Solutions Maths by Amiya

Maths By Amiya, 3E Learning, www.facebook.com/MathsByAmiya

TARGET - CAT 13, QUESTIONS

Till 16th June

If a milkman has 25 cows and he marked all cows from 1-25 , all different numbers. After this marking suddenly one magic and one tragedy happens, any cow which has a number "n"(1 ( 25) start giving "n" litter(s) of milk a day, e.g. Cow no 6 gives 6 litters of milk a day.

And tragedy, his all 5 sons start fighting and ask for separation. They demanded equal number of cows and total equal litter of milk a day. If each sons donate his all cows to 5 saints (one cow to each saint), then saints have equal litters of milk per day.

1. How many different ways cows can be distributed among sons-

a] 50 b] 60 c]120 d] 240 e] None of these

2. If one son get Cow number 3, 16 then rest three cows he has, are

a] 14, 22 and 10 b] 15, 22 and 10 c] 17, 21 and 8 d] 15, 22 and 9

e] None of these

3. Which pair of cows one cannot get

a] 7, 13 b] 4,11 c] 16, 23 d] 15, 10 e] None of these

Solution : - Magic Box :

https://www.youtube.com/watch?v=a8yW0KR4Y1I

4. From the given options, choose the one that completes the paragraph in the most appropriate way.

While healthy people expect the future to be slightly better than it ends up being, people with severe depression tend to be pessimistically biased: they expect things to be worse than they end up being. People with mild depression are relatively


accurate when predicting future events. They see the world as it is._______________________________________________.

a. In other words, in the absence of a neural mechanism that generates unrealistic optimism, it is possible all humans would be mildly depressed.

b. In other words, these findings are particularly fascinating because these precise regions show abnormal activity in depressed individuals.

c. In other words, the more pessimistic a person is the more likely he is to be severely depressed.

d. In other words, the more optimistic a person is the more likely he is to predict an accurate future.

Sol: 4. [a]

Since all four choices begin with in other words, what follows should be some sort of a summary or conclusion of what has been talked about. The passage basically says:

Healthy people-expect future to be better better better better than it ends up being

Severely Depressed people expect future to be worse worse worse worse than it ends up being

Mildly Depressed people expect future to be asasasas it ends up being

From this [a] easily follows; without unrealistic optimism (i.e. optimism that doesnt turn out to be accurate) people would be accurate about the future and so be mildly depressed.

b] is unrelated to the paragraph. Nothing about any precise regions is talked about.

c] can confuse you but remember. all As are Bs does not imply All Bs are As. People with severe depression tend to be pessimistically biased does not imply that the reverse is also true i.e. pessimistically biased people are likely to be severely depressed.

d] is inaccurate as per the passage; optimistic people predict a future that is slightly better that it ends up being i.e. they predict as lightly wrong future.

5. A completes a work in 6 days. He can take any one person out of B, C, D or E with him. B, C, D or E with him. B, C, D, E work with 100%, 80%, 75% and 62.5% efficiencies respectively as compared to A. Then the work can be completed in which of the following number of days?

a. 3 KLM b. 3MN c. 3 days

d. All the above e. None of these


Sol: 5. [d]

Let the work be 120 units A works 20 units/day; B works 20 units/day; C works 16 units/day; D works 15units/days; E works 12.5 units/day

A + B finish in LPQRQ = 3 daysA + C Tinish in LPQMU = 3M

L days

A + D finish in LPQMV = PRN = 3N

M days

A + E finish in LPQ PUV = RXLM = 3LM

K days

6. A two digit number is divided by absolute value of difference of their digits What is the maximum possible remainder?

a. 5 b. 6 c. 9 d. 3 e. None of These

Sol: 6 [a]

Diff of digits Numbers Possible remainders

9 90 0

8 91, 80, 19 0 or 3

7 92, 81, 70, 18, 29 0 or 1 or 4

6 93, 82, 71, 60 17, 28, 39 0 or 3 or 4 or 5

Now when difference of digits is less than or equal to 5 then remainder cannot be greater than 4.

Directions Directions Directions Directions for questions 7for questions 7for questions 7for questions 7 to to to to 9999: : : : The following table gives details of population, the gender ratio (females per 100 males), percentage of population staying in rural areas and the literacy rates for 4 different states of India.

State

Population

(in mn)

Gender

Ratio

Percent staying in

Rural areas

Literacy Rate

Male Female Urban


Maharashtra 117 95 55VK% 85% 66PM% 75%

Gujarat 60 92 60.00% 80% 60% 80%

Kerala 34 104 50.00% 96% 90% 90%

Punjab 28.5 90 60.00% 75% 64% 70%

7. What is the literacy rate among the rural dwellers in Gujarat?

a. 60% b. 64% c. 68%

d. Cannot be determined

8. In the state of Punjab, what is the approximate ratio of the literate rural dwellers to the number of literate urban dwellers?

a. 3 : 2 b. 2 : 1 c. 1 : 0.8 d. Cannot be determined

9. What is the literacy rate among the urban dwellers across all the above four states?

a. 72% b. 75% c. 78% d. 81%

Sol 7. [b] Women in Gujarat = KPLKP 60 =

PMRX 60 =

LLVR = 28.75

Men in Gujarat = 60 28.75 =31.25 Total number of literates in Gujarat = 80% of 31.25 + 60% of 28.75 = 25 + 17.25 = 42.25. Urban Dwellers in Gujarat = 40% of 60 = 24. Literates among them = 80% of 24 = 19.2 Hence Rural dwellers in Gujarat = 36 of which the number of literates = 42.25 19.2 = 23.05 Required percentage =PM.QVMU 100 =

PM.QVPVK 64%

Alternate Solution: Using weighted average, total percentage of literates in Gujarat = XQ% LQQ]UQ% KPLQQ]KP =

XQ% RQ]^% UQLQQ

Which in the first step, on reducing the weights can be directly written as: XQ% PV]UQ% PMPV]PM =

XQ% P]^% MP]M

i.e. 80% (125 96) + 60% 115 = x% 3 48 23.2 + 69 = 144 `a. b. ` = KP.PLRR 64%


Sol 8. [a] Using weighted average and writing the weights in the most reduced form, NV% LQ]UR% KLQ]K =

NQ% P]^% MP]M

i.e. 75% (50 38) + 64% 45 = `% 3 19 Dividing by 3 and finding the percentages, 3 + 9.6 = x 19 i. e. x = LP.U LK 73.7% We need to find the ration x% 3 70% 2, a. b. 221.1 140, which among the given options is closed to 3 : 2 Alternately as explained in above question, one can also find the exact number of literates in rural areas and urban areas. Sol 9. [c] Literates among urban dweller in Maharashtra = MR d

RK 117e =

MR 52 = 39

RV 24 = 19.2 KLQ 17 = 15.3 NLQ d

PV 28.5e =

NLQ 11.4 = 7.98

Total literates in urban dwellers = 39 + 19.2 + 15.3 + 8 = 81.5. And total urban population = 52 + 24 + 17 + 11.4 = 104.4 Thus, the answer is XL.VLQR.R and the two close choices are 78% and 81%. Now 80% of 104.4 will be more than 83. Thus, answer has to be less than 80% i.e. 78% is the answer.

10. If fgh (27,33,43) = iklm(PN,MM,RM)klm(PN,MM)klm(MM,RM)klm(RM,PN) then N= ? a. 1001 b. 37313 c. 1 d. 38313 e. None of These Sol: 10. [d]

nop (q, r, s) = (q r s) tou(q, r, s)tou(q, r) tou(r, s) tou(s, q) S0, N = 27*33*43 = 38313. 11. Arun and Tarun start running simultaneously from end A towards other end B of a

straight track of length 1000 m with speeds in the ratio 3 : 2. Whenever either of


them reach any end (A or B) and/or whenever they meet, each one turns direction in which he is running and continues to run at the same speed. How far is Tarun from end A when he meets Arun for the 3rd time?

a. 0 m b. 200 m c. 600 m d. 800 m e. None of These Sol: 11. [d] Let their speeds be 3x and 2x (m/s) respectively First Meeting: They meet 1st time after LQQQPM^]P^ =

PQQQV^ =

RQQ^ wbx

In this time, Tarun covers RQQ^ 2` = 800 y. Second Meeting: In the time Tarun returns back to the starting point A, i.e. covers 800 m, Arun will cover 1200 m i.e. Arun will reach other end (200 m) and turn and reach back at A. Thus, second meeting is at A. This is exactly like the beginning. Thus, every odd number of meeting will occur at 800 m from A and every even meeting will occur at A. Directions: Four sentence of a paragraph are given. One or more of these is/are

grammatically incorrect. Choose the option with the incorrect sentence or sentences.

12. A. The physical and psychological unrest of the working

B. class was explored often in the plays of Arthur Miller,

C. for who the subject of the American Dream, and its

D. achievability for ordinary Americans never got stale.

a. A and B b. A and C c. C and D d. C only

Sol: [c] C should have whom and not who as it is the object of the preposition for D and its achievability for ordinary Americans is a parenthetical phrase and so should have a coma at each end. So, you need a coma after Americans. 13. Let m1, m2, m3, ...... be the numbers which can be written as a sum of one or more

different powers of 3 with m1 < m2 < m3


a. 972 b. 981 c. 999 d. 1002 e. None of These Sol: [b] Every positive integer has a unique representation in base 2. This is the same as saying that each positive integer can be written uniquely as a sum of different powers of 2. For example, 1 = 20, 2 = 21, 3 = 20 + 21, and so on. We are interested in the numbers beginning m1 = 30, m2 = 31, and m3 = 30 + 31. We can find the nth term in this sequence by writing n as a sum of different powers of 2 and then replacing each 2j in the sum by 3j. Since n = 100 can be written in base 2 as n = (1100100)2, we get 100 = 22 +25 +26 so that m100 = 32 +35 +36 = 9 + 243 + 729 = 981. 14. If 3 roots of the polynomial x - 3px + qx + r are in an A.P. then which of the

following is true? a] p + q + r = 0 b] p + pq + n = 0 c] 3p + 2pq r = 0 d] 2p - pq + r = 0 e] None of these

Sol: [e] Let roots are a - d , a and a+d, Sum of roots taken one at a time , (a - d) + a + (a+d) = 3p => 3a=3p => a=p .................(A) Sum of roots taken twice at a time , (a - d)* a + a*(a+d)+ (a+d)* (a-d) = q => 3~P P = => P = 3~P ......... (B) Sum of roots taken all at a time , a*(a+d)*(a-d) = -r => p (~P P) = => p (~P 3~P + ) = => 2~M + ~ = => 2~M ~ = 0 15. How many integral solutions exist for the given set of inequalities? |x + 3| + |x 2| 5 and x + y 9 a. 28 b. 30 c. 27 d. 29 e. None of these Sol. 15 [a] |x+3| + |x 2| 5 suggests -3 ` 2 x + y 9 If x = -3, y = 0 a. b. 1 set of values of (x, y) If x = 2, y = 0 to 2 a. b. 10 values of (x, y) If x = 1, = 0 2 a. b. 10 set of values of (x, y) If x = 0, y = 0 or 1 2 3 a. b. 7 set of values of (x, y) Total possible set of values of (x, y) = 28.


16. In an examination, questions carry 5, 8 or 11 marks only. A person scores either full

marks for a question or 0 marks in the question. There is no negative marking. Further the maximum marks of the test are 100 marks. What is the maximum number of students who can take the test such that no two students scored the same marks?

a. 80 ways b. 85 ways c. 90 ways d. 91 ways e. None of these

Sol : 16 [d] The lowest possible score could be 0. We could get every possible score by adding 5 or 8 or 11 to any of the previous score Thus, adding 5, 8 or 11 to 0, we get possible scores as 0, 5, 8, 11. Adding 5, 8 or 11 to 5, the list possible scores get expanded to include 10, 13, 16. We keep doing the above process till the time we get 5 consecutive numbers. Adding 5, 8 or 11 to 8, the list possible scores get expanded to include 19. Adding 5, 8 or 11 to 10, the list possible scores get expanded to include 15, 18, 21. Adding 5, 8 or 11 to 11, the list possible scores get expanded to include 22. Adding 5, 8 or 11 to 13, the list possible scores get expanded to include 21. Adding 5, 8 or 11 to 15, the list possible scores get expanded to include 20, 23 and 26.on consecutive numbers and so on. Thus, hereafter all possible marks can be scored. Thus, the only scores that cannot be scored are 1, 2, 3, 4, 6, 7, 9, 12, 14, 17 only (i.e. 10 nos) From 0 to 100, there are 101 whole numbers of which 10 are not possible. Thus, there are 91 different scores possible (including 0).

17. In how many ways can 72 be written as a product of 3 positive integers? a. 12 b. 16 c. 60 d. 4 e. None Of These Sol 17 [a] (2* 3*) (2* 3*) (2* 3*) =23 The powers (indices) of 2 should add up to 3. This can be done in 3 ways viz. {3, 0, 0}, {2, 1, 0}, {1, 1, 1}. The powers (Indices) of 3 should add up to 2. This can be done in 2 ways viz. viz. {2, 0, 0} or {1, 1, 0}. 3P, 3Q, 3Q can be clubbed with 2M, 2Q, 2Q in a total of 2 ways. 3L, 3L, 3Q can be clubbed with 2M, 2Q, 2Q in a total of 2 ways. 3P, 3Q, 3Q can be clubbed with 2P, 2L, 2Qin a total of 3 ways. 3L, 3L, 3Q can be clubbed with 2P, 2L, 2Q in a total of 3 ways. 3P, 3Q, 3Q can be clubbed with 2L, 2L, 2L in a total of 1 ways. 3L, 3L, 3Q can be clubbed with 2L, 2L, 2L in a total of 1 ways. Thus, total of 12 ways.


18. 2P(Px)+L/P(P22x) = 1 is valid for what values of x? a. 8 b. LP c. 2 d. 4 e. None of These

Sol. [a]

L/P = P = P Thus, the expression can be re-written as

P(P`)P PP22` = 1 P (^) P PP^ = 1

Thus, (^)

PP^ = 2

(P`) = 2 P22` (P`) = P8` = P` + P8 (P`) = 2P` + 3 Let P` = p. Thus, p-2p-3=0i.e. (p-3) (p+1) =0i.e. p=3 or -1. P` = 1 > ` = LP P` = 3 > ` = 8 But for x=1/2, P(P^) is not defined since it because P(1). Thus, x=8

Short cut:

Work by plugging options, it is far more easier.

19. If L, P, .., LQ be 10 arithmetic means between 13 and 67, then what is

maximum possible value of L P M . LQ? a. 40LQ b. 800 c. 10RQ d. 400 e. None of these

Sol [a]

13, L,P, . , LQ, 67 will be a AP with Arithmetic Mean equal to LM]UNP = 40. Thus, L, P, . . , LK will also be a AP with Arithmetic Mean equal to 40. Using AMGM, L P LQ 40LQ

20. From the given options, choose the one that completes the paragraph in the most

appropriate way. Here is the classical world, the temples and groves, the measured walks, the echoes of the classical authors learned at school and in wanders a figure quite at odds with all of it. He comes from another world. He has no possessions. He is the Green Man, personification of the ancient forest, he is Enkidu the wild man who was the friend of Gilgamesh; at the same time he is the holy man in the desert. He is Ishmael


the outcast and he is the disenchanted leader Timon of Athens, the Emperor Charles V he is even perhaps the ultimate reproach, the shadow over the bright clarity of the classical world, the voice in the wilderness which says blessed are the meek. Whatever he is, he has always attracted attention, sometimes awe, sometimes envy, usually respect._______________________________________________.

a. To our extremely gregarious species, the solitary is a challenge. b. The eighteenth century hermitage was seen as a rustic retreat for those

moments when its proprietor or his guests felt like indulging. c. The holy hermit has been there since time immemorial. d. If it was more than simply a small step sideways into a childhood never

quite lost, what was it that the hermit stood for in the play of ideas which so enchanted the eighteenth-century gentleman?

Sol : 20. [a] In these questions, the second last sentence (the one before the blank) is the best clue you can have. In this case, the last sentence talks about how the kind of person being talked about elicits conflicting feelings or emotions. That is aptly summed up by choice a which says that the solitary is a challenge.challenge.challenge.challenge. b is wrong because hermitage is not being talked about. c makes some sense but is very general as compared to 1 which carries on the last sentence. d is talking about something not related to the paragraph. It evidently belongs to some other paragraph whose topic would be the interest (not conflicting feelings) that hermits arouse.

21. Card games requiring more than one player, such as poker and bridge, employ strategies

aimed at out-witting the opponent; however, card games that are played alone, such as

games of solitaire, do not. Hence, strategies that aim at outwitting an opponent are not an

essential feature of all card games.

Which of the following most closely parallels the reasoning in the above passage?

a. Games of chance, such as roulette and craps, employ odds that are detrimental to the

player, but favourable to the house. Since these are the only kinds of games that are

found in gambling casinos, having odds that favour the house is an essential feature of

all games played in gambling casinos.

b. Most aircraft have wings, but others, such as helicopters, do not. Hence, having

wings is not an essential property of all aircraft.

c. Chez Bons most celebrated features are its great food and extensive wine list. But,

since these are features of many other fine restaurants as well, they are not the only

features which define the essence of this outstanding restaurant.


d. It has been reliably reported that deer occasionally eat meat, but if deer were

not primarily vegetarians they would have much different shaped teeth than they do.

Hence, being a vegetarian is an essential feature of being a deer.

Sol: [b] The given argument has the following structure: Some card games involve outwitting opponents, but others do not. Therefore, outwitting an opponent is not essential to all card games. It should be fairly clear that the reasoning process in (b) is closest to that of the given argument. Thus, the answer is (b).

22. If 2 (`) + 3 d L^L]^ e = 2` + 3, where x is a real number not equal to 1, then what is the value of (0) ? a. KV b. 0 c.

PN d. Cannot be determined

e. None of these Sol: [a] Putting x=0, 2*(0) + 3 (1) = 3 (a) Putting x=1, 2 (1) + 3 (0) = 5 . . (aa) Solving (i) and (ii) simultaneously we get, 5 (0) = 9a. b. (0) = KV Directions for questions 2Directions for questions 2Directions for questions 2Directions for questions 23333 to to to to 25252525:::: In the following questions, in addition to some data and a question asked, there are two statements labeled I and II. You have to identify data in which of the statements I and/or II is sufficient to answer the question asked. Mark you answer as:

a if the question can be answered by using one of the statements alone, but cannot be answered using the other, statement alone.

b if the question can be answered by using either statement alone. c if the question can be answered by using both statements together, but

cannot be answered by using either statement alone. d if the question cannot be answered.

23. a, b, c, d, e and f are 6 consecutive odd integers, in order. What is the average of c, d,

e and f? I. The average of a and b is 16. II. The sum of the six numbers is 120.


24. The number 165 is split into three parts two perfect squares and one perfect cube. If all the three parts are distinct, what are the three parts? I. One of the perfect squares is 36. II. One of the perfect cubes is 8.

25. If my house number is a multiple of 3, then it is a number from 50 to 59. What is my house number? I. If my house number is not a multiple of 4, then it is a number from 60 to 69. II. If my house number is not a multiple of 6, then it is a number from 70 to 79. Sol: 23 [b] The six numbers can be represented using just one variable, x,(x+2), (x+4)+. Either of the statements individually uniquely identifies the 6 numbers because each will give a unique value of x. Sol: 24 [a] The only way the split can happen is 1, 64,100 4, 36, 125 8, 36,121 Thus, knowing that the perfect square is 36 still leaves out two possibilities for the three part. But knowing that the cube is 8, only on way exists of splitting 165. Thus, using statement II alone, we can find the three parts. Sol : 25 [c] If the house number is a multiple of 3, then it has to be 51, 54 or 57. Using statement I: House number will be a multiple of 4 OR a number from 60 to 69. However the number cannot be any multiple of 3 or of 4 in the range 60 to 69. Thus, using only this statement the house number can be: Any multiple of 4 OR 61, 62, 65, 67. Using statement II: House number will be a multiple of 6 OR a number from 70 to 79. However any number that is a multiple of 6 also has to be a multiple of 3. In that case, house number has to be only 54. Also the number cannot be any multiple of 3 in the range 70 to 79. Thus, using only this statement the house number can be: 54 OR 70, 71, 73, 74, 76, 77, 79. Using statement I and II: None of 61, 62, 65, 67 is valid for the second condition. Thus, the house number will be a multiple of 4. The only such number satisfying the second condition is 76. Thus, using both the conditions, the house number can be ascertained to be 76.

Maths By Amiya, 3E Learning,

26. Choose the option in which the usage of the word is incorrect or inappropriate. ComeComeComeCome

a. Not many people bought tickets for the concert in advance, but quite a few came along and bought tickets a the

b. The antique picture frame just came apart in my hands.c. A nurse was with me when I came round after the operation.d. If this story comes about the Prime Minister, hell have to resign.

Sol: [d] The correct phrasal verb in d should be comes out about the Prime Minister, hell have to resign.Come alongCome alongCome alongCome along means to arrive at a placeCome apartCome apartCome apartCome apart means separate into piecesCome aroundCome aroundCome aroundCome around or come roundcome roundcome roundcome round

27. In the figure shown AC = 3 unitAD is bisector of angle BAC. What is the length of BD?

a. 23

e. None of These Sol : [d] Triangle ABC is a 30-60-opposite 30 degree is MM Now, the angle bisector divides BC in the ratio of adjacent sides. Thus, BD= MM]M 23 =

UM]M =


Choose the option in which the usage of the word is incorrect or inappropriate.

Not many people bought tickets for the concert in advance, but quite a few came along and bought tickets a the door. The antique picture frame just came apart in my hands. A nurse was with me when I came round after the operation.If this story comes about the Prime Minister, hell have to resign.

The correct phrasal verb in d should be come outcome outcome outcome out followed by about: If this story comes out about the Prime Minister, hell have to resign.

means to arrive at a place means separate into pieces

come roundcome roundcome roundcome round means to become conscious again.

In the figure shown AC = 3 units, angle B = 30 degrees, angles B = 60 degrees and AD is bisector of angle BAC. What is the length of BD?

b. ML ]M c. PM]L

P

-90 triangle with side opposite 60 degree is 3 units, AB side = 3 Units . And BC, side opposite 90 degrees, is 2

Now, the angle bisector divides BC in the ratio of adjacent sides. Thus, = 3 3

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Choose the option in which the usage of the word is incorrect or inappropriate.

Not many people bought tickets for the concert in advance, but quite a

A nurse was with me when I came round after the operation. If this story comes about the Prime Minister, hell have to resign.

lowed by about: If this story

s, angle B = 30 degrees, angles B = 60 degrees and

d. 3 - 3

90 triangle with side opposite 60 degree is 3 units, AB side . And BC, side opposite 90 degrees, is 23 units.

Now, the angle bisector divides BC in the ratio of adjacent sides. Thus,


Directions for questions 28Directions for questions 28Directions for questions 28Directions for questions 28 to to to to 30303030: : : : Each of five friends was born in a different year, over the period 1968 to 1972. Each one stays in a different city and practices a different profession.

The teacher was born in 1971. The architect was born a year before Rohit. Ramesh, who stays at Delhi, is a year younger than the nurse.

The friend staying at Mumbai, who was not born in 1970, is not Rajesh who is now a lawyer.

The friend staying at Chennai, who is an author, is neither Rahul nor the oldest among the five.

The friend staying at Kolkata, who was born in 1969, is not Rohit or Rajesh.

28. What is the current profession of Rahul? a. Nurse b. Architect c. Teacher d. Author

29. Who stays at Mumbai? a. Ramesh b. Rahul c. The teacher d. Both (a) and (c)

30. Who is the youngest in the group? a. Rajesh b. Rohit c. The lawyer d. The author From the data given, the following table can be filled from direct clues. . . 1968 1969 1970 1971 1972

Name

Profession Teacher

City Kolkata

We can also get the following block of data that needs to be fitted into the above table:

Rohit

Architect


Ramesh

Nurse

Delhi

Rajesh

Lawyer

The second block of Nurse Delhi can be at either 1969 1970 1971 Whatever be the case, since the Author is not the oldest, the only position for the fourth block is the last column. And once this is fixed, the only position for Rajesh will be the first column. Thus, the table will not look: 1968 1969 1970 1971 1972

Name Rajesh

Profession Lawyer Teacher Author

City Kolkatta Chennai

Since The guy staying at Mumbai is not born in 1970 an is not Rajesh, Mumbai has to be the city for the teacher Fill this in the above table for next explanation to make sense. Now the only possible column for Ramesh staying at Delhi is the middle column. Now all the other cells can be filled easily as we have those two block-firsts and second to fit in the middle three columns. 1968 1969 1970 1971 1972

Name Rajesh Ramesh Rohit

Author

Chennai


Profession Lawyer Nurse Architect Teacher Author

City Kolkatta Delhi Mumbai Chennai

28.a 29.c 30.d.

31. Para-Jumble A. As a recent New York Times op-ed notes: Plain-vanilla Top 40, once the chief

vehicle for hit songs, is now the format for only 5 percent of the nations 10,000-plus

stations.

B. To some extent, the feeling of marginalization may be the result of the very real

process of cultural fragmentation.

C. A few crossover hits notwithstanding, a young singer who wants to incorporate her

faith into her music is now likely to narrowcast to a Christian rock audience because,

well, she can.

D. But its perceived as niche culture, in large part because cultural products are

increasingly tailored to niches.

E. There is probably now as rich and varied a marketplace of Christian media-from

Veggie Tales cartoons to the apocalyptic fantasy of the Left Behind series and its

spinoffs-as theres ever been.

a. BCDEA b. ABCDE c. BEDAC d. CEABD

Sol 31 [c]

The But its perceived as niche culture of (D) gives us the clue that something should

be preceding the but Which can then be found in (E) s as rich and varied a

marketplace The concomitant pair ED in present in only one answer choice.

32. If p, q, r be root of equation `M + `P + bx c = 0 then find the equation whose

roots are ~P, P, P? a. `M (P 2b)x2 + (P + 2ac) x xP = 0 b. `M (P 2)`P + (P 2x)` + xP = 0 c. Both a & b d. None of these Sol: [a]


p + q + r = - a; pq + qr + pr = b; Required equation is x3 - p2 + q2+ r2 = (p + q + r)p2q2 +q2r2 + p2r2 = (pq + qr + pr) 2pqr( q + r+ p ) = b2 + 2ac p2q2r2 = (pqr)2 = c2 Thus required equation =

33. If ` L = then ` + + = ? a. 6 b. 9

34. If (1 + `P + `R) =

QL a. 0 b. (1) Sol: [a], the expansion have only even power of x so all odd


pq + qr + pr = b; pqr = c. (p2 + q2 + r2) x2 + ( p2q2 + q2r2 + p2r2) x

= (p + q + r)2 2(pq +qr + rp ) = a2 2b = (pq + qr + pr) 2 2 ( pq2r + qpr2 + p2 qr) = (pq + qr + pr)

+ 2ac

Thus required equation = x3 ( a2 -2b) x2 +( b2 +2ac)x-c2

L = L^ =

NR

c. 12. d. None of these

Q + L` + P`P + M`M + + R`R

LP + PM MR + . . RLR

( ) c. 1 d. None of These

Sol: [a], the expansion have only even power of x so all odd would be 0.


) x - p2q2r2

qr) = (pq + qr + pr)2

R = ?

d. None of These

would be 0.


35.

Sol: [b]

use sine rule and get g =

36. Sol: f(20)=f(f(16) ) .(i) f(16)=f(14)=f(12)=f(10)=13 Plugging in (i); f(20)=f(13)=f(11)=f(9)=12 Next, f(21)=f(f(17) ) (ii) f(17)=f(15)=f(13)=f(11)=f(9)=12 Plugging in (ii); f(21)=f(12)=f(10)=13 Next, f(22)=f(f(18)) ..(iii)


ABC is an isosceles triangle, with AB=BC BD=AC, = 30,

then g = ??? a. 100 b. 200

c. 300 d. None of These

=200 then DCA=700

f(20)=f(f(16) ) .(i)

f(16)=f(14)=f(12)=f(10)=13

Plugging in (i); f(20)=f(13)=f(11)=f(9)=12

Next, f(21)=f(f(17) ) (ii)

f(17)=f(15)=f(13)=f(11)=f(9)=12

Plugging in (ii); f(21)=f(12)=f(10)=13

Next, f(22)=f(f(18)) ..(iii)


ABC is an isosceles triangle, with AB=BC , & g = 10

d. None of These


f(18)=f(16)=f(14)=f(12)=f(10)=13 Plugging in (iii); f(22)=f(13)=f(11)=f(9)=12 Observe that for all values of x more than 10, there will be only 2 values of f(x),12 or 13. For x20: For all even x, f(x)=12 and for all odd x, f(x)=13. Thus, f(100)=12

37. There are how many integral pairs of (x, y) exist, such that `P 2` P = 1 a. 0 b. 1 c. 2 d. None of these

38. Find the remainder when 67!NM is divided by 71 a. 67 b. 46 c. 24 d. None of these

39. A series is written on the board as: 20, 28, 36, 44, 52 .324. One number of the series is erased and the average decreases by 4. Find the number erased. a. 20 b. 176 c. 324 d. multiple values are possible. Sol: [c] Short-cut: In an AP, by erasing one number, the maximum possible movement of the average can only be d/2, where d is the common difference. And this maximum difference can come only when a number from the end is erased. Since average in this case moves by 4 i.e. half of the common difference, and the average decreases, the number erased has to be 324. Theoretical: Current Average = PQ]MPRP = 172 and number of terms

MPRPQX + 1 = 39.

Now the new average = 176, when one number is erased (leaves). Thus, the number erased contributes 4 to each of the 38 number i.e. contributes a total of 4 38 = 152. Thus; the number erased would have been 172 + 152 = 324. Or one can find the original sum and the new sum and then subtract.


40. What would be degree of the expression whose graph is given below, and it it known that all coefficients of the expression are real numbers. a. 3 b. 4 c. 5 d. 8 What would be General expression of the given below graph for natural number "n" & "m" a. ((x^2 - 1)^n)*(x-2)^m b. ((x^2 - 1)^2n)*(x-2)^(2m-1) c. ((x^2 - 1)^2n)*(x-2)^(2m) d. ((x^2 - 1)^2n)*(x-2)^(2m+1)


Directions for questions Directions for questions Directions for questions Directions for questions 41414141 to to to to 44443333: A management trainee as a part of his summer project was handed the following graph of price index and quantify index to analyze the trends in the revenue and quantity sold by M/s Apex Pharmaceuticals, which is into the business of selling the bulk drug Proplaxcillin. The price index shows the selling price (Rs/kg) in a particular year, as a percentage of the price in the year 2004. The quantity index shows the quantity sold (in tons) in a particular year, as a percentage of the quantity sold in the year 2004.

41. If the revenue of Apex Pharmaceuticals by selling Proplaxcillin was Rs. 41.4 lacs in the year 2006, find the revenue of the company in the year 2004. a. Rs. 40 lacs b. Rs. 37.5 lacs c. Rs. 34.8 lacs d. Rs. 32.5 lacs

42. What was the percentage increase in the revenues (earned through selling Proplaxcillin) of Apex Pharmaceuticals in the year 2009, over the previous year? a. 2.2% decrease b. 2.2% increase c. 1.6% decrease d. 1.6% increase

43. If the revenue of Apex Pharmaceuticals by selling Proplaxcillin was Rs. 66.24 cr in the year 2006, and the selling price in the year 2007 was Rs. 3840/kg, find the quantity (in tons) of Proplaxcillin sold in the year 2008. a. 150 tons b. 160 tons c. 180 tons d. 200 tons Sol Sol Sol Sol For For For For 41414141 to to to to 44443:3:3:3: Let the prices (Rs./kg) in successive years be 100k, 108k, 120k, 96k, 112k and 106k. Similarly let the quantity sold (in tons) be 100n, 88n, 92n, 108n, 120n, 124n.

80

85

90

95

100

105

110

115

120

125

130

2004 2005 2006 2007 2008 2009

Price Index Quantity Index


41414141.b.b.b.b One need not get into adjusting units between tons and kgs and changing Rs. lacs. All one needs to realize is that 120k 92( will be 41.4 lacs once units are converted. And we need to find 100n 100. Thus required value will be 41.4 LQQLQQLPQKP a. b.

RLRVPVLQUPM a. b.

UKPVPPM a. b.

NVP a. b. 37.5 lacs

42424242.a.a.a.a Required percentage can be found from the ratio LQULPRLLPLPQ a. b.

VMMLVUMQ a. b.

LURMLUXQ i.e. a decrease of

MNLUXQ expressed as a percentage. Since 1%

of 1680 is 16.8, the numerator will surely be more than 2% decrease. 44443.c3.c3.c3.c 120k 92( 1000 = 66.24 10N. 96k = 3840. Thus, 120k = VR 3840 = 5 960 = 4800. Substituting this value in the above, we have 92n = UUPRRX i.e.

XPXU i.e. 138 tons

And we need to find 120n. Thus, required value = 138 LPQKP a. b.

LMXMQPM a. b. 6 30 a. b. 180 (w.

44. In a triangle with base being 12 units and having an area of 90 sq. units, a rectangle is placed such that one side of the rectangle lies along the base and the other two vertices of the rectangle lie on the other two sides of the triangle. What is the maximum area of the rectangle? a. 25 b. 36 c. 45 d. 49 Sol Sol Sol Sol 44444. 4. 4. 4. .c.c.c.c Height of the triangle can be found as 15 using LP 12 = 90 Let the base and height of the rectangle be x and (15 y) as shown:


Using similarity, we have We need to maximize x This will be maximum when RV

LVP

LVP = 45.

45. In the parallelogram as shown below, DC = 4 the respective sides of the parallelogram. Find the ratio of the areas,

a. 1 : 1 b. 1 : 3 Sol: 44.aSol: 44.aSol: 44.aSol: 44.a Joining the other vertices with the midpoints we get:


Using similarity, we have ^LP =

LV ` =

RV .

(15 )a. b. RV ( 15) This will be maximum when = ( 15) = LVP and the maximum area will be

In the parallelogram as shown below, DC = 4 AD. P, Q and R are the midthe respective sides of the parallelogram. Find the ratio of the areas,

c. 3 : 1 d. 2 : 1

Joining the other vertices with the midpoints we get:


and the maximum area will be

AD. P, Q and R are the mid-points of the respective sides of the parallelogram. Find the ratio of the areas, ()() .


Since triangle DRS is similar to triangle DAT and ratio of corresponding sides is 1 : 2, the areas of triangle DRS and that of quadrilateral RSTA can be taken as a and 3a respectively. By the same logic area of triangle ATP and quadrilateral TPBU can be taken as b and 3b. Also triangle BTQ is congruent to triangle DRS and hence its area will be a.Now area of triangle DAP (i.e. 4a + b) will be oneAlso area of triangle ABQ (i.e. 4b + a) will also be oneparallelogram. Thus, 4a + b = 4b + a i.e. a = b. Hence required ratio is 1 : 1.

46. What is the remainder when 25 a. 1 b. 3 Sol Sol Sol Sol .d.d.d.d M + M + xM = 3x a In this case a =25, b = -16 and c = 25 16 9. Cancelling out 30 from both dividend and divisor, the division can be reduced to (5 8 9) divided by 7, which gives a remainder of 3. Thus, the remainder in toriginal division will be 3

47. Two non-congruent triangles have equal areas. If the sides of one of the triangles is 13, 21 and 32 and the sides of the other triangle are 13, 21 and x. Find x. a. 16 b. 20

48. If ai where i assumes values 1, 2, 3 and 4, be four real numbers of the same sign, then the minimum value of and 4 but i j is: (the symbol a. 8 b. 12


RS is similar to triangle DAT and ratio of corresponding sides is 1 : 2, the areas of triangle DRS and that of quadrilateral RSTA can be taken as a and 3a

By the same logic area of triangle ATP and quadrilateral TPBU can be taken as b and

Also triangle BTQ is congruent to triangle DRS and hence its area will be a.Now area of triangle DAP (i.e. 4a + b) will be one-fourth the area of parallelogram. Also area of triangle ABQ (i.e. 4b + a) will also be one-fourth the area of

hus, 4a + b = 4b + a i.e. a = b. Hence required ratio is 1 : 1.

What is the remainder when 25 - 16 - 9 is divided by 210?

c. 60 d. 90

+ + x = 0. 16 and c = -9. Thus, a + b + c = 0 and 25 -

Cancelling out 30 from both dividend and divisor, the division can be reduced to (5 divided by 7, which gives a remainder of 3. Thus, the remainder in t

original division will be 3 30 = 90.

congruent triangles have equal areas. If the sides of one of the triangles is 13, 21 and 32 and the sides of the other triangle are 13, 21 and x. Find x.

c. 14 d. 25 e. None of these

assumes values 1, 2, 3 and 4, be four real numbers of the same sign, then for all possibilities where i = 1, 2, 3 and 4 and

is: (the symbol refers to adding up all possibilities) c. 6 d. 16 e. None of these


RS is similar to triangle DAT and ratio of corresponding sides is 1 : 2,

the areas of triangle DRS and that of quadrilateral RSTA can be taken as a and 3a

By the same logic area of triangle ATP and quadrilateral TPBU can be taken as b and

Also triangle BTQ is congruent to triangle DRS and hence its area will be a. fourth the area of parallelogram.

fourth the area of

- 16 - 9 = 3

Cancelling out 30 from both dividend and divisor, the division can be reduced to (5 divided by 7, which gives a remainder of 3. Thus, the remainder in the

congruent triangles have equal areas. If the sides of one of the triangles is 13, 21 and 32 and the sides of the other triangle are 13, 21 and x. Find x.

e. None of these

assumes values 1, 2, 3 and 4, be four real numbers of the same sign, then = 1, 2, 3 and 4 and j = 1, 2, 3

refers to adding up all possibilities) e. None of these


49. If the (~ + ) term of a geometric series is m and the (~ ) term is n, then the ~ term is a. y( b. mn c. ]P d.

P e. None these

50. In an Arithmetic Progression, let represents the term in the progression.

If N = 9, and if the product of L P N is the least possible, then the common difference is? . LLMQ .

XV x.

KV .

MMPQ e. None of these

51. Sood Maruti Sanket (SMS) was studying for his CAT exam when the lights went off at

11:00 p.m. He lit two uniform candles of equal length but different thickness. The thick candle would last 6 hours and the thin one two hours lesser. He studied till the thick candle was twice as long as the thin one and then he immediately went to sleep. What time did SMS go to sleep? a. 12:30 am b. 2:00 am c. 1:00 am d. 1:30 am e. None of These

Sol: b

Let the lengths of the candle be L. Thus, if he studies for t hours,

f U = 2 df R e 1

U = 2

P

M = 1 a. b.

SMS slept at 11:00 +3hrs i.e. 2:00am.

Directions for questions Directions for questions Directions for questions Directions for questions 52525252 & & & & 53 53 53 53 : : : : On each face of a cube, positive integers are written (one integer on each face). At each vertex, the products of the numbers on the faces that meet at the vertex are calculated. The sum of the numbers so calculated (by products of faces) equals 2004. Let p denote the sum of the integers on all the faces of the cube. 52. How many possible values can p take?

a.1 b. 2 c. 3 d. 4 e. None of These

53. Which of the following could be a possible value of p? a. 167 b. 175 c. 150 d. 200 e. None of These


For 52 & 53:

Let a & c be written on two opposite faces, b and d be written on another pair of opposite

face and let e and be written on the last pair of opposite face.

Then the products written at the corners would be

The sum of these 8 numbers =

= (b + ) + x(b + ) + x=(b + )( + x + x + 2004 = 2P 3 167 Now, none of the factors (e +), (a + c) or (b + d) can be 1 (sum of 2 positive integers)

So, (e + (a + c) (b + d) could be 4

2 501.

52.d

The values of ~ = + + x +4 + 3 + 167 = 174

2 + 3 + 334 = 339 or

There are 4 possible values of p viz. 174, 175, 339, 505

53.b

The only option value that appears in the above is 175.

54. A function is defined such that (x + y) = (xnumbers, each greater than 3. If (8) = 9, then what is the value of (9)? a. 9 b. 8

54. a

We need to try and break (9) into smaller elements (keeping the smaller elements more than

3) and relate it to the given value of (8) = 9.



face and let e and be written on the last pair of opposite face.

Then the products written at the corners would be b, b, , , xb,The sum of these 8 numbers = b + + xb + x + xb + x + b

x(b + ) + (b + ) ) = (b + )( + x)( + ) = 2004 (ab(

e +), (a + c) or (b + d) can be 1 (sum of 2 positive integers)

So, (e + (a + c) (b + d) could be 4 3 167 2 6 167 2

+ + b + x b 2 + 6 + 167 = 175

2 + 2 + 501 = 505

There are 4 possible values of p viz. 174, 175, 339, 505

The only option value that appears in the above is 175.

A function is defined such that (x + y) = (x y) for all x and y being natural numbers, each greater than 3. If (8) = 9, then what is the value of (9)?

c. 10 d. cannot be determined


3) and relate it to the given value of (8) = 9.



xb, x ( x. b +

ab()

e +), (a + c) or (b + d) can be 1 (sum of 2 positive integers)

3 334 2

y) for all x and y being natural numbers, each greater than 3. If (8) = 9, then what is the value of (9)?

d. cannot be determined



(9) = (4 + 5) = (4 5) = (20) = (16 + 4) = (16 4) = (64) = (8 8) = (8 + 8) = (16) = (4 4) = (4 + 4) = (8) = 9

55. If ~ = = , and = , then the value of

+

= ?

a. 1 b. 0 c. 2 d. 3 e. None of These

55 .C

Let ~ = = = . w, ~ =

, =

, =

Since ~P = ,

=

a. b. ] =

P

Next, we want to find + =

(]) , which from the above data can be easily found as 2.

56. If a = b + c, b = c + a and c = a + b, where a, b, c are real numbers not necessarily

identical, then which of the following would be the value of L]L +L

]L +L

]L ? a. 1 b. 0 c. -1 d. 1 or -1 56. 56. 56. 56. .a.a.a.a Adding a to both sides of first relation, P + = + + x a. b. ( + 1) = + + x a. b. L]L =

]]

Similar operation can be done on the other two relations to identify L]L =

]] (

L]L =

]]. Now the required sum can easily deduced as 1, if a + b +

c is not equal to 0. If a + b + c = 0 then b + c = and hence P = a. b. = 1 in which case the required expression will not be defined. 57. If L = 1 and L + P+. + = (P for all natural numbers n > 1, find LQ

. 55 . 155 x.

1110 .

166

Sol Sol Sol Sol .b.b.b.b L = 1 L + P = 4 P a. b. 3P = L = 1 a. b. P = LM L + P + M = 9 M a. b. 8M = 4P = RM a. b. M =

LU

L + P + M + R = 16 R a. b. 15R = 9M =MP a. b. R = L

LQ


Thus, the denominators of L, P, M, R are 1, 3, 6, 10 i.e. sum of the first n natural numbers. Since sum of first 10 natural numbers is 55, LQ will be LVV 58. The women and men who run small independent presses, often on a frayed

shoestring, know in their bones that poetry is if cultural work. a. important..significant b. necessaryunderappreciated c. necessary.indispensable d. important..pathetic

SolSolSolSol.... bbbb The if serves as the biggest clue; it means that words of its either side are going to be opposites or somewhat like that. a and c are ruled out because both words are synonymous. d is awkward because its unlikely that something is important and pathetic at the same time, esp. poetry. b fits in the best.

59. Recessions are often a good time to launch new products, as old are questioned and consumer shift.

a. certainties.tastes b. products.decline c. tastes..certainties d. certainties.convictions

Sol Sol Sol Sol .a.a.a.a In such questions, you have to take care of which words go together and which dont. For example certainties or convictions dont shift, they could be questioned or changed. This is what makes c and d wrong. b is absurd; consumer decline shift has no meaning. 60. In a particular calendar, 7 days make a week, 4 weeks make a month and 12 months

make a year. In a family are three male members grand-father, father and son. The age of the grandfather in years is same as the age of the son in months. The age of the father in weeks is same as the age of the son in days. If the sum of ages of the three is 120 years, find the different in age of the father and grand-father (in years)

a. 24 years b. 30 years c. 36 years d. cannot be determined Sol [Sol [Sol [Sol [bbbb]]]]


Let the age of grandfather in years be x years. Then age of son = x months i.e. x/12 years. Age of son in days = x 4 7 = 28` days. Thus, father will be 28x weeks old i.e. 28x/(4 *12) years i.e. 7x/12 years. Since sum of their ages in years is 120 years, ` + N^LP +

^LP = 120 a. b. 12` + 7` + ` = 12 120 a. b. 20` = 12 120 a. b. ` = 72.

Thus, grandfather is 72 years, father is 42 years and son is 6 years old.

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Solutions Maths by Amiya