following data.

A teacher found that the performance of her students in the mid-term exams, comprising 6 subjects – A, B, C, D, E andF, is as follows:

59. If the number of students who passed in all the six subjects is 10, then find the number of students who passed inexactly five subjects.(1) 10 (2) 15 (3) 20 (4) Cannot be determined.

A

B E F C

D

Subjects passed in Number of students

A, B, C, D and E

B, C, D, E and F

C, D, E, F and A

D, E, F, A and B

E, F, A, B and C

F, A, B, C and D

10

15

20

10

15

10

C3

C1

C2

60. If the total number of students in the class is 60, then find the number of students passing in exactly three or less thanthree subjects, given that the sum of the number of students who passed in exactly four subjects and those whopassed in exactly six subjects together is 15. (Use data from the above question if necessary)(1) 20 (2) 25 (3) 30 (4) Cannot be determined

DIRECTIONS for questions 61 to 63: These questions are based on the data given below.

Everyday, Saddam, the office attender fetches water for the office in container A which has certain rated capacity.However, because of a dent at the bottom of the container, only 80% of the rated capacity of the container can be used tofill water. This water is transferred periodically into a smaller container B - for people in the office to use this water fordrinking. There is an outlet (a faucet) in B from which water is let out. Since the faucet is fixed at a level above the baseof B, water upto 10% of the rated capacity of B cannot be let out through the faucet. Everyday in the morning, afterSaddam fetches water in container A, he cleans B and fills B to the brim by pouring water from A into B. Whenever thewater level falls to the faucet level in B, he again fills B to the brim by pouring water from A into B. The questions in thisset are independent of each other.

61. On a particular day, Saddam finds that he filled B five times (including the first time) and at the end of the day, A wasempty. The water level in B reached the faucet level. What is the ratio of the rated capacities of A and B?(1) 4.6 : 1 (2) 5 : 1 (3) 5.75 : 1 (4) 6.25 : 1

62. If Saddam gets the dent in container A removed (so that water can be fetched in this container to its rated capacity)how many times can he fill container B (including the first time in the morning) given that the rated capacities of thetwo containers are in the ratio 10 : 1?(1) 9 times (2) 10 times (3) 12 times (4) 11 times

63. Saddam gets the dent in container A removed. He also gets the faucet in container B refixed so that all the water filledinto B can be used. He keeps filling B from A everytime B gets emptied. After he pours out water from A into B the lasttime (i.e., A gets emptied), what percentage of B is empty? The ratio of the rated capacities of A and B is 7.5 : 1?(1) 0% (2) 331/

3% (3) 25% (4) 50%

DIRECTIONS for questions 64 and 65: These questions are based on the following data.

Amar, Akbar and Anthony sold their three cycles manufactured in different years to Mr.Kishanlal. Mr.Kishanlal gave atotal of Rs.1700 to the three and said that Amar should get about one-half of the total amount as his cycle was used less.Akbar’s cycle being used more than Amar’s, he should get about one-third of the total amount and the last one gets aboutone-ninth. Each individual gets his amount only in denominations of Rs.100.

64. What is the difference between the amounts received by Amar and Anthony?(1) Rs.900 (2) Rs.700 (3) Rs.800 (4) Rs.600

65. The amount that Amar has is how much more than what Akbar and Anthony together have?(1) Rs.200 (2) Rs.300 (3) Rs.100 (4) Rs.400

Directions for questions 66 to70: Select the correct alternative from the given choices.

66. A, B and C start running simultaneously from the points P, Q and R respectively on a circular track. The distance(when measured along the track) between any two of the three points P, Q and R is L and the ratio of the speeds ofA, B and C is 1 : 2 : 3. If A and B run in opposite directions while B and C run in the same direction, what is thedistance run by C before A , B and C meet for the first time?

(1) 310L (2) 3

11L

(3) All three of them will never meet. (4) Cannot be determined

67. A circle of radius 1cm circumscribes a square. A dart is thrown such that it falls within the circle. What is theprobability that it falls outside the square?(1) 1/2π (2) (2π - 1) /2π (3) (π - 1) /π (4) (π - 2) /π

68. Fifteen boys went to collect berries and returned with a total of 80 berries among themselves. What is the minimumnumber of pairs of boys that must have collected the same number of berries?(1) 0 (2) 1 (3) 2 (4) 3

69. A cube of edge 12 ft is placed on the floor with one of its faces touching a wall. A ladder of length 35 ft is restingagainst that wall and is touching an edge of the cube. Find the height at which the top end of the ladder touches thewall, given that it is more than the distance of the foot of the ladder from the wall?(1) 11 ft (2) 23 ft (3) 21 ft (4) 28 ft

70. Two circles touch each other externally. One of the circles is 300% more in area than the other. If A is the centre ofthe larger circle and BC is the diameter of the smaller circle and either AB or AC is a tangent to the smaller circle,then find the ratio of the area of the triangle ABC to that of the smaller circle?(1) 2 : π (2) 3 : π (3) π:22 (4) 24:π

DIRECTIONS for questions 71 and 72: Select the correct alternative from the given choices.

71. a1, a

2, a

3, a

4 and a

5 are five natural numbers. Find the number of ordered sets (a

1, a

2, a

3, a

4, a

5) possible such that a

1 +

a2 + a

3 + a

4 + a

5 = 64.

(1) 64C5

(2) 63C4

(3) 65C4

(4) None of these

72. In the above question if a1, a

2, a

3, a

4 and a

5 are non-negative integers then find the number of ordered sets (a

1, a

2, a

3,

a4 and a

5) that are possible.

(1) 64C5

(2) 63C4

(3) 68C4

(4) None of these

DIRECTIONS for questions 73 to 75: Each question gives certain information followed by two quantities A and B.Compare A and B, and thenMark 1 if A > BMark 2 if B > AMark 3 if A = BMark 4 if the relationship cannot be determined from the given data.

73. A baker had a certain number of boxes and a certain number of cakes with him. Initially he distributed all the cakesequally among all the boxes and found that there was no cake left without a box. He later found that he had one morebox with him and so he redistributed all the cakes equally among all the boxes and found that there was one cake lessper box than initially and one cake was left without a box with the baker.A. The number of cakes per box in the first case.B. The total number of boxes with the baker.

74. A trader gives a discount of r% and still makes a profit of r%. A second trader marks up his goods by r% and givesa discount of r%.A. The cost price of the first trader.B. The cost price of the second trader.

75. A piece of work is carried out by a group of men, all of equal capacity, in such a way that on the first day one manworks and on every subsequent day one additional man joins the work. A group of women, all of equal capacity isengaged to carry out a second piece of work with ten women starting the work on the first day and one womanleaving the work at the end of everyday. The second piece of work is thrice as time consuming as the first piece ofwork while each man is thrice as efficient as each woman. It is known that one man working alone can complete thefirst piece of work in 6 days.A. Number of days in which the first piece of work is completed.B. Number of days in which the second piece of work is completed.

DIRECTIONS for questions 76 and 77: Select the correct alternative from the given choices.

76. A number when divided by a certain divisor, left a remainder of 8. When the same number was multiplied by 12 andthen divided by the same divisor, the remainder is 12. How many such divisors are possible?(1) 1 (2) 2 (3) 4 (4) 5

77. Consider the equation x² + y² + z² = 1. Let (x1, y

1, z

1) and (x

2, y

2, z

2) be two sets of values of (x, y, z) satisfying the given

equation and let A = (x1 – x

2)² + (y

1 – y

2) ² + (z

1 – z

2)². What is the maximum possible value that A can assume?

(assume that all the quantities involved are real numbers)(1) 1 (2) 2 (3) 4 (4) 6