It is said that a cat has nine lives. Hats off to the lives and times of people who have lived the nine lives and are currently leading a fulfilling life after their MBA - "The Tenth Life". e.g. Major (Retd.) Devashish Chakravarty who achieved the impossible-becoming India's only two time CAT 100 percentiler (CAT 2004 and CAT 2006)and postgraduating from IIM Ahmedabad as the Best All Round Student of the Batch 2005-07.
CAT 08 has ended with a resolve to improve at least through online implementations in CAT 09. It had been a weird CAT, not to sound insipid. There was a sense of deja vu when the guesstimates were abuzz about the result of CAT 08 being announced earlier than expected.
That done, now keeping CAT 09 in mind, I believe that tenaday.com concept might suffice for any section. We can try cracking a ten good questions in all the three sections of CAT on a daily basis. I have tried to collate a wee bit of data on Quant questions that can be of some help in cracking MBA exams like CAT, XAT etc. (I have not cracked any and that is my candid confession!). The questions have been taken from varied sources like pagalguy.com, totalgadha.com, tenaday.com, Arun Sharma's book, materials of CL, TIME, PT, IMS, catindiaonline.com, minglebox.com, peakseekers.in etc etc. I hope I am no infringing on any IPR issue. Firstly, I am a tyro. Secondly, I believe that information is best handy when it is spread all across with altruistic motives. For that Internet has become a good global medium and I am trying for the synergism of the technological and conceptual resources. No book can be a one-stop shop for any aspirant. At best, it can be a caricature of a sublime innovation. Same is the case here too. Nonetheless, the aspirants can find it okay as a soul food. The intent is clear - to help the CAT aspirants solve quality question(s) and develop skills in quant to tame the dreaded CAT. The questions have been carefully designed to make one fundamentally and conceptually sound. Right now one should attempt answering the questions without fretting and sulking much about the speed and the time. Yes, speed is important but once the basics are in place, any problem can be cracked in a stipulated time span. CAT has been consistently over the years moving away from speed game like earlier years to the concept game. The focus is more on accuracy, so try getting the correct answer in the very first attempt. The problems here are slightly tougher than actual CAT quant problems. Ergo, one should be a bit patient while solving them :-)
Many face the problem of sticking to a particular question for too long, doing some activity and coming out with zero or even negative results. This can happen with any one of us. The best way to deal with this is to expand your skill set areas in quant by practising problems of varied kinds. Another way is to divide the problems into a) funda problem b) calculation problem or c)mixture of both. Calculation problems are the best bet (if you are sure to ace each of these in 3 minutes then attempt it else leave). Funda problems that involves litle or no calculation should be thought of and understood well in its completeness. The aim of the examiner is not to kill you in the exam but to see the best of you. Again, if you are unable to think of a line to solve a funda problem in 3 minutes then come out immediately. Funda + calculation problems are actually less tougher than pure funda problems, so device a strategy yourself how to deal with this, but at any point no problem should be given more than 4 minutes. It's a SIN not to read all the 25 questions in the section to decide which one's should be attempted and which one should be left untouched. Like I read and wrote somewhere, these few years CAT is now shifting more towards conceptual clarity and the focus is clearly lesser on speed techniques. It is not a paradigm shift for the speed lovers who want a literal F1 race in the exams. But that is the way it is now as we perhaps have enough time and resource to take managerial decisions. Or God knows what can be the reason behind that! Maybe this year will prove to be a different ballgame altogether. I am now trying to give the solutions of few questions in the end. Few questions can be wrong too because of typo or any other error because of lack of proofreading facility. Mea Culpa!
Enjoy a bit of Math now (I too enjoy it sometimes!)
Remember, practice doesn't make a man perfect, only a perfect practice makes a man perfect. It is not for the incorrigible ones who are perfectly imperfect! Joking...
Sample tough questions that can help in belling the CAT:
1. From a circular sheet of paper with a radius of 20 cm, four circles of radius 5cm each are cut out. What is the ratio of the uncut to the cut portion?
(1) 5 (2) 3 (3) 6 (4) 4 (5) none of these
2. Let S be the set of first 14 natural numbers. A special subset of S is a subset S' which satisfies the following three properties
a) S' has exactly 8 elements
b) If x belonging to S is even, then x is in S' if and only if x/2 is in S'
c) If y belonging to S is odd, then y is in S' if and only if (y+15)/2 is in S'
Let X denotes elements of S that cannot be the part of special subset. Then n(X) (i.e. number of elements in X) equals
(1)2 (2) 3 (3) 5 (4) 6 (5) none of these
New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.
3. What is the number of projects in which Medha alone is involved?
[1] Uniquely equal to zero
[2] Uniquely equal to 1
[3] Uniquely equal to 4
[4] Cannot be determined uniquely
[5] None of the above.
4. What is the number of projects in which Gyani alone is involved?
[1] Uniquely equal to zero [2] Uniquely equal to 1
[3] Uniquely equal to 4 [4] Cannot be determined uniquely [5] None of the above.
5. How many four-digit numbers are there with less than 6 different prime factors?
(1) 1224 (2) 8476 (3) 9000 (4) 7613 (5) none of these
6. By giving one rubber free with 4 pencils, it means that a discount of 10% is given on the sale of pencils. Then by giving 1 pencil free with 6 rubbers, it means that a discount of x% is given on the sale of rubbers. Then x (approximately) equals
(1) 57% (2) 61% (3) 64% (4) 73% (5) none of these
7. What is the minimum value of (a+b+c) when its given that (a^2) *(b^3)*c= 256/27 and a,b,c are all real positive numbers?
(1)5 (2)3 (3) 6 (4) 4 (5) none of these
8. I have 3 rs. and the types of stamps are 2,7,10,15,20 paise. I should buy 6 each of 2 types and 5 each of remaining 3 types exactly. What should be the types of stamps in the 5 each lot and 6 each lot?
(1) 300 (2) 400 (3) 500 (4) 200 (5) none of these
The 5th quotient of a division is 5, 6, 10, 9 and the 6th quotient for the same division is x, y, z, 93 where x, y, z are unknowns
9. In the above division, if the divisor is (x – A), then the
value of A is
(1) 1 (2) 2 (3) 3 (4) 4 (5) None of these
10. x = ?
(1) 5 (2) 6 (3) 10 (4) 9 (5) None of these
11. y = ?
(1) 11 (2) 6 (3) 1 (4) 4 (5) None of these
12. z = ?
(1) 16 (2) 23 (3) 28 (4) 42 (5) None of these
13. Mr. & Mrs. shaitan singh travel from Bombay to Goa and break journey at khandala in between. Somewhere between Bombay & Khandala, Mrs. Shaitan Singh asks "How far have we travelled?" Mr. Shaitan singh replies, "Half as far as the distance from here to khandala". Somewhere between Khandala & Goa, exactly 200 Km from the point where she asked the first question, Mrs. shaitan singh asks "How far do we have to go ?" Mr. shaitan singh replies "Half as far as the distance from Khandala to here." What is the distance between the cities Bombay to Goa?
(1) 200 (2) 300 (3) 400 (4) 500 (5) None of these
14. A point on a circle inscribed in a square is 1 and 2 units from the two closest side of the square. What is the area of the square?
(1) 36 (2) 80 (3) 100 (4) can not be determined (5) none of these
15. Amirchand is selling some articles. Amirchand is offering a discount of 33.33% if one pays by credit card. Amirchand has marked on the article in such a way that after giving a discount, he still manages to get a profit of 25%. Garibchand uses false weighing balance and deceives Amirchand by 20% and he also pays the amount by credit card. If Garibchand gives the same article to another customer at 40% discount on marked price of Amirchand and Garibchand has a profit of Rs 20, then what is the cost price of the article in rupees?
(1) 160 (2) 200 (3) 240 (4) 300 (5) none of these
16. The area bounded by the region |x| + |y| + |x+y| <= 2 is
(1) 2 (2) 3 (3) 5 (4) 4 (5) none of these
17. A set of 3 distinct elements which are in arithmetic progression is called a fundoo-trio. What is the largest number of fundoo-trios that can be subsets of a set of 15 distinct real numbers?
(1) 42 (2) 45 (3) 49 (4) 75 (5) none of these
18. Once I had been to the post-office to buy stamps of five rupees, two rupees and one rupee. I paid the clerk Rs 20, and since he did not have change, he gave me three more stamps of one rupee. If the number of stamps of each type that I had ordered initially was more than one, what was the total number of stamps that I bought?
[1] 10 [2] 9 [3] 12 [4] 8 [5] None of the above.
19. The base 6 representation of 0.33333..... is
(1) 3/5 (2) 2/3 (3) 1/3 (4) 3/4 (5) none of these
20. Let in a triangle ABC, AD, BE and CF be the altitudes intersecting at H. If AH = 3, AD = 4, BH = 2 then BE =
(1) 3 (2) 3.5 (3) 4 (4) 4.5 (5) none of these
21. There are z digits in the decimal expression of the natural number N, while there are y digits in the decimal expression of N^3. Then which of the following cannot be equal to y+z?
(1) 20 (2) 26 (3) 35 (4) 45 (5) none of these
22. Two cars A and B started from P and Q respectively towards each other at the same time. Car A was travelling at a speed of 54km/h but due to some problem reduced its speed by 1/3rd after travelling for 60 minutes. Car B was travelling at a speed of 36km/h. Had the technical problem in car A had arisen 30 minutes later, they would have met at a distance which is (1/30*PQ) more than towards Q than where they met earlier (PQ > 120km). Another car C starts from P, 90 minutes after car B started at Q, and car C travels towards Q with a speed of 36km/h, at what distance from P will cars B and C meet?
(1) 63 km (2) 54 km (3) 40.5 km (4) 36 km (5) none of these
23. The four numbers a < b < c < d can be paired in exactly 6 different ways. If each pair has a different sum and if the four smallest sums are 1, 2, 3, 4 then what is the sum of all the possible values of d?
(1) 4 (2) 35/6 (3) 15/2 (4) 7 (5) 11
24. A test has exactly 10 questions and is each question is either answered in True or False. If Deepika answers 5 questions "true" and five "false", her score is guaranteed to be at least 4. How many answer keys are there for which this is true?
(1) 12 (2) 13 (3) 16 (4) 19 (5) 22
25. A car travels downhill at 72 kmph (kilometers per hour), on the level at 63 kmph, and uphill at only 56 kmph The car takes 4 hours to travel from town A to town B. The return trip takes 40 minutes more. What is the distance between the two towns in kilometers?
(1) can not be determined (2) 191 (3) 255 (4) 273 (5) none of these
26. The remainder when 100*(99^10) is divided by 100*99 + 1 is
(1) 0 (2) 1 (3) 100 (4) 9900 (5) none of these
27. Two kinds of rice are mixed in the ratio 1:2 and 2:1 and then they are sold fetching the profit of 10% and 20% respectively. If they are mixed in equal ratio and the individual profit percents on them are increased 4/3 and 5/3 time respectively, then the profit % would be
(1) 18 (2) 20 (3) 21 (4) 25 (5) none of these
28. For how many integral k does the inequality log2 + log (2x^2 + 2x + 7/2) >= log(kx^2 + k) possesses at least one solution?
(1) 6 (2) 8 (3) 11 (4) 5 (5) 10
29. Five former beauty contest winners Sushmita, Aishwarya, Diana, Lara, Priyanka were placed 1 to 5 in a contest with no ties. One prediction was that the result would be the order Sushmita, Aishwarya, Diana, Lara, Priyanka. But no contestant finished in the position predicted and no two contestants predicted to finish consecutively did so. For example, the outcome for Diana and Lara was not 1, 2 (respectively), or 2, 3, or 3, 4 or 4, 5. Another prediction was the order Lara, Sushmita, Priyanka, Diana, Aishwarya. Exactly two contestants finished in the places predicted and two disjoint pairs predicted to finish consecutively did so. Who finished immediately next after Priyanka?
(1) Sushmita (2) Aishwarya (3) Diana (4) Lara (5) Impossible to determine
30. Let ABC be a triangle and D be the midpoint of AC. Point E lies internally on BD such that BE = 2. Also, AB = 3, BC = 4 and < AEC = 90˚. Then AC =
(1) √21 - 1 (2) √20 - 1 (3) √18 - 1 (4) √24 – 1
31. Let the product of four consecutive integers be a five-digit integer pq0pq, where p and q are single digit positive integers. Then p+q is
(1) 4 (2) 6 (3) 8 (4) 9 (5) none of these
32. A person is said to be n years old, where n is a non-negative integer, if the person has lived at least n years and has not lived n+1 years. At some point in time, Anupam is 4 years old and Nilesh is three times as old as Shrikant. At some other time, Shrikant is twice as old Anupam, and Nilesh is 5 times as old as Anupam. At yet another time, Nilesh is twice as old as Shrikant and Anupam is Y years old. There are different possibilities of what Y can be. The largest possible Y is in the range
(1) [15, 18] (2) [21, 25] (3) [27, 32] (4) [35, 39] (5) none of these
33. Let three positive integers p, p^2 + 2, p^3 + 2 be given. Which among the following is always true?
(1) All the three numbers are prime for atleast 2 values of p
(2) Exactly 2 of these numbers are perfect squares for some p
(3) The product of 2 of these numbers can be expressed as a 6 digit number in the base of the third number
(4) atleast 2 of the foregoing
(5) none of these
34. Triangle ABC is right-angled at A. D is a point on AB such that CD = 1. AE is the altitude from A to BC. If BD = BE = 1, what is the length of AD?
(1) 2^1/3 - 1 (2) (5^1/2 - 1)/2 (3) (5^1/2 + 1)/4 (4) 2^1/2 - 1 (5) none of these
35. Each of the 10 digits from 0 to 9 are used exactly once altogether to form 3 positive integers. One of these numbers is the sum of the other two. What is the difference between the largest possible and the smallest possible of the largest of these 3 numbers?
(1) 4995 (2) 5775 (3) 6858 (4) 7632 (5) none of these
36. Kaizen and Warrior are 1 km apart when they decide on phone to meet after some time. Warrior starts moving at 60 degrees to the line joining them initially and at the same time Kaizen starts moving at an angle of 45 degrees to the line joining them initially. It is known that Warrior and Kaizen move at a constant velocity which is 10m/s in case of Warrior. Both of them reach the meeting point at the same time.After the meeting they retrace the path that they took to reach the point in order to go back from where they came. Assuming that they retrace as soon as they meet each other, what is the time taken (in mins) by Warrior to cover the whole journey?
(1) 2.44 (2) 2.86 (3)9.107 (4) 10.274 (5)None of the foregoing
37. Let f(a+x) - f(a-x) = pf(2a) + qf(2x) for all real x and non-zero f(2a). Which of the following is always true?
(1) p+q = 0 (2) q = 1 (3) both (1) and (2) (4) either (1) or (2) (5)none of these
38. Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle in meters be, if we wish to have a flower bed with the greatest possible surface area?
(1) 2√2 (2) 2√5 (3) 5 (4) 4√2 (5) none of these
39. S is a region bounded by 5|y|=2(x-2) and x=7.There is an infinite plane mirror of negligible width at x=-3 perpendicular to the x-axis. Consider a region T bounded by x(x+6) =16-y^2 for x<=-3 and the mirror. Find the area of the region contained within y=+/-2 and x=+/-14 which does not include S,T or their images as formed by the mirror.
(a)13.46 (b)15.75 (c)52.73 (d)53.875 (e)None of the foregoing
40. To offset the increase in price of sugar and rice, either Sargam has to reduce the consumption of sugar by 20% or rice by 25%. How much percent consumption of sugar Sargam must reduce if she reduced the consumption of rice by 10%?
(1) 12% (2) 12.5% (3) 15% (4) 16% (5) can not be determined
41. Let the cost of 3 apples and 4 oranges be Rs 21. If Anjali can buy at most 4 apples and 3 oranges in Rs 20, then the maximum amount that could be left with Anjali will be about
(1) Rs 1 (2) Rs 1.50 (3) Rs 2 (4) Rs 2.50 (5) none of these
42. Consider the following system of equations:
r + x = l ; l + p = n ; n + r = k ; r = 8 ; x + p + k = 30
The value k is
(1) 17 (2) 22 (3) 23 (4) 15 (5) 11
43. Within a 5X5 table, a box is marked at the intersection of second row and third column. How many rectangles formed by the boxes of the table do not contain the marked box?
(1) 129 (2) 135 (3) 152 (4) 153 (5) 165
Each question is followed by 2 statements, A and B. Anwer each question using the following instructions
Choose 1 if the question can be answered using A alone
Choose 2 if the question can be answered using B alone
Choose 3 if the question can be answered using either A or (exclusive) B
Choose 4 if the question can be answered using A and B together
Choose 5 if the question can be answered neither using A nor B
44. ABCD is a rectangle and M and N are the points on AB and BC respectively. AN and DM intersect at P, AN and CM at Q, and CM and DN at R. Area of triangle APM is 3, and area of triangle CRN is 2. What is the area of the region PQRD?
(A) Area of region MBNQ is 20
(B) Area of triangle APD+ Area of triangle DRC = 18
45. Let s be the sum of squares of 6 consecutive odd integers. Which of the following cannot be the final quotient when S is successively divided by 2 and 4?
(1) 35 (2) 56 (3) 308 (4) 253 (5) none of these
46. In a hockey match India beat Pakistan 5-4. India scored first and kept the lead until the end. In how many different orders could the goals been scored?
(1) 13 (2) 15 (3) 14 (4) 16 (5) none of these
47. The necessary and sufficient condition for the equations x+y = a and x^4 + y^4 = b to have real roots is
(1) b >= a^4 (2) a >= 4b^4 (3) a >= b^4 (4) b >= 4a^4 (5) none of these
48. Let (10+x)/(110+x) = (20+y)/(120+y) = (30+z)/(130+z) = 1/n, where x, y, z and n are positive integers. The number of distinct possible value of n is
(1) 2 (2) 4 (3) 3 (4) 1 (5) none of these
48. N represents a series in which all the terms are consecutive integers and the sum of all the terms of N is 100. If the number of terms of N is greater than one , find the difference between the maximum and minimum possible number of terms of N.
a) 20 b) 30 c) 45 d) 125 e)195
49. The savings of Akash are 30 % of his earnings while the savings of Ranjit are 150 % of the savings of Akash. The combined earnings of Akash and Ranjit as a percentage of Akash's earnings cannot be
a) 150% b) 160% c)120% d)180% e)200%
50.If [log 1] +[log2] +[log3]+.....................+[log n]= n where [x] denotes the greatest integer less than or equal to x then (note:- all are to the base 10)
a) 96<=n <=104 b) 104<=n<=107 c) 107<=n<=111 d) 111<=n<=116 e) 116<=n<=120
50. Three vessels A, B and C have different concentrations of alcohol. If the contents of pairs of vessels (A, B), (B, C) and (C, A) are mixed, then the concentrations of alcohol become 30%, 40% and 50% respectively. If the concentration of alcohol in vessel A in % is 30 less than twice the concentration in % in vessel B, then the concentration of vessel C is
(1) 42% (2) 45% (3) 48% (4) can not be determined (5) none of these
51. A fruitseller gives 1 orange free with 4 apples bought, 1 gauva free with 6 oranges bought, and 1 apple free with 9 gauvas bought. If the three schemes are economically same, then it can be concluded that one apple should be given free with the sale of every
(1) 6 apples (2) 8 oranges (3) 8 apples (4) 6 oranges (5) none of these
52. Three squares each of area 16 sq.units have their one diagonal along a common line. Let the area common to first and second square be 9 sq. units while that common to second and third be 4 sq. units.The area (in sq. units) common to first and third squares will be
(1) 0 (2) 1 (3) 2 (4) 3 (5) none of these
53. RSZ is an acute-angled triangle. S' is a point on the perpendicular bisector of RZ on the opposite side of RZ to S such that angle RS'Z= 2R. R' and Z' are defined similarly (with
54. Which one of the following is the remainder when x + x^7 + x^16 + x^37 is divided by x^4 - x?
(1) 4x (2) 2x(x-1) (3) x^2 + 2x (4) x(x-1) (5) none of these
55. A graph has p points. The degree of a point in the graph is the number of other points it is connected to by edges. Each point has degree at most 3. If there is no edge between two points then there is a third point joined to them both. What is the maximum possible value of p?
(1) 12 (2) 7 (3) 10 (4) 9 (5) 8
56. Divya and Raveena can do a work alone exactly in 20 and 25 days respectively. However, when they work together, they do 25% more work than is expected. If they work for a few days alone and for few days together (both being integers only), then the work could not have been completed in exactly
(1) 10 days (2) 14 days (3) 16 days (4) 17 days (5) either none or at least 2 of these
57. Sara, Kyna, Riddhi can complete the work W1, W2, W3 alone in 6, 9 and 15 days respectively. If (Kyna, Riddhi), (Riddhi, Sara) and (Sara, Kyna) can do the work W1, W2, W3 respectively in n days each, then n lies in
(1) (3, 3.5) (2) (3.5, 4) (3) (4, 4.5) (4) (4.5, 5) (5) either none or at least 2 of these
58. the graph of |2x| + |3y| <=18 is plotted on X-Y axis. The number of points with integer coordinates lying inside the area enclosed by the graph?
(1) 115 (2) 103 ( 3) 102 (4) 76 5)none of these
59. Find the sum of last 3 digits of S where S= 871* 873 * 875*878 *881 *883?
(1) 12 (2) 5 (3) 15 (4) 9 (5) 7
60. A student could not see the full polynomial equation and could see only x^5-11x^4+……-13=0. He also knows that 1 is a root of the polynomial. What is the sum of squares of the other roots?
(1) 170 (2) 171 (3) 172 (4) 173 (5) 174
61. The number of employees in Obelix Menkir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of mployess who are not, can possibly be:
(1) 101:88 (2) 87:100 (3) 110:111 (4) 85:98 (5) 97:84
62. Let N=1421*1423*1425. What is the remainder when N is divisible by 12?
(1) 0 (2) 9 (3) 3 (4) 6 (5) 11
63. Let N=1*2*3*4*5…..*N for integer n greater than or equal to 1.
If p= 1!+2*2!+3*3!+4*4!+…..+10*10!. Find the remainder when p+2 is divided by 11!.
(1) 10 (2) 0 (3) 7 (4) 3 (5) 1
64. Two kinds of Vodka are mixed in the ratio 1:2 and 2:1 and they are sold fetching the profit 10% and 20% respectively. If the vodkas are mixed in equal ratio and the individual profit percent on them are increased by 4/3 and 5/3 times respectively, then the mixture will fetch the profit of
(1) 18% (2) 20% (3) 21 % (4) 23% (5) Cannot be determined
65. The question is followed by two statements X and Y. Answer each question using the following instruction:
Choose 1 if the question can be answered by X only
Choose 2 if the question can be answered by Y only
Choose 3 if the question can be answered by either X or Y
Choose 4 if the question can be answered by both X and Y
Choose 5 if the question can be answered by neither X nor Y
The positive integers are such that p < q ≤ r < s < 100, ps = qr and √s - √p ≤ 1. What is the value of p?
(X) The last digit of s is either 1, 2 or 3
(Y) 50 < p and r < 90
66. In a test taken by 100 students, 60 cleared cut-off in DI, 44 cleared cut-off in mathematics, 38 cleared cut-off in English and 27 students cleared cut-off in GK. 20 students cleared cut-off in all 4 sections. How many maximum students could have failed to clear the cut-off in all four sections?
(1) 38 (2) 41 (3) 47 (4) 50 (5) none of these
67. Let ABCD be a rectangle with AB = a, and BC = b. Suppose x is the length of the radius of the circle passing through A and B and touching CD, and y be the length of the circle passing through B and C and touching AD. If x + y ≥ k.(a+b) for all a and b, then k =
(1) √3/2 (2) 5/8 (3) 1/√2 (4) 1/2 (5) none of these
68. Vineet has Rs 600 with him. Each day he buys either beer for Rs 100 or vodka for Rs 200 or whisky for Rs 200. In how many ways can Vineet spend all his money?
(1) 20 (2) 24 (3) 30 (4) 32 (5) none of these
69. The number of those subsets of {1, 2, 3, 4, 5, 6} such that the equation x+y = 7 has no solution in it is
(1) 18 (2) 21 (3) 27 (4) 36 (5) none of these
70.ABCD is a square, point E is inside triangle ACD and point F is inside triangle ACB. < EAF =
71. Let a sequence S(n) be defined for positive integers n such that S(0) = 1 and S(1) = 1. If S(n+2) = 2S(n+1) + S(n), and S(n+1)/S(n) approaches a finite number R as n -> Infinity, then R equals
(1) √2 + 1 (2) 5/2 (3) √5 (4) 3√3/2 (5) none of these
72. S=1+1/2+1/3+1/4+1/5+1/6+.......ad infinitum
(1) 2 (2) 3 (3) 4 (4) 5 (5) none of these
73. What is the remainder when 5^37 is divided by 63?
(1) 22 (2)1 3 (3) 5 (4)62 (5) none of these
74. A piece of equipment cost a certain factory Rs. 600,000. If it depreciates in value, 15% the first year, 13.5 % the next year, 12% the third year, and so on, what will be its value at the end of 10 years, all percentages applying to the original cost?
(1) 2,00,000 (2) 1,05,000 (3) 4,05,000 (4) 6,50,00075 (5) none of these
75.Karan and Arjun run a 100m race, where Karan beats Arjun by 10 meters. Doing a favor, Karan starts 10 meters behind the starting line in the second race. They both run at their earlier speeds. Which of the following is true in connection with the second race?
(1)Both reach the finishing point simultaneously (2) Arjun beats Karan by 1m (3) Arjun beats Karan by 11m (4) Karan beats Arjun by 1m (5) none of these
76. Reema tries to prove her conjecture that f(a)/(a-b) is an integer for two distinct integers a and b for a given polynomial f(x) which satisfies the following properties I and II
(I) Every coefficient in f(x) is an integer (II) f(a)f(b) = -(a-b)^2
Find out the mistake in her proof if there is any
Proof
(a) For a natural number n, f(a) - f(b) is divisible by a-b since a^n-b^n is divisible by a - b (from I)
(b) Thus, [f(a) - f(b)]/(a-b) is an integer. The quadratic equation that has 2 solutions f(a)/(a-b) and -f(b)/(a-b) is x^2 - x[f(a) - f(b)]/(a-b) + 1 = 0 (from II)
(c) Since, f(a)/(a-b) is rational, (from I) and [f(a) - f(b)]/(a-b) is integer => f(a)/(a-b) is an integer
(1) (a) (2) (b) (3) (c) (4) exactly 2 of the foregoing (5) none of these
77. There are Martian amoebae of three types (A, B and C) in a test tube. Two amoebae of any two different types can merge into one amoeba of third type. After several such merges only one amoeba remains in the test tube. What is its type, if initially there were 20 amoebae of type A, 21 amoebae of type B, and 22 amoebae of type B?
(1) A (2) B (3) C (4) can be exactly two of the foregoing (5) can not be determined
78. Find the four digit perfect square so that the first two digits and the last two digits are the same
(1)6644 (2)7744 (3)8844 (4)9944 (5) none of these
79. What is the minimum value of mod(x+1) +mod(x+3)+mod(x+5)?
(1)3 (2)5 (3)6 (4)8 (5) none of these
80. What is the rightmost non-zero digit in 15!?
(1)1 (2)4 (3)8 (4)9 (5) none of these
81. The time in a clock is 20 minute past 2. Find the angle between the hands of the clock.
(1) 60 degrees (2) 120 degrees (3) 45 degrees (4) 50 degrees (5) none of these
82. A 20 liter mixture of milk and water contains milk and water in the ratio 3 : 2. 10 liters of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removals and replacement, what is the ratio of milk and water in the resultant mixture?
(1) 17: 3 (2) 9: 1 (3) 3: 17 (4) 5: 3 (5) none of these
83. How many numbers of times will the digit ‘7' be written when listing the integers from 1 to 1000?
(1) 271 (2) 300 (3) 252 (4) 304 (5) none of these
84. What are the last three digits of 57^802?
(1)000 (2)00 3 (3) 142 (4)249 (5) none of these
85. Some birds settled on the branches of a tree. First, they sat one to a branch and there was one bird too many. Next they sat two to a branch and there was one branch too many. How many branches were there ?
(1)3 ( 2)4 (3)5 (4)6 (5) none of these
86. The sum of all the divisors of 19^88 - 1 which are of the form (2^a).(3^b) with a, b > 0 is
(1) 168 (2) 224 (3) 360 (4) 744 (5) 1080
87. A cargo ship circles a lighthouse at a distance 20 km with speed 1500 km/h. A torpedo launcher fires a missile towards the ship from the lighthouse at the same speed and which moves so that it is always on the line between the lighthouse and the ship. How long does it take to hit?
(1) 37.7 secs (2) 56.57 secs (3) 75.43 secs (4) 94.29 secs (5) 113.14 secs
88. he marks scored by a student in three subjects are in the ratio of 4 : 5 : 6. If the candidate scored an overall aggregate of 60% of the sum of the maximum marks and the maximum marks in all three subjects is the same, in how many subjects did he score more than 60%?
(1) 1 (2) 2 (3) 3 (4) None of the subjects (5) none of these
89. Three variants of CAT paper are to be given to 12 students. In how any ways can the students be placed in 2 rows of 6 each so that there should be no identical variants side by side and that the student sitting behind should get the same variant? Find the number of ways it can be done.
(1) 6! ^2 (2) 6 x 6! X 6! (3) 6! ^3 (4) 30 x (12 C 6) x 6! x 6! (5) None of these
90. Find the no. of numbers between 100 to 400 which r divisible by either 2, 3, 5 and 7
(1) 1 (2)3 (3) 130 (4)200 (5) none of these
91. What is the remainder when 5^37 is divided by 63?
(1) 22 (2)1 3 (3) 5 (4)62 (5) none of these
92. If 15x+20y=375, then find the minimum value of (x^2+y^2), where x and y are natural numbers
1)5*13^1/2 (2)17 (3) 14 (4)16 (5) none of these
93. I purchased a lottery ticket, but since I did not have rs 100 with me, I took a loan of rs 50 from a friend. The next week when the results were out I found that the first 5 digits tallied with the first prize but since the last digit was torn, I could not find out if I was the winner. When I contacted my friend, he told me that the six-digit number was such that its first 6 multiples had the same 6 digits in different orders. But for this piece of information and the loan he demanded half the prize money from me. What was the last digit?
(1) 2 (2) 5 (3) 4 (4) 7 (5) none of these
94. The harmonic mean of two positive integers x and y is 14^6 such that x is always less than y. Find how many such pairs (x,y) are possible.
(1) 68 (2) 69 (3)70 (4)71 (5) none of these
95. Each question is followed by two statements X and Y. Answer each question using the following instructions:
Choose 1 if the question can be answered by X only
Choose 2 if the question can be answered by Y only
Choose 3 if the question can be answered by either X or Y
Choose 4 if the question can be answered by both X and Y
Choose 5 if the question can be answered by neither X and Y
Let x and y be positive real numbers. Is x^2 + y^2 < 1?
(X) y^3 + y <= x - x^3
(Y) |x| + |y| < √2 and |x| < 1 and |y| < 1
96. A cylinder of radius √6 cm and height 3√3 cm is inscribed inside a cube such that the axis of cylinder is along a diagonal of the cube. The length of side of the cube is
(1) 6 cm (2) 7 cm (3) 8 cm (4) 9 cm (5) none of these
97. X^2-7x+12 < |x-4|
(1) X<2 (2) x>4 (3). 2
(1)15624 (2)16384 (3)6144 (4)240 (5) none of these
99. If a+b+c=30; & a>=3; b>4; c>=5.In how many ways can this be done
(1)190 (2)210 (3)171 (4)153 (5) none of these
100. Given a point P inside of a quadrilateral ABCD where
101. LMN is a triangle. MO is the angle bisector. The point P on LM is such that
102. A cylinder of radius √6 cm and height 3√3 cm is inscribed inside a cube such that the axis of cylinder is along a diagonal of the cube. The length of side of the cube is
(1) 6 cm (2) 7 cm (3) 8 cm (4) 9 cm (5) none of these
103. A tank is fitted with 8 pipes, some of them that fill the tank and others that are waste pipe meant to empty the tank. Each of the pipes that fill the tank can fill it in 8 hours, while each of those that empty the tank can empty it in 6 hours. If all the pipes are kept open when the tank is full, it will take exactly 6 hours for the tank to empty. How many of these are fill pipes?
(1)2 (2)4 (3)6 (4)5 (5) none of these
104. If r is the root of f(x) = x^4 + ax^3 - 6x^2 - ax + 1 = 0, then which among the following is also a root of f(x) = 0?
(1) -1/r (2) (1+r)/(1-r) (3) (r-1)/(r+1) (4) All of these (5) Exactly two of these
105. Find the number of zeroes in (1^1)*(2^2)*(3^3)*(4^4)*(5^5)…………..(100^`100)
(1)120 (2) 200 (3) 1300 (4) 1420 (5) none of these
106. Garibchand decided to sell 89 articles (of same kind). After achieving break even (on exact no. of articles), he decided to give 10% discount and when he achieved exact 10% profit, he decided to give 20% discount. Garibchand sold all articles with 20% net profit. Had he not given discount, he would have made profit of
(1)23.6% (2) 25.3% (3) 27.8% (4) 29.2% (5) 31.4%
107. Find the number of five-digit multiples of 4, which can be formed using the digits from 1 to 7, using the digits at most once:
(1) 720 (2) 84 (3) 840 (4) 90 (5) none of these108. Sarah and Neha start running simultaneously from the diametrically opposite ends of a circular track towards each other at 15km/h and 25km/h respectively. After every 10 minutes their speed reduces to half of their current speeds. If the length of the circular track is 1500 m, how many times will Sarah and Neha meet on the track?
(1) 6 (2) 9 (3) 11 (4) 7 (5) 8
109. A cone of radius 1 unit and height 2 units just fits inside a cylinder with their axis perpendicular. The radius of the cylinder in unit is
(A) 1 (B) 1.25 (C) 1.5 (D) 2 (E) more than 2
(110) There are a certain facts about the software engineers working with an IT firm. Total number of software engineers is 70 out of which 30 are females. 30 people are married. 24 software engineers are above 25 years of age. Out of all married software engineers, 19 are above 25 years, of which 7 are males. 12 males are above 25 years and overall 15 males are married. How many unmarried females are there which are above 25?
(A) 12 (B) 8 (C) 7 (D) 0 (E) 11
DIRECTIONS for Questions 4 and 5: Each question is followed by two statements X and Y. Answer each question using the following instructions:
Choose A if the question can be answered by X only
Choose B if the question can be answered by Y only
Choose C if the question can be answered by either X or Y
Choose D if the question can be answered by both X and Y
Choose E if the question can be answered by neither X and Y
(111) Let p(x) = x^2 + 40. Then for any two positive integers i and j where i > j, is p(i) + p(j) a composite number?
(X) p(i) – p(j) is not a composite number
(Y) p(2i) + p(2j) is a composite number
(112) What is the length of the side AB of triangle ABC?
(X) AB <= AC = 2, and area of triangle ABC is 2
(Y) Exactly two sides have integer length
(113) The numerical value of f(1/10) + f(2/10) + …. + f(9/10), where f(x) = 9^x/(3+9^x) is
(A) 10/3 (B) 4 (C) √10 (D) 5 (E) 9/2
(114) A blackboard bears a half-erased mathematical calculation exercise:
2 3 ? 5 ?
+ 1 ? 6 4 2
-------------
4 2 4 2 3
In which number system was this calculation performed?
(A) 4 (B) 9 (C) 8 (D) 5 (E) 7
(115) Given points P1, P2, P3, …, P7 on a straight line, in the order stated (not necessarily evenly spaced). Let P be an arbitrary point selected on the line and let s be the sum of undirected lengths PP1, PP2, PP3, …, PP7. Then s is smallest if and only if the point P is
1. Midway between P1 and P7 2. Midway between P2 and P6 3. Midway between P3 and P5 4.
At P4 5. At P1
(116) A new Ducati is designed for the Indian market such that its mileage at a particular speed follows a certain relationship with that speed. Also, the speed decreases linearly with the mass of the rider while the petrol consumption per km increases linearly with the mass of rider. Ideally, when the mass of the rider is negligible, the speed is 100km/hr and mileage is 100km/l .When the speed of the Ducati is 50 km/hr, the mileage is 50 km/l. When the speed of the Ducati is 75 km/hr, the mileage will be
(a) 60 km/l (b) 67 km/l (c) 72 km/l (d) 75 km/l (e) Cannot be determined
(117) Let S be a 6 element set. Then the number of pairings (3 pairs) of S is
(A) 9 (B) 12 (C) 15 (D) 20 (E) 27
Directions for questions 118 to 119:
The cost of 1 kg of sugar is Rs 20 while the cost of 1 liter of pure milk is Rs 15. Sweetened milk is prepared by adding a fixed amount of sugar in a liter of milk.
(118) If the cost of sweetened milk is Rs 15 per kg, then it can be concluded that the weight of x liter pure milk is y kg more than a liter of sweetened milk where (x, y) is
(A) (4, 5) (B) (4, 3) (C) (5, 3) (D) (2, 1) (E) (3, 2)
(119) If the cost of 1 liter of sweetened milk is Rs 16 and its weight is 1.25 kg, then the weight of 1 liter of pure milk is
(A) 1.05 kg (B) 1.04 kg (C) 1.10 kg (D) 1 kg (E) none of these
(120) Katrina, her brother, her son and her daughter are chess players (all relations by birth). The worst player’s twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are of the same age. Who is the worst player?
(A) Katrina (B) Her brother (C) Her son (D) Her daughter (E) Cannot be determined
(121) An elastic string laying along the interval [-2, 2] on the x-axis is stretched uniformly and displaced so that it lays along [3, 9]. What is the new location of the point of the string which was formerly at x = 1?
(A) 8 (B) 6 (C) 4.5 (D) 4 (E) 7.5
122. A dishonest hairdresser uses a mixture having 5 parts after shave lotion and 3 parts water. After taking out some portion of mixture, he adds equal amount of water to the remaining portion of mixture such that amount of after shave lotion and water become equal. Find part of mixture taken out?
a) 4/5 b) 1/3 c) 3/5 d)1/5 e)1/2
123. The largest number amongst the following that will perfectly divide 101100 - 1 is
(1) 100 (2) 10,000 (3) 100100 (4) 100,000 (5) none of these
124. The equation 2x2 + 2(p + 1)x + p = 0, where p is real, always has roots that are
(1) Equal (2) Equal in magnitude but opposite in sign (3) Irrational (4) Real (5) Complex Conjugate
125. If a/ (b+c) =b/(c+a) =c/ (a+b) =r then r cannot take any value except:
(1) 1/2 (2) -1 (3) ½ or -1 (4) -½ or -1 (5) none of
(126) The number of values of k for which the roots of the equation kx^3 + 2x^2 – 3x + 1 = 0 are in harmonic progression is
(A) 0 (B) 1 (C) 2 (D) 3 (E) more than 3
(127) Consider a regular polygon of p sides .The number of values of p for which the polygon will have
angles whose values in degrees can be expressed in integers?
(a) 24 (b) 23 (c ) 22 (d) 20 (e)21
128. The number of ordered (x, y) such that 1/√x + 1/√y = 1/√20 is
(1) 1 (2) 3 (3) 5 (4) 7 (5) none of these
129. A circle passes through the vertex C of rectangle ABCD and touches its sides AB and AD at P and Q respectively. If the distance from C to the line segment PQ is equal to 4 units, then the area of the rectangle ABCD in sq. units (is)
(a) 20 (b) can not be determined (c) 16 (d) greater than 20 (e) none of the foregoing
130. There are three runners viz , Nishant , Deepak and Mohit who jog on the same path. Nishant goes jogging every two days. Deepak goes jogging every four days. Mohit goes jogging every seven days. If it’s the first day that they started this routine, what is the total number of days that each person will jog by himself in the next seven weeks?
(a) 12 (b) 13 (c) 14 (d) 15 (e) 16
131. Shravya invests some amount of money in a firm M .This amount grows upto 5000 in 2 years and upto 5500 in 3 years on R% compound interest . Then she goes to another firm N and borrows Rs 7000 at a compound interest of R%. At the end of each year she pays back Rs 3000 to firm N. Then, the amount she should pay to firm N at the end of 3 years to clear all the dues is?
(a) 2387 (b) 2550 (c) 2667 (d) 2477 (e) None of the foregoing
132. The area of the region bounded by the graph mod(x+y) +mod(x-y) =4 is
(A) 8 (B) 12 (C) 16 (D) 18 (E) none of these
133. A 33 rpm record which normally plays for 30 minutes was inadvertently started at 45 rpm, then switched to 33 rpm when mistake was realized. Altogether the record played for 26 minutes. For how many minutes was it playing at 45 rpm?
(A) An Indian king was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. Then the year he could have been born in was
(a) 1764 (b) 1600 (c) 1444 (d) 1936
135. 20 teams of five archers each compete in an archery competition. An archer finishing in kth place contributes k points to his team, and there are no ties. The team that wins will be the one that has the least score. Given that, the 1st position team's score is not the same as any other team, the number of winning scores that are possible is?
(a) 236 (b) 237 (c) 238 (d) 239 (e) none of the foregoing
136. Given that f(1)=1 and f(2)=1. If f(n)=f(n+1)-f(n-1), then find the value of f(8)-f(7)+f(5)/f(7)-f(6)-f(4)
(1)13/5 (2) 9/2 (3) 16/5 (4) 13/2 (5) none of these
137. Pavan had 6 friends in a B-school. At a certain restaurant, he met each of them 12 times, every 2 of them 6 times, every 3 of them 4 times, every 4 of them 3 times, every 5 twice and all 6 only once. Pavan had dined out alone 8 times without meeting any of them. How many times had he dined out altogether?
(1) 36 (2) 22 (3) 32 (4) 26 (5) none of these
138. 2^36 -1 is exactly divisible by two numbers between 510 and 520. The sum of these two numbers is:
(1) 1026 (2) 1022 (3) 1024 (4) 1028 (5) none of these
139. Evaluate: 1/3+1/15+1/35+……..+1/483
(1) 21/44 (2) 23/44 (3) 27/44 (4) 1/2 (5) none of these
140. Let f(k) be defined on integer k as f(k) = [k](3) + [2k](5) + [3k](7) - 6k,
where [k](2n+1) denotes the multiple of (2n+1) closest to k. How many values can f(k) assume?
(1) 4 (2) 7 (3) 10 (4) 13 (5) none of these
141. Two cars A and B started from P and Q respectively towards each other at the same time. Car A was travelling at a speed of 54km/h but due to some problem reduced its speed by 1/3rd after travelling for 60 minutes. Car B was travelling at a speed of 36km/h. Had the technical problem in car A had arisen 30 minutes later, they would have met at a distance which is (1/30*PQ) more than towards Q than where they met earlier(PQ > 120km). Another car C starts from P, 90 minutes after car B started at Q, and car C travels towards Q with a speed of 36km/h, at what distance from P will cars B and C meet?
(a) 63 km (b) 48 km (c) 40.5 km (d) none of the foregoing
142. N people vote for one of 27 candidates. Each candidate's vote % is at least one less than his/her number of votes. What is the smallest possible value of N?
(a) 108 (b) 127 (c) 134 (d) 162 (e) none of these
143. Choose 1 if the question can be answered by X only
Choose 2 if the question can be answered by Y only
Choose 3 if the question can be answered by either X or Y
Choose 4 if the question can be answered by both X and Y
Choose 5 if the question can be answered by neither X and Y
For positive reals x, y, z, Is 1/x + 1/y + 1/z <= 1?
(X) For every quadrilateral with sides a, b, c, d, x.a^2 + y.b^2 + z.c^2 > d^2
(Y) √x + √y +√ z >= √x.√y.√z
144. Consider two cones of heights 1 and 8 units having the same base radii. It is found that their height is increased by x keeping their vertex angle unchanged, their volume becomes equal. Then x equals
(1) 2/3 (2) 4/3 (3) 8/3 (4) 16/3 (5) none of these
145. The question is followed by two statements X and Y. Answer using the following instructions:
Choose 1 if the question can be answered by X only
Choose 2 if the question can be answered by Y only
Choose 3 if the question can be answered by either X or Y
Choose 4 if the question can be answered by both X and Y
Choose 5 if the question can be answered by neither X and Y
Two vessels A and B having different capacities are partly filled with spirit of different concentrations. If the content of A is poured into vessel B till it is full, then the % concentration of B increases by 5%. Is the difference in concentrations of spirits in containers more than 5%?
(X) initial level of spirit B is greater than that in A
(Y) If the content of B is poured into vessel A till it is full, then the % concentration of A decreases by 10%
Let S he the set of all pairs (i,j) where 1≤ i < j ≤ n Any two distinct members of S are called 'friends' if they have one constituent of the pairs in common and 'enemies' otherwise. For example if n = 4, then S = {(1, 2), (1, 3), (I, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
146.For general n, how many enemies will each member of S have?
(A) n-3(B) 1/2 (n2- 3n - 2 ) (C) 2n - 7(D) 1/2(n2 -5n + 6) (E) 1/2(n2 -7n + 14)
147.For general n, consider any two members of S that are friends. How many other members of S will be common friends of both these members?
(A) 1/2(n2 -5n + 8) (B) 2n - 6 (C) 1/2 n (n -3) (D) n -2 (E) 1/2(n2 - 7n + 16)
148. At what time between 2 and 3 o’clock will the hands of a clock be in a straight line facing away from each other?
(1) 2:42:9/11 (2) 2:42:49/77 (3) 2:43:9/11 (4) 2:43:49/77 (5) none of these
149. If the average marks of 17 students in a class is A. The marks of the students when arranged in either an ascending or descending order were found to be in Arithmetic Progression. The class teacher found that the students who were ranked 3rd, 7th, 9th, 11th, and 15th had copied in the exam and hence got all of them rusticated. The average of the remainder of the class was B. Then
(1) A=B (2) A>B (3) A150. The area bounded by the curves y=mod(x)-2 and y=-mod(x-1) is
(1) 1.75 (2) 1.5 (3) 2.5 (4) 2.75 (5) none of these
151. Let ABCD be a rectangle and E be a point beyond C on AC extended. If < DEB = < CBE, and AB/BC = 3, then BE/CE equals
(1) √3: √2 (2) √2: 1 (3) √2 + 1: √ 3 (4) 4: √5 + 1 (5) √3+1: 2
152. The number of real roots of the equation |1 - |x|| - (1.01) ^ (1.01x) = 0 is/are
(a) 1 (b) 2 (c) 3 (d) 0 (e) none of these
153. Two budding MBAs who are also mathematicians, Srikar and arbit_rageur, play a game. The computer selects some secret positive integer N < 60 (both Srikar and arbit_rageur know that, but that they don't know what the value of N is). The computer tells Srikar the unit digit of N, and it tells arbit_rageur the number of divisors of N. Then, Srikar and arbit_rageur have the following dialogue:
Srikar: I don't know what N is, and I'm sure that you don't know either. However, I know that N is divisible by at least two different primes.
arbit_rageur: Oh, then I know what the value of N is.
Srikar: Now I also know what N is.
Assuming that both Srikar and arbit_rageur speak truthfully and to the best of their knowledge, how many possible values of N are there?
(a)0 (b) 1 (c) 2 (d) 3 (e) none of these
154. Which of the following is not a perfect square in scale of 8?
(1) 121 (2) 14641 (3) 100 (4) 58 (5) none of these
155. In how many ways can 2004 be written as a sum of two or more consecutive positive integers?
(a)0 (b) 1 (c)2 (d)3 (e) none of these
156. Three positive integers a, b, and c are consecutive terms in an arithmetic progression. Given that n is also a positive integer, for how many values of n below 1000 does the equation a^2 - b^2 -c^2 = n have no solutions?
(a) 458 (b) 493 (c) 524 (d) 559 (e) 596
157. Given p and q be positive such that 2 >= p-q, the min value of 2/(p+q) + q/2 is
(1) √2 - 1/2 (2) (√2 + 1)/2 (3) 1 (4) 1/√2 (5) none of these
158. Mr. X bought 10 identical chocolates. He ate 2 chocolates and sold the remaining 8 chocolates for Rs. 60 making a net profit of 32%. Find the profit percent had he eaten 6 chocolates and sold the remaining 4 chocolates for Rs. 48.
(1) 4.8% (2) 5.6%2 (3) 6.4% (4) 7.2% (5) none of these
159. Out of 200 fish in an aquarium, 99% are red. How many red fishes must be removed in order to reduce the percentage of red fishes to 98%?
(1) 50 (2) 100 (3) 10 (4)2 (5) none of these
160. What is the remainder when S = 1! + 2! + 3! +… + 19! + 20! is divided by 20?
(a)0 (b) 1 (c) 3 (d)13 (e) none of these
161. Each question is followed by two statements X and Y. Answer each question using the following instructions:
Choose 1 if the question can be answered by X only
Choose 2 if the question can be answered by Y only
Choose 3 if the question can be answered by either X or Y
Choose 4 if the question can be answered by both X and Y
Choose 5 if the question can not be answered by combining X and Y also
If 1 < a < 2 and k is an integer, then what is [ak/(2 - a)], where [x] denotes the greatest integer not larger than x.
(X) [a[k/(2 - a)] + a/2] = p
(Y) [a[k/(2 - a)] + (a+1)/2] = q and k is even
162. Which among 3^1/2, 4^1/3, 5^1/4, 7^1/6, and 13^1/12 is the largest?
(1) 3^1/2 (2) 4^1/3 (3) 5^1/4 (4) 13^1/12 (5) none of these
163. Given a trapezium ABCD with AB || CD , CD = 2AB and DB perpendicular to BC. Let E be the intersection of lines DA and CB, and F be the midpoint of DC. Which among the following is not true?
(1) ABFD is a rhombus
(2) Triangle CDE is isosceles
(3) If AF and BD meet at G, and GE and AB meet at H then the line DH bisects segment EB
(4) At least two of the above
(5) none of the above
164. The function f(x)=mod(x-1)+mod(2.8-x)+mod(x-3), where x is a real number can attain a maximum value of
(1) 1 (2) 2 (3) 3 (4) 3.8 (5) none of these
:
165. Let xy, yz, zx be three 2-digit numbers. If 9xy+yz+zx) is divisible by 15, then its value can be:
(1) 220 (2) 150 (3) 330 (4) 165 (5) none of these
166. If cb*c*3=ccc then the value of b*c is:
(1) 10 (2) 7 (3) 21 (4) 12 (5) none of these
167. Find the number of integral solutions of mod(x) +mod(y) +mod (z) =15.
(1) 902 (2) 728 (3) 734 (4) 904 (5) none of these
168. A number has 28 factors. If n of these are perfect squares, n cannot be:
(1) 4 (2) 6 (3) 7 (4) 8 (5) none of these
169. A set of four consecutive two-digit odd numbers are added. The sum when divided by 16 results in a perfect square. How many such set of numbers are there?
(1) 1 (2) 6 (3) 4 (4) 5 (5) none of these
170. Nbangalorekar bought a 10 kg of water-melon in Bangalore that had 99% water. After the water-melon was left outdoors for a day, it was 95% water. What was the weight of the dehydrated water-melon?
(1) 9.6 kg (2) 9.5 kg (3) 2 kg (4) cannot be determined (5) none of these
171. If x=2+2^1/3+2^2/3 the value of x^3-6x^2+6x is
(1) 0 (2) 1 (3) 2 (4) 3 (5) none of these
171. A number is of the form abba and it divides a number of the form xyzxyz. What is the number of the vehicle?
(1) 1221 (2) 1001 (3) 4334 (4) 5445 (5) none of these
172. 5 professors decide to hold daily meetings such that (i) at least one professor attend each day (ii) a different set of professors must attend on different days. (iii) on day N for each 1 <= d < N, at least one professor must attend who was present on day d. How many maximum days can meetings be held?
(1) 14 (2) 16 (3) 20 (4) 24 (5) none of these
173. Crime master Gogo, riding on a motorcycle, starts at the back of a 2 km train as its front enters a 4 km tunnel. Both GoGo and the train travel at constant speed and GoGo exits the tunnel just as the train is entirely in the tunnel. When the front of the train emerges from the tunnel, Gogo turns instantly and heads back toward the train. How many meters from the tunnel does Gogo meet the front of the train?
(1) 2 (2) 3 (3) 2.5 (4)1.5 (5) none of these
174. The sum of two natural numbers p and q is 4. The minimum value of the expression (p+1/p)^2+(q+1/q)^2 is
(1) 14 (2) 15.1 (3) 12.5 (4) 4 (5) none of these
175. 10^10+10^100+10^1000+……. +10^10000000000 when divide by 7 leaves the remainder
(1) 0 (2) 2 (3) 4 (4) 5 (5) none of these
176. The 228th digit of the following number is 1222333334444444…………. Is
(1) 0 (2) 1 (3) 4 (4) 5 (5) none of these
:
177. 1/3+2/21+3/91+4/273+………………………..=
(1) 1/2 (2) 2/3 (3) 4/5 (4) 7/10 (5) none of these
178. Let F be mini 4X4 chessboard => it has 16 fields in all. In how many ways is it possible to select two fields of F such that the midpoint of the segment joining the centres of the two fields should also be the centre of a field?
(1) 15 (2) 18 (3) 24 (4) 32 (5) none of these
179. Find the largest value of x for which 12^x leaves zero as the remainder when it divides 50!/10!
(1) 16 (2) 17 (3) 18 (4) 19 (5) none of these
Find the largest value of x for which 12^x leaves zero as the remainder when it divides 50!/10!
180. In how many ways can the letters of the word JUPITER be arranged in a row so that the vowels appear in alphabetic order?
(1) 736 (2) 768 (3) 792 (4) 840 (5) none of these
181. If x, y, and z are positive integers, such that x+y+z=60 and x^2+y^2=z^2, then how many such triplets (x,y,z) exists
(1) 0 (2) 1 (3) 2 (4) 3 (5) none of these
182. What is the minimum value of the sum of the squares of the roots of the equation x^2-(@-2)+(@-5)=0 is
(1) 0 (2) 2 (3) 5 (4) cannot be determined (5) none of these
183. How many integral pairs (x,y) exist such that x^2+10y^2-4x*y+2x+8y-18=0?
(1) 2 (2) 3 (3) 4 (4) 6 (5) none of these
184. In a triangle ABC, D is a point on the side BC and X, Y are length of perpendicular dropped on line AD from the vertices B and C respectively. Is X > Y?
(A) BD < DC
(B) AB > AC
185. Three distinct numbers are randomly selected from the first 20 natural numbers. Find the probability that the selected random numbers are in geometric progression.
(1) 2/285 (2) 11/1140 (3) 3/285 (4) 1/114 (5) none of these
186. Three natural numbers form an arithmetic progression, the common difference being 13. If the first number is decreased by 2, the second is decreased by 3 and the third is doubled, the resulting numbers are in geometric progression. Find the sum of the numbers in arithmetic progression.
(1) 63 (2) 84 (3) 76 (4) 80 (5) none of these
187. In a triangle ABC, AB = AC, (1) 20˚ (2) 40˚ (3) 50˚ (4) 60˚ (5) none of these ˚
188. Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be
[1] 85.5 [2] 92.5 [3] 90.5 [4] 87.5 (5) none of these
189. A test has 50 questions. A student scores 1 mark for a correct answer, -1/3 for a wrong answer,
and –1/6 for not attempting a question. If the net score of a student is 32, the number of questions
answered wrongly by that student cannot be less than
[1] 6 [2] 12 [3] 3 [4] 9 (5) none of these
190. The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero?
[1] 1st [2] 9th [3] 12th [4] 13th [5] (5) none of these
191. When the curves y = log10 x and y = x-1 are drawn in the x-y plane, how many times do they intersect for values x ? 1?
[1] Never [2] Once [3] Twice [4] More than twice [5] none of these
192. At the end of year 1998, Shepherd bought nine dozen goats. Henceforth, every year he added p% of the goats at the beginning of the year and sold q% of the goats at the end of the year where p>0 and q>0. If Shepherd had nine dozen goats at the end of year 2002, after making the sales for that year, which of the following is true?
[1] p = q [2] p < q [3] p > q [4] p = q/2 [5] none of these
193. The function f(x) = |x – 2| + |2.5 – x| + |3.6 – x|, where x is a real number, attains a minimum at
[1] x = 2.3 [2] x = 2.5 [3] x = 2.7 [4] x=2.9 [5] None of the above.
194. In a 4000 meter race around a circular stadium having a circumference of 1000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5th minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race?
[1] 20 min [2] 15 min [3] 10 min [4] 5 min [5] None of the above.
195. Find the sum of the factors of 8! which are odd and of the form 3m+2,where m is a natural no.
(1) 35 (2) 40 (3) 5 (4) 98 (5) none of these New Age Consultants have three consultants
196. The digits of a four-digit number form an arithmetic progression, not necessarily in the same order. How many such four digit numbers are possible if the arithmetic mean of all the digits is an integer?
(1) 72 (2) 84 (3) 90 (4) 96 (5) none of these
197. Let the sum S = 20 of four natural numbers a, b, c, d be such that a (a+1) + b (b+1) + c(c+1) + d (d+1) = 312. Which among the a, b, c, d is/are uniquely determinable?
(1) None if a = b (2) At least 2 if a ≠b (3) All if a > b (4) All of the foregoing (5) Exactly 2 of the foregoing
198. Two liquids A and B are in the ratio 5 : 1 in container 1 and in container 2, they are in the ratio 1 : 3. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1 : 1?
[1] 2 : 3 [2] 4 : 3 [3] 3 : 2 [4] 3 : 4 [5] None of the above.
199. Out of two -thirds of the total number of basket-ball matches, a team has won 17 matches and lost 3 of them. What is the maximum number of matches that the team can lose and still win three-fourths of the total number of matches, if it is true that no match can end in a tie?
[1] 4 [2] 6 [3] 5 [4] 3 [5] None of the above.
200. A closed wooden box of thickness 0.5 cm and length 21 cm, width 11 cm, and height 6 cm, is panted on the inside. The cost of painting is Rs 70. What is the rate of painting in rupees per sq. cm?
[1] 0.7 [2] 0.5 [3] 0.1 [4] 0.2 [5] None of the above.
201. If a number 774958A96B is to be divisible by 8 and 9, the values of A and B, respectively, will be:
[1] 7,8 [2] 8,0 [3] 5,8 [4] 6,9 [5] None of the above.
202. How many real numbers r exist such that the roots of the equation x^2 + rx + 6r = 0 are both integers?
(1) 6 (2) 10 (3) 4 (4) 8 (5) none of these
203. Given the quadratic equation x^2 - (A - 3) x - (A - 2), for what value of A will the sum of the squares of the roots be zero?
[1] - 2 [2] 3 [3] 6 [4] 8 [5] None of the above.
204. I sold two watches for Rs. 300 each, one at a loss of 10% and the other at a profit of 10%. What is the percent loss (- ) or the percent profit (+) that resulted from the transaction?
[1] (+) 10 [2] ( -) 1 [3] (+) 1 [4] 0 [5] None of the above.
205. What is the minimum value of (a+b+c) when its given that (a^2) *(b^3)*c= 256/27 and a,b,c are all real positive numbers?
(1)5 (2)3 (3) 6 (4) 4 (5) none of these
206. Let |2x-1| - 3|x+1| = a has two real solutions p and q satisfying 2 <= |p-q| <= 10, then Max(a) - Min(a) equals
(a) 19/3 (b) 15/2 (c) 35/6 (d) 11/2 (e) none of these
207. In the quadrilateral ABCD, AD = DC = CB, and < ADC = 100˚, < ABC = 130˚. Then the measure of < ACB (is)
(1) 20˚ (2) 30˚ (3) 50˚ (4) can not be determined (5) none of these
208. A student wrote all the natural no. from 2 to 10000 on a blackboard, one after the other. Another student came and erased all perfect cubes. If student come this way and erase all the higher powers, find the no. of students who erase at least one number.
(1) 6 (2) 7 (3) 12 (4) 13 (5) none of these
209. Which of the following value of x do not satisfy the inequality (x² - 3x + 2 > 0) at all?
[1] 1 ? ? ? 2 [2] - 1 ? x ? - 2 [3] 0 ? x ? 2 [4] 0 ? x ? - 2 [5] None of the above.
210. A man travels three-fifths of distance AB at a speed of 3a, and the remaining at a speed of 2b. If he goes from B to A and back at a speed of 5c in the same time, then:
[1] 1/a + 1/b = 1/c [2] a + b = c [3] 1/a + 1/b = 2/c [4] a + b = c^2 [5] None of the above.
211. Square ABCD has side length 6. Circle Q is tangent to sides AB and BC, and is externally tangent to circle P. Circle P is tangent to sides CD and DA, and is externally tangent to circles O1 and O2. Circle O1 is tangent to side CD, circle O2 is tangent to side DA, and circles O1 and O2 are externally tangent to each other and to circle P. If the radius of circle P is twice the radius of circle Q, and if circles O1 and O2 both have radius r, then r is (up to 2 places of decimal)
(1) 0.29 (2) 0.36 (3) 0.47 (4) 0.54 (5) none of these
212. Given a dart board divided in two regions, one red, one green. If you hit the red region you get 5 points, if you hit the green region you get y > 2 points. If gcd(5, y) = 1 and let R be the maximum number of points you can not get for a given choice of y, but can get R+1 points for same choice of y, then R can not be a
(1) prime (2) composite (3) perfect square (4) two of the foregoing (5) none of the foregoing
213. A vertical tower OP stands at the center O of a square ABCD. Let h and b denote the length OP
and AB respectively. Suppose ? APB = 60o then the relationship between h and b can be expressed as
[1] 2b2 = h2 [2] 2h2 = b2 [3] 3b2 = 2h2 [4] 3h2 = 2b2 (5) none of the foregoing
214. How many three digit positive integers, with digits x, y and z in the hundred’s, ten’s and unit’s
place respectively, exist such that x < y, z < y and x ? 0?
[1] 245 [2] 285 [3] 240 [4] 420 (5) none of the foregoing
Choose [1] if the question can be answered by one of the statements alone but not by the other.
Choose [2] if the question can be answered by using either statement alone.
Choose [3] if the question can be answered by using both the statements together, but cannot be answered
by using either statement alone.
Choose [4] if the question cannot be answered even by using both the statements together.
215. Is a44 < b11, given that a = 2 and b is an integer?
A. b is even
B. b is greater than 16
216. What are the unique values of b and c in the equation 4x2 + bx + c = 0 if one of the roots of the
equation is (-1/2)?
A. The second root is 1/2
B. The ratio of c and b is 1
217. AB is a chord of a circle. AB = 5 cm. A tangent parallel to AB touches the minor arc AB at E.
What is the radius of the circle?
A. AB is not a diameter of the circle
B. The distance between AB and the tangent at E is 5 cm.
218. D, E, F are the mid points of the sides AB, BC and CA of triangle ABC respectively. What is the
area of DEF in square centimeters?
A. AD = 1 cm, DF = 1 cm and perimeter of DEF = 3cm
B. Perimeter of ABC = 6 cm, AB = 2 cm, and AC = 2 cm
A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. Once, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs. 1148, but the inventory reduced by 54.
219. What is the actual price per piece?
[1] 82 [2] 41 [3] 56 [4] 28
220. What is the actual quantity sold?
[1] 28 [2] 14 [3] 82 [4] 41
DIRECTIONS for Questions 221 and 222: In a locality, there are five small towns, A, B, C, D and E. The
distances of these towns from each other are as follows:
AB = 2km AC = 2 km AD > 2 km AE > 3 km BC = 2km
BD = 4 km BE = 3 km CD = 2 km CE = 3km DE > 3 km
221. If a ration shop is to be set up within 2 km of each city, how many ration shops will be required?
[1] 2 [2] 3 [3] 4 [4] 5 (5) none of the foregoing
222. If a ration shop is to be set up within 3 km of each city, how many ration shops will be required?
[1] 1 [2] 2 [3] 3 [4] 4 (5) none of the foregoing
223. The number of real solutions of the equation 2(mod(x))^2-5mod(x)+2=0 is:
[1] 0 [2] 4 [3] 2 [4] infinitely many (5) none of the foregoing
224. A can finish a work in 12 days and B can do it in 15 days. After A had worked for 3 days, B also joined A to finish the remaining work. In how many days, the remaining work will be finished:
[1] 3 [2] 4 [3] 5 [4] 6 (5) none of the foregoing
225. 2^x=4^y=8^z and xyz=288, then value of 1/2x+1/4y+18z is:
[1] 11/12 [2] 11/96 [3] 29/96 [4] 31/96 (5) none of the foregoing
226. A man travels form A to B at a speed of x kmph. He then rests at B or x hours. He then travels from B to C at a speed of 2x kmph and rests at C for 2x hours. He moves further to D at a speed twice as that between B and C. He thus reaches D in 16 hours. If distances A-B, B-C, C-D are all equal to 12 km, the time for which he rested at B could be:
[1] 3 hours [2] 6 hours [3] 2 hours [4] 4 hours (5) none of the foregoing
227. Instead of a metre scale, a cloth merchant uses a 120 cm scale while buying, but uses an 80 cm scale while selling the same cloth. If he offers a discount of 20 percent on cash payment, what is his overall percent profit?
[1] 20% [2] 25% [3] 40% [4] 15% (5) none of the foregoing
228. A man has nine friends, four boys and five girls. In how many ways can he invite them, if there have to be exactly three girls in the invitees?
[1] 320 [2] 160 [3] 80 [4] 200 (5) none of the foregoing
229. In a watch, the minute hand crosses the hour hand for the third time exactly after every 3 hrs 18 min 15 seconds of watch time. What is the time gained or lost by this watch in one day?
[1] 14 min 10 seconds lost [2] 13 min 50 seconds lost
[3] 13 min 20 second gained [4] 14 min 40 second gained (5) none of the foregoing
230. In a mile race Akshay can be given a start of 128 metres by Bhairav. If Bhairav can given Chinmay a start of 4 metres in a 100 metres dash, then who out of Akshay and Chinmay will win a race of one and half mile, and what will be the final lead given by the winner to the loser? (One mile is 1600 metres).
[1] Akshay, 1/12 miles [2] Chinmay, 1/32 miles [3] Akshay, 1/24 miles [4] Chinmay, 1/16 miles (5) none of the foregoing
231. A stockist wants to make profit by selling grains. Which of the following would maximize his profit?
I. Sell grains at 20% profit.
II. Use 800 g of weight instead of 1 kg.
III. Mix 20% impurities in grains and sell it at cost price.
IV. Increase the price by 10% and reduce weights by 10%
232. Priyanka has a triangular garden whose corners lie on the edge of a circular track. One of the sides of the garden forms a diameter of the plot and one of the other sides is 20 m. She builds the largest possible circular pond in the garden. The area of the pond is 36*pi sq. meters. What is the diameter of Priyanka's circular plot?
(A) 18√2 m (B) 28 m (C) 25 m (D) (9√2 + 10) m (E) 29 m
233. A permutation (a, b, c, d, e) of A = (1, 2, 3, 4, 5) is good if a + b < d +e. The number of good permutations of A is
(A) 36 (B) 40 (C) 48 (D) 60 (E) 72
234. Aamir would take hours to build a chimney alone, and Saif would take hours to build it alone. When they work together they talk a lot, and their combined output is decreased by bricks per hour. Working together, they build the chimney in hours. How many bricks are in the chimney? The inhabitants of Planet-X measure time in hours and minutes which is different from out measurement of earth. The day in Planet-X consists of 36 hours with each hour having 120 minutes. The dials of their clock shows 36 hours. What is the angle (in Planet-X degrees) between the hours and the minute hands when it shows a time of 11:24? The angle around a point in planet-X is 720 degrees.
(A) 40 (B) 100 (C) 120 (D) 60 (E) 80
235. A function has domain and range. (The notation denotes.) What are the domain and range, respectively, of the function defined by?
(A) [-1, 1], [-1, 0] (B) [-1, 1], [0, 1] (C) [0, 2], [-1, 0] (D) [1, 3], [-1, 0] (E) none of these
236. The Principal of a college issues an order that, for administrative reasons, every student of the college must take exactly four of the seven courses offered in a semester. In addition, he also wants to ensure that each course has the same number of takers. If the actual number of students is 53, how many of the students would have to leave the college to make the Principal’s plan feasible?
(A) 5 (B) 6 (C) 4 (D) 3 (5) none of these
237. Two subsets of the set are to be chosen so that their union is and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
(A) 20 (B) 40 (C) 60 (D) 160 (5) 320
238. A man robs a bank at time t = 0 (in hours), jumps onto his scooter and rides along a highway at a speed u(t) in kilometres per hour (kmph) given by
u(t) = 240t² 0 <= t <= 1/2
= 120t t > 1/2
The police on being informed reach the scene of the crime 20 minutes after the robbery, and pursue the robbers along the same highway at a speed v(t) kmph given by
v(t) = 180(t - 1/3) t > 1/3
How long after the crime will the police catch up with the robber?
(A) Within thirty minutes.
(B) Within one hour but not within thirty minutes.
(C) Within two hours but not within one hour.
(D) Never
(E) None of the above
239. The state income tax where Obama lives is levied at the rate of p % of the first 28000 dollars plus (p+2) %of annual income plus of any amount above 28000. Obama noticed that the state income tax he paid amounted to (p+0.25) %of his annual income. What was his annual income in dollars?
(A) 28000 (B) 32000 (C) 35000 (D) 42000 (5) 56000
240. a, b, c are positive integers forming an increasing geometric sequence, b-a is a square, and (loga + logb + logc)/log6 = 6. Then a + b + c is
(A) 54 (B) 63 (C) 78 (D) 93 (E) 111
241. Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.ab is expressed as a fraction in lowest terms. How many different denominators are possible?
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9
242. Sahib has five red cards numbered 1 through 5 and four blue cards numbered 3 through 6. He stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
243. P(x) is a third degree polynomial and the coefficients of P(x) are rational. If the graph of P(x) touches the x-axis, then how many rational roots does P(x) = 0 have?
(A) 3 (B) 1 (c) 2 (d) 0 (e) none of the foregoing
244. Jamal wants to store 30 computer files on floppy disks, each of which has a capacity of 1.44 megabytes (MB). Three of his files require 0.8 MB of memory each,12 more require 0.7 MB each, and the remaining 15 require 0.4 MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
(A) 12 (B) 13 (c) 14 (d) 15 (e) 16
245. The sum of 5 natural numbers (not necessarily distinct) is 12. The positive difference between the highest and the least possible value of LCM of these 5 numbers is
(A) 26 (B) 24 (C) 31 (D) 35 (E) none of these
Instructions for 24-25: Each question is followed by 2 statements X and Y. Answer each question using the following instructions
Choose A if the question can be answered using X alone
Choose B if the question can be answered using Y alone
Choose C if the question can be answered using either X or (exclusive) Y
Choose D if the question can be answered using X and Y together
Choose E if the question can not be answered using X and Y also
245. Let f(x) be a polynomial, and p and q be integers. Is f(p)/(p-q) an integer?
(X) Every coefficient in f(x) is an integer
(Y) f(p).f(q) = -(p-q)^2
246. There are insects of three types (A, B and C) in a test tube. Two insects of any two different types can merge into one insect of third type. After several such merges only one insect remains in the test tube. What is its type, if initially there were 3 insects of type A, and at most 4 insects of type B and C each?
(X) There are 2 more insects of type B than type C
(Y) C has at least 2 insects
A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E1, the east end point of OR to N2, the north end point of IR; from N1, the north end point of OR to W2, the west end point of IR; from W1, the west end point of OR, to S2, the south end point of IR; and from S1, the south end point of OR to E2, the east end point of IR. Traffic moves at a constant speed of 30 ? km/hr on the OR road, 20 ? km/hr on the IR road, and 15 5 km/hr on all the chord roads.
247. Amit wants to reach N2 from S1. It would take him 90 minutes if he goes on minor arc S1 – E1 on OR, and then on the chord road E1 – N2. What is the radius of the outer ring road in kms?
[1] 60 [2] 40 [3] 30 [4] 20 [5] none of the foregoing
248. Amit wants to reach E2 from N1 using first the chord N1 – W2 and then the inner ring road. What will be his travel time in minutes on the basis of information given in the above question?
[1] 60 [2] 45 [3] 90 [4] 105 [5] none of the foregoing
249. The ratio of the sum of the lengths of all chord roads to the length of the outer ring road is
[1] 5: 2 [2] 5: 2? [3] 5: ? [5] None of the above.
250. If the base 8 representation of a perfect square is ab3c, where a is non-zero, then c equals
(1) 0 (2) 1 (3) 3 (4) 4 (5) cannot be uniquely determinable
251. Let T be the set of integers {3, 11, 19, 27,…….451, 459, 467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is
[1] 32 [2] 28 [3] 29 [4] 30 [5] None of the above.
252. Let 0 ≤ m ≤ n ≤ k ≤ 9 be three integers such that mn + nk + km = 60. The least possible value of m is
(1) 1 (2) 2 (3) 3 (4) 4 (5) none of these
253. If p, q, r be positive numbers satisfying p + 1/q = 4, q + 1/r = 1, r + 1/p = 7/3, then pqr =
(1) 2/3 (2) 1 (3) 4/3 (4) 2 (5) 7/3
254. The numbers +1 and -1 are positioned at the vertices of a regular 12-gon so that all but one of the vertices are occupied by +1. It is permitted to change the sign of the numbers in any k successive vertices of the 12-gon. It is possible to shift the only -1 to the adjacent vertex if k =
(1) 3 (2) 4 (3) 6 (4) at least two of the foregoing (5) none
255. Bus A leaves the terminus every 20 minutes, it travels a distance 1 km to a circular road of length 10 km and goes clockwise around the road, and then back along the same road to the terminus (a total distance of 12 km). The journey takes 20 minutes and the bus travels at constant speed. Having reached the terminus it immediately repeats the journey. Bus B does the same except that it leaves the terminus 10 minutes after Bus A and travels the opposite way round the circular road. The time taken to pick up or set down passengers is negligible. A man wants to catch a bus a distance 0 < x < 12 km from the terminus (along the route of Bus A). Let f(x) the maximum time his journey can take. The value of x for which f(x) is a maximum is
(1) 3 (2) 5 (3) 8 (4) 10 (5) none
256. For which positive integer values of n the set {1, 2, 3, ..., n} can be split into n disjoint elements subsets {a, b, c, d} such that a = (b+c+d)/3?
(1) 6 (2) 12 (3) 16 (4) 36 (5) at least two of the foregoing
257. Let ABC be a triangle and D and E be internal points on BC and AC respectively. BD/DC = EA/CE = 1/2. If the area of triangle ABC is 2 sq. unit, then area of quadrilateral ODCE (in sq. unit) is
(1) 4/5 (2) 14/15 (3) 16/15 (4) 6/5 (5) none of these
258. Katrina and Deepika have some marbles with each of them, such that the number of marbles with Deepika is thrice that with Katrina. If Katrina distributes her marbles equally among certain number of bags, then she is left with 31 extra marbles. If Katrina and Deepika were to pool the marbles and then distribute the total marbles equally among the same number of bags as Katrina did, they will be left with 16 marbles. The number of marbles with Deepika is the largest possible three digit number. How many bags are needed to equally divide all the marbles with Deepika, if the number of those bags is the smallest possible two digit number?
(1) 11 (2) 13 (3) 29 (4) 31 (5) none of these
259. If all palindromes (positive integers which is unchanged if you reverse the order of its digits) are written in increasing order, what is the possible number of prime values can the difference between successive palindromes take?
(1) 0 (2) 1 (3) 2 (4) 3 (5) none of these
260. Find the product of all the factors of 3^16
(1) 3^33 (2) 3^68 (3) 3^136 (4) 3^128 (5) none of these
261. If 1 ltr. Of water is added to 5 ltrs of a 20% solution of alcohol in water, then the strength of alcohol in water, then the strength of alcohol is
(1) unaltered (2) 5% (3) 12.5% (4) 16.66% (5) none of these
262. Let -2 < x < 3, 0 < y < 4, 2 < z < 5. If (3-x)(4-y)(5-z)(3x+4y+5z) achieves the maximum possible value then which among the following is not true?
(a) 3x+4y = 0 (b) |x| < |y| (c) z = 5/2 (d) two of the foregoing (e) none
263. If a, b, c, d, e, f are six positive real numbers such that a, b, c are in AP, d, e, f are in HP, and ad, be, cf are in GP (with common ratio not equal to 1) then a : b : c is
(a) 1/d : 1/e : 1/f (b) f : e : d (c) d^2 : e^2 : f^2 (d) (e+f) : (d+f) : (d+e) (e) none
264. In a chess tournament each player plays every other player once. A player gets 1 point for a win, ½ point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. The total number of players in the tournament can be
(a) 18 (b) 25 (c) 32 (d) 42 (e) 45
265. A circle C1 of radius x touches other two circles C2 and C3 of radii y and z (both < x), the centres of 3 circles being on the line (C1 being in the middle). If the common tangents of C1, C2 and C1, C3 are perpendicular, then (1+ √(y/x))(1+√(z/x)) =
(a) √2 (b) √3 (c) 2 (d) can not be determined (e) none
266. For a balanced diet Khiladi Akshay Kumar needs to take pulse between 1/2 kg and 1 kg for every 3 kg of rice. Let there be an unequal percent rise in the prices of rice and pulse. Assuming that Akshay doesn't take unbalanced diet and keeps his total amount of consumption of rice and pulse same as earlier, the maximum percent rise in the consumption of rice Akshay can make will be about
(a) 8% (b) 14% (c) 22% (d) can not be determined (e) none
267. In a certain class of 300 students, the number of students who either do not study at home or do not attend classes is a third more than of those who either study at home or attend classes. the number of students who do not study at home but attend classes is two fifths more than those who study at home but do not attend classes, while the number of students who study at home as well as attend classes is half of those who neither study at home nor attend classes. If the number of students who only study at home or only attend classes is a third less than those who do either, then how many students who either do neither or do both?
(1) 120 (2) 150 (3) 180 (4) 210 (5) none of these
268. Yuvraj, Rohit and Mahender each had age (always considered an integer) less than 100, such that sum of the ages of any two of them is same as reverse of third's age. Which among the following can not be true?
(1) The sum of the ages of the three is always 99.
(2) Yuvraj's age, strictly middle in age among the three, can assume 8 values
(3) If Rohit was older than either of the others, the youngest he could be is 45
(4) At least two of the above
(5) none of the above
269. N < 100 Miss Universe contestants of 2010 are standing in a circle and numbered from 1 to N. Starting counting from 1 initially, in succession, every second one is removed from the contest and eliminated, and the last one is declared the winner. If contestant number 25 wins eventually, then how many values can N take?
(1) 3 (2) 1 (3) 2 (4) 0 (4) 4
270. A rectangular block L x 100 x H, with L ≤ 100 ≤ H is cut into two non-empty parts by a plane parallel to one of the faces, so that one of the parts is similar to the original. How many possibilities are there for (L, H)?
(1) 10 (2) 12 (3) 20 (4) 24 (5) none of these
271. ABCDE is a pentagon such that < A = < B = < D = 120˚, and < C = < E. Let BC = 1 and CD = √3. If a circle can be inscribed in the pentagon, then its radius will be
(1) 4 - 2√3 (2) 3 - √3 (3) 3√3 - 4 (4) 4√3 - 6 (5) none of these
272. A car gives the mileage of 60km/L, 50km/L and 40km/L when driven at the speeds of 40km/hr, 50km/hr and 60km/hr respectively. Assume that each car is driven only at the three speeds mentioned above. The car is driven for 3 hours using 2 Litres of petrol. The distance covered by the car
(a) > 120 km (b) < 120 km (c) = 120 km (d) can not be determined (e) none of the foregoing
273. 7 IIMs participate in a B-schools sports meet, where the use of expletives is officially prohibited. Each team from an IIM plays against the other exactly once. What is the possible minimum number of matches that could have been played so far such that among every 3 teams, at least two have played against each other?
(1) 9 (2) 10 (3) 14 (4) 15 (5) none of these
274. In a triangle ABC, AC = 3AB. From C, CD is drawn perpendicular to the bisector of < A. If AD intersects BC at X, then AX/XD is
(1) 1:1 (2) √2: 1 (3) √2 + 1: 2 (4) √3 - 1: 1 (5) √3: √2
275. Gripened comes back to India after earning good bucks from Korea and heads for a shop in Bangalore having expensive shirts. The price of each shirt is a multiple of ten in Rs and is marked in K (in Rs) i.e. multiple of thousand e.g. if the price is Rs 2670, then it's marked as 2.67 K. Gripened buys 4 shirts and goes to the counter for the bill. The counter-boy multiplies the marked price of 4 shirts and hands gripened the bill of 7.11 K. Gripened being smart realizes the mistake and asks for the new bill in which the marked price of each shirt is added. To Gripened's astonishment, the bill this time also comes as 7.11 K. What was the marked price in K of the least expensive shirt that Gripened bought?
(1) 1 (2) 1.2 (3) 1.5 (4) can not be determined (5) none of these
276. Let x, y, z be three distinct integers such that x < y < z and 3x^2 – (y+z)x + 2(y-z)^2 = 0. The minimum positive value of z will be
(1) 2 (2) 3 (3) 5 (4) 8 (5) none of these
277. Let a, b, c be real such that a+b+c ≠ 0. If (a^5+b^5+c^5)/((a^2+b^2+c^2)(a^3+b^3+c^3)) = 5/6, then ab(a+b) + bc(c+a) + ca(c+a) - 3abc equals
(1) 1/2*(a^3 + b^3 + c^3)
(2) (a^2 + b^2 + c^2)^2/(a+b+c)
(3) a^3 + b^3 + c^3
(4) 3*(a^3 + b^3 + c^3)
(5) (a^4 + b^4 + c^4)/(a + b +c)
278. A manufactured cloth piece comes in a fixed length, say L. The piece can be sold
in at most two parts, however it becomes difficult to sell if the cloth is cut.
A shopkeeper sells it with selling price which is directly proportional to the length x of cloth piece bought upto L/2 and after that (x >= L/2) he also charges the cost price of the remaining unsold part. If the selling price of cloth piece bought of length x >= L/2 is directly proportional to (x+L), then for x <= L/2, the profit % on the sale of the cloth is
(1) 50% (2) 75% (3) 100% (4) can not be determined (5) none
279. Five students Implex, Slam, Sanyo, dewan and nbangalorekar are wearing caps of Blue or Green color without knowing the color of his own cap. It is known that the students wearing the Blue cap always speaks the truth while the ones wearing Green always tell lies. If the students make the following statements
Implex: I see 3 blue caps and one Green
Slam: I see 4 Green caps
Sanyo: I see 1 Blue cap and 3 Green
dewan: I see 4 Blue caps
Then, which among the following (Student, Cap Color) combination is correct?
(1) (Implex, Blue) (2) (Slam, Green) (3) (dewan, Blue) (4) at least two of the foregoing (5) none of these
280. If f(x) = x^2 - 2x then for how many distinct real α is f(f(f(f(α)))) = 3?
(1) 3 (2) 6 (3) 5 (4) 9 (5) none of these
281. In triangle ABC, M is the mid-point of BC. If < AMB = 45˚, and < ACM = 30˚, then < BAM is
(1) 30˚ (2) 45˚ (3) < 30˚ (4) > 45˚ (5) none of these
283. Let S be a subset of {1, 2, 3, ... , 15} such that no two subsets of S have the same sum. What is the largest possible sum for S?
(1) 55 (2) 58 (3) 61 (4) 63 (5) none of these
284. Two identical marked dices are brought together and kept with one of their faces in full contact. How many different arrangements are possible?
(1) 36 (2) 60 (3) 72 (4) 84 (5) none of these
284. Let the quadratic ax^2 + bx + c be such that a, b, c are distinct and each of a, b, c belong to {1, 2, 3, ..., n} such that x+1 divides ax^2 + bx + c.
If the number of such quadratic polynomials are < 99, then max (n) is
(1) 14 (2) 15 (3) 16 (4) 18 (5) none of these
285. Yana and Urvashi normally work for 8 hours but do extra four hours on overtime. A job was done by 2 ladies together in 12 days in which Yana and Urvashi did overtime for few days (not necessarily same number of days). It is calculated that had Yana not done her overtime, then the work would have taken 15 days to complete and corresponding value for Urvashi is 20 days. How many days it would have taken for the work to complete had both ladies not done overtime?
(1) 20 (2) 23 (3) 30 (4) can not be determined (5) none of these
286. An Elective on strategy is attended by 15 students sitting on adjacent chairs in a circle. The Prof. enters the class with 6 copies of material on BSC. The prof. wishes to distribute the copies in such a manner that each student gets at most 1 copy and anyone who hasn't should be able to read from his immediate (left or right) neighbour's copy. In how many ways can the copies be distributed?
(1) 60 (2) 75 (3) 90 (4) 125 (5) 150
287. Let x= p/11 satisfies log(2x-3/4)/log x > 2, where p is an integer. Then the number of possible p is
(1) 2 (2) 6 (3) 3 (4) 5 (5) none of these
288. How many natural numbers n are there such that out of all the positive divisors of number n (other than both 1 and n) the largest one is 15 times than the smallest one?
(1) 1 (2) 2 (3) 3 (4) There are no such numbers (5) Infinitely many
289. For how many pair of primes (p, q) does there exist natural number n such that
(p^2+1).(q^2+1) = n^2+1?
(1) 2 (2) 3 (3) 5 (4) 6 (5) none of these
290. The average value of |a - b| + |c - d| + |e - f| for all possible permutations a, b, c , d, e, f of 1, 3, 5, 7, 9, 11 is
(1) 21 (2) 18 (3) 12 (4) 14 (5) none of these
291. A gathering of a certain number of families consists of people belonging to two generations only. It is known that the number of families is less than the number of girls, the number of girls is less than the number of boys and that the
number of boys is less than the number of parents. If the minimum number of single parent families is two, then what is the minimum number of families, given that no family has more than 3 children?
(1) 3 (2) 4 (3) 5 (4) 7 (5) none of these
292. Let f(x + f(x)) = x for all real x, and if f(ax + bf(x)) = cx + df(x), then which
among the following is necessarily true?
(1) b = c (2) b = d+1 (3) a = d (4) at least 2 of the foregoing (5) none
293. How many equilateral triangles of side 2/√3 are formed by the lines y = k, y = x(√3) + 2k, y = -x(√3) + 2k for |k| <= 10 where k is an integer?
(1) 600 (2) 660 (3) 720 (4) 780 (5) none of these
294. A village has n residents, named P(1), P(2), • • • , P(n). Each either tells the truth or lies all the time. For each k: If k is a perfect square, P(k) says that P(k+1) is lying. Otherwise, P(k) says that P(k+1) is telling the truth. (P(n) talks about P1.) What is the minimum number of residents, given that n > 100?
(1) 109 (2) 110 (3) 111 (4) 121 (5) none of these
295. CAT 200X quant had two sections, each with at least one question and 28 questions in total. Each student in a certain class attempted 7 questions. Each pair of questions was attempted by just two students. Which among the following is true?
(1) Each question was attempted by 9 students
(2) One student attempted either nil or at least 4 questions in the first section
(3) There were 36 students in the class
(4) All of the above
(5) Exactly two of the above
296. Let a(1), a(2), a(3), ..., a(n) be a sequence of integers such that -1 <= a(i) <= 2 for all i = 1, 2, 3, ..., n. It is given that a(1) + a(2) + a(3) + ... + a(n) = 19, and (a(1))^2 + (a(2))^2 + (a(3))^2 + ... + (a(n))^2 = 99.
Let m and M be the minimum and maximum possible values of (a(1))^3 + (a(2))^3 + (a(3))^3 + ... + (a(n))^3 respectively, then M/m equals
(1) 3 (2) 4 (3) 7 (4) 9 (5) none of these
297. A point (x, y) is called lattice iff both x and y are integers. How many lattice points
are inside the quadrilateral whose four sides are on the lines x = 100, x = 300, y = x/3 + 0.1 and y = x/3 + 0.6?
(1) 67 (2) 60 (3) 50 (4) 100 (5) none of these
298. I) A committee has met 40 times, with 10 members at every meeting. No two people have met more than once at committee meetings => There are more than 60 people on the committee.
II) One cannot make more than 30 subcommittees of 5 members from a committee of 25 members with no two subcommittees having more than one common member. Which of the above is not true?
(1) only I (2) only II (3) I && II (4) none of the foregoing
299. Let p, q be non-zero integers. Then minimum possible value of |5p^2 +11pq - 5q^2| is
(1) 2 (2) 3 (3) 4 (4) 1 (5) none of these
300. The area enclosed by the graph of |x - 60| + |y| = |x/4| is
(1) 120 (2) 240 (3) 360 (4) 480 (5) none of these
301. Let X be the set of non-negative integers. Let functions f: X -> X such that x f(y) + y f(x) = (x + y) f(x^2 + y^2) for all x, y. Then which among the following is always true?
(1) f(x) >= 0 (2) f(x) = f(1) (3) f(0) = 0 (4) At least 2 of the foregoing (5) none of these)
302. For all positive integers n, let nf(n+1) = n^2 + f(n). If f(1) is a positive integer and f(n) is an integer only for 1 <= n <= 6, then the minimum value of f(1) is
(1) 25 (2) 49 (3) 81 (4) 121 (5) none of these
303. n+2 students played a tournament. Each pair played each other once. A student scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other students all had equal total scores. How many values of n are possible?
(1) 0 (2) 1 (3) 2 (4) more than 3 (5) none of these
Each question is followed by two statements A and B. Answer each question using the following instructions
Choose 1 if the question can be answered by A alone
Choose 2 if the question can be answered by B alone
Choose 3 if the question can be answered by A and B combined
Choose 4 if the question can be answered by either A or B
Choose 5 if the question can be answered by neither A and B
304. A N-sided regular polygon was inscribed in a circle. The consecutive vertices of the polygon were numbered from 1 to N. If vertex number n lies on one end of the circle’s diameter, then the other end of the diameter lies on vertex number 3n+1. What is the value of N?
A) 2N -4 = n2
B) 14 < N < 22
305. A set S = {a, b, c, d} consists of prime numbers between 20 and 100, such that no two elements of S have any digit in common. What is the value of a+b+c+d?
A) The difference of some pairs of elements in S is prime
B) If a > b > c > d, then the number of distinct ways in which the elements of S can be chosen is 14
306. Implex purchased Honda Civic recently, but the faulty car odometer of Civic proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005 (starting with 000000), how many km has Civic actually travelled?
(1) 1284 (2) 1462 (3) 1542 (4) 1604 (5) none of these
307. Manu has a machine into which he can put any number of one rupee coins. If he inserts n rupees, the machine returns 2n rupees. Each time he uses the machine, however, he must insert more money than he did on the previous use. If he starts with exactly Rs 1 and use the machine once, he will have Rs 2. On his next use of the machine, he is forced to insert Rs 2 yielding Rs 4, and on his third use of the machine, he can insert either Rs 3 or Rs 4 yielding a total of Rs 7 or Rs 8. The largest 2 digit integer Z such that it is impossible to obtain exactly Z rupees with the machine, starting with Rs 1 is
(A) 13 (B) 14 (C) 10 (D) 9 (E) none of the foregoing
308. The cost of 6 mangoes is equal to the cost of 3 apples, 4 oranges and 5 peaches. The cost of a peach is between that of an apple and an orange. After 5 mangoes are bought with the price of 6 mangoes, it is found that with the remaining money two of the fruits (both different) can be bought. Which of the two fruits can be bought?
(A) an apple and a peach (B) an apple and an orange (C) an orange and a peach (D) exactly two of the foregoing (E) can not be determined
309. How many pairs of positive integers (a,b) are there such that gcd(a,b)=1 and (a/b)+ (14b/9a) is an integer?
310. Call a set of integers OK if it contains no more than one out of any three consecutive integers. How many subsets of {1,2,3,………….12} including the empty set, are OK?
311. The longest stick that can completely fit into the larger cuboidal box is 24cm long. The biggest sphere that can fit into smaller a smaller cuboidal box has a surface area of 12*pi sq. cm. What is the maximum number of small boxes that can fit into the larger box?
(A) 36 (B) 48 (C) 64 (D) 256 (E) none of the foregoing
312. There is one real root r of 4x^100 – 2x^2 + 3x -1 satisfying 0.15 < r <= 0.65. Which of the following is nearest in value to r?
(A) 0.2 (B) 0.4 (C) 0.5 (D) 0.6 (E) 0.65
313. In the non-decreasing sequence of odd integers (A1, A2, A3)= (1,3,3,3,5,5,5,5,5) each odd positive integer k appears k times. It is a fact that there are integers b, c, and d such that for all positive integers n, An= b[sqrt(n+c)]+d where [x] denotes the largest integer not exceeding x . The sum b+c+d equals
(A) 4 (B) 5 (C) 1 (D) 3 (E) 2
314. The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
315. In a hockey match Germany beat Spain 5-4. Germany scored first and kept the lead until the end. In how many different orders could the goals been scored?
(A) 13 (B) 15 (C) 16 (D) 14 (E) none of these
316. Ten teams of five runners each compete in a cross-country race. A runner finishing in nth place contributes n points to his team, and there are no ties. The team with the lowest score wins. Assuming the first place team does not have the same score as any other team, how many winning scores are possible?
(A) 111 (B) 110 (C) 127 (D) 125 (E) 112
317. An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
318. There are six sticks with lengths 2, 4, 4, 10, 22, 37. How many isosceles trapeziums can be constructed, each time using all six sticks?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
319. Let f(x ,y) be a function satisfying f(x ,y)=f(2x + 2y, 2y -2x) for all x , y. Define g(x)=f(2^x , 0). What is the minimum positive integer p if g(x+p)=g(x) for all x?
(A) 4 (B) 8 (C) 16 (D) 6 (E) 12
320. Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 is written at a distinct corner of a cube. Assume that the sum of any three numbers written on a face of the cube is no less than 10. What is the minimum value of the sum of numbers written on a face of the cube?
(A) 13 (B) 14 (C) 15 (D) 16 (E) none of these
321. A, B and C start running simultaneously from the points P, Q and R respectively on a circular track. The distance between any two of the three points P, Q and R is L and the ratio of the speeds of A, 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 the distance run by A before A, B and C meet for the third time?
(A) A, B and C never meet (B) 9 L (C) 15 L
(D) 10 L (E) none of these
321. What is the largest positive number n such that | √x - 2| < 0.1 whenever |x - 4| < n?
(A) 0.39 (B) 1 (C) 0.41 (D) 0.40 (E) none of these
Each question is followed by 2 statements, X and Y. Answer each question using the following instructions
Choose A if the question can be answered using X alone
Choose B if the question can be answered using Y alone
Choose C if the question can be answered using either X or (exclusive) Y
Choose D if the question can be answered using X and Y together
Choose E if the question can be answered neither using X nor Y
322. ABCD is a rectangle and M and N are the points on AB and BC respectively. AN and DM intersect at P, AN and CM at Q, and CM and DN at R. Area of triangle APM is 3, and area of triangle CRN is 2. What is the area of the region PQRD?
(X) Area of region MBNQ is 20
(Y) Area of triangle APD+ Area of triangle DRC = 18
323. In a triangle ABC, D is a point on the side BC and X, Y are length of perpendicular dropped on line AD from the vertices B and C respectively. Is X > Y?
(X) AB > AC
(Y) BD < DC
324. The number of integral k such that log2 + log(2x^2 + 2x + 7/2) >= log(kx^2 + k), possesses at least one real solution is
(a) 3 (b) 4 (c) 7 (d) 8 (e) none of these
325. The 150 Quant devils of QQAD are given individual numbers from 1 to 150, and a contest happens in multiple rounds to select the Ultimate QQAD devil. The elimination follows a weirdo pattern. In the 1st round starting from first devil, every 3rd devil is eliminated i.e. 1st, 4th, 7th, .... This repeats again from the first numbered (among the remaining) devil in the next round (leaving 3, 5, 8, 9, ...). This process is carried out repeatedly until there is only the winner left. What is the number of the Ultimate QQAD Devil?
(a) 93 (b) 48 (c) 119 (d) 38 (e) 140
326. Let X = {1, 2, 3, 4}. How many functions f: X->X are there satisfying (f.f)(x) = x for all x in X?
(1) 13 (2) 15 (3) 14 (4) 16 (5) none of these
327. Akshita is standing at the midpoint of the points where her two other friends Simran and Aahna are standing. They start moving simultaneously with constant speeds along the line on which they are standing and they pass through a common point X after 10, 30 and 40 minutes. If they all meet simultaneously after time T from the start, then T in minutes is
(1) 60 (2) 70 (3) 90 (4) 100 (5) none of these
328. Let [x] denotes the greatest integer that is less than or equal to x, e.g. [5.43] = 5.Let x be any number selected randomly such that [√x] = 10. The probability that [√(100x)] = 100 is
(1) 1/7 (2) 1 (3) 67/700 (4) 1/2 (5) none of the foregoing
329. Which one of the following is the remainder when x + x^7 + x^16 + x^37 is divided by x^4 - x?
(1) 4x (2) 2x(x-1) (3) x^2 + 2x (4) x(x-1) (5) none of these
330. Sara, Kyna, Riddhi can complete the work W1, W2, W3 alone in 6, 9 and 15 days respectively. If (Kyna, Riddhi), (Riddhi, Sara) and (Sara, Kyna) can do the work W1, W2, W3 respectively in n days each, then n lies in
(1) (3, 3.5) (2) (3.5, 4) (3) (4, 4.5) (4) (4.5, 5) (5) either none or atleast 2 of these
331. Four articles are sold at Rs 50, 60, 70 and 80 fetching profits 20%, 25%, 40% and 50% (may not be in that order). Given that the total cost price of the articles is an integral number of rupees, the net profit % on the articles together is
(a) 36% (b) 34% (c) 32% (d) 31% (e) none of these
332. Let three positive integers p, p^2 + 2, p^3 + 2 be given. Which among the following is always true?
(1) All the three numbers are prime for atleast 2 values of p
(2) Exactly 2 of these numbers are perfect squares for some p
(3) The product of 2 of these numbers can be expressed as a 6 digit number in the base of the third number
(4) atleast 2 of the foregoing
(5) none of these
333. There are two squares of side length unity .They are arranged in a way that their centers coincide .These two squares have their inside area as an octagon whose one side is 43/99. The area of the octagon is
(a) 1 (b) 7/9 (c) 78/99 (d) 86/99 (e) none of these
334. Consider the following system of equations:
r + x = l ; l + p = n ; n + r = k ; r = 8 ; x + p + k = 30
The value k is
(1) 17 (2) 22 (3) 23 (4) 15 (5) 11=>
335. Five former beauty contest winners Sushmita, Aishwarya, Diana, Lara, Priyanka were placed 1 to 5 in a contest with no ties. One prediction was that the result would be the order Sushmita, Aishwarya, Diana, Lara, Priyanka. But no contestant finished in the position predicted and no two contestants predicted to finish consecutively did so. For example, the outcome for Diana and Lara was not 1, 2 (respectively), or 2, 3, or 3, 4 or 4, 5. Another prediction was the order Lara, Sushmita, Priyanka, Diana, Aishwarya. Exactly two contestants finished in the places predicted and two disjoint pairs predicted to finish consecutively did so. Who finished immediately next after Priyanka?
(1) Sushmita (2) Aishwarya (3) Diana (4) Lara (5) Impossible to determine
336.Let [x] denotes the greatest integer that is less than or equal to x, e.g. [5.43] = 5.Let x be any number selected randomly such that [√x] = 10. The probability that [√(100x)] = 100 is
(1) 1/7 (2) 1 (3) 67/700 (4) 1/2 (5) none of the foregoing
337.The 150 Quant devils of QQAD are given individual numbers from 1 to 150, and a contest happens in multiple rounds to select the Ultimate QQAD devil. The elimination follows a weirdo pattern. In the 1st round starting from first devil, every 3rd devil is eliminated i.e. 1st, 4th, 7th, .... This repeats again from the first numbered (among the remaining) devil in the next round (leaving 3, 5, 8, 9, ...). This process is carried out repeatedly until there is only the winner left. What is the number of the Ultimate QQAD Devil?
(a) 93 (b) 48 (c) 119 (d) 38 (e) 140
338.Let a fruitseller sells an apple, an orange, a mango and a peach at loss of 10%, loss of 40%, profit of 20% and profit of 50% respectively. Let the loss made on an apple and an orange together be as much as in amount the profit made on a mango and a peach together. Suppose you have the money to buy 2 apples and 12 oranges but instead you buy 3 mangoes. How many peaches can you buy with rest of the money?
(1) 3 (2) 5 (3) 6 (4) 8 (5) none of these
Short cu:
Let the initial costs of Apple|Orange|Mango|Peach be 10|10|10|6
The costs at which he sells Appl|orange|Mango|Peach = 9|6|12|9
He has money to buy 2 apples and 12 oranges = 2*9 + 12 *6 = 90
He buys 3 mangoes instead : 12 * 3 = 36 - Left with 54 Rs
He can buy 54/9 = 6 Peaches with that amount !!
Conventional
Apple - CP = a , SP = 0.9a
Orange - CP = r, SP = 0.6r
Mango - CP = m, SP = 1.2m
Peach - CP = p, SP = 1.5p
0.1a + 0.4r = 0.2m + 1.5p => a+4r = 2m + 5p --(1)
Money left
= 2(0.9a) + 12(0.6r) - 3(1.2m)
= 1.8(a+4r) - 3.6m
= 1.8(2m +5p) - 3.6m = 9p = 6 (1.5p) => 6 peaches.
=> Choice (3) is the right answer
339. A positive integer p is called "CAT number" if p^3 + 7p = q^3 + 133 for some positive integer q. The sum of all such "CAT numbers" is
(a) 24 (b) 26 (c) 30 (d) 34 (e) 38
Solution:
We consider the size of p relative to q. If p = q, then q is a CAT number if and only if
7q = 133. In this case, we get 19 is a good number. Now, suppose that p >= q + 1. Then
q^3 + 7q − 133 = p^3 >= (q + 1)^3 = q^3 + 3q^2 + 3q + 1
so that 3q^2 − 4q + 134 <= 0. For positive integers q, it is easy to see that this is impossible. It remains to consider p <= q − 1. For such p, we have q^3 + 7q − 133 = p^3 <= (q − 1)^3 = q^3 − 3q^2 + 3q − 1 which implies 3q^2 + 4q − 132 < 0. It follows here that q < 6. One checks that 6^3 + 7 • 6 − 133 = 5^3 and 5^3 + 7 • 5 − 133 = 3^3,
so 6 and 5 are CAT numbers. For q <= 4, we see that q^3 +7q−133 < 0 and, hence, q^3 +7q−133 cannot equal p^3 for a positive integer p. Therefore, the sum of all CAT numbers is 19+6+5 = 30.
=> Choice (c) is the right answer
340.Let n be such that the number of integers less than or equal to n and divisible by 3 equals the number divisible by 5 or 7 (or both). Then the largest n is divisible by
(a) 3 (b) 5 (c) 7 (d) all of the foregoing (e) none of the foregoing
Solution:
Let f(n) = [n/3] - [n/5] - [n/7] + [n/35]. We are looking for the largest n with f(n) = 0.
Now [n/5] + [n/7} <= [n/5 + n/7] = [12n/35] = [n/3 + n/105]. So for [n/5] + [n/7] to exceed [n/3]
we certainly require n/105 = 1/3 or n = 35. Hence f(n) = 0 for n = 35. But f(n+35) = [n/3 + 11 + 2/3] - [n/5 + 7] -
[n/7 + 5] + [n/35 + 1] = [n/3 + 2/3] - [n/5] - [n/7] + [n/35] = f(n) (*). Hence f(n) = 0 for all n.
But f(n+105) = [n/3 + 35] - [n/5 + 21] - [n/7 + 15] + [n/35 + 3] = f(n) + 2. Hence f(n) = 2 for all n = 105.
Referring back to (*) we see that f(n+35) > f(n), and hence f(n+35) > 0, unless n is a multiple of 3.
But if n is a multiple of 3, then n + 35 is not and hence f(n+70) > f(n+35) > 0. So f(n) > 0 for all n = 70.
f(70) = 1. So f(69) = 2 (we have lost 70, a multiple of 7). So f(68) = f(67) = f(66) = 1 (we have lost 69, a multiple of 3).
Hence f(65) = 0 (we have lost 66, a multiple of 3).
=> Choice (b) is the right answer.
341.The quadratic polynomial f(x) has the following properties: f(x) >= 0 for all real numbers x, f(1) = 0 and f(2) = 2. What is the value of f(0) + f(4)?
(a) 12 (b) 16 (c) 20 (d) 22 (e) none of these
Solution:
Given that x = 1 is a root of f(x) = 0 and f(x) >= 0 for all x => f(x) = a*(x-1)^2, putting f(2) = 2, we get a = 2.
=> Choice (c) is the right answer
342. The system of linear equations x+y+z = 6, x+2y+3z = 14 and 2x+5y+pz = q is consistent if
(1) p = 8, q = 36 (2) q ≠ 36 (3) p = 8, q ≠ 36 (4) p ≠ 8 (5) more than one of the foregoing
Solution:
The system of linear equations is consistent if the determinent of the 3X3 matrix formed by the coefficients of x, y, z is non-zero. Solving we get p ≠ 8.
If p = 8 we get x = z-2 and y=8-2z. Putting this in the last equation we get q = 36. In this case we have infinite solutions, generated by any choice of z. OR we can see this as 3*(eq. 2) - eq. 1 = 2x + 5y + 8z = 36 = 2x + 5y + pz, which is generates infinite solutions for p =8, q = 36 as the 3rd equation can be expressed as linear representation of 1st two equations.
=> Choice (5) is the right answer
343. One day Vikram was out bicycling. After entering a one-way tunnel and after having ridden one-fourth of the distance through it, he looked back over his shoulder and saw a bus approaching the tunnel entrance at a speed of 80 miles/hr. Doing a quick mental exercise, Vikram realized that if he accelerated immediately to his top speed, he could just escape with his life, whichever direction he rode. What is Vikram's top biking speed in miles/hr?
a) 32 (b) 36 (c) 40 (d) 48 (e) none of the foregoing
Solution:
Let d=distance truck is in front of tunnel entrance, L=length of tunnel, x=Vikram's speed.
Case 1: Vikram turns around and heads for entrance, a distance of L/4.
Vikram and truck get to entrance at same time T1=d/80=(L/4)/x.
Case 2: Vikram streaks for exit, a distance of 3L/4. Vikram and truck get to exit at same time
T2=(L+d)/80=(3L/4)/x.
Solving both equations for x and setting them equal, x=(L/4)*80/d=(3L/4)*80/(L+d)
After simplifying, d=L/2, hence x=40.
=> Choice (c) is the right answer
344. Two footballs are coming together along two perpendicular lines with same speed but they are at 10 m and 20 m away from the point of intersection of the lines along which they are moving. At this instant, one of the football is deflected perpendicular to its direction of motion with speed 5m/s (its net motion being the superposition of the motion in two directions) and they collide. The time of collision (in seconds) of the footballs after one of them is deflected is
a) 2 (b) 2√2 (c) 4 (d) 5 (e) none of the foregoing
Solution:
Let s be the speed of two balls.
now distance to be covered in
vertical direction = 10 m
horizontal = 20 m
ball at 10m away given a speed of 5m/s in horizontal direction.
so relative speeds of approachin in
vertical direction = s m/s
horizontal = (s+5) m/s
time taken => 10/s = 20/ (s+5)
s =5
so time taken = 10/s = 20/ (s+5) = 2 sec
=> Choice (a) is the right answer
345. A real number x is chosen at random in the interval [-21/2, 21/2]. What is the probability that the closest integer to x is odd?
a) 1/21 (b) 5/11 (c) 10/21 (d) 11/21 (e) 1/2
Solution:
From [-10.5,10.5] There can be 21 intervals [-10.5,-9.5],[-9.5,-8.5].....[9.5,10.5]..and there are 10 intervals [-9.5,-8.5]....[8.5,9.5]..in which any x selected in those intervals is closest to an odd integer.So the probability is 10/21
=> Choice (c) is the right answer
346. Slam and dewan are playing a game with the following rules: If one of them says an integer n, the opponent then says an integer of their choice between 2n and 3n, inclusive. Whoever first says 2007 or greater loses the game, and their opponent wins. Dewan must begin the game by saying a positive integer less than 10. With how many of them can Dewan guarantee a win?
a) 2 (b) 3 (c) 4 (d) 5 (e) 6
347.Given a triangle area of area 6 and perimeter 12, let S be the set of all points a distance 5 or less from a point of the triangle. What is the area of S to the nearest integer?
(1) 126 (2) 134 (3) 145 (4) 157 (5) 168
Solution:
The region consists of 3 rectangles outside the triangle who one side is 5 and the other side as the length of the side of the triangle. Also the region consists of 3 sectors of radius 5 each around the three vertices.
Let ABC be the triangle. Let the rectangles be ABEF, ACFG and CBJI. Then
=> Choice (3) is the right answer
348. Three vessels A, B and C have different concentrations of alcohol. If the contents of pairs of vessels (A, B), (B, C) and (C, A) are mixed, then the concentrations of alcohol become 30%, 40% and 50% respectively. If the concentration of alcohol in vessel A in % is 30 less than twice the concentration in % in vessel B, then the concentration of vessel C is
(1) 42% (2) 45% (3) 48% (4) can not be determined (5) none of these
Solution:
Let L1, L2, L3 be the amount of alcohol in vessels A, B and C while n1%, n2%, n3%be their respective concentration. Given that n1 = 2n2 - 30. Also, (n1L2+n2L1)/(L1+L2) = 30 => L1/L2 = (30-n2)/(n1-30).
Similarly, L2/L3 = (40-n3)/(n2-40) and L3/L1 = (50-n1)/(n3-50).
solving we get n3 = 45% but that gives n1 > 50 and n2 > 40 which is inconsistent with the given data.
Hence, choice (4) is the right option
349. classroom consists of a 5 X 5 array of desks, to be filled by anywhere from 0 to 25
students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
(a) 624 (b) 625 (c) 961 (d) 962 (e) none of these
Solution:
The set of empty desks must be of the form (non-full rows) £ (non-full columns): each empty desk is in a non-full column and a non-full row, and the given condition implies that each desk in such a position is empty. So if there are fewer than 25 students, then both of these sets are nonempty; we have 2^5-1 = 31 possible sets of non-full rows, and 31 sets of non-full columns, for 961 possible arrangements. Alternatively, there may be 25 students, and then only 1 arrangement is possible. Thus there are 962 possibilities altogether.
=> Choice (d) is the right answer
350. The system of linear equations x+y+z = 6, x+2y+3z = 14 and 2x+5y+pz = q is consistent if
(1) p = 8, q = 36 (2) q ≠ 36 (3) p = 8, q ≠ 36 (4) p ≠ 8 (5) more than one of the foregoing
Solution:
The system of linear equations is consistent if the determinent of the 3X3 matrix formed by the coefficients of x, y, z is non-zero. Solving we get p ≠ 8.
If p = 8 we get x = z-2 and y=8-2z. Putting this in the last equation we get q = 36. In this case we have infinite solutions, generated by any choice of z. OR we can see this as 3*(eq. 2) - eq. 1 = 2x + 5y + 8z = 36 = 2x + 5y + pz, which is generates infinite solutions for p =8, q = 36 as the 3rd equation can be expressed as linear representation of 1st two equations.
=> Choice (5) is the right answer
351. There are z digits in the decimal expression of the natural number N,while there are y digits in the decimal expression of N^3. Then which of the following cannot be equal to y+z?
(a) 20 (b) 26 (c) 35 (d) 45 (e) none of these
Solution:
As per the question,
10^(z−1) <= N < 10^z =>10^(3z−3)<= N^3 < 10^3z.
So, y can be {3z − 2, 3z − 1, 3z} and y+z can be {4z-2 ,4z-1,4z)
Clearly all the natural numbers except those of type 4k+1 satisfy the value for y+z.
So, 35 cannot be the value.
=> Choice (c) is the right answer
352. Let S be the set {1, 2, 3, . . . , n} consisting of the first n positive integers.
What is the maximum value of n for which every 100-element subset of
S contains two integers which differ by 25 ?
(a) 149 (b) 150 (c) 174 (d) 175 (e) none of these
353. A wooden unit cube rests on a horizontal surface. A point light source a distance x above an upper vertex casts a shadow of the cube on the surface. The area of the shadow (excluding the part under the cube) is 35.
Let ABCD be a square. P, Q, R, S are on AB, BC, CD, DA extended respectively. A, B, C, D are the midpoints of DS, AP, BQ, CR respectively. Area(ABCD)/area(PQRS) = y.
Which among the following is true about x and y?
(a) x > y (b) 1/x = y (c) x < 1/y (d) x = y (e) none of these
Solution:
By similar triangles the total shadow (including the part under the cube) is a square with side (1+x)/x. So (1+x)2/x2 = 36, x = 1/5.
Plot the square ABCD on the x-y axis such that A(0, 0), B(1, 0), C(1, 1), D(0, 1). We get P(2, 0), Q(1, 2), R(-1, 1) and S(0, -1). Area(PQRS) = 5 => y = 1/5.
=> Choice (d) is the right answer
354.The number of integral k such that log2 + log(2x^2 + 2x + 7/2) >= log(kx^2 + k), possesses at least one real solution is
(a) 3 (b) 4 (c) 7 (d) 8 (e) none of these
Solution:
log (4x^2 + 4x + 7) >= log(kx^2 + k)
=> for 4x^2 + 4x + 7 >= kx^2 + k, we must have atleast one real solution for k > 0.
=> (k-4)x^2 -4x + (k-7) <= 0
Multiply both the sides by (k-4).
=> LHS of the inequality becomes ((k-4)x - 2)^2 + (k-8 )(k-3).
For k >=4 we have ((k-4)x - 2)^2 + (k-8 )(k-3) <= 0 ... we have k = 4, 5, 6, 7, 8
For k < 4 we have ((k-4)x - 2)^2 + (k-8 )(k-3) >= 0 ... we have k = 1, 2, 3
=> Choice (4) is the right answer
355. QQAD team decides to go on vacation for 8 days trusting their NL software system. The problems are fed into the system and date timer set in advance for each 8 days for the questions to be delivered to the subscribers daily. However, the software follows a weird rule. It doesn't always deliver the NL daily in those 8 days, but never misses 3 consecutive deliveries. How many possible ways are there for the NLs delivery in those 8 days?
(a) 162 (b) 138 (c) 117 (d) 176 (e) 149
On each day NL can take 2 states - it either delivers or it doesn't. Let f(n) be the possible ways for n days with given conditions.
=> f(1) = 2, f(2) = 2^2, f(3) = 2^3-1 (1 subtracted as we can't have 3 misses). Let n > 3. Then, NL either delivers on day 1 or it doesn't. When it does it can have f(n-1) ways from there; on day 2 also it either delivers or it doesn't, when it does it can have f(n-2) ways from there. Now on day 3, it has to deliver as we have missed first 2 days, thus after delivering on day 3 it can have f(n-3) ways => f(n) = f(n-1)+f(n-2)+f(n-3) => f(4) = 13, f(5) = 24, f(6) = 44, f(7) = 81 and f(8 ) = 149
Hence, choice (e) is the right option
356. Four dices(6-faced) are thrown. How many different arrangements can one get if the order is unimportant(e.g. 2356 is same as 2635 or 1124 is same as 4211)?
(a) 96 (b) 108 (c) 126 (d) 156 (e) none of these
Solution:
No. of combinations with 4 Alike (Identical) Results = 6C1 = 6
No. of combinations with 3 Alike and 1 Non Alike = 6C1*5C1 = 30
No. of combinations with 2 Alike and 2 Non Alike = 6C1*5C2 = 60
No. of combinations with 2 Alike of 1 Kind and 2 Alike of another kind i.e.1122, 2233 etc = 6C2 = 15
No. of combinations with all 4 different = 6C4 = 15
In general if we have N dices which are r-faced, distinct enumerations such as what we want in our question would be (N+r-1)C(r-1).
=> Choice (C) is the right answer
357. A car travels downhill at 72 kmph (kilometers per hour), on the level at 63 kmph, and uphill at only 56 kmph The car takes 4 hours to travel from town A to town B. The return trip takes 40 minutes more. What is the distance between the two towns in kilometers?
(1) can not be determined (2) 191 (3) 255 (4) 273 (5) none of these
Let the total distance travelled downhill, on the level, and uphill, on the outbound journey, be x, y, and z, respectively. The time taken to travel a distance s at speed v is s/v.
Hence, for the outbound journey x/72 + y/63 + z/56 = 4
While for the return journey, which we assume to be along the same roads x/56 + y/63 + z/72 = 14/3
Multiplying both equations by the least common multiple of denominators 56, 63, and 72, we obtain
7x + 8y + 9z = 4 • 7 • 8 • 9
9x + 8y + 7z = (14/3) • 7 • 8 • 9
Now it is clear that we should add the equations, yielding
16(x + y + z) = (26/3) • 7 • 8 • 9
Therefore x + y + z = 273; the distance between the two towns is 273 kms.
=> Choice (4) is the right answer
358. 100 numbers are written around a circle. The sum of every 8 consecutive numbers is 25/2. The 9th number is -1/2, the 19th number is 3/4 and 20th number is 2. What is the 50th number?
(a) 4 (b) 5 (c) 1 d) 3(e) none of these
Solution:
Let the numbers be a(1), a(2), a(3), ..., a(100), a(101), a(102), ... where a(101) = a(1) and a(102) = a(2).
a(n) + a(n+1) + ... + a(n+7) = a(n+1) + a(n+2) + ... + a(n+8) => a(n) = a(n+8)
=> a(1) = a(9) = a(105) = a(5) => a(n) = a(n+4)
=> 2*(a(1) + a(2) + a(3) + a(4)) = 25/2 => a(50) = a(2) = 4
=> Choice (a) is the right answer
359. Triangle ABC is right-angled at A. D is a point on AB such that CD = 1. AE is the altitude from A to BC. If BD = BE = 1, what is the length of AD?
(1) 2^1/3 - 1 (2) (5^1/2 - 1)/2 (3) (5^1/2 + 1)/4 (4) 2^1/2 - 1 (5) none of these
Solution:
Let AD = x, CE = y, and < ABC = t. Let AE and CD meet at F.
Since BCD is isosceles, < BCD = t.
Hence < CFE = 90° − t, and so < DFA= 90° − t.
Since also < FAD = < EAB = 90° − t, DFA is isosceles, and so DF = AD = x.
Hence CF = 1 − x.
Triangles ABE and CFE are similar, as each contains a right angle, and < ABC = < ECF.Hence y/(1 − x) = 1/(1 + x), and so
y = (1 − x)/(1 + x)(1)
Triangles ABC and ABE are similar, as each contains a right angle, and < ABC = < ABE.
Hence (1 + x)/(1 + y) = 1/(1 + x), and so (1 + x)^2 = 1 + y.
Substituting for y from (1), we obtain (1 + x)^2 = 1 + (1 − x)/(1 + x) = 2/(1 + x).
Hence (1 + x)^3 = 2.
=> Choice (1) is the right answer
360. Mid-points of each side of a cube are marked and then all the vertices are chipped off by planes passing through the mid-points of the sides passing through the vertices. The ratio of the volume of the new figure to that of original cube is
(a) 1/2 (b) 2/3 (c) 3/4 (d) 5/6 (e) none of these
Solution:
The part that is cut out is a pyramid with base as an equilateral triangle of side x*rt(2), where (2x) is the side of the cube.
Now, we want the volume of this pyramid = 1/3 * (base area) * (height)
base area = rt(3)/4 * (x*rt(2) ) ^2
For height,
the vertex of the pyramid is right above the centroid of the base => vertex(A), centroid(B) and (line perpendicular from centroid to side of base (BC)) form a right angled triangle.
Now,
BC = (1/3) * rt(3)/2 * x rt(2) = x/rt(6)
CA = (x * x )/ (x rt(2) ) = x / rt(2)
=> AB^2 = (x^2)/3
=> AB = x/ rt(3) = height.
=> volume of tetrahedron = (1/3) * rt(3)/4 * (2x^2) * x / rt(3) = (1/6) x^3
=> ratio = 1 - (volume * 8 / 8x^3)
= 5/6
=> Choice (d) is the right answer
361.At Pizza-Hut pizzas are made only on an automatic pizza-making machine. The machine continually makes different sorts of pizzas by adding different sorts of toppings on a common base. The machine makes the pizzas at the rate of 1 pizza per minute. The various toppings are added to the pizza in the following manner. Starting from every pizza, every fifth pizza is topped with pepperoni, every seventh with olive and baby corn, every eigth with mushroom, and the rest with chesse and tomatoes. The machine works for 13 hours per day without any breaks in between. How many pizzas per day are made with cheese and tomatoes as topping?
(1) 418 (2) 438 (3) 458 (4) 468 (5) 498
Solution:
In 13 hours exactly 780 pizzas are made out of which we need all those numbers which are neither multiples of 5 nor 7 nor 8.
A U B U C = A + B + C - A/B - A/C - B/C + A/B/C
Multiples of 5 are 780/5 = 156
Multiples of 7 are [780/7] = 111
Multiples of 8 are [780/5] = 97
Multiples of 5 and 7 are [780/35] = 22
Multiples of 5 and 8 are [780/40] = 19
Multiples of 7 and 8 are [780/56] = 13
Multiples of 5, 7 and 8 are [780/270] = 2
Since, numbers are counted from 1st pizza
=> A U B U C = 156 + 112 + 98 - 23 - 20 - 14 + 3 = 312
=> Choice (4) is the right answer
362. The year is 2008 and a new system(base n > 2) is used in place of decimal for making calculations. Priyanka, one day receives twice the number of questions in "Quant questions a week" b'coz the digits of the original number of questions which was a two digit number got reversed. Let x be the least possible value of n and y is the next possible value of n. The 4 digit number abab is a perfect cube in base z. Let z be the smallest possible such base. Then y+z equals
(a) 11 (b) 15 (c) 19 (d) 21 (e) none of the foregoing
Solution:
Let the original number in base n Priyanka was to receive is pq. => 2*(pn+q) = qn + p. => p/q = (n-2)/(2n-1) => x = 5 and y = 8.
The 4 digit number abab in base z when converted to decimal system is az^3 + bz^2 + az + b = (az+b)*(b^2+1) is a perfect cube. Checking for values z = 2, 3, ..., we see that 2626 is 1000 in base z = 7.
=> Choice (b) is the right answer
363. Let f(x) be a polynomial of degree 4, such that f(2) = f(-2) = f(-3) = -1, and f(1) = f(-1) = 1, then f(0) is
(a) 2 (b) 3 (c) 4 (d) 5 (e) 6
364. (A) S: Implex found 3 different solutions to the equation f(x - 2/x) = 0 => I: He could have definitely found 1 more different solution.
(B) S: Sanyo decided to solve a problem where a, b, c are non-zero real numbers such that a^2 + b^2 + c^2 = 1 and a(1/b + 1/c) + b(1/c + 1/a) + c(1/a + 1/b) = -3 => I: Exactly 3 distinct values of a+b+c exist.
(C) S: By giving one rubber free with 4 pencils, it means that a discount of 10% is given on the sale of pencils. Then by giving 1 pencil free with 6 rubbers, it means that a discount of x% is given on the sale of rubbers. => I: x is approx 27
(1) only A (2) only C (3) B and A (4) C and B (5) none of these
Solution:
(A) If x - 2/x = a => x^2 - ax - 2 = 0 => D = a^2 + 8 => the equation has unequal roots for the same value of a => the number of roots of f(x - 2/x) if finite are always even.
(B) If a+b+c is x and ab+ac+bc = y then x^2 = 1 + 2y and xy = 0
=> x = 0, 1, -1
(C) is easy to see that x = 41%
Choice (2) is the right answer
365. The number of roots of the equation x + | x^2 - 1 | = k, where k is a real constant is/are
(1) 4 for k = 6/5 (2) 3 for k = 5/4 (3) 2 for -1 < k < 1 (4) All of the foregoing (5) Exactly two of the foregoing
Solution:
Plot the graph of f(x) = x + | x^2 - 1 | by taking cases when X > 1, when X < -1 and when x lies in [-1, 1], the graph will be W shaped.
It is clear from the graph that there are no roots for k < -1, and one root for k = -1 (namely x = -1).
Then for k > -1 there are two roots except for a small interval [1, 1+h]. At k = 1, there are 3 roots (x = -2, 0, 1).
The upper bound is at the local maximum between 0 and 1. For such x, y = x + 1 - x^2 = 5/4 - (x - 1/2)^2, so the local maximum is at 5/4. Thus there are 3 roots at k = 5/4 and 4 roots for k belonging to (1, 5/4).
=> Choice (4) is the right answer
366. Each of the 10 digits from 0 to 9 are used exactly once altogether to form 3 positive integers. One of these numbers is the sum of the other two. What is the difference between the largest possible and the smallest possible of the largest of these 3 numbers?
(1) 4995 (2) 5775 (3) 6858 (4) 7632 (5) none of these
Solution:
Since the sum has at least as many digits as either of the summands, the sum must have at exactly 4 digits.
A 4-digit sum can arise either as the sum of two three-digit numbers or as the sum of a 4-digit and a 2-digit number. In the former case, the sum must exceed 1000 and be less than 2000 and, in the latter case, it must be at least 2000.
Thus, the smallest possible sum must be obtained by adding two three-digit numbers to get a 4-digit sum. Since, the digits of the sum are all distinct, the smallest sum is atleast 1023. Since, 589+437 = 1026, the smallest sum is atmost 1026. We may assume that each digit in the first summand exceeds the corresponding digit in the second summand. The only possibilities for a lower sum are 5pq + 4rs = 1023, 6pq + 3rs = 1024, 6pq + 3rs = 1025. None of these work.
For the largest sum, let the first summand have 4-digits and the second two. The hundereds digit of the first summand is 9 and the thousands digit of the sum exceeds the thousand digit of the first summand by 1. Since 5987 + 34 = 6021, the largest sum is atleast 6021. The only possibility to consider for a larger sum are 79ab + cd = 80ef, 69ab + cd = 70ef, 59ab + cd = 60ef. None of these work.
Thus, max is 6021 and the min is 1026.
=> Choice (1) is the right answer
367. A test has exactly 10 questions and is either answered in True or False. If Deepika answers 5 questions "true" and five "false", her score is guaranteed to be at least 4. How many answer keys are there for which this is true?
(1) 12 (2) 13 (3) 16 (4) 19 (5) 22
Solution:
Suppose that either 9 or 10 of the questions have the same answer. Then no matter which 5 questions Deepika picks to have this answer, she will be right at least 4 times. Conversely, suppose that there are at least two questions with each answer; we will show that she can get a score less than 4. By symmetry, assume there are at least five questions who answer is true. Then if we label five of these false, not only will Deepika get these 5 wrong, but she will also have answered all the false questions with true, for a total of at least 7 incorrect. There are 2 ways for all the questions to have the same answer and 20 ways for one question to have a different answer from the others, for a total of 22 ways.
=> choice (5) is the right answer
368. A non-empty set of elements G is said to form a Group if in G there is defined a binary operation * such that (€ denotes belonging to)
(1) a, b € G implies a*b € G
(2) a, b, c € G implies a*(b*c) = (a*b)*c
(3) There exists an element e € G such that a*e = e*a = a for all a € G
(4) For every a € G, there exists an inverse element b € G, such that a*b = e
Also, the group is called commutative iff a*b = b*a for all a, b € G.
Which among the following is not true?
(1) G = set of all rational numbers with odd denominator such that for every a, b € G, a*b = a+b, forms a group
(2) Let G be a group such that (a*b)*(a*b) = (a*a)*(b*b), for all a, b € G then G is commutative.
(3) If G is a group such that every element is its own inverse then G is commutative
(4) Atleast 2 of the above
(5) None of the above
Solution:
(1) Let 3 rationals be such that these are, a/b, c/d, e/f where b, d, f are odd then the sum of any 2 elements in G will also yield a rational whose denominator is odd. Property (2) is easy to see ... e = 0 here, and the inverse of any element x in G is -x. Thus, G satisfies all the properties of a Group
(2) Let x I denote the inverse element of x. Left inverse both the LHS and RHS with a I , and right inverse both the LHS and RHS with b I, we get a*b = b*a.
(3) (a*b) I * (a*b) = e from (3)
=> (a*b ) I * a*b*b I = e* b I from (2)
=> (a*b ) I * a = b I
=> (a*b ) I * a * a I = b I * a I
=> (a*b ) I = b I * a I
Given that for all x in G, x = x I
=> a * b = a I * b I = (b * a) I = b * a
=> Choice (5) is the right answer
369. In the quadrilateral ABCD, AD = DC = CB, and < ADC = 100˚, < ABC = 130˚. Then the measure of < ACB (is)
(1) 20˚ (2) 30˚ (3) 50˚ (4) can not be determined (5) none of these
Solution:
A, B, C will lie on a circle with centre at D (as the angle subtended by the arc at the centre i.e. 260˚ is twice subtended at the circle i.e. 130˚ ) => in triangle DAC, < DAC = < DCA = 40˚.
Let < ADB = 2x => < ACB = x, and let < BDC = < CDB = y => 2x+y = 100˚, and 2y+x = 140=> Choice (1) is the right answer
370. The number of positive integral solutions to x + y/2 + z/3 = 7(1 + 1/2 + 1/3) is
(a) 66 (b) 72 (c) 78 (d) 84 (e) none of these
Solution:
The equation can be re-written as (x-1) + (y-1)/2 + (z-1)/3 = 11. Putting x-1 = X, (y-1)/2 = Y, and (z-1)/3 = Z, we need non-negative solutions to X + Y + Z = 11 which is (11+3-1)C2.
=> Choice (c) is the right answer
371. A particle moves inside a square with vertices (0, 0), (1, 0), (1,1) and (0,1). It begins at the point (2/3, 0), travels to some point on the edge BC, then travels to some point on the edge CD, then travels to some point on the edge DA, and then travels to (1/6, 0). The minimum distance that the particle could have travelled in such a journey is
(a) √6 (b) 5/2 (c) 8/3 (d) 3 (e) none of these
Solution:
Let the points on BC be (1, p), on CD be (q, 1) and DA be (0, r). The distance the particle covers before it comes to stop at (1/6, 0) is √(1/9+p^2) + √(q-1)^2+(p-1)^2 + √q^2+(1-r)^2 + √1/36+r^2 = F(p, q, r), where 0 < p, q, r < 1. Differentiating F(p, q, r) each wrt to p, q, r and solving we get p = 4/9, q = 7/12 and r = 2/9. The distance covered by the particle is 5/9 + 25/36 + 35/36 + 5/18 = 5/2.
=> Choice (b) is the right answer
372.A circle passes through the vertex C of rectangle ABCD and touches its sides AB and AD at P and Q respectively. If the distance from C to the line segment PQ is equal to 4 units, then the area of the rectangle ABCD in sq. units (is)
(a) 12 (b) 16 (c) 20 (d) can not be determined (e) none of these
Solution:
Let M be the feet of perpendicular from C to PQ. Now, by alternate segment theorem we have
=> < CQM = < CPB and < CPM = < CQD. Thus, right triangles CQM and CPB are similar and also CPM and CQD. Thus, CQ/CP = CM/CB and CP/CQ = CM/CD. Thus, CB.CD = CM^2 = 16.
Alternate Solution:
Assume the vertices of A, B, C, D as (0, b), (a, b), (a, 0) and (0,0). Since, AP = AQ, let AQ = m. Thus P = (0, b-m) and Q = (m, b). Now x = 0 is tangent to the circle at P. The eq. of circle is x^2 + (y - b+m)^2 + kx = 0, here k is variable. But, it's also tangent at Q = (m, b). Thus we get k = -2m.
Now, the circle x^2 + (y-b+m)^2 -2mx passes through (a, 0) => a^2 + (b-m)^2 = 2am. Equation of PQ is x-y = b-m. Also |(a-m+b)/v2| = 4. Squaring and using the relation in bold we get 2ab = 32, ab = 16.
373. The number of positive inegral solutions where 1 <= a1 <= a2 <= a3 <= a4 <= a5 <= a6 <= 6 is
(a) 360 (b) 432 (c) 462 (d) 498 (e) none of these
Solution:
Arrange 6 black balls and 5 white balls in a row. Then the 5 white balls divide the line into 6 parts corresponding to ai = 1, 2, ... , 6. The number of black balls in each part gives the number of ai with that value.
Thus we have in all (6+6-1)C6 such possibilities
=> Choice (c) is the right answer
374.A triangular piece of paper of area 1 is folded along a line parallel to one of the sides
and pressed flat. What is the minimum possible area of the resulting geometrical figure?
(a) 1 (b) 1/√2 (c) √3/2 (d) 2/3 (e) none of these
Solution:
Consider any triangle of area 1. Now suppose a line is drawn parallel to the base let the height of it from the vertex be k. Now, it's area would be k^2. Now when it is folded downwards, the part of the triangle that would extra is (k-(1-k))^2= (2k-1)^2. Now the total area of the figure is 1-k^2+(2k-1)^2 = 3*(k-2/3)^2 + 2/3 Now we can easily observe that the minimum occurs at k=2/3.
=> Choice (d) is the right answer
375.ABCD is an isosceles trapezium with AB || CD. The length of AB, BC, CD, DA is 7 cm , 3 cm , 5 cm , 3 cm respectively. The exterior bisectors of (a) 8 cm (b) 9 cm (c) 10 cm (d) 12 cm (e) none of these
Solution:
By symmetry we see that PQ passes through the midpoints of BC and DA. Let PQ meet BC at R and DA at S. RS = 1/2(7+5) = 6. PS=QR=3/2 (WHY?).
=> Choice (3) is the right answer
376.The set M consists of p consecutive integers with sum 2p. The set N consists of 2p consecutive integers with sum p. The difference between the largest elements of M and N is 9. Then p is
(A) 17 (B) 36 (C) 9 (D) 27 (E) 21
Solution:
Let the first term of series M and N is a and b respectively
For the series M we have, (a+1) + ... + (a+p) = p(2a+p+1)/2 = 2p
For the series N we have, (b+1) + ... + (b+2p) = p(2b+2p+1) = p
Also, b+p-a = 9
Solving we get, b = -p, a= -9, p = 21
Hence, choice (E) is the right option
377.Consider two different cloth cutting processes. In the first one, n circular cloth pieces are cut from a square cloth piece of side s in the following steps: the original square of side s is divided into n smaller squares, not necessarily of the same size; then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from the square of side s and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total scrap cloth generated in the former to that in the latter is: (∏ = circumference of the circle/diameter of the circle)
(A) 1:1 (B) √2:1 (C) n(4-∏)/(4n-∏) (D) (4n-∏)/n(4-∏) (E) 1:√2
Solution:
Take n = 4 and verify.
Choice (A) is the right option
378.Two stations A and B are 920 km apart. A train T1, which stops for 5 minutes in every town-station and for 3 minutes in every village-station started from A with a speed of 60km/h towards B and at the same time a train T2 with a speed of 80km/h which does not stop in any intermediate station started from B towards A. They met at C which is 560 km away from B. If the number of town-stations between A and C is less than the number of village-stations, then at least how many stations - town or village - are there between A and C? Assume T1 stops only at town or village-stations.
(A) 12 (B) 13 (C) 14 (D) 16 (E) 20
Solution:
As the train T2 covered 560km, both trains travelled for 7 hours, but T1 travelled only 360 kms.
=> Halting time = 60 minutes
Let the number of town-stations be x and number of village stations be y,x < y
=> 5x + 3y = 60 => x=3, y=15, x=6, y = 10, or x=0, y = 20
=> least is 16
Choice (D) is the right option
379.Given a set of n rays in a plane, define a reversal as the operation of reversing precisely one ray and obtaining a new set of rays. If all the rays are reversed after 42 operations, then n can be
(a) 21 (b) 23 (c) 41 (d) 24 (e) At least two of the foregoing
Solution:
For a ray to get back to its original orientation, it require even # of operations. Since, all the ray get reversed after 42 operations and let there be k rays => 2n+k = 42 => k has to be even.
Hence, choice (D) is the right answer
380.To complete the table, each cell must contain either 0 or 1, and the total of each row and column must be 2. What is the pair (X, Y) for the values of entries X and Y?
0 0
0
X 1
Y
(A) (1, 0) (B) (0, 0) (C) (0, 1) (D) (1, 1)(E) can not be
381.An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
(I) the selection of four red marbles;
(II) the selection of one white and three red marbles;
(III) the selection of one white, one blue, and two red marbles; and
(IV the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition?
Solution:
Rc4=W*Rc3=W*B*Rc2=R*W*B*G
where R=no. of red balls, W=no. of white balls, B=no. of blue balls, G=no. of green balls therefore, R=3B+2, R=2G+1, R=4W+3 for W=1 will not satisfy(B=5/3 which is not an integer) for W=2 we get smallest no. of marbles(R=11,B=3,G=5)
Hence, choice (B) is the right answer.
382. Two different infinite geometric progressions both have sum 1 and the same second term. One has third term 1/8. The second term of the progression upto 2 places of decimal is
(A) 0.40 (B) 0.30 (C) 0.25 (D) 0.20 (E) 0.15
Solution:
a + ak + ak^2 + ... = a/(1-k). So a = 1-k and |k| < 1. If we have another sequence (1-h), (1-h)h, (1-h)h^2 ... with same second term, then (1-k)k = (1-h)h implies h-k=0 or h+k=1. h=k gives same sequence, so must have h = 1-k. Third term = 1/8 gives 8k^3 - 8k + 1 = 0, or (2k-1)(4k^2-2k-1) = 0. Hence 4k^2-2k-1=0. So k = (1+√5)/4 or (1-√5)/4. Corresponding h=1-k are (3-√5)/4 and (3+√5)/4. Latter is > 1. So k = (1+√5)/4 and second term (√5-1)/8 =~ 0.15
Hence, choice (E) is the right answer.
383. Let x, y, z be real, consider a system of equations 2x + y + 4z = 2, x + z = -3, x + 2y + Mz = 13. Which of the following is not true?
I. There is a value of M for which the system has more than one solution
II. The system will become inconsistent for at least one value of M
(A) I (B) II (C) I && II (D) either I or II (E) none of these
Solution:
2x + y + 4z = 2, x + z = -3 => 2x+y+4(-3-x) = 2 => y-2x = 14. Also, x + 2y + Mz = 13 => x+2y + M(-3-x) = 13 => 2y +x(1-M) = 13+3M
13+3M +x(3M-1) – 4x = 28 => x(M-5) = 3(5-M) => when M = 5 the system has more than one solution. For M = 6, the system will become inconsistent.
Hence, choice (E) is the right answer.
384.An unlimited number of coupons bearing the letters A, B and C are available. What is the possible number of ways of choosing 3 of these coupons so that they can not be used to spell BAC?
(A) 15 (B) 18 (C) 21 (D) 24 (E) 27
Solution:
Each of the 3 places can take 3 letters => 27. But we don't want the combination (A, B, C) => 3! = 6 are out => 27-6 = 21.
Hence, choice (C) is the right answer.
385. The number of rational points x = p/5 satisfying log(2x-3/4)/logx > 2, where p is an integer and gcd(p, 5) = 1 is/are
(A) 3 (B) 2 (C) 5 (D) 1 (E) 4
Solution:
(log(2x-3/4) – 2logx).log x > 0 solving
Case 1: When x > 1 => (2x-3/4) > x^2.
Case 2: When x < 1 => (2x-3/4) < x^2 && x > 3/8
=> x lies in (3/8, ½) U (1, 3/2)
Hence, choice (A) is the right answer.
386. How many natural numbers n have such a property that out of all the positive divisors of number n, which are different from both 1 and n, the greatest one is 15 times greater than the smallest one?
() 1 (2) 2 (3) 3 (4) There are no such numbers (5) Infinitely many
Solution:
If S is the smallest asked factor and L the largest then S.L = N => 15S^2 = N. But S can take values less than or equal to the smallest prime factor of 15 => S = 2, 3.
Hence, choice (B) is the right answer.
387. Let a and b be such prime numbers that a>b and numbers a – b and a + b are also prime numbers. Then the number S = a + b + (a – b) + (a + b) is:
(1)An even number(2)Divisible by 3(3Divisible by 5(4)Divisible by 7(5) A prime number
Solution:
b has to be 2 => a, a-2, a+2 are all primes. Now, a can be of any 3 forms 3k, 3k+1, 3k+2. For a = 3k, all of 3k, 3k-2, 3k+2 are primes or for a = 3k+1, all of 3k+1, 3k-1, 3k+3 are primes or for a = 3k+2, all of 3k+2, 3k, 3k+4 are primes => k can take just one value = 1 => a = 5.
388.A square tin sheet of 12 inches is converted into a box with open top in the following steps: The sheet is placed horizontally. Then, equal sized squares each of side x inches are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If x is an integer, then what value of x maximizes the volume of the box?
(A) 3 (B) 4 (C) 1 (D) 2 (E) 6
Solution:
Volume ; V = (12 - 2x)^2 * x
= 4x^3 - 48 x^2 + 144x
for maxm. or min. dV/dx = 0;
=> 144+12x^2-96x = 0 => x=2 or x= 6
Hence, choice (D) is the right answer.
DIRECTIONS for Questions 16 and 17: Each question is followed by two statements X and Y. Answer each question using the following instruction:
Choose A if the question can be answered by X only
Choose B if the question can be answered by Y only
Choose C if the question can be answered by either X or Y
Choose D if the question can be answered by both X and Y
Choose E if the question can be answered by neither X and Y
389. Vineet has 9 crayons in a box. From every 4 crayons at least two are of the same color, and from every 5 crayons at most three are of the same color. How many blue crayons are in this box?
(X) At least one of them is blue
(Y) At least one of them is red, yellow, green each
Solution:
Given,there are 9 crayons
Since, there can be at max 3 crayons of same colour out of every 5,
so same coloured crayons cannot be more than 3.
Now, if we take 2 crayons of 1 colour,3 crayons each of another 2 colours and 1 more crayon of another colour,then condition 1 gets violated.(for example,lets have 3G,3R,2Y and 1B crayons....then if we 1 crayon each from the 4 different colours,d condition that there should be atleast 2 crayons of same colour out of every4 gets violated)
So, we should have 3 crayons of 1 color and total we should have crayons of 3 colors
Now, take X atleast one of them is blue...so we can say that there are 3 blue crayons
take Y- atleast 1 each of Red, Green, yellow...so actually we have 3 crayons each of Red, green and yellow and therefore 0 blue crayons.
Hence, choice (C) is the right answer.
390. A family has only one kid. The father says “after n years, my age will be 4 times the age of my kid”. The mother says “after n years, my age will be 3 times that of my kid”. Which of the following statements can answer the combined ages of parents after n years?
(X) The age difference between the parents is 10 years
(Y) After n years the kid is going to be twice as old as she is now
Solution:
let age of the kid after n years be x then age of father after n years is 4x and that of mother is 3x years. now the age difference will remain constant always and given that
(X) 4x-3x = 10 => x = 10.
after n years the combined ages of parents will be 7x =70 years
(Y) => k = n => F+M+2n = 14n
Hence, choice (C) is the right answer.
391. The negation of the proposition "For all pairs of real numbers a ,b , if a=0 , then ab=0" is: There are real numbers such that
A) a=0, ab=/0 (b) a=/0, ab=/0 (c) a=/0, ab=0 (d)a=/0, ab=/0(e)a=0, ab=0
Solution:
The negation of if A then B is A and not B.
Hence, choice (C) is the right answer
392. Consider the sequence defined recursively by u1=a(any positive integer), and
u n+1= (-1)/(un +1), n=1,2,3,... For which of the following values of n must un=a ?
(A) 15 (b) 18 (c) 16 (d) 14 (e) 17
Solution:
u1 = a; u2 = (-1)/(a+1); u3 = (-1-1/a); u4 = a.
Hence the pattern u1 = u4 =u7 = u10 = u13 = u16 = u19= ...............= a
Hence, choice (A) is the right answer
393. A function from the integers to the integers is defined as follows:
f(x)=n+3 if n is odd
=n/2 if n is even
Suppose k is odd and f(f(f(k)))=27 . What is the sum of the digits of k?
Solution:
since k is odd then f(f(f(k))) =27
=> f(f(k+3)) = 27------------(1) since k is odd , k+3 will be even.
hence, expanding (1) we get, f((k+3)/2) = 27--------------(2)
here (k+3)/2 can either be even or odd
case 1: let (k+3)/2 is even
then (2) reduces to (k+3)/4 = 27 => k = 105 here sum of digits is 6
case 2 : let (k+3)/2 is odd
then (2) reduces to ((k+3)/2)+3 = 27 => k = 45
substituting in (k+3)/2 it comes to be 24 which is not odd hence there is a contradiction hence k = 105 here sum of digits is 6 is the only solution.
Hence, choice (B) is the correct answer
394. If p, q are real numbers such that p^2 + pq + q^2 = 1, then the greatest value of the expression (p^3q+ q^3p) is
(A) 1/4 (B) 2 (D) 2/5 (D) 2/9 (E) none of the foregoing
Solution:
We need to find the greatest value of pq(p^2+q^2) given p^2 + pq + q^2 = 1, but p^2 + q^2 >= 2pq => p = q.
Hence, choice (D) is the correct answer
395.The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Then m equals
(A) 450 (B) 400 (C) 350 (D) 250 (E) 150
396. In a narrow alley of width w, a ladder of length a is placed with its foot at point P between the walls. Resting against one wall at Q, the distance k above the ground makes a 45 degree angle with the ground. Resting against the other wall at R, a distance h above the ground, the ladder makes a 75 degree angle with the ground. The width is equal to
(A) a (B) RQ (C) k (D) (h+k)/2 (E) h
Solution:
tan(75 degree) = tan (45 degree + 30 degree) = (tan45 + tan 30)/(1- tan45*tan30)
= [1+1/v3] / [ 1-1/v3] = 2+v3.
now, w = k+[h/(2+v3)] = k+(2-v3)*h
also a =[2*v2*h]/[v3+1]
k = a/v2 = 2h/(v3+1) = (v3-1)h
hence w = k+(2-v3)*h = (v3-1)h + (2-v3)*h = h
Hence, choice (E) is the right answer
397.a, b, c are reals such that if p1, p2 are the roots of ax^2 + bx + c = 0 and q1, q2 are the roots of cx^2 + bx + a = 0, then p1, q1, p2, q2 is an arithmetic progression of distinct terms. The value of a + c = equals
(a) 0 (b) -1 (c) 1(d) can be either of two of the foregoing (e) none of these
Solution:
Put p1 = h-k, q1 = h, so p2 = h+k, q2 = h+2k. Then h^2-k^2 = c/a, 2h = -b/a, h^2+2hk = a/c, 2h+2k = -b/c.
So h = -b/2a, k = b/2a - b/2c and b^2/2ac - b^2/4c^2 = c/a, b^2/2ac - b^2/4a^2 = a/c. Subtracting, (b^2/4)(1/a^2 - 1/c^2) = c/a - a/c, so (c^2-a^2)(b^2/4 - ac)/(a^2*c^2) = 0. Hence a = c or a + c = 0 or b^2 = 4ac. If b^2 = 4ac, then p1 = p2, whereas we are given that p1, p2, q1, q2 are all distinct. Similarly, if a = c, then {p1,p2} = {q1,q2}. Hence a + c = 0.
=> Choice (a) is the right answer
398.The year is 2008 and a new system(base n > 2) is used in place of decimal for making calculations. Priyanka, one day receives twice the number of questions in "Quant questions a week" b'coz the digits of the original number of questions which was a two digit number got reversed. Let x be the least possible value of n and y is the next possible value of n.
The 4 digit number abab is a perfect cube in base z. Let z be the smallest possible such base. Then y+z equals
(a) 11 (b) 15 (c) 19 (d) 21 (e) none of the foregoing
Solution:
Let the original number in base n Priyanka was to receive is pq. => 2*(pn+q) = qn + p. => p/q = (n-2)/(2n-1) => x = 5 and y = 8.
The 4 digit number abab in base z when converted to decimal system is az^3 + bz^2 + az + b = (az+b)*(b^2+1) is a perfect cube. Checking for values z = 2, 3, ..., we see that 2626 is 1000 in base z = 7.
=> Choice (b) is the right answer
399.The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10n, on the n th day of 2007 (n = 1, 2, ….., 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the nth day of 2007 (n = I, 2, ..., 365). On which date in 2007 will the prices of these two varieties of tea be equal?
(A) June 30 (B) May 21 (C) April 11 (D) May 20 (E) April 10
400. A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f(x) at x = 10?
(A)-105 (B)-119 (C)-159 (D)-110 (E)-180
401. Two circles with centers P and Q cut each other at two distinct points A and B. The circles have the same radii and neither P nor Q falls within the intersection of the circles. What is the smallest range that includes all possible values of the angle AQP in degrees?
(A) Between 0 and 45 (B) Between 0 and 90 (C) Between 0 and 30 (D) Between 0 and 60
(E) Between 0 and 75
401.I was at a restaurant for lunch the other day with some QQAdites. The bill came and I wanted to give the beautiful waiteress a multiple of 100 number of rupees, with the difference between what I give her and the bill being the tip. I always like to tip between 10 and 15 percent of the bill (ofcourse to a good looking damsel only). But if I give her a certain number of rupees (mutiple of 100), the tip would have been less than 10% of the bill, and if instead I give her 100 rupee more, the tip would have been more than 15% of the bill. The exact bill was a whole number. Let Rs PQRS be the largest possible amount of the bill where P, Q, R and S are each single digit numbers. Then P+Q+R+S is
(1) 14 (2) 15 (3) 17 (4) 22 (5) none of these
Solution:
Let the bill amount be B
100k < 1.1B and 100(k+1) >1.15B (where k is an integer)
from the above 2 inequalities we get b<2000 1.1b <2200 and 1.15b>2300
Hence maximum value of k=21
when k = 21 1.1B>2100 => B>1909.xx 1.15B<2200 => B<1913.xx
Greatest value for B of the form PQRS = 1913
P+Q+R+S = 14
=> Choice (3) is the right answer
402.Let's consider all irreducible fractions below 1, of type p/q where q =3007 and p/q is positive .If S denotes the sum of all such fractions denoted by p/q ,the the value of S is best represented by
(a) 1001 (b) 2002 (c) 1234 (d) 1405 (e) None of the foregoing
Solution:
A fraction a/b is called irreducible iff a and b are coprime--GCD(a,b)=1
3007 has prime factors 31 and 97
3007=31x97 so in series sum where we obtains multiple of 31 and 97--that nos we should not take
sum of all nos (1+2+3+...+3006)/3007=1503
sum of multiples of 31--->(31+62+...+2976)/3007=48
sum of multiples of 97---->(97+194+...+2910)/3007=15
ans. is==1503-(48+15)=1503-63=1440
Hence, choice (e) is the right option
403.The product of 114 and 21 in a certain number system is 2444. Find the decimal value of the number represented as 4231 in that number system.
(1) 566 (2) 1398 (3) 944 (4) 676 (5) none of these
404. 32^32^32 when dived by 7 leaves the remainder
(1) 4 (2)2 (3) 1 (4) 3 (5) none of these
405.Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?
(A) 1 (B) 3 (C) 2 (D) 4 (E) 0
406. A cube has vertices (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1). At t=0, a ball is at ( 1/2 ,1/2,0). After one second, the ball travels to (9/16,5/8,1/13). The ball bounces until it hits an edge or a corner at which point it stops. If the ball travels in straight lines with uniform speed, find the time (in seconds) when the ball stops motion
(a) 46 (b) 52 (c) 60 (d) 66 (e) 70
Solution:
Let the sector be of angle x
The ball moves 1/16 per second in the x direction, 1/8 per second in the y direction, and 1/13 per second in the z direction.
So it takes the ball 16 seconds for a "lap" in the x direction, 8 seconds for a lap in the y direction, and 13 seconds for a lap in the z direction.
The ball has an 8 second "headstart" in the x direction and a 4 second "headstart" in the z direction.
We want the ball to simultaneously finish laps in either 2 dimensions or 3 dimensions, whichever comes first.
We want two or more of:
t = 8 (mod16)
t= 4 (mod8)
t=0 (mod13)
The first two clearly cannot happen together, so the ball will not reach a corner, and we need a multiple of 13. In order to fulfill either of the other two criteria, it must also be a multiple of 4, making the lowest value (13)(4)=52. This satisfies the second and third conditions, so t=052
=> Choice (b) is the right answer
407. Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle in meters be, if we wish to have a flower bed with the greatest possible surface area?
(1) 2√2 (2) 2√5 (3) 5 (4) 4√2 (5) none of these
Solution:
Area of Sector, A = x/360*pi*r^2
Circumference of the sector = 20
=> x/360*2*pi*r +2r= 20
=> 2A/r+2r=20
=> A= r(10-r)
=> Choice (3) is the right answer
408.On January 01, 1989, the difference between the ages of Mr. and Mrs. Naidu was equal to the age of their son Shrinivas. (Mr. Naidu is older than his wife.) After a few years, when Mr. Naidu turns 35, Mrs. Naidu's age was half the square of Shrinivas' age. Ten years after this, the difference between the sum of the ages of the parents and that of their son was 69. How old will Mrs. Naidu be on January 01, 2014?
(1)49 (2) 50 (3) 54 (4) 52 (5)53
409.The triangle ABC has AB and AC unequal. The angle bisector of A meets the perpendicular bisector of BC at X. The line joining the feet of the perpendiculars from X to AB and AC meets BC at D. Then BD/DC is
(1) 1 (2) < 1/2 (c) > 1/2, and < 1 (d) > 1 (e) can not be determined
Solution:
Let the perpendiculars from X to the lines AB, AC meet them at Z, Y respectively. Triangles XBZ, XYC are congruent because XB = XC (X lies on angle bisector), XZ = XY (X lies on perpendicular bisector) and < BZX = < CYX = 90 degrees.
Hence BZ = CY. Also AZ can be > or < AY (since AB and AC are unequal). By Ceva's theorem, (AZ/ZB) (BD/DC) (CY/YA) = 1. Hence BD/DC is < 1 or > 1 depending on the ratio AB/AC.
=> Choice (5) is the right answer
410.To offset the increase in price of sugar and rice, either Sargam has to reduce the consumption of sugar by 20% or rice by 25%. How much percent consumption of sugar Sargam must reduce if she reduced the consumption of rice by 10%?
(1) 12% (2) 12.5% (3) 15% (4) 16% (5) can not be determined
Solution:
Let the initial prices of sugar and rice be x and y per kg and their consumption be a and b per kg respectively. Also, let there been an increase of 100p1% and 100p2% in their prices respectively. Then, ax+by = 0.8(1+p1)ax + (1+p2)by and ax+by = (1+p1)ax + 0.75(1+p2)by.
Let ax+by = (1-k)(1+p1)ax + 0.9(1+p2)by where 100k% is reduction in consumption of sugar when consumption of rice is reduced by 10%. Solving we get k = 0.12
=> Choice (1) is the right answer
411.In the nineteenth century a person was X years old in the year X2. How old was he in 1884?
a. 78 b. 68 c. 58 d. 43 e. none of these
411.How many sets of three or more consecutive odd numbers can be formed such that their sum is 500?
a.10 b.0 c.3 d.4 e.5
412.N has f factors, 2N has 2f factors, 6N has 4f factors, 15N has 3f factors. How many factors 30N has?
a. 8f b. 6f c. 5f d. 4f e. none of these
413.S is a six digit number beginning with 1. If the digit 1 is moved from the leftmost place to the rightmost place the number obtained is three times of S. Then the sum of the digits of S is
a. 26 b. 21 c. 27 d. 24 e. none of these
414.When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?
Choose one answer.
a. 6 b. 8 c. 5 d. 10 e. 7
415. How many pairs of (x,y) satisfy x^1/2+y^1/2=1332^1/2(x, y are integers)
a. 5 b. 3 c. 9 d. 7 e. none of these
416.If S = 5^2n + 1 + 11^2n + 1 + 17^2n + 1 where n is any whole number, then S is always divisible by
a. 33 b. 7 c. 17 d. 19 e. none of these
417. What is the smallest number which has divisors ending in every digit from 0 to 9?
114
a. 5 b. 3 c. 9 d. 7 e. none of these
418.For single digit numbers a,b,c when (abc)7=(cba)9, the value of a + b + c
a. cannot be determined b. 11 c. 8 d. 16 e. none of these
419. If a, a + 2, and a + 4 are prime numbers, then the number of possible solutions for a is
a. 3 b. 2 c. 1 d. more than 3 e. none of these
420.If f(x) = ax4 + bx3 + cx2 + dx + e and m, n, o and p are the roots of the equation f(x) = 0, then what is the value of (m + n + o + p + mn + mo + mp + no + np + op + mno + mnp + mop + nop + mnop)? Given f(0) = 1, f1(0) = 1, f2(0) = 2, f3(0) = 6 and f4(0) = 24, where fn(x) is the nth derivative of f(x).
a.1 b.22 c.9 d.18 e.none of these
421. If the roots of the equation(x+a)/(x+a+c) + (x+b)/(x+b+c)=1 are equal in magnitude and opposite in sign then
(1) c>=a (2) a>=c (3) a+b=0 (4) a=b (5) None of these
422. How many numbers between 1 to 1000 (both excluded) are both squares and cubes?
(1) 0 (2)1 (3) 2 (4) 3 (5) None of these
423Jamuna Express runs between Kashi and Prayagand has five stoppages in between. Find the number of different kinds of one way second class ticket that Indian Railways have to print to service all types of passengers who might travel by Jamuna Express?
(1) 49 (2)42 (3) 21 (4) 7 (5) None of these
424.If A(1,1), B((sqrt(3)+1,2), C(sqrt(3), sqrt(3)+2) be three vertices of a square, then the diagonal through B is
(1) y=0 (2 y=x (3) y= (sqrt(3)-2)x+(3- sqrt(3)) (4) y=3x+5 (5) None of these
425. A conical tent of given capacity is constructed. The ratio of height to the radius of the base for the minimum amount of canvas required for the tent is
(1) 1:2 (2) 2:1 (3) sqrt (2):1 (4) 1: sqrt (2) (5) None of these
426.In a triangle ABC, right angled at B, the median through C bisects the angle between CA and the bisector of < C. Which among the following best describes the range of AC/CB?
(1) ((√5 -1)/2, 2) (2) (2, 5/2) (3) (√2+1, 2√3) (4) (5/2, 3) (5) (√5, 4)g
Solution:
Let E be the mid-point of AB such that AE = EB = c/2.
CD bisects < C => BD = ac/(a+b). Since, CE bisects < ACD, CD/CA = DE/EA.
CD^2 = a^2 + a^2c^2/(a+b)^2, DE^2 = (c/2 - ac/(a+b))^2
Simplifying, we get a^2{(a+b)^2 + c^2} = b^2(b-a)^2. Using b^2 = a^2 + c^2 and eliminating c we get b^3 - 2ab^2 -a^2b - 2a^3 = 0. Put p = b/a, we have p^3 - 2p^2 - p - 2 = 0.Let f(p) = p^3 - 2p^2 - p - 2. The 1st derivative of f(p) is 3p^2 - 4p - 1 > 0 for p > 2 and thus a increasing function for p > 2. Also f(2).f(3) < 0 => p lies in (2, 3). Looking at the options, we have to find the narrowest interval which fits the range of p.
f(2.5) < 0, and thus this works. p = b/a lies in (5/2, 3).
=> Choice (4) is the right answer
427.The number of values of z for which the equation zx^2 + (z-7/2) x + 1/(16z)=0 has a unique solution is
(a) 2 (b)3 (c)5 (d) 6 (e) none of the foregoing
Solution: For the solution to be unique D = 0 => (z-7/2)^2 = 1/4 => z = 3, 4.
Hence, choice (a) is the right option.
428.How many distinct positive integral solutions of the equation a+b+c+ab+bc+ac =89 are possible?
(a) 2 (b)3 (c)5 (d) 6 (e) none of the foregoing
Solution: a+b+c+ab+bc+ca = 89 => (90-ab)/(a+b+1) = c+1 => 90-ab must divide a+b+1 => 90+a+a^2 must divide (a+b+1).
Let's enumerate a bit now.
For a = 1, we must have 92 div by b+2 => b = 2.
For a = 2, we must have 96 div by b+3, but 96 = (2^5)*3 => b = 3, 5, 9, 13, 21
For a = 3 we must have 102 = 2.3.17 div by (4+b), thus no new solution
For a = 4 we must ahve 110 = 2.5.11 div by (5+b) => b = 5, 6
For a = 5 we must have 120 div by (b+6) => b = 6, 9
For a = 6 we must have 132 = 2^2.3.11 div by (7+b), thus no new solution
And we stop here. WHY?
Thus, in all 10 unordered solutions.
Hence, choice (e) is the right option
429.n+2 students played a tournament. Each pair played each other once. A student scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other students all had equal total scores. How many values of n are possible?
(1) 0 (2) 1 (3) 2 (4) more than 3 (5) none of these
Solution:
There are (n+2)(n+1)/2 matches, so the total score is (n+2)(n+1)/2. Let the other students score k each. Then 8 + nk = (n+2)(n+1)/2, so n^2 - (2k-3) - 14 = 0. We know this equation has one root which is a positive integer. The product of the
roots is -14, so the possibilities for the roots are: 1, -14; 2, -7; 7, -2; 14, -1. Hence the sum of the roots is -13, -5, 5, or 13 (respectively). Hence k = -5, -1, 4 or 8 (respectively). But k must be non-negative, so n = 7 or 14 is a necessary condition.
We need to check that these values can be achieved. Take n = 7, so there are 9 students in total. If every match is a draw, then every student draws 8 matches and scores 4, which satisfies the conditions. Take n = 14, so there are 16 students in
total. Suppose one student loses to everyone, and all the other games end in a draw. Then the first student scores 0 and all the other students score 1 + 14/2 = 8. That also satisfies the conditions
=> Choice (3) is the right answer.
Each question is followed by two statements A and B. Answer each question using the following instructions
Choose 1 if the question can be answered by A alone
Choose 2 if the question can be answered by B alone
Choose 3 if the question can be answered by A and B combined
Choose 4 if the question can be answered by either A or B
Choose 5 if the question can be answered by neither A and B
430. A N-sided regular polygon was inscribed in a circle. The consecutive vertices of the polygon were numbered from 1 to N. If vertex number n lies on one end of the circle’s diameter, then the other end of the diameter lies on vertex number 3n+1. What is the value of N?
A) 2N -4 = n2
B) 14 < N < 22
According to question As vertex number n and 3n +1 are opposite ends of the diameter hence, number of sides of the polygon is 2n+2n+2 = 4n + 2 = N
or, N-2 = 4n
=>2N-4 = 8n---------------(1)
A) says that 2N-4 = n^2---------------(2) from (1) and (2) we have N=34
So question can be answered by A)
B)
14 < N < 22
We have N = 4n+2,
=> N = 18 is the only possibility
Hence, choice (4) is the right option
431. A set S = {a, b, c, d} consists of prime numbers between 20 and 100, such that no two elements of S have any digit in common. What is the value of a+b+c+d?
A) The difference of some pairs of elements in S is prime
B) If a > b > c > d, then the number of distinct ways in which the elements of S can be chosen is 14
Solution:
A) gives 2 set (23, 47, 59, 61) and (47, 59, 61, 83)
B) is redundant
Hence, choice (5) is the right option
432.The number of real pairs (p, q) such that whenever α is the root of x^2 + px+q = 0 then α^2 - 2 is also its root is
(1) 2 (2) 4 (3) 6 (4) 3 (5) none of these
Solution:
Case 1: a = b => (a^2-2) = (b^2-2)
=> a = a^2 - 2 => a = 2 or a = -1.
a = b =2 => (p,q) = (-4,4)
a = b = -1 => (p,q) = (1,1)
Case 2: a = (a^2-2) ; b = (b^2-2)
=> (a = 2 or a = -1 )and (b = 2 or b = -1)
=> a = 2, b = -1 or vice versa => (p,q) = (1,-2)
Case 3: a = (b^2-2) ; b = (a^2-2)
=> a = (a^2-2)^2 -2 (substituting the value of b from above)
=>a^4 - 4a^2 - a + 2 = 0
=> (a-2) (a^3 + 2a^2 - 1) = 0
=> (a-2) (a+1) (a^2 + a -1) = 0
a = 2 and a = -1 have already been handled.
a^2 + a - 1 = 0
a = (-1 + root(5))/2 => b = a^2 - 2 = (-1 - root(5))/2
and vice versa.
This gives (p,q) as (1,-1)
Case 3: when a and b are different and a = a^2 - 2 = b^2 - 2
This gives (p, q) as (0, -1) and (0, -4). Try this yourself!
Thus in all we have 6 pair of (p, q) -> (-4, 4), (2, 1), (-1, -2), (1, -1), (0, -4), (0, -1).
=> Choice (3) is the right answer
433.What is the greatest integer n for which there exists a simultaneous solution x to the inequalities k < x^k < k+1, k = 1, 2, 3, ..., n?
(a) 2 (b) 3 (c) 4 (d) 6 (e) none of the foregoing
Solution:1 < x < 2, 2 < x^2 < 3 is satisfied by all x in (√2, √3), if we have one more inequality with the former two i.e. 3 < x^3 < 4 is satisfied by all x in (3^1/3, 4^1/3). If we also have 4 < x^4 < 5; then 4 simultaneous are satisfied for x in (3^1/3, √√5). If we also have 5 < x^5 < 6, there will be no x satisfying 5 equations as 3^1/3 > 6^1/5 (243 > 216).
Hence, choice (c) is the right option
434.Priyanka does daily work for a fixed time, and her efficiency would reduce by 10% on each subsequent day. However, when she took a break for a day her efficiency would come to normal the next day. Assume Priyanka worked for a long time, then after how many days of consecutive work does she needs to take a break to have the maximum output?
(1) 3 (2) 4 (3) 5 (4) 7 (5) none of these
Solution:
Let her efficiency be E per day. Then for 2nd day Priyanka's efficiency would be 0.9 E. Thus, if she were to take break after 2 days of work then Output will be (1+0.9+0)E/3 = 0.633 E. Her efficiency in day 3 without break is 0.81 => if she were to take break after 3 days her output is (1+0.9+0.81+0)E/4 = 0.67 E.
If she were to take break after 4 days her output is (1+0.9+0.81+.729+0)E/5 = 0.68 E.
But, her efficiency without break on 5th day is .6561 E => from here on the average output above can only go down.
=> Choice (2) is the right answer
435.If three unequal positive real numbers p, q, r are in GP and q-r, r-p, p-q are in HP, then the value of p+q+r is independent of
(a) p (b) q (c) r (d) all of the foregoing (e) none of the foregoing
Solution:since p,q,r are in GP => q^2 = p*r
also since q-r,r-p and p-q are in HP
So 1/q-r,1/r-p and 1/p-q are in AP
=> 2(q-r)(p-q) = - (r-p)^2
=> p^2 + r^2 - 2q^2 +2 pq +2qr-4pr=0
=> p+q+r = -3q => data is inconsistent with the given information
Hence, choice (e) is the right option
436.A function f(x) is defined for all real numbers x. For all non-zero values of x, 3f(x) +2f (1/x) =2x+1. If S is the sum of all the values of x for which f(x) =350, then find the value of S.
a. 285.5 b. 291.5 c. 293.5 d. 287.5 e. 295.5
437.What is the remainder when 8^643 is divided by 132?
a. 28 b. 20 c. 116 d. 80 e. 112
438.How many multiples of 18 are less than 3500 and are2 more than the square of a natural number?
a. 6 b. 7 c. 5 d. 8 e. 9
439.An eight digit telephone number consists of exactly two zeroes. One of the digits is repeated twice. Remaining three digits are distinct. If the first three digits are 987(from left to right), then find the probability of having only one 9, one 8 and one 7 in the telephone number
a. 245/8 b. 225/8 c. 225/16 d. 245/16 e. none of these
440.Between 1350 and 2450, how many numbers when divided by 11, 8 and 6 gives remainder 2, 3, and 5 respectively?
a. 1 b. 3 c. 5 d. 4 e. 2
441.There are three cards each with a different positive integer. In each round of a game between 3 players, the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, Abhinav has received 20, Sushil 10 and Vijender 9 counters. In the last round Sushil received the largest number of counters. Who received the middle number in the first round?
(1) Abhinav(2) Sushil (3) Vijender (4) Abhinav or Sushil (5) Sushil or Vijender
Solution:
The total of the scores, 39, must equal the number of rounds times the total of the cards. But 39 has no factors except 1, 3, 13 and 39, the total of the cards must be at least 1 + 2 + 3 = 6, and the number of rounds is at least 2. Hence there were 3 rounds and the cards total 13.
The highest score was 20, so the highest card is at least 7. The score of 10 included at least one highest card, so the highest card is at most 8. The lowest card is at most 2, because if it was higher then the highest card would be at most 13 - 3 - 4 = 6, whereas we know it is at least 7. Thus the possibilities for the cards are: 2, 3, 8; 2, 4, 7; 1, 4, 8; 1, 5, 7. But the only one of these that allows a score of 20 is 1, 4, 8. Thus the scores were made up: 8 + 8 + 4 = 20, 8 + 1 + 1 = 10, 4 + 4 + 1 = 9. The last round must have been 4 to Abhinav, 8 to Sushil and 1 to Vijender. Hence, on each of the other two rounds the cards must have been 8 to Abhinav, 1 to Sushil and 4 to Vijender.
=> Choice (3) is the right answer
442.GOI directive proposed formation of more new IIMs. Fearing lack of good faculty and geographical distances as disadvantageous, IIMs decide to have one for all campus. The School of Planning and Architecture proposed 2 new projects for the new Housing campus. In each project, the campus is designed to have several identical dorimiatory buildings, with the same number of 1 bedroom apartment in each building. In the first project there are 12096 apartments in total. There are 8 more buildings in the second project than the the first and each building has more apartments which raises the total of apartments in the project to 23625. Let N be the number of buildings the second project require. Then N is divisible by
(a) 5 (b) 7 (c)9 (d) at least two of the foregoing (e) none of the foregoing
Solution:
12096 = (2^6)*(3^3)*7 and 23625 = (5^3)*(3^3)*7; well you don't have to be a genius to observe this; how numbers move in tandem is best demonstrated by this example. For 12096, the last 3 digits tell us it's div by 8, thus 12096/8 = 1512, again 512 is div by 8; for 23625 the last 3 digits tell it's div by 125. So divide!
12096/n < 23625/(n+8). Now, comes a bit of data manipulation.
n must divide 12096 and n+8 should divide 23625. And n has to be odd => n = 3^3 and # of buildings 2nd project requires is 5*7 =3^3 + 8 = 35
Hence, choice (d) is the right option
443.How many ordered pairs of positive integers m, n satisfy m <= 2n <= 50, n <= 2m <= 50?
(1) 313 (2) 325 (3) 337 (4) 349 (5) none of these
Solution:
There are 625 points (m,n) such that 1 <= m,n <= 25. Out of these 2m < n and 2n < m have to be excluded.
The number of points satisfying 2n < m is 0 for m=1,2; 1 for m=3,4;...;11 for m=23, 24; 12 for m=25
=> total points are 2(1+2+3+ ... + 11) + 12 = 144
By symmetry 2m < n also gives 144 points.
=> in all 625 - 2*144 = 337 pairs satisfy m <= 2n <= 50, n <= 2m <= 50
=> Choice (3) is the right answer
444.Arvind and Swarnali are team mates and they can do a project in certain number of days.If Arvind is on holiday for x days,then they take y more days to complete the work while if Swarnali is on holiday for x days,then they take z more days to complete the same project.Then
(a)y, x/2,z are in AP (b) y, x/2,z are in GP (c)1/x=1/y+1/z (d) Cannot be determined (e) None of the foregoing
Solution:
Let Aravind complete 1/a portion of the work in a day
and Swarnali 1/b portion of the work in a day
Aravind on holiday for x days implies loss of (1/a)*x fraction of the work This loss is compensated by arvind and swarnali in the coming extra y days
so ((1/a)+(1/b))*y =(1/a)*x
Similarly, for Swarnali's holiday of x days
((1/a)+(1/b))*z= (1/b)*x
Adding the two equations
((1/a)+(1/b))*(y+z)=x*((1/a)+(1/b)) implying y+z=x
x/2=(y+z)/2
Hence y,x/2,z are in AP
Hence, choice (a) is the right option
445.12 students are sitting on chairs, one on each at most, in 3X5 array i.e. 3 rows and 5 columns, such that the top and bottom rows are all filled and middle row chairs are empty alternatively. The teacher wishes to re-assign the seats such that every student moves by 1 seat either left, right, forward or backward e.g. for student at seat (1, 1) , only 2 choices are possible and for student at seat (2, 5) 3 choices are possible. In how many ways is this re-assignment possible?
(1) 0 (2) 12C3 (3) 15C3 (4) 12P3 (5) none of these
Solution:
Color each chair alternately Black and Red => We have 8 Black and 7 Red chairs and 8 Black and 4 Red are occupied. In the re-assignment every black should come to Red and every Red should come to black chair => our final configuration should have 8 Red occupied, but only 7 are available!
=> Choice (1) is the right answer
446.A candy company makes 5 colours of jelly-beans which comes in equal proportions. If Divya grabs a random sample of 5 jellybeans, what is the probability that she gets exactly 2 distinct colours?
(a) 2/5 (b) 20/63 (c) 8/625 (d) 1/2 (e) 2/21
Solution:
Let there be A,B,C,D,E number of candies with five colours
So total ways is number of solutions of the equation
A+B+C+D+E = 5
which is 9C4 = 126
when the candies are of exactly two different colors than
Total number of ways = Number of ways of selecting two colors out of five x ( nunmber of solutions of equation X+Y = 5 , where X, Y are natural numbers )
= 5C2 x 4C1
= 40
Hence Probability = 40/126= 20/63
Hence, choice (b) is the right option
447. A bag contains 5 jelly-beans and it is not known how many of them are blue in colour. If 2 jelly-beans are drawn and found to be blue, then the probability that all the jelly-beans in the bag are blue is
(a) 2/5 (b) 20/63 (c) 8/625 (d) 1/2 (e) 2/21
Solution:
This is a case of conditional probability:
Required probability = (probability that 2 jelly beans drawn being blue when all are blue)/((probability that 2 jelly beans drawn being blue when all are blue) + (probability that 2 jelly beans drawn being blue when 4 are blue) + (probability that 2 jelly beans drawn being blue when 3 are blue) + (probability that 2 jelly beans drawn being blue when 2 are blue))
= 1/(1+ 4/5.3/4 + 3/5.2/4 + 2/5.1/4)
= 1/2
Hence, choice (d) is the right option
448.The sum of the series 1 + (1+1/2).1/3 + (1+1/2+1/2^2).1/3^2 + (1+1/2+1/2^2+1/2^3).1/3^3 + .... is
(1) 3/2 (2) 9/5 (3) 2 (4) 5/2 (5) none of these
Solution:
The given series can be re-written as
(1 + 1/3 + 1/3^2 +.....) + 1/2(1/3 + 1/3^2 +.....) + 1/2^2( 1/3^2 + 1/3^3 .....) etc
=> 3/2(1+ 1/6/ 1/36 + ....)
= 9/5
=> Choice (2) is the right answer
449. If the remainder when 281^508 is divided by 153 is P, and p^(46P) when divided by 83 gives remainder Q. Then sum of the digits of Q is
(a) 1 (b)3 (c)7 (d)11 (e)None of the foregoing
Solution:
281^508 by 9 gives remainder 16 and by 17 16 => 281^508 by 153 gives remainder 16.
16^82 when divided by 83 gives remainder as 1 => 16^ (46*16) mod 83 == 16^80 mod 83.
If 16^80 = x mod (83) => 256x by 83 gives 1 => (249+7) x by 83 gives 1 => 7x by 83 gives 1 => x = 12
=> Choice (2) is the right answer
450.Spiderman has to chase a gang of robbers riding on a car after looting a nearby bank . Spiderman is having coffee at a distance 36 kms south of the car initially when his spider senses give him an inkling. So he heads for the car immediately at a velocity of 50 km/hr as soon as the car also starts at a velocity of 40 km/hr. The gangsters drive the car towards the east with 40 km/hr. Spiderman keeps moving towards the car following his inkling. (i.e. the direction of Spiderman keeps changing and he keeps following the shortest path to the car.) The time in hours after which Spiderman reaches upto car and gives gangster the treatment is
(1) 1.5 (2) 2 (3) 2.5 (4) 3 (5) none of these
Solution:
Let at some point of time spiderman is following the car with his velocity making angle a with east direction...
Now horizontal velocity of spider man = vcos(a)
The integrals below used are definite from 0 to t.
Horizontal distance covered by Spiderman= ∫vcos(a)dt= v*∫cos(a)dt = Horizontal distance covered by robbers= 40t
So ∫cos(a)dt= 4t/5
Spiderman approaches robbers at a relative speed of 50-40cos (a)…so in time t this distance = 36
∫(50-40cos(a))dt = 36 => 50t- 40∫cosadt =36
Putting value of cos(a)dt
50t-40*4t/5 = 36 =>18t=36 or t= 2hrs
=> Choice (2) is the right answer
451.Salman does pushups in sets each consisting either of 6, 9, or 20 pushups. By doing 2 sets of 6 pushups, he can do 12 pushups. But he cannot do 13 pushups, as one cannot get 13 by using 20,6 or 9. What is the maximum number of pushups that Salman cannot do?
(1) 39 (2) 43 (3) 45 (4) 57 (5) None of the foregoing
Solution:
9mod6=3 . Hence all multiples of 3 greater than 6 can be arrived at
20mod6=2. Hence all numbers >25 which leave remainder of 2 when divided by 3 can be arrived at
40mod6=1. Hence all numbers >43 which leave remainder of 1 when divided by 3 can be arrived at
Hence, choice (3) is the right option
452.A gardener was given a Rs. 100 note by his landlord and asked him to buy exactly 100 flowers from the nursery by spending all the money. A piece of Daffodil costs Rs15. Canterbury Bells cost Re 1, and Marigold are 25 paise each.
He was asked to buy at least one flower of each of the three varieties. How many Canterbury Bells did the gardener buy from the nursery?
(1) 39 (2) 43 (3) 45 (4) 57 (5) None of the foregoing
Solution:
Let the number of Daffodils be 'd' Canterbury be 'c' and marigold be 'm'
Then according to question
15d + c + (.25)m = 100
or, 60d + 4c + m = 400 --------------- (1)
also, d+c+m = 100 -------------------(2)
subtracting (2) from (1) we get,
59d + 3c = 300
or, c = 100 - (59d / 3)
As c can only be an integer and d has to be atleast 1
only d =3 satisfies.
so c = 41 and m = 56.
so answer is 41
Hence, choice (5) is the right option
453.Let f(x) be a polynomial with integral coefficients such that f(x) = f(-x) > 0 for all x > 0. If f(5) = 6, and f(x) is distinct for all X > 0. Then minimum possible value of f(6) will be
(1) 7 (2) 9 (3) 13 (4) 17 (5) none of these
Solution:
(1) f(x) is an even polynomial => it will only have even powers of x.
(2) f(5) = 6 => f(x) = 6 + (x^2 - 25).g(x) where g(x) is an even function
(3) f(x) > 0 for all x -> by putting g(x) = x^2 - 23, we have f(x) that satisfies conditions 1, 2, 3.
(4) f(x) is distinct for all x > 0.
g(x) will have its co-efficients as integers => f(6) = 6 + 11.g(6) can't be each 7, 9, 13 as g(6) is an integer. If g(6) = 1 , not all of g(1), g(2), g(3), g(4) can be < 1 for f(x) > 0 for all x > 0
=> Choice (5) is the right answer
454.BC is a triangle with AB = 14, BC = 10 and CA = 6. D and E are points on BC and CA respectively such that CD = 3 and CE = 2.5. A line passing through C and the point of intersection of AD and BE cut the side AB at F. Then AF =
(a) 5 (b) 5.25 (c) 6 (d) 6.25 (e) 7.5
Solution:
CD = 3, DB = 7, AC = 6 and AB = 14.
=> CD/DB = AC/AB => AD is the angle bisector of angle A.
Similarly: CE/EA = BC/AB => BE is the angle bisector of angle B.
So CF is the angle bisector of angle C
AF/FB = 6/10 anf AB = 14. => AF = 5.25.
Hence, choice (b) is the right option.
455.ABC is a triangle with AB = 14, BC = 10 and CA = 6. D and E are points on BC and CA respectively such that CD = 3 and CE = 2.5. A line passing through C and the point of intersection of AD and BE cut the side AB at F. Then AF =
(a) 5 (b) 5.25 (c) 6 (d) 6.25 (e) 7.5
Solution:
CD = 3, DB = 7, AC = 6 and AB = 14.
=> CD/DB = AC/AB => AD is the angle bisector of angle A.
Similarly: CE/EA = BC/AB => BE is the angle bisector of angle B.
So CF is the angle bisector of angle C
AF/FB = 6/10 anf AB = 14. => AF = 5.25.
Hence, choice (b) is the right option.
456.ABCDE is a pentagon such that < A = < B = < D = 120˚, and < C = < E. Let BC = 1 and CD = √3. If a circle can be inscribed in the pentagon, then its radius will be
(1) 4 - 2√3 (2) 3 - √3 (3) 3√3 - 4 (4) 4√3 - 6 (5) none of these
Solution:
The construction of the pentagon is very much valid here. Let's see if we can inscribe a circle in this pentagon.
tan (< DBC) = √3/1 => < DBC = 60 degrees.
Let O be the centre of the circle and P and Q be tangency points of CB and AB respectively => By RHS, triangles OQB and OPB are congruent and hence < OBP = 60 degrees, but since < DBP = 60 degrees, we are effectively saying that DB passes through the centre of the circle.
What you get is R > 1 [3/2(√3-1)], when clearly it should be < 1. Thus, no such construction of the circle is possible.
=> Choice (5) is the right answer
457.A rectangular block L x 100 x H, with L ≤ 100 ≤ H, where L and H are integers, is cut into two non-empty parts by a plane parallel to one of the faces, so that one of the parts is similar to the original. How many possibilities are there for (L, H)?
(1) 10 (2) 12 (3) 20 (4) 24 (5) none of these
Solution:
We must cut the longest edges, so the similar piece has dimensions L x 100 x k for some 1 ≤ k < H. The shortest edge of this piece cannot be L, so it must be k. Thus L x 100 x H and k x L x 100 are similar. Hence H = 100^2/L, k = L^2/100. Now 100 = 2^4•5^4, so 100^2 has 25 factors, of which (25-1)/2 = 12 are < 100.
=> Choice (2) is the right answer
458.(i) The sum of real numbers x and y is 1. The maximum value of xy(x^3 + y^3) is p.
(ii) Let the equation [n/2] + [n/4] = n has q possible solutions, where [n] denotes the greatest integer less than or equal to n e.g. [3.21] = 3.
What is the value of p*q?
(a) 1/4 (b) 3/16 (c) 1/3 (d) 1/2 (e) none of the foregoing
Solution:
(i) If n is positive then [n/2] + [n/4] can at the max be 3n/4. Thus, no solutions for positive n. For non-negative n, n = 0 is a trivial solution. For n < -5 LHS > RHS. For -5 <= n < 0, we have 3 solutions, n = -2, -3, -5. Thus p = 4.
(ii) xy(x^3 + y^3) = xy(1-3xy) = 1/3*(3xy*(1-3xy)). Thus, 3xy + (1-3xy) is constant => max of 3xy*(1-3xy) occurs when 3xy = 1-3xy => xy = 1/6 => q = 1/12
=> option (c) is the correct answer.
459.A car gives the mileage of 60km/L, 50km/L and 40km/L when driven at the speeds of 40km/hr, 50km/hr and 60km/hr respectively. Assume that each car is driven only at the three speeds mentioned above. The car is driven for 3 hours using 2 Litres of petrol. The distance covered by the car
(a) > 120 km (b) < 120 km(c) = 120 km(dcan not be determined(e) none of the foregoing
Solution:
Let the distance traveled with 40km/hr, 50km/hr, 60km/hr is A, B and C respectively. => A/40 + B/50+ C/60 = 3 and A/60 + B/50 + C/40 = 2. Eliminating A, we get 2B/5 + 5C/6 = 0 => B=C=0 => A = 120 km
=> option (c) is the correct answer.
460.Let [x] denotes the greatest integer less than or equal to x. The number of positive reals x such that 9x/10 = [x]/( x - [x] ) is a
(1) perfect square (2) prime (3) perfect cube (4) perfect number(5) none of these
Solution:
Let x = I + f, where I is non-negative integer and 0 < f < 1 [for f = 0, 1 we have no solution]
Thus, what you get is 9(I+f)/10 = I/f. This is a quadratic in f whose only one root will be in (0, 1), as the product of the roots of 9f^2 + 9If - 10I = 0 is negative, the other root is negative.
The positive root is [-9I + root(81I^2 + 400I)]/18 and is < 1. Solving for this, we get eight legitimate I from 1 to 8.
=> Choice (3) is the right answer
461.Let f(x) be an algebric expression of odd degree (> 1). If the degree of f(x) + f(1-x) is at least two less than the degree of f(x) and the coefficients of x and x^2 are equal in magnitude but opposite in sign in the given expression, then the highest possible degree of f(x) is
(a) 9 (b) 13 (c) 5 (d) 7 (e) none of the foregoing
Solution:
Taking f(x) = ax^n + bx^n-1 + cx^n-2 + .... , where the third last coefficient and the second last coefficient are equal in magnitude but opposite in sign and n is odd.
f(x) + f(1-x) has at least first 2 terms as 0 => 2b + na = 0. Thus, we can have such f(x) for any n, thus n-> Infinity.
=> option (e) is the correct answer.
462.Gripened comes back to India after earning good bucks from Korea and heads for a shop in Bangalore having expensive shirts. The price of each shirt is a multiple of ten in Rs and is marked in K (in Rs) i.e. multiple of thousand e.g. if the price is Rs 2670, then it's marked as 2.67 K. Gripened buys 4 shirts and goes to the counter for the bill. The counter-boy multiplies the marked price of 4 shirts and hands gripened the bill of 7.11 K. Gripened being smart realizes the mistake and asks for the new bill in which the marked price of each shirt is added. To Gripened's astonishment, the bill this time also comes as 7.11 K. What was the marked price in K of the least expensive shirt that Gripened bought?
(1) 1 (2) 1.2 (3) 1.5 (4) can not be determined (5) none of these
Solution:
Let the MP in Tens (Rs) of the four shirts be a, b, c, d. Then a, b, c, d are whole numbers with
a + b + c + d = 711 = 32 × 79 and (a/100).(b/100).(c/100).(d/100) = 711/100.
=> abcd = 711 × 10^6 = 2^6 × 3^2 × 5^6 × 79. Exactly one marked price is a multiple of 79, and at most three prices (in Tens Rs) are even or are a multiple of 5.
It is not possible for three prices to be a multiple of 25. Otherwise, the remaining price would be the multiple of 79, and the sum of the three remaining prices would also be a multiple of 79 as well as of 25.
But 79 × 25 > 711, and this is not possible. Hence, at least one of the prices is a multiple of 5^3 = 125; this price is clearly not a multiple of 79.
Case 1: One of the prices is 5×79 = 395. Suppose that a = 5×79 = 395. Suppose that b is a multiple of 5^3 = 125. Since, not all four MPs can be a multiple of 5, one price, c, say, must be a multiple of 5^2 = 25. If (a, b) = (395, 125), then, modulo 25, a+b+c = 20. Since d can have only 2, 3, 5 as prime divisor, d = 16. But this leads to c = 175 = 7×5^2, which is not possible. If (a, b) = (395, 250), again d = 16 so that c = 50 = 2×2×5^2. But then abcd is not divisible by 3. Since a+b < 711, this exhausts the possibilities and Case 1 cannot occur.
Case 2. One of the prices, say a is one of the multiples 79, 158, 231, 316, 474 of 79 and another, say b is one of the multiples 125, 250, 375, 500, 625 of 125. Examining the cases and conducting an analysis similar to that of Case 1, we arrive at the unique solution (a, b, c, d) = (316, 125, 150, 120) = (2^2 × 79, 5^3, 2 × 3 × 5^2, 2^3 × 3 × 5)
Therefore, the MP of four shirts is 1.2 K, 1.25 K, 1.50 K and 3.16 K.
=> Choice (2) is the right answer
463.The value of 1/(1+1^2+1^4)+ 2/(2+2^2+2^4) + 3/(3+3^2+3^4) + ...up to infinity is
(a) 1/2 (b) 2/3 (c) 1 (d) 3/2 (e) None of these
Solution:
Each term in the summation is n/(n^4 + n^2 + n). Had each term be n/(n^4 + n^2 + 1), then the summation would become 1/2(1 - 1/3 + 1/3 - 1/7 + .... up to infinity) = 1/2. Note that n^4 + n^2 + 1 can be factored as (n^2+1)^2 - n^2 = (n^2+n+1)*(n^2-n+1).
n/(n^4 + n^2 + n) < n/(n^4 + n^2 + 1) for all n > 1. Thus, our answer is < 1/2. All options among the first 4 are >=1/2.
=> option (e) is the correct answer.
464.7 IIMs participate in a B-schools sports meet, where the use of expletives is officially prohibited. Each team from an IIM plays against the other exactly once. What is the possible minimum number of matches that could have been played so far such that among every 3 teams, atleast two have played against each other?
(1) 9 (2) 10 (3) 14 (4) 15 (5) none of these
Solution:
Treat each team as a point (vertex), and denote matches between two teams as the segment joining these two points. If we isolate these teams into a set of 4 and 3 each, and draw the complete graphs (a quadrilateral with diagonals also joined, and another disjoint triangle), then we have 4C2 + 3C2 = 9 edges (matches) in all. Our condition is satisfied. Can we do better than 9? The answer is NO, and left as an exercise for students to prove.
=> Choice (1) is the right answer
465. 100 ex-students of the same batch meet for alumni meet in the campus of a B-school. The benches of Fr Prabhu Hall in the B-school are arranged in a rectangle of 10 rows of 10 seats each. All the 100 have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. 4 members at most) how much they earn. They feel a lot of envy towards each other: a person is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of ex-students who are satisfied with their salaries?
(a) 72 (b) 32 (c) 48 (d) 64 (e) none of the foregoing
Solution:
Let us represent the ex-students with a square grid of 10x10 points, and label each point with the salary of the ex-student.
Have a look at the figure in the document by following the undermentioned link.
Let us draw arrows between neighbouring points such that the arrow is directed from the smaller to the larger number:
www.pagalguy.com/forum/814259-post1030.html
(Satisfied alumni are marked in green, and dissatisfied ones in red.)
Let a be the number of satisfied EX-STUDENT's sitting in the corners, b the number of those sitting at the sides of the square, and c the number of those sitting inside.
The number of arrows is 180. There is at most one arrow originating at any satisfied EX-STUDENT, and there will be at least one point where no arrow originates, the EX-STUDENT with the largest salary (obviously satisfied). Hence the number of arrows originating at satisfied EX-STUDENT's is at most a+b+c-1.
There are at most (4-a).2 arrows from the 4-a dissatisfied EX-STUDENT's in the corners, at most (32-b).3 from the 32-b dissatisfied EX-STUDENT's along the sides, and at most (32-b).3 from those (64-c).4 sitting inside. The total number of arrows is thus
180 <=(a+b+c-1)+(4-a).2+(32-b).3+(64-c).4,
that is, a+2b+3c <=179.
The one with the lowest salary out of the 36 EX-STUDENT's around the circumference is necessarily dissatisfied, thus a+b<=35. It is also obvious that a <=4. By adding the inequalities, we have
3(a+b+c)=(a+2b+3c)+(a+b)+a <=179+35+4=218,
that is, a+b+c <=72. Hence, the number of satisfied EX-STUDENT's cannot be greater than 72.
The diagram shows the case when there are exactly 72 EX-STUDENT's who are content with their salaries.
=> option (a) is the correct answer.
466.What is the value of x?
x=1/1+1/1+1/1+1/1+1/.....ad infinitum
a) 0 (b) -1 (c) 2 (d) -2 (e) none of the foregoing
Solution:
Replace x in the RHS by whole expression in RHS, thus we get the same case when the sum approaches Infinity in RHS. We get the equation x = 1 + 1/x.
Thus, option (e)i.e. 1 is the correct answer.
467.cube root(2+sqrt(5))+cube root(2-sqrt(5))equals
a) sqrt(5)-1 (b) cube root(2(c) sqrt(5)-cube root(2 d) -2 (e) none of the foregoing
Solution:
Let the number be x and cube both sides now => 4 +3(-1)x = x^3, solving we get x = 1 as the valid value.
Thus, option (b) is the correct answer.
468.The number of triplets (A, B, C) where A, B, C are subsets of {1, 2, 3} and
A/B/C is NULL, A/B is not NULL, and B/C is not NULL is (A/B denotes intersection set of A and B)
(1) 12 (2) 18 (3) 27 (4) 36 (5) 48
Solution:
Assume the Veinn-Diagram with 3 subsets A, B and C. As usual Universal set gets divided into 8 regions. The region A/B/C is NULL. Start from here.
Any element can go in the 7 regions in 7 ways, 3 elements will go in 7^3 ways. Subtract some cases as given in the question and we will have our answer.
A - Total Ways 7 regions can be filled = 7*7*7 = 343
B - Total ways A/B is null 6*6*6 = 216
C - Total ways B/C is null 6*6*6 = 216
D - Total ways A/B is null and B/C is Null = 5*5*5 = 125
E= Total way A/B or B/C is null = B+C-D = 307
=> our desired answer, A- E = 36
=> Choice (4) is the right answer
469.Divya works for Google that claims only 4 working days. In the 1st week of joining, she went for movies after 4 working days. On Friday, she noticed that the fraction p of the line is in front of her, while 1/q of the line was behind her. On Saturday, the same fraction p of the line was in front of her, while 1/(q+1) of the line was behind her. On Sunday, the same fraction p of the line was in front of her, while 1/(q+2) of the line was behind her. For how many values of p is this possible?
(a) 3 (b) 1 (c) 2 (d) no such value (e) infinitely many
Solution:
Let p = (x-1)/x for some natural number x. Let q = 1/(x+y), Divya herself represents the line => 1 - ((x-1)/x + 1/(x+y)) = y/x(x+p) => x(x+p)/y is the length of the line. To make it an integer we can make x to be a multiple of 6. For q = x+1, we want to get an integer for y=1, 2, 3.
More generally, for x(x+1), x(x+2)/2, and x(x+3)/3 to be integers we require x to be a multiple of 6.
On Friday, there are 36k^2+6k people in the queue, on saturday 18k^2+6k and on Sunday 12k^2+6k.
Hence, choice (e) is the right option
470.What is the sum of the reciprocals of all the numbers in the form (2^a)*(3^b), where a and b are non-negative integers?
(a) 3 (b) 3.5 (c) 2√3 (d) √3 + 1 (e) none of the foregoing
Solution:
Let the asked sum be x => 1+(1/2+1/4+1/8+...)+(1/3+1/9+1/27+...)+1/6(x) = x => 1+1+1/2+1/6(x) = x => x = 3
Hence, choice (a) is the right option.
471.S is a set of positive integers containing 1 and 99. No elements are larger than 99. For every n in S, the arithmetic mean of the other elements of S is an integer. What is the largest possible number of elements of S?
(1) 8 (2) 10 (3) 1 (4) 15 (5) none of these
Solution:
Let us tackle this problem with small example.
Let we have 4 numbers in S, and let these be a, b, c, d
Thus, (a+b+c)/3, (b+c+d)/3, (a+c+d)/3 and (a+b+d)/3 are each integers.
=> subtract each term from any other => each of a, b, c, d are equal modulo (3) i.e. each on division by 3 leave same remainder.
Extending this when we have n numbers with us then each will be equal modulo (n-1).
This was the funda part.
We start the calculation now -> Let we have n numbers [the modulo (n-1) is 1 as the 1st term is 1].
1, 1+(n-1).f, 1+(n-1).2f, .... 1+(n-1).(n-1)f
The last term is 1+(n-1).(n-1)f = 99 => (n-1)^2.f = 98 which yields integer value of n for f = 2. => we can at max have 8 numbers in our set.
=> Choice (1) is the right answer
472.How many integral values of p are there for which the inequality 3 - |x-p| > x^2 is satisfied by at least one negative x?
(a) 3 (b) 4 (c) 5 (d) 6 (e) none of the foregoing
Solution:
If x > p then 3-x+p > x^2 => p > -13/4. If x < p then 3-p+x > x^2 => p < 3. Thus, p lies in (-13/4, 3)
Hence, choice (d) is the right option
473.How many integral values of p are there for which the inequality x^2 + px + p^2 + 6p < 0 is satisfied for all x in (1, 2)?
(a) 3 (b) 4 (c) 5 (d) 6 (e) none of the foregoing
Solution:
If f(x) = x^2 + px + p^2 + 6p then f(1) < 0 && f(2) < 0 => p lies in ((-7-√45)/2, -4+2√3))
Hence, choice (d) is the right option.
474.For a positive integer i define p(i) as the product of base 4 digits of i e.g. 27 in base 4 is 123 => p(27) = 1*2*3 = 6. What is the numerical value of p(1) + p(2) + p(3) + ... + p(255)?
(1) 1496 (2) 1554 (3) 1592 (4) 1636 (5) 1684
Solution:
The sum of all p(i) where i has 1 digit in base 4 is (0+1+2+3) = 6
The sum of all p(i) where i has 2 digits in base 4 is (0+1+2+3)(0+1+2+3) = 6^2
The sum of all p(i) where i has 3 digits in base 4 is (0+1+2+3)(0+1+2+3)(0+1+2+3) = 6^3
The sum of all p(i) where i has 4 digits in base 4 is (0+1+2+3)(0+1+2+3)(0+1+2+3)(0+1+2+3) = 6^4
Note that 255 = 3.(1+4+4^2+4^3) is the largest decimal number that can be expressed as 4 digit number in base 4
=> Choice (2) is the right answer
475.A tower stands on a horizontal plane. From a mound 14 m above the plane and at a horizontal distance of 48 m from the tower, an observer notices a loophole and finds that the portions of the tower above and below the loophole subtend equal angles. If the height of the loophole is 30 m, then the height of the tower is
(a) 56 (b) 64 (c) 78 (d) 92 (e) data insufficient
Answer:
Choice (c) is the right option
476.Two straight roads OA and OB intersect at O. A tower is situated within the angle formed by them and subtends angles of 45˚ and 30˚ at the points A and B where the roads are nearest to it. If OA = 81 m and OB = 17 m, then what is the height of the tower?
(a) 56 (b) 64 (c) 78 (d) 92 (e) data insufficient
Answer:
Choice (a) is the right option
477.The inscribed circle of an isosceles triangle ABC is tangent to side AB at point D and bisects the segment CD. If CD = 6√2. Which among the following can not be true about ABC?
(a) The perimeter is 24
(b) It's obtuse angled
(c) The bisector segment of the smallest angle is 6√2
(d) The perimeter is 28
(e) none of the foregoing
Solution:
Let in triangle ABC the incircle meet AB at D, BC at F and AC at E.
Case 1: AB = AC
Now CE^2 = 6√2*3√2 = 36 => CE = CF = 6,
Also in an isosceles triangle AF will contain the incentre, circumcentre , centroid and orthocentre.
=> AF will bisect BC hence BF = 6 = BD.
In triangle CDB BD = 6 CD = 6√2 and BC = 12,
=> cos(< B) = 3/4 -> FB / AB = 3/4 -> AB =8.
=> the perimeter is 28 and angle A is obtuse (square of the largest side is greater than sum of square of other 2 sides) so it is an obtuse angled triangle. So (2) and (4) are true here.
Case 2: When AC = BC
we already have CE = BF = 6 and CD is also perpendicular to AB. Let AD = AE = a, applying pythagorean
theorem for triangle ADC, we get a = 3 => AC = BC = 9, and AB = 6
=> (1) and (3) are also true.
=> Choice (5) is the right answer
478.Divya works for Google that claims only 4 working days. In the 1st week of joining, she went for movies after 4 working days. On Friday, she noticed that the fraction p of the line is in front of her, while 1/q of the line was behind her. On Saturday, the same fraction p of the line was in front of her, while 1/(q+1) of the line was behind her. On Sunday, the same fraction p of the line was in front of her, while 1/(q+2) of the line was behind her. For how many values of p is this possible?
(a) 3 (b) 1 (c) 2 (d) no such value (e) infinitely many
Solution:
Let p = (x-1)/x for some natural number x. Let q = 1/(x+y), Divya herself represents the line => 1 - ((x-1)/x + 1/(x+y)) = y/x(x+p) => x(x+p)/y is the length of the line. To make it an integer we can make x to be a multiple of 6. For q = x+1, we want to get an integer for y=1, 2, 3.
More generally, for x(x+1), x(x+2)/2, and x(x+3)/3 to be integers we require x to be a multiple of 6.
On Friday, there are 36k^2+6k people in the queue, on saturday 18k^2+6k and on Sunday 12k^2+6k.
Hence, choice (e) is the right option
479.What is the sum of the reciprocals of all the numbers in the form (2^a)*(3^b), where a and b are non-negative integers?
(a) 3 (b) 3.5 (c) 2√3 (d) √3 + 1 (e) none of the foregoing
Solution:
Let the asked sum be x => 1+(1/2+1/4+1/8+...)+(1/3+1/9+1/27+...)+1/6(x) = x => 1+1+1/2+1/6(x) = x => x = 3
Hence, choice (a) is the right option.
480.The number of triplets (A, B, C) where A, B, C are subsets of {1, 2, 3} and
A/B/C is NULL, A/B is not NULL, and B/C is not NULL is (A/B denotes intersection set of A and B)
(1) 12 (2) 18 (3) 27 (4) 36 (5) 48
Solution:
Assume the Veinn-Diagram with 3 subsets A, B and C. As usual Universal set gets divided into 8 regions. The region A/B/C is NULL. Start from here.
Any element can go in the 7 regions in 7 ways, 3 elements will go in 7^3 ways. Subtract some cases as given in the question and we will have our answer.
A - Total Ways 7 regions can be filled = 7*7*7 = 343
B - Total ways A/B is null 6*6*6 = 216
C - Total ways B/C is null 6*6*6 = 216
D - Total ways A/B is null and B/C is Null = 5*5*5 = 125
E= Total way A/B or B/C is null = B+C-D = 307
=> our desired answer, A- E = 36
=> Choice (4) is the right answer
481.The number of real pairs (p, q) such that whenever α is the root of x^2 + px+q = 0 then α^2 - 2 is also its root is
(1) 2 (2) 4 (3) 6 (4) 3 (5) none of these
482.How many ordered pairs ( P,Q) are there such that the unit’s digit of PP QQ and are same ?( Given that P and Q are natural numbers less than 10)
1)12 (2) 14 (3) 10 (4) 15 (5)18
483.In a box containing 15 apples, exactly 6 are rotten. Each day one apple is taken out from the box. What is the probability that after four days there are exactly 8 apples in the box that are not rotten ?
1)12/91 2)1/7 (2) 2/13 (3) 2/7 (4)5/17 (5)none of these
484.78 identical cubes each with 2 cm edge are joined together to form a cuboid. If the perimeter of the base of cuboid is 64 cm, then the number of cubes along the height of the cuboid is
1)3 (2) 6 (3) 2 (4) 4 (5)cannot be determined
485.Given that a5 + b5 + c5 = 91849 , where a,b,c are distinct digits. What is the remainder when a six digit number 2a5b1c is divided by 11 ?
1)4 (2) 6 (3) 2 (4) 4 (5)none of these
486.Sarah and Neha start running simultaneously from the diametrically opposite ends of a circular track towards each other at 15km/h and 25km/h respectively. After every 10 minutes their speed reduces to half of their current speeds. If the length of the circular track is 1500 m, how many times will Sarah and Neha meet on the track?
(A) 6 (B) 9 (C) 11 (D) 7 (E) 8
Solution:
Total distance Shrikant can cover = 10/60*(15 + 15/2 + 15/4 + ...) = 5 km
Total distance Sachin can cover = 10/60*(25 + 25/2 + 25/4 + ...) = 25/3 km
First time they cover 750 m and subsequently they cover a distance of 1500 m to meet.
The total distance they cover together is 40/3 km.
Number of meetings possible is 1 + (40000/3 - 750)/1500 = 9.4
=> choice (B) is the right answer
487.A cone of radius 1 unit and height 2 units just fits inside a cylinder with their axis perpendicular. The radius of the cylinder in unit is
(A) 1 (B) 1.25 (C) 1.5 (D) 2 (E) more than 2
Solution:
Triangle ABC be the cross section of cone and circle K be the cross section Of Cylinder
Let AM be perpendicular to BC
clearly BC=2( diameter of cone)
AM=2 (Height of cone)
Let O be the center of K , so O lies on AM
In right traingle OBM , OB=r, BM=1 and OM=AM-AO=2-r
r^2=1+(2-r)^2
r=1.25
=> choice (B) is the right answer
488.There are a certain facts about the software engineers working with an IT firm. Total number of software engineers is 70 out of which 30 are females. 30 people are married. 24 software engineers are above 25 years of age. Out of all married software engineers, 19 are above 25 years, of which 7 are males. 12 males are above 25 years and overall 15 males are married. How many unmarried females are there which are above 25?
(A) 12 (B) 8 (C) 7 (D) 0 (E) 11
Solution:
15 males are married, so 15 females are married
now 7 males above 25 are married. so (19-7)=12 females above 25 are married
given that 12 males are above 25 years
so 24-12 =12 females are above 25 years
so all females above 25 years are married
this means 0 unmarried female above 25
=> choice (D) is the right answer
DIRECTIONS for Questions 4 and 5: Each question is followed by two statements X and Y. Answer each question using the following instructions:
Choose A if the question can be answered by X only
Choose B if the question can be answered by Y only
Choose C if the question can be answered by either X or Y
Choose D if the question can be answered by both X and Y
Choose E if the question can be answered by neither X and Y
489.Let p(x) = x^2 + 40. Then for any two positive integers i and j where i > j, is p(i) + p(j) a composite number?
(X) p(i) – p(j) is not a composite number
(Y) p(2i) + p(2j) is a composite number
Solution:
p(i) – p(j) is not a composite number
=>i^2-j^2 is a prime as i ,j are positive integers and i >j, ( i^2-j^2) can't be 1
=>(i+j)(i-j)= prime
so i-j=1
let p be the prime so i=(p+1)/2
j=(p-1)/2
clearly p is not 2 hence all p is odd
p(i) + p(j)=80 +(p^2+1)/2
now p^2=6k+1 ( can be easily proved) [ p(p-1)(p+1) is divisible by 6 now p is a prime so p^2-1=6k p^2=6k+1, for any prime p greater than 3]
therefore
p(i) + p(j)=80 +(p^2+1)/2
becomes
80+(6k+2)/2=81+3k=3(27+k)
so not a prime => can be answered by using X
now, p(2i) + p(2j) is a composite number
4(i^2+j^2+20) is composite
now i and j can be anything
can't make any conclusions
=> choice (A) is the right answer
490. What is the length of the side AB of triangle ABC?
(X) AB <= AC = 2, and area of triangle ABC is 2
(Y) Exactly two sides have integer length
Solution:
Area = ½*(AB).(AC).sin A => 2 = ½*AB.2.sinA => ABsinA= 2.
But AB <=2 and sin A <= 1 => AB = 2.
=> choice (A) is the right answer
491 The numerical value of f(1/10) + f(2/10) + …. + f(9/10), where f(x) = 9^x/(3+9^x) is
(A) 10/3 (B) 4 (C) √10 (D) 5 (E) 9/2
Solution:
f(x) = 9^x/(3+9^x)=1-3/(3+9^x)
now look at this
3/(3+9^x) +3/(3+9^(1-x))= [9+9+3(9^x++9^(1-x))]/[9+9+3(9^x++9^(1-x))]=1
f(1/10) + f(2/10) + …. + f(9/10)=9-4 -3/(3+sqrt(9))=9-4-1/2=9/2
=> choice (E) is the right answer
492. A blackboard bears a half-erased mathematical calculation exercise:
2 3 ? 5 ?
+ 1 ? 6 4 2
-------------
4 2 4 2 3
In which number system was this calculation performed?
(A) 4 (B) 9 (C) 8 (D) 5 (E) 7
Solution:
23A5B
1c642
-------
42423
clearly number system is greater than 6
so we get 7, 8, 9, now clearly B=1
now the moment B is 1, there is no carry
gives 5+4=9, 9=2(mod7)
hence number system is 7
=> choice (E) is the right answer
493. Given points P1, P2, P3, …, P7 on a straight line, in the order stated (not necessarily evenly spaced). Let P be an arbitrary point selected on the line and let s be the sum of undirected lengths PP1, PP2, PP3, …, PP7. Then s is smallest if and only if the point P is
(A) midway between P1 and P7
(B) midway between P2 and P6
(C) midway between P3 and P5
(D) at P4
(E) at P1
Solution:
Let P1, P2, P3, …, P7 be points in a co-ordinate plane such that P1 = (0, 0), P2 = (a, 0), P3 = (b, 0), …., P7 = (f, o) such that a < b < … < f.
Let X be arbit point on P1P7 such that X = (x, o) where a <= x <= f.
We want to minimize s = x + |x-a| + |x-b| + |x-c| + |x-d| + |x-e| + |x-f|
If X = P4 then s = c + (c-a) + (c-b) + 0 + (d-c) + (e-c) + (f-c) = d+e+f-a-b
If X = P1 then s = a + 0 + (b-a) + (c-a) + (d-a) + (e-a) + (f-a) = d=e+f-2a > d+e+f-a-b
For choice (A) -> X = (a+f)/2, (B) -> X = (b+e)/2, (C) -> X = (c+d)/2 we see that none is < d+e+f-a-b
=> choice (D) is the right answer
494. A 33 rpm record which normally plays for 30 minutes was inadvertently started at 45 rpm, then switched to 33 rpm when mistake was realized. Altogether the record played for 26 minutes. For how many minutes was it playing at 45 rpm?
(A) 9 (B) 10 (C) 11 (D) 8 (E) 12
Solution:
33*30=45x+33(26-x)
33.4=12x
x=11
=> choice (C) is the right answer
495. Let S be a 6 element set. Then the number of pairings (3 pairs) of S is
(A) 9 (B) 12 (C) (D) 20 (E) 27
Solution:
{(1, 2), (3, 4), (5, 6)} is one pairing of 3 pairs.
Keeping (1, 2) as constant we can have {(3, 5), (4, 6)} or {(3, 6), (4, 5)}
ð for (1, 2) we have 3 set of pairs to map.
ð For (1, x) where x varies from 2 to 6 we will have in all 5*3 = 15 pairing.
=> choice © is the right answer
Directions for questions 496 to 497 :
The cost of 1 kg of sugar is Rs 20 while the cost of 1 litre of pure milk is Rs 15. Sweetened milk is prepared by adding a fixed amount of sugar in a litre of milk.
496. If the cost of sweetened milk is Rs 15 per kg, then it can be concluded that the weight of x litre pure milk is y kg more than a litre of sweetened milk where (x, y) is
(A) (4, 5) (B) (4, 3) (C) (5, 3) (D) (2, 1) (E) (3, 2)
Solution:
Let the weight of pure milk be a kg/liter
assume b kgs of sugar is added to 1 liter of pure milk
so now cost is 15+20b=15(a+b), 3+b=3a
now y=ax-a-b=a(x-1)-3a+3
if we put x=4 y is 3 (x-1=3)
=> choice (B) is the right answer
497. If the cost of 1 litre of sweetened milk is Rs 16 and its weight is 1.25 kg, then the weight of 1 litre of pure milk is
(A) 1.05 kg (B) 1.04 kg (C) 1.10 kg(D) 1 kg (E) none of these
Solution:
This is sitter;
=> choice (D) is the right answer
Directions for questions 498 to 499 :
A point X is taken on a circle having its centre at O and a chord is drawn at an angle 50 ˚ to OX clockwise to cut the circle at Y. Again a chord is drawn at an angle 50 ˚ to OY clockwise and this process continues until a chord in subsequent process intersects the circle at X.
498. After how many revolution(s) of the circle, the process is completed?
(A) 1 (B) 2 (C) 4 (D) 8 (E) 9
Solution:
so we need 360*k=80.m
k =1 gives no integer value of m
clealry for k=2 we can find m=9, so no of revolutions is 2
=> choice (B) is the right answer
499. At how many points the chords drawn during the process intersect each other inside the circle?
(A) 12 (B) 6 (C) 7 (D) 8 (E) 9
Solution:
clearly the first 4 chords won't intersect
the fifth chord cuts the first
teh sixth cuts the first and second
seventh cuts second and third
eighth cuts third and fourth
ninth cuts fourth and fifth
so 9 internal intersections
=> choice (E) is the right answer
500. If a is a nonzero integer and b is a positive number such that ab^2=log b what is the median of the set {0,1, a,b,1/b} ?
(A) 0 (B) 1 (C) a (D) b (E)1/b
Solution:
Clearly, 0 < b < 1 => a < -1 (always). Thus, a < 0 < b < 1 < 1/b
=> choice (D) is the right answer
501. A wooden cube n unit on a side is painted red on all six faces and then cut into n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n ?
(A) 4 (B) 3 (C) 2 (D) 6 (E) none of these
Solution:
Total cubes is 6n^3
colored cubes is (1/4)6n^3
now only the six outer faces are colored = 6n^2
=> 6n^2= (1/4)6n^3 => n=4
=> choice (A) is the right answer
501. Katrina, her brother, her son and her daughter are chess players (all relations by birth). The worst player’s twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are of the same age. Who is the worst player?
(A) Katrina(B) Her brother(C) Her son(D) Her daughter (E) Can not be determined
Solution:
If Katrina is worst player , the best player is female , but this defies other condition that the best player and worst player are of same age
now if her brother is worst player best player is then her son. this means katrina her son and her brother are of same age. not possible again
if her son is worst player, the best player is male so he must be katrina's brother
now if we assume that her son and her brother are of same age this can happen.
all other cases are impossible
=> choice (C) is the right answer
502. An elastic string laying along the interval [-2, 2] on the x-axis is stretched uniformly and displaced so that it lays along [3, 9]. What is the new location of the point of the string which was formerly at x = 1?
(A) 8 (B) 6 (C) 4.5 (D) 4 (E) 7.5
Solution:
length of string previously= 4
now = 6, so unit length of string will now become= 6/4= 3/2.....(i)
now part of string at x= 1 means 3 units from starting of string....
means now it will be at 3x3/2 units from starting = 4.5 units from starting..
starting is at x=3.. hence now it will shift to x= 3+4.5 = 7.5...answer
503. A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length x. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is x?
(A) 2sqrt5 (B) 1 (C) 2 (D) 2sqrt3 (E) none of these
504. The number of values of k for which the roots of the equation kx^3 + 2x^2 – 3x + 1 = 0 are in harmonic progression is
(A) 0 (B) 1 (C) 2 (D) 3 (E) more than 3
Solution:
if we put y=1/x
we will get an equation which has roots in AP
y^3-3y^2+2y+k=0
let roots be a-b ,a, a+b => 3a=3 => a=1
Also, a(a-b)+a(a+b)+(a^2-b^2)=2
3-b^2=2
b^2=1
k=-a(a^2-b^2)=-1(1-1)=0
but for this value of k we get one of the roots as zero !
which makes the root of original equation -> infinity!
=> No solution
Hence, choice (A) is the right answer
505. Vineet attends an IPL game in Delhi and estimates that there are 50,000 fans in attendance. Rahul attends an IPL game in Bangalore and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
i. The actual attendance in Delhi is within of Vineet's estimate.
ii. Rahul's estimate is within of the actual attendance in Bangalore.
To the nearest 1,000, the largest possible difference between the numbers attending the two games is
(A) 10,000 (B) 11,000 (C) 20,000 (D) 21,000 (E) 22,000
Solution:
Actual attendance in Delhi can at least be 45, 000
Actual attendance in Bangalore can at most be 60, 000/(0.9) = 66, 666
=> choice (E) is the right answer
Directions for questions 506 to 507:
A Latin square of order n is an arrangement of n symbols in n rows and n columns such that each symbol appears exactly once in each row and each column. For example, two Latin squares of order three are shown below.
0 1 2 1 0 2
1 2 0 2 1 0
2 0 1 0 2 1
Two Latin squares of order n are said to be orthogonal if, upon superimposition of one on the other, each of the n² possible ordered pairs of symbols appears in exactly one cell. For example, the Latin squares L and M, as shown above, are orthogonal since superimposing M on L, we get the structure:
01 10 22
12 21 00
20 02 11
Where in each cell the first entry comes from L and the second entry comes from M. Note that each of the nine possible ordered pairs 00, 01, 02, 10, 11, 12, 20, 21, 22, appears exactly once in each cell. This shows that L and M are orthogonal.
506. The number of distinct Latin squares of order three, with symbols 0, 1, 2 each of which is orthogonal to both L and M shown above is
(A) 0 (B) 1 (C) 2 (D) 3 (E) 6
507. The numbers of distinct Latin squares of order four, with symbols 0, 1, 2, 3 that can be formed by completing the following incomplete structure equals
0 2
1
2
3
(A) 0 (B) 1 (C) 3 (D) 6 (E) 24
508. Consider the following two Latin squares of order four:
0 1 2 3 x y z w
1 0 3 2 q r s t
2 3 0 1 2 3 1 0
3 2 1 0 1 0 2 3
Here each of x, y, z, w, q, r, s, t belongs to the set {0, 1, 2, 3}. If the two Latin squares are orthogonal, then x equals
(A) 0 (B) 1 (C) 2 (D) 3 (E) can not be determined
509. Suppose we have two Latin squares P and Q which are orthogonal. Both P and Q are of order n(>2) and both of them have the symbols 0, 1, . . . , n – 1. Now suppose, a new square R is formed from Q by replacing the symbols 0, 1, 2 in Q by 1, 2, 0 respectively and keeping the other symbols in Q unchanged. Then the P and R are orthogonal Latin squares
(A) For no choice of n (> 2) (B) Only for odd n (> 2)
(C) Only for even n (> 2) (D) For every n (> 2)
(E) None of the above
510. Consider the set S = {2, 3, 4, …., 2n+1}, where n is a positive integer larger than 2007. Define X as
the average of the odd integers in S and Y as the average of the even integers in S. What is the value of
X - Y?
(1) 0 (2) 1 (3) 1/2*n (4) {n +1}/2n (5) 2008
511. Ten years ago, the ages of the members of a joint family of eight people added up to 231 years.
Three years later, one member died at the age of 60 years and a child was born during the same year.
After another three years, one more member died, again at 60, and a child was born during the same
year. The current average age of this eight- member joint family is nearest to
(1) 23 years (2) 22 years (3) 21 years (4) 25 years (5) 24 years
f (1) + f (2) + ... + f (n ) = n 2 f ( n) , for all positive
512. A function f (x ) satisfies f (1) = 3600, and
integers n > 1. What is the value of f (9) ?
(1) 80 (2) 240 (3) 200 (4) 100 (5) 120
513. Suppose you have a currency, named Miso, in three denominations : 1 Miso, 10 Misos and 50
Misos. In how many ways can you pay a bill of 107 Misos?
(1) 17 (2) 16 (3) 18 (4) 15 (5) 19
514. A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja,
giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise,
Shailaja noticed that she was left with exactly three times as much as the amount on the cheque.
Which of the following is a valid statement about the cheque amount?
(1) Over Rupees 13 but less than Rupees 14
(2) Over Rupees 7 but less than Rupees 8
(3) Over Rupees 22 but less than Rupees 23
(4) Over Rupees 18 but less than Rupees 19
(5) Over Rupees 4 but less than Rupees 5
515. How many pairs of positive integers m, n satisfy
1/m plus 4/n equals 1/12, where n is an odd integer less than 60?
(1) 6 (2) 4 (3) 7 (4) 5 (5) 3
Directions for Questions 516 through 519: Each question is followed by two statements A and B.
Indicate your responses based on the following directives:
Mark (1) if the question can be answered using A alone but not using B alone.
Mark (2) if the question can be answered using B alone but not using A alone.
Mark (3) if the question can be answered using A and B toge ther, but not using either A or B alone.
Mark (4) if the question cannot be answered even using A and B together.
516. The average weight of a class of 100 students is 45 kg. The class consists of two sections, I and II,
each with 50 students. The average weight, WI , of Section I is smaller than the average weight, WII ,
of Section II. If the heaviest student, say Deepak, of Section II is moved to Section I, and the lightest
student, say Poonam, of Section I is moved to Section II, then the average weights of the two sections
are switched, i.e., the average weight of Section I becomes WII and that of Section II becomes WI .
What is the weight of Poonam?
A: WII − WI = 1.0
B: Moving Deepak from Section II to I (without any move from I to II)
makes the average weights of the two sections equal.
517. ABC Corporation is required to maintain at least 400 Kilolitres of water at all times in its factory,
in order to meet safety and regulatory requirements. ABC is considering the suitability of a spherical
tank with uniform wall thickness for the purpose. The outer diameter of the tank is 10 meters. Is the
tank capacity adequate to meet ABC’s requirements?
A: The inner diameter of the tank is at least 8 meters.
B: The tank weighs 30,000 kg when empty, and is made of a material with density of
3 gm/cc.
518. Consider integers x, y and z. What is the minimum possible value of x 2 + y 2 + z 2 ?
x + y + z = 89
A:
B: Among x, y, z two are equal.
519. Rahim plans to draw a square JKLM with a point O on the side JK but is not successful. Why is
Rahim unable to draw the square?
A: The length of OM is twice that of OL.
B. The length of OM is 4 cm.
Directions for Questions 520 and 521:
Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes
the schedule of an airline operating non-stop flights between A and B. All the times indicated are
local and on the same day.
City Departure Time City Arrival Time
B 8:00 am A 3:00 pm
A 4:00 pm B 8:00 pm
Assume that planes cruise at the same speed in both directions. However, the effective speed is
influenced by a steady wind blowing from east to west at 50 km per hour.
520. What is the time difference between A and B?
(1) 1 hour and 30 minutes (2) 2 hours (3) 2 hours and 30 minutes
(4) 1 hour (5) Cannot be determined
521. What is the plane’s cruising speed in km per hour?
(1) 700 (2) 550 (3) 600 (4) 500 (5) Cannot be determined.
Directions for Questions 522 and 523:
Shabnam is considering three alternatives to invest her surplus cash for a week. She wishes to
guarantee maximum returns on her investment. She has three options, each of which can be utilized
fully or partially in conjunction wit h others.
Option A: Invest in a public sector bank. It promises a return of +0.10%.
Option B: Invest in mutual funds of ABC Ltd. A rise in the stock market will result in a return of +5%,
while a fall will entail a return of –3%.
Option C: Invest in mutua l funds of CBA Ltd. A rise in the stock market will result in a return of
–2.5%, while a fall will entail a return of +2%.
522. The maximum guaranteed return to Shabnam is
(1) 0.25% (2) 0.10% (3) 0.20% (4) 0.15% (5) 0.30%
523. What strategy will maximize the guaranteed return to Shabnam?
(1) 100% in option A
(2) 36% in option B and 64% in option C
(3) 64% in option B and 36% in option C
(4) 1/3 in each of the three options
(5) 30% in option A, 32% in option B and 38% in option C
524. If F(x, n) be the number of ways of distributing "x" toys to "n" children so that each child
receives at the most 2 toys then F(4,3)=_____?
A.2 B.3 C.4 D.5 E.6
525. In a cricket match. Team A scored 232 runs without losing a wicket. The score consisted of byes, wides and runs scored by two opening batsmen: Ram and Shyam. The runs scored by the two batsmen are 26 times wides. There are 8 more byes than wides. If the ratio of the runs scored by Ram and Shyam is 6:7, then the runs scored by Ram is
A. 88
B. 96
C. 102
D. 112
E. None of the above.
For questions 526 & 527, a statement is followed by three conclusions. Select the answer from the
following options.
A. Using the given statement, only conclusion I can be derived.
B. Using the given statement, only conclusion II can be derived.
C. Using the given statement, only conclusion III can be derived.
68
Y=29
9
Y 4 5 2 X
D. Using the given statement, all conclusions I, II and III can be derived.
E. Using the given statement, none of the three conclusions I, II and III can be derived.
526. An operation "#" is defined by a # b = 1 – b/a
Conclusion I. (2 # 1) # (4 # 3) = -1
Conclusion II. (3 #1) # (4 #2) = -2
Conclusion III. (2#3) # (1#3) = 0
527. A, B, C and D are whole numbers such that
A+B+C =118
B + C + D = 156
C + D + A = 166
D + A + B = 178
Conclusion I. A is the smallest number and A=21.
Conclusion II. D is the largest number and D=88.
Conclusion III. B is the largest number and B=56.
528. Let X= {a, b, c} and Y= {l, m}. Consider the following four subsets of X x Y.
F1 = { (a, l), (a, m), (b, l), (c, m)}
F2={(a, l), (b, l), (c, l)}
F3={(a, l), (b, m), (c, m)}
F4={(a, l), (b, m)}
Which one, amongst the choices given below, is a representation of functions from X to Y?
A. F1, F2 and F3
B.F2, F3 and F4
C. F2 and F3
D. F3 and F4
E. None of the above
529. ABCD is a rectangle with AD = 10. P is a point on BC such that angle APD = 90° If DP = 8 then the length of BP is __?
A.6.4
B.5.2
C.4.8
D.3.6
E. None of the above
530. Four digits of the number 29138576 are omitted so that the result is as large as possible. The largest omitted digit is
A.9
B.8
C.7
D.6
E.5
531. Rajiv is a student in a business school. After every test he calculates his cumulative average. QT and OB were his last two tests. 83 marks in QT increased his average by 2. 75 marks in OB further increased his average by 1. Reasoning is the next test, if he gets 51 in Reasoning, his average will be ___?
A.59
B.60
C.61
D.62
E.63
532.ABCD is a square. P is the midpoint of AB. The line passing through A and perpendicular to DP intersects the diagonal at Q and BC at R.
If AB = 2 then PR = ___?
A. 1/2
B.3
C.4
D. 1
E. None of the above.
533. For each p>1, a sequence {An} is defined by Ao = 1 and An = pn + (-1)n An-1 for each n >= 1. For how many integer values of p, 1000 is a term of the sequence?
A.8
B.7
C.5
D.4
E. None of the above.
534. If 0
B. Imaginary
C. Real and both positive
D. Real and both negative
E. Real and of opposite sign.
535.Consider a sequence –6, –12, 18, 24, –30, –36, 42,..... . If sum of the first n terms of thensequence is 132, then the value of n is ___?
A. 11
B. 13
C. 18
D. 22
E. 24
536. The number of possible real solution(s) of y in equation y^2 – 2y cos x + 1 = 0 is ___?
A. 0
B. 1
C. 2
D. 3
E. None of the above.
537. In a triangle ABC, AB = 3, BC =4 and CA = 5. Point D is the midpoint of AB, point E is on segment AC and point F is on segment BC. If AE = 1.5 and BF = 0.5 then angle DEF =
A. 30°
B. 45°
C. 60°
D. 75°
E. Cannot be determined
538. If 3f(x + 2) + 4f(1/x+2)) = 4x, x ≠ –2, then f(4) =
A. 7
B. 52/7
C. 8
D. 56/7
E. None of the above.
539. A train left station X at A hour B minutes. It reached station Y at B hour C minutes on the same day, after traveling for C hours A minutes (clock shows time from 0 hours to 24 hours). Number of possible value(s) of A is __.
A. 3
B. 2
C. 1
D. 0
E. None of the above.
Questions 540-541: A, B, C, D, E and F are six positive integers such that
B+C+D+E = 4A
C+F = 3A
C + D + E = 2F
F = 2D
E + F = 2C + 1
If A is a prime number between 12 and 20, then
540. The value of C is
A. 13
B. 17
C. 19
D. 21
E. 23
541. The value of F is
A. 14
B. 16
C. 20
D. 24
E. 28
542. Which of the following must be true?
A. D is the lowest integer and D = 14
B. C is the greatest integer and C = 23
C. F is the greatest integer and F = 24
D. B is the lowest integer and B = 12
E. A is the lowest integer and A = 13
543. ABCD is a quadrilateral. The diagonals of ABCD intersect at the point P. The area of the triangles APD and BPC are 27 and 12, respectively. If the areas of the triangles APB and CPD are equal then the area of triangle APB is
A.12
B.15
C.16
D.18
E.21
544. Two circles of radius 1 cm. touch at point P. A third circle is drawn through the points A, B, and C such that PA is the diameter of the first circle, and BC –perpendicular to AP – is the diameter of the second circle. The radius of the third circle is __ cms.
A. 9/5
B. 7/4
C. 5/3
D. 1/2
E. 2
In the questions 545-546, one statement is followed by three conclusions. Select the appropriate
answer from the options given below.
A. Using the given statement, only conclusion I can be derived.
B. Using the given statement, only conclusion II can be derived.
C. Using the given statement, only conclusion III can be derived.
D. Using the given statement, conclusions I, II and III can be derived.
E. Using the given statement, none of the three conclusions I, II and III can be derived.
545. A0, A1, A2 ,..…… is a sequence of numbers with A0= 1, A1 =3,
and At = (t +1)At-1 –t At - 2 for t = 2, 3, 4,…
Conclusion I. A8 =77
Conclusion II. A10 = 121
Conclusion III. A12 = 145.
546. A, B, C be real numbers satisfying A < B < C, A + B + C = 6 and AB + BC + CA = 9
Conclusion I. 1 < B < 3
Conclusion II. 2 < A < 3
Conclusion III. 0
A. If Statement I alone is sufficient to answer the question.
B. If Statement II alone is sufficient to answer the question.
C. If Statement I and Statement II together are sufficient but neither of the two alone is sufficient
to answer the question.
D. If either Statement I or Statement II alone is sufficient to answer the question
E. Both Statement I and Statement II are insufficient to answer the question.
547. The base of a triangle is 60 cms, and one of the base angles is 60°. What is length of the
shortest side of the triangle?
I. The sum of lengths of other two sides is 80 cms.
II. The other base angle is 45°.
548. A, B, C, D, E and F are six integers such that E < F, B > A, A < D < B. C is the greatest integer. Is A the smallest integer?
I. E+B
A. 4
B. 3
C. 2
D. 1
E. 0
Questions 550-551; In second year, students at a business school can opt for Systems,
Operations, or HR electives only. The number of girls opting for Operations and the number of
boys opting for Systems electives is 37. Twenty-two students opt for operations electives. Twenty
girls opt for Systems and Operations electives. The number of students opting for Systems
electives and the number of boys opting for Operations electives is 37. Twenty-five students opt
for HR electives.
550. The number of students in the second year is_?
A.73
B.74
C.75
D.76
E.77
551. If 20% of the girls opt for HR electives, then the total number of boys in the second year is _?
A.50
B.51
C.52
D.53
E.54
Answer the question nos. 552 to 553 on the basis of the data given below:
Area/Month January February March
Sale in Bistupur
Television
900 1050 1200
Ipods 15750 16800 17850
Sales in Sakchi
Television 1800 2100 2400
Ipods 9450 10080 10710
Sales in Kadma
Television 6300 7350 8400
Ipods 6300 6720 7140
Units ordered=Units Sold + Ending Inventory – Beginning Inventory
All sales figures are in Rupees thousand.
All other things are constant
All Rupees figures are in thousands
552. In the period from January to March, Jamshedpur Electronics sold 3150 units of television, having started with a beginning inventory of 2520 units and ending with an inventory of 2880. What was value of order placed (Rupees in thousands) by Jamshepur Electronics during the three month period? [Profits are 25% of cost price, uniformly.]
A. 28080
B. 2808
C. 26325
D. 22320
E. 25200
553. What was the total value of surcharge paid – at the rate of 14% of sales value – by Jamshedpur Electronics, over the period of three months?
A. Cannot be calculated
B.18522
C.18548
D.18425
E.18485
554. 10% of sales price of Ipods and 20% of sales price of television contribute to the profits of Jamshedpur Electronics. How much profit did the company earn in the month of January from Bistupur and Kadma from the two products?
A. 513
B. 4410
C. 3645
D. 5230
E. 5350
555. In the period from January to March, consider that Jamshedpur Electronics ordered 7560 units of Ipods for all three areas put together. What was unit sales price of Ipod during the period? The ending inventory was 6120 units and the beginning inventory stood at 5760.
A. 14.00
B. 14.65
C. 14.80
D. 13.00
E. 13.60
556. For Jamshedpur Electronics beginning inventory was 720 for televisions and 1800 for Ipods and ending inventory was 840 for televisions and 1920 for Ipods in the month of January. How many units of televisions and Ipods did Jamshedpur Electronics order for the month of January?
Additional data: In the month of February, 1050 units of televisions and 2400 units of Ipods were sold in all three areas put together.
A.1020,2270
B.1020,2370
C.2270,1030
D.1030,2370
E.1020, 2280
Answer question nos. 557 through 82 on the basis of the data given below.
Gender bias is defined as disproportion in percentage of drop-out rate of the two genders.
Drop-Out Rates, in percentage, at Primary, Elementary and Secondary Classes in
India
Primary
(I - V) Classes
Elementary
(I-VIII) Classes
Secondary
(I- X) Classes
Year Boys Girls Total Boys Girls Total Boys Girls Total
1996-97 39.7 40.9 40.2 54.3 59.5 56.5 67.3 73.7 70.0
1997-98 37.5 41.5 39.2 53.8 59.3 56.1 66.6 73.0 69.3
1998-99 40.9 41.3 41.5 54.2 59.2 56.3 64.5 69.8 66.7
1999-00 38.7 42.3 40.3 52.0 58.0 54.5 66.6 70.6 68.3
2000-01 39.7 41.9 40.7 50.3 57.7 53.7 66.4 71.5 68.6
2001-02 38.4 39.9 39.0 52.9 56.9 54.6 64.2 68.6 66.0
2002-03 35.8 33.7 34.8 52.3 53.5 52.8 60.7 65.0 62.6
2003-04 33.7 28.6 31.5 51.9 52.9 52.3 61.0 64.9 62.7
2004-05 31.8 25.4 29.0 50.4 51.2 50.8 60.4 63.8 61.9
557. Based on the data above, choose the true statement from the following alternatives:
A. Gender bias in primary education has consistently decreased over the years.
B. Gender bias decreases as students move from primary to secondary classes.
C. Total drop-out rate decreased consistently for primary classes children from 1996-97 to 2004–
05
D. Gender bias was consistently highest for secondary classes.
E. None of the above.
558. Assume that girls constituted 55% of the students entering school. In which year, as
compared to the previous year, number of boys in secondary education would be more than the
number of girls?
A.1997-98
B. 1996-97
C.2000-01
D.1998-99
E. 2001-02
559. Suppose, every year 7,000 students entered Class I, out of which 45% were boys. What was
the average number (integer value) of girls, who remained in educational system after elementary
classes, from 1996-97 to 2004-05?
A. 1475
B. 1573
C. 1743
D. 1673
E. 3853
560. Suppose the total number of students in 1996-97 were 1,000 and the number of students increased every year by 1000, up to 2004-05. The total number of drop-outs from primary classes, from 1996-97 to 2004-05, were (approximately) _?
A. 18500
B. 19500
C. 16000
D. 24500
E. 11500
Question No 561-564:
There are exactly ten stores and no other buildings on a straight street in
Bistupur Market. On the northern side of the street, from West to East, are stores 1, 3, 5, 7, and 9; on the southern side of the street, also from West to East, are stores 2, 4, 6, 8, and 10. The stores on the northern side are located directly across the street from those on the southern side, facing each other in pairs, as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the following colors: green, red and yellow. The stores
have been decorated with lights according to the following conditions:
No store is decorated with lights of the same colors as those of any store adjacent to it. No store is decorated with lights of the same color as those of the store directly across the street from it. Yellow lights decorate exactly one store on each side of the street. Red lights decorate store 4.Yellow lights decorate store 5.
561. Which one of the following could be an accurate list of the colors of the lights that decorate stores 2, 4, 6, 8 and 10, respectively?
A. green, red, green, red, green
B. green, red, green, yellow, red
C. green, red, yellow, red, green
D. yellow, green, red, green, red
E. yellow, red, green, red, yellow
562. If green lights decorate store 7, then each of the following statements could be false EXCEPT:
A. Green lights decorate store 2
B. Green lights decorate store 10
C. Red lights decorate store 8
D. Red lights decorate store 9
E. Yellow lights decorate store 2
563. Which one of the following statements MUST be true?
A. Green lights decorate store 10
B. Red lights decorate store 1
C. Red lights decorate store 8
D. Yellow lights decorate store 8
E. Yellow lights decorate store 10
564. Suppose that yellow lights decorate exactly two stores on the south side of the street and exactly one store on the north side. If all other conditions remain the same, then which one of the following statements MUST be true?
A. Green lights decorate store 1
B. Red lights decorate store 7
C. Red lights decorate store 10
D. Yellow lights decorate store 2
E. Yellow lights decorate store 8
Question No 565-568: Professor Mukhopadhay works only on Mondays, Tuesdays, Wednesdays,Fridays, and Saturdays. She performs four different activities - lecturing, conducting quizzes,evaluating quizzes, and working on consultancy projects. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule MUST satisfy the following restrictions:
She conducts quizzes on exactly three mornings. If she conducts quizzes on Monday, she does not conduct a quiz on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She evaluates quizzes on exactly one morning and three afternoons. She works on consultancy project on exactly one morning. On Saturday, she neither lectures nor conducts quizzes.
565. On Wednesdays, the Professor could be scheduled to
A. Work on a consultancy project in the morning and conduct a quiz in the afternoon
B. Lecture in the morning and evaluate quizzes in the afternoon
C. Conduct a quiz in the morning and lecture in the afternoon
D. Conduct a quiz in the morning and work on consultancy project in the afternoon
E. Evaluate quizzes in the morning and evaluate quizzes in the afternoon
566. Which one of the following statements must be true?
A. There is one day on which she evaluates quizzes both in the morning and in the afternoon.
B. She works on consultancy project on one of the days on which she lectures.
C. She works on consultancy project on one of the days on which she evaluates quizzes.
D. She lectures on one of the days on which evaluates quizzes.
E. She lectures on one of the days on which she conducts quiz.
567. If the Professor conducts a quiz on Tuesday, then her schedule for evaluating quizzes could be
A. Monday morning, Monday afternoon, Friday morning, Friday afternoon
B. Monday morning, Friday afternoon, Saturday morning, Saturday afternoon
C. Monday afternoon, Wednesday morning Wednesday afternoon, Saturday afternoon
D. Wednesday morning, Wednesday afternoon, Friday afternoon, Saturday afternoon
E. Wednesday afternoon, Friday afternoon, Saturday morning, Saturday afternoon
568. Which one of the following must be a day on which Professor lectures?
A. Monday
B. Tuesday
C. Wednesday
D. Friday
E. Saturday
Question Nos 569-572:
During a four-week period, each one of seven previously unadvertised
products - G, H, J, K, L, M am O - will be advertised. A different pair of these products will be advertised each week Exactly one of the products will be a member of two of these four pairs. None of the other products gets repeated in any pair. Further, the following constraints must be observed:
J is not advertised during a given week unless H is advertised during the immediately preceding week.
The product that is advertised twice is advertised during week 4 but is not advertised during week 3.
G is not advertised during a given week unless either J or O is also advertised that week.
K is advertised during one of the first two weeks.
O is one of the products advertised during week 3.
569. Which one of the following could be the schedule of the advertisements?
A. Week 1: G, J; week 2: K, L; week 3: O, M; week 4: H, L
B. Week 1: H, K; week 2: J, G; week 3: O, L; week 4: M, K
C. Week 1: H, K; week 2: J, M; week 3: O, L; week 4: G, M
D. Week 1: H, L; week 2: J, M; week 3: O,G; week 4: K, L
E. Week 1: K, M; week 2: H, J; week 3: O, G; week 4: L, M
570. If L is the product that is advertised during two of the weeks, which one of the following is a product that MUST be advertised during one of the weeks in which L is advertised?
A.G
B.H
C.J
D.K
E.M
571. Which one of the following is a product that could be advertised in any of the four weeks?
A.H
B.J
C.K
D.L
E.O
572. Which one of the following is a pair of products that could be advertised during the same
week?
A. G and H
B. H and J
C. H and O
D. K and O
E. M and O
Question Nos. 572-576:
In a game, “words” (meaningful or meaningless) consist of any
combination of at least five letters of the English alphabets. A “sentence” consists of exactly six words and satisfies the following conditions:
The six words are written from left to right on a single line in alphabetical order. The sentence can start with any word, and successive word is formed by applying exactly one of three operations to the preceding word: delete one letter; add a letter; replace one letter with another. At the most
three of the six words can begin with the same letter. Except for the first word, each word is formed by a different operation used for the preceding word.
572. Which one of the following could be a sentence in the word game?
A. Bzaeak blaeak laeak paeak paea paean
B. Crobek croeek roeek soeek sxoeek xoeek
C. Doteam goleam golean olean omean omman
D. Feted freted reted seted seteg aseteg
E. Forod forol forols forpls orpls morpls
573. The last letter of the English alphabet that the first word of a sentence in the word game can
begin with is
A. t
B. w
C. x
D. y
E. z
574. If the first word in a sentence is "illicit" and the fourth word is "licit", then the third word can be
A. Implicit
B. Explicit
C. Elicit
D. Enlist
E. Inlist
575. If “clean” is the first word in a sentence and “learn” is another word in the sentence, then
which one of the following is a complete and accurate list of the positions “learn” could occupy?
A. Third
B. Second, third, fourth
C. Third, fourth
D. Third, fourth, fifth
E. Third, fourth, fifth, sixth
576. If the first word in a sentence consists of five letters, then the maximum number of letters that the fifth word in the sentence could contain is
A. Four
B. Five
C. Six
D. Seven
E. Eight
Question No 577-580:
Six square states having equal area in a country are located in North-
South direction in two columns next to each other. States are located in the given order, State 1, State 3, and State 5 are on the western side and State 2, State 4, and State 6 are on the eastern side. Within the six states, there are exactly four medical institutes, two management institutes, and two technical institutes. These eight institutions are located as follows:
No institution is in more than one of the states.
None of the states contain more than one management institute, and none contains more than one technical institute.
None of the states contain both a management institute and a technical institute.
Each management institute is located in a state that contains at least one medical institute.
The technical institutes are located in two states that do not share a common boundary.
State 3 contains a technical institute, and State 6 contains a management institute.
578. Which one of the following could be true?
A. State 1 contains exactly one medical institute
B. State 1 contains exactly one technical institute
C. State 2 contains exactly one management institute
D. State 5 contains exactly one technical institute
E. State 6 contains exactly one technical institute
579. A complete and accurate list of the states, any one of which could contain the management institute that is not in state 6, would be __.
A. 1, 4
B. 2, 4
C. 4, 5
D. 1, 4, 5
E. 1, 2, 4, 5
580. If each of the six states contain at least one of the eight institutions, then which one of the following must be true?
A. There is a management institute in state 1
B. There is a medical institute in state 2
C. There is a medical institute in state 3
D. There is a medical institute in state 4
E. There is a management institute in state 4
581. If one of the states contains exactly two medical institutes and exactly one technical institute, then which combination of three states might contain no medical institute
A. 1,3,5
B. 1, 4, 5
C. 2, 3, 5
D. 2, 4, 6
E. 4, 5, 6
Question nos. 582-583: Go through the situation and pick up the best alternative to answer.
There are five sets of digits – Set A, Set B, Set C, Set D and Set E as
shown in given diagram. Set A contains one digit, Set B contains two digits, Set C contains three digits, Set D contains two digits, and Set E contains one digit.
Rearrange the digits, across the sets, such that the number formed out of digits of Set C is a multiple of the numbers formed from digits in sets on either side. For example; in the given diagram, SET C is a multiple of digits in SET A and SET B but not of SET D and SET E.
SET A SET B SET C SET D SET E
7 28 196 34 5
582. What is the minimum number of rearrangements required to arrive at the solution? A rearrangement is defined as an exchange of positions between digits across two sets. For example: when 1 from set C is exchanged with 5 of set E, it is counted as one rearrangement.
A. 2
B. 5
C. 8
D. 3
E. 7
583. Which of the following pair of digits would occupy set A and E?
A. 2 and 4
B. 2 and 6
C. 3 and 6
D. 3 and 9
E. 4 and 8
584. Point P has coordinates (3, 2) with reference to a rectangular frame in two-dimensional space. This coordinate frame is rotated in the clockwise direction through an angle of 30° radians(pi/6.The coordinates of P with reference to the rotated frame are:
A. (3sqrt(3)/2)-1,3/2+sqrt(3))
B. (3sqrt(3)/2)+1,3/2+sqrt(3))
C. (3sqrt(3)/2)-1,3/2-sqrt(3))
D. (-3sqrt(3)/2)-1,3/2+sqrt(3))
E. None of the above
585. If 5 log 27 (y) + 2 log 9 (81y) = 20, then y is equal to:
a. 81 b. 2187 c. 729 d. 59049 e. None of the above
Solution:
5 log 27(y) + 2 log 9(81y)= 5/3log3(y)+2/2log3(y) 3
On solving:
log 3 (y) = 6
y = 3^6 = 729
Hence, (c).
586. An employee joined a company on 1.4.2004 in the salary grade of Rs. 8000 – 500 – 9500 – 750 –12500 with a basic salary of 9000. He is due to retire on 31.3.2007. He contributes 10% of his basic salary to an EPF scheme. His employer contributes an equal amount. If, on retirement, he gets full amount of his share of EPF and 50% of the employer’s share, the amount of EPF received by him, ignoring any interest earned on the deposits, will be:
a. 34500 b. 17250 c. 43500 d. 51750 e. None of the above
Clue: d
Find his EPF and 50% of employer’s share from 1-4-2004 to 31-3-2005
= Rs. 16200
His amount from 1-4-2005 to 31-3-2006
= Rs. 17100
His amount from 1-4-2006 to 31-3-2007 = Rs. 18450
Total amount = 16200 + 17100 + 18450 = Rs. 51750
587. lim n tend to infinity(2^n+7^n)^(1/n)is equal to
a. 7e b. 7 c. 2e d. 2 e. None of the above
Clue:b
lim n tend to infinity(2^n+7^n)^(1/n)is equal to 7*lim n tend to infinity(1+(2/7)^n)^(1/n
588. A complex number z lies on the curve z+6=3. The largest magnitude of z+3 will be:
a. 6 b. 3 c. 36 d. 12 e. None of the above
clue:a
mod(z+6)=3 represents circle with centre (–6, 0) and radius 3.
z lies as the boundary of circle. |z + 3| is maximum when it is at (–9, 0) and max |z + 3| = 6
589. An equilateral triangle is inscribed in a circle such that its vertices lie on the circumference of the circle. A point is selected at random from within the circle. The probability of finding the point inside the triangle is:
a.sqrt(3)/2pi b. 3sqrt(3)/4pi c.2pi/sqrt(3) d. 2/3sqrt(3)pi e. None of the above
Clue: b
The required probability is the ratio of area of triangle to the area of the circle.
590.If A, B and C are the angles of a triangle and e^iA, e^iB and e^iC are in Arithmetic Progression, then the triangle is:
a. Right angled but not isosceles b. Isosceles but not right angled
c. Equilateral d. Right angled isosceles
Clue: c 2e^iB=e^iA+e^iC
2cosB=cosA+cosC
2sinB=sinA+sinC
On solving:
Let A = B = C = 60°
then e^A-e^iC=1/2+isqrt(3)/2 form a constant AP
591. Let (x) and [x] represent the fractional and integral components of x belongs to R. We define f : R ® R; g : R ® R by f(x) = (x); g(x) = sin[x]pi. The range of gof is:
a. (–1, 1) b. {0} c. [1, 1] d. {2,2} e. None of the above
Clue b
gof=sin[(x)]pi
0£(x)<1
[(x)] = 0
gof(x) = sin0pi = 0
592. If z belongs to C lies on the circle whose equation is mod(z–3i)=3sqrt(2), then the argument of (z – 3 )/(z+3 ) is:
a.pi/4 b. tan inverse 3 c. tan inverse 3sqrt(2) d. pi/2 e. None of the above
593. A toy consists of a base that is the section of sphere and a conical top. The volume of the conical top is 30p sq. units and its height is 10 units. The total height of the toy is 19 units. The volume of the sphere (in cubic units) from which the base has been extracted is:
a.256/3#pi b.64/3*pi c.108/3*pi d.500/3*pi e. None of the above
Clue. d
Height of the cone = 10 units
Volume of the cone = 30 pi cubic units.
Þ diameter of the cross section from where sphere has been sectioned = 6 units.
Height of the sectioned sphere = 19 – 10 = 9 units.
Þ r + sqrt(r^2– 3^2) = 9
Þ r = 5 units
Þ volume of the original sphere from which the base has been sectioned = 4/3*pi*r^3
=500/3*pi
594. If you have 3 tickets to a lottery for which 10 tickets were sold and 5 prizes are to be given, the probability that you will win at least one prize is:
a.7/12 b.9/12 c.1/12 d.11/12 e. None of the above
Clue:d
Probability that you will win at least one prize = 1 – probability that you will not win any prize.
= 1 –(5c3/10c3)=11/12
595.The adjacent sides AB, BC of a square ABCD of side ‘a’ units are tangent to a circle. The vertex D of the square lies on the circumference of the circle. The radius of the circle could be:
a. a(2 – sqrt(2)) b. a( sqrt(2) – 1) c. a(3/2+sqrt(2)) d.a( sqrt(2) + 1) e. None of the above
596. A 300 room motel can rent all its rooms at Rs. 150 per room per day. For every one rupee increase in tariff, the occupancy falls by 2 rooms. Normal maintenance, independent of occupancy is Rs. 120 per room per day. Occupancy entails additional costs of Rs. 16 per room per day on the units occupied. The optimal tariff (in Rupees) to maximize annual profits is:
a. 150 b. 162 c. 158 d. 160 e. None of the above
597. Now, suppose that rooms of the motel can be permanently shut down in blocks of 5, if desired, resulting in saving of normal maintenance of Rs. 120 per room per day. The optimal tariff (in Rupees) to maximize annual profits in this case would be:
a. 158 b. 160 c. 162 d. 156 e. None of the above
For questions 596 and 597:
Revenue = (300 – 2x) (150 + x)
Maintenance = 120 × 300 + 16 (300 – 2x)
Þ Profit = Revenue – Maintenance
= 4200 + 32x – 2x2
= 4328 – 2 (8 – x)2
Þ Maximum profit occurs when x = 8
104. c Optimal tariff = 150 + 8 = Rs. 158
105. b Now,
Profit = 4200 + 32x – 2x2 + 600 [2x/5]
From the options, x = 10 gives us maximum profit Þ optimal tariff = 150 + 10 = Rs. 160
598. The position vector of the mirror image of the point represented by the position vector r=2ˆi+3ˆj+4kˆ r
across the plane mirror x + y = 0 is:
a. -3ˆi-2ˆj+4kˆ b. 2ˆi+3ˆj-4kˆ c. -2ˆi-3ˆj+4kˆ d. 3ˆi+2ˆj-4kˆ e. None of the above
599. A man 6 feet tall standing 50 feet away from the base of a tower observes that the angle subtended by the tower at his eye level is tan–1 2. A flagstaff atop the tower subtends an angle of tan inverse(0.1) at then same point. The height (in feet) of the flagstaff is closest to:
a. 18 b. 20 c. 25 d. 22 e. None of the above
600. A, B and C are assigned a piece of work which they can complete by working together in 15 days. Their efficiencies (measured in terms of rate of doing work) are in the ratio 1 : 2 : 3. After 1/3 of the work is completed, one of them has to be withdrawn due to budget constraint. Their wages per day are in the ratio 3 : 5 : 6. The number of days in which the remaining two persons can complete the
work (at optimal cost) is:
a. 18 b. 20 c. 15 d. 12 e. None of the above
Clue:d
A, B and C together in 15 days.
Þ A alone in 90 days
B alone in 45 days
C alone in 30 days
Wages per day per unit work for A, B and C are
3/1: 5/2: 6/3
Hence, A is the least efficient and hence, must be done away with. For B and C, the whole work can be finished in 18 days and hence, remaining 2/3 rd of the work can be finished in 12 days only.
601. If F(x, n) be the number of ways of distributing "x" toys to "n" children so that each child receives at the most 2 toys then F(4,3)=_____?
A.2 B.3 C.4 D.5 E.6
602. In a cricket match. Team A scored 232 runs without losing a wicket. The score consisted of byes, wides and runs scored by two opening batsmen: Ram and Shyam. The runs scored by the two batsmen are 26 times wides. There are 8 more byes than wides. If the ratio of the runs scored by Ram and Shyam is 6:7, then the runs scored by Ram is
A. 88
B. 96
C. 102
D. 112
E. None of the above.
or questions 603 & 604, a statement is followed by three conclusions. Select the answer from the
following options.
A. Using the given statement, only conclusion I can be derived.
B. Using the given statement, only conclusion II can be derived.
C. Using the given statement, only conclusion III can be derived.
68
Y=29
9
Y 4 5 2 X
D. Using the given statement, all conclusions I, II and III can be derived.
E. Using the given statement, none of the three conclusions I, II and III can be derived.
603. An operation "#" is defined by a # b = 1 – b/a
Conclusion I. (2 # 1) # (4 # 3) = -1
Conclusion II. (3 #1) # (4 #2) = -2
Conclusion III. (2#3) # (1#3) = 0
604. A, B, C and D are whole numbers such that
A+B+C =118
B + C + D = 156
C + D + A = 166
D + A + B = 178
Conclusion I. A is the smallest number and A=21.
Conclusion II. D is the largest number and D=88.
Conclusion III. B is the largest number and B=56.
605. Let X= {a, b, c} and Y= {l, m}. Consider the following four subsets of X x Y.
F1 = { (a, l), (a, m), (b, l), (c, m)}
F2={(a, l), (b, l), (c, l)}
F3={(a, l), (b, m), (c, m)}
F4={(a, l), (b, m)}
Which one, amongst the choices given below, is a representation of functions from X to Y?
A. F1, F2 and F3
B.F2, F3 and F4
C. F2 and F3
D. F3 and F4
E. None of the above
606. ABCD is a rectangle with AD = 10. P is a point on BC such that angle APD = 90° If DP = 8, then the length of BP is __?
A.6.4 B.5.2 C.4.8 D.3.6 E. None of the above
607. Four digits of the number 29138576 are omitted so that the result is as large as possible. The largest omitted digit is
A.9 B.8 C.7 D.6 E.5
608. Rajiv is a student in a business school. After every test he calculates his cumulative average. QT and OB were his last two tests. 83 marks in QT increased his average by 2. 75 marks in OB further increased his average by 1. Reasoning is the next test, if he gets 51 in Reasoning, his average will be ___?
A.59 B.60 C.61 D.62 E.63
609. ABCD is a square. P is the midpoint of AB. The line passing through A and perpendicular to DP intersects the diagonal at Q and BC at R.If AB = 2 then PR = ___?
A. 1/2 B.0 C.-1 D. 1 E. None of the above.
610. For each p>1, a sequence {An} is defined by Ao = 1 and An = pn + (-1)n An-1 for each n >= 1. For how many integer values of p, 1000 is a term of the sequence?
A.8 B.7 C.5 D.4 E. None of the above.
611. If 0
B. Imaginary
C. Real and both positive
D. Real and both negative
E. Real and of opposite sign.
612. Consider a sequence –6, –12, 18, 24, –30, –36, 42,..... . If sum of the first n terms of the sequence is 132, then the value of n is ___?
A. 11 B. 13 C. 18 D. 22 E. 24
613. The number of possible real solution(s) of y in equation y^2 – 2y cos x + 1 = 0 is ___?
A. 0 B. 1 C. 2 D. 3 E. None of the above.
614. In a triangle ABC, AB = 3, BC =4 and CA = 5. Point D is the midpoint of AB, point E is on segment AC and point F is on segment BC. If AE = 1.5 and BF = 0.5 then angle DEF =
A. 30° B. 45° C. 60° D. 75° E. Cannot be determined
615. If 3f(x + 2) + 4f(1/x+2)) = 4x, x ≠ –2, then f(4) =
A. 7 B. 52/7 C. 8 D. 56/7 E. None of the above.
616. A train left station X at A hour B minutes. It reached station Y at B hour C minutes on the same day, after traveling for C hours A minutes (clock shows time from 0 hours to 24 hours). Number of possible value(s) of A is __.
A. 3 B. 2 C. 1 D. 0 E. None of the above.
Questions 617-19: A, B, C, D, E and F are six positive integers such that
B+C+D+E = 4A
C+F = 3A
C + D + E = 2F
F = 2D
E + F = 2C + 1
If A is a prime number between 12 and 20, then
617. The value of C is
A. 13 B. 17 C. 19 D. 21 E. 23
618. The value of F is
A. 14 B. 16 C. 20 D. 24 E. 28
619. Which of the following must be true?
A. D is the lowest integer and D = 14
B. C is the greatest integer and C = 23
C. F is the greatest integer and F = 24
D. B is the lowest integer and B = 12
E. A is the lowest integer and A = 13
620. ABCD is a quadrilateral. The diagonals of ABCD intersect at the point P. The area of the triangles APD and BPC are 27 and 12, respectively. If the areas of the triangles APB and CPD are equal then the area of triangle APB is
A.12 B.15 C.16 D.18 E.21
621. Two circles of radius 1 cm. touch at point P. A third circle is drawn through the points A, B,and C such that PA is the diameter of the first circle, and BC –perpendicular to AP – is the diameter of the second circle. The radius of the third circle is __ cms.
A. 9/5 B. 7/4 C. 5/3 D.1 E. 2
In the questions 623-624, one statement is followed by three conclusions. Select the appropriate
answer from the options given below.
A. Using the given statement, only conclusion I can be derived.
B. Using the given statement, only conclusion II can be derived.
C. Using the given statement, only conclusion III can be derived.
D. Using the given statement, conclusions I, II and III can be derived.
E. Using the given statement, none of the three conclusions I, II and III can be derived.
623. A0, A1, A2 ,..…… is a sequence of numbers with A0= 1, A1 =3,
and At = (t +1)At-1 –t At - 2 for t = 2, 3, 4,…
Conclusion I. A8 =77
Conclusion II. A10 = 121
Conclusion III. A12 = 145.
624. A, B, C be real numbers satisfying A < B < C, A + B + C = 6 and AB + BC + CA = 9
Conclusion I. 1 < B < 3
Conclusion II. 2 < A < 3
Conclusion III. 0
these statements are sufficient to conclusively answer the question. Choose the appropriate
answer from options given below:
A. If Statement I alone is sufficient to answer the question.
B. If Statement II alone is sufficient to answer the question.
C. If Statement I and Statement II together are sufficient but neither of the two alone is sufficient
to answer the question.
D. If either Statement I or Statement II alone is sufficient to answer the question
E. Both Statement I and Statement II are insufficient to answer the question.
625. The base of a triangle is 60 cms, and one of the base angles is 60°. What is length of the shortest side of the triangle?
I. The sum of lengths of other two sides is 80 cms.
II. The other base angle is 45°.
626. A, B, C, D, E and F are six integers such that E < F, B > A, A < D < B. C is the greatest integer. Is A the smallest integer?
I. E+B
A. 4 B. 3 C. 2 D. 1 E. 0
Questions 628-629; In second year, students at a business school can opt for Systems,
Operations, or HR electives only. The number of girls opting for Operations and the number of boys opting for Systems electives is 37. Twenty-two students opt for operations electives. Twenty girls opt for Systems and Operations electives. The number of students opting for Systems electives and the number of boys opting for Operations electives is 37. Twenty-five students opt for HR electives.
628. The number of students in the second year is_?
A.73 B.74 C.75 D.76 E.77
629. If 20% of the girls opt for HR electives, then the total number of boys in the second year is _?
A.50 B.51 C.52 D.53 E.54
630. If x>0, the minimum value of ((x+1/x)^6-(x^6+1/x^6))/((x+1/x)^3+(x^3+1/x^3))
is ______?
A.1 B.2 C.3 D.6 E. None of the above.
631. Consider obtuse angled triangles with sides 8 cm,15 cm,x cm. If 'x' is an integer ,then how many such triangles exist?
A.5 B.21 C.10 D.15 E.14
632. What is the number of distinct terms in the expansion of (a+b+c)^20?
A.231 B.253 C.242 D.210 E.228
633.In a certain system product of 44 X 11 is represented as 2124 find how 1453 can be represented in decimal system? (
A) 248 (B) 346 (C) 393 (D) 410 (E) none of these.
634.All positive consecutive integers starting from 1 were written on the blackboard. One of the students entered the class and erased one of the numbers. Now the average of the remaining numbers is 35 7/17.The erased number is
(A) 17 (B) 8 (C) 9 (D) 10 (E) none of these
635.In a four-digit number, the sum of digits in the first and fourth place is twice the sum of the digits in the third and fourth place. The difference in the first and third digit is equal to the second digit. What is the digit in the third place?
(A) 7 (B) 8 (C) 9 (D) 5 (E) none of these
636.A person had to multiply two numbers. Instead of multiplying by 35, the person multiplied by 53, and the product went up by 540.What was the raised product?
(A) 780 (B) 1040 (C) 1590 (D) 1670 (E) none of these
637. Euclid had a triangle in his mind .The longest side is 20 and the other side is 10.Area of the triangle is 80.The third side is__
(A)root 260 (B)root 240 (C)root 250 (D)root 210 E) none of these.
638. 09/12/2001 is Sunday. What was the day on 09/12/9071?
(A)Thursday (B)Wednesday (C)Saturday (D)Sunday (E) none of these.
639.In a fibonacci series, difference between squares of 6th and 7th terms was 517. What is the 8th term?
(A) 32 (B) 65 (C) 83 (D) 89 (E) none of these
640. There is a circular field having four doors in North, East, West and South. A person walks 3 meters from the North door. Another person comes out of the South door and walks 9 meter towards East so that he is just able to see the first man. What is the diameter of the field.
(A) 12 (B) 15 (C) 13 (D) 9 (E) none of these
641. Ram has 128 boxes with him. He has to put at least 120 oranges in one box and 144 at the most. Find the least number of boxes which will have the same number of oranges.
(A) 5 (B) 6 (C) 103 (D) Can't determined (E) none of these.
642.There are two persons A and B who joined an organization on 1st Jan 1970. A joined at Rs. 300 and annual increment of Rs.30. B gets salary of Rs. 200 and hike of Rs. 50 per six month. Find the total amount distributed at the end of 31st December 1979.
(A) 91200 (B) 92800 (C) 97200 (D) Can't determined (E) none of these.
643.What will be the remainder when ( 60! - 5 )( 61! - 4 )( 62! - 3)(63! - 2) ..... (this way till) .... (70! + 5 )is divided by 67 ?
(A) 52 (B) 66 (C) 0 (D) Can't determined (E) none of these
Clue:
Use Wilson's theorem
If p is prime then, (p-1)!+1 is multiple of p
hence remainder is Zero.
644.Find the remainder when ( 15! ^ 222! )*( 2 ^ 222! ) is divided by 221.
(A) 52 (B) 220 (C) 0 (D) 17 (E) none of these
645. Remainder obtained when 777777777....7777 (101 digits) is divided by 440.
(A)337 (B277 (C)377 (D)227 (E) none of these
Chin Up!
Raktabh Mahesh
