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
3.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
9. 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 5y=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
44.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 (1) 3 (2) 3.5 (3) 4 (4) 4.5 (5) None of these
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 = (1) 5 (2) 2√3 (3) 7/2 (4) 24/7 (5) none of these
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 <>4 (3). 2<4 (4)2<=x<=4 (5) none of these
98. There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song? (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 (1)KL = LM (2) triangle KLM is equilateral (3) triangle KLM is isosceles (4) a and b (5) a and c. 101. LMN is a triangle. MO is the angle bisector. The point P on LM is such that < po =" ON" 1 =" 0,"> 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 < n =" 4," s =" {(1," a="B">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 =" <" bc =" 3," 2 =" n">= 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 < cd =" 2AB" 3="ccc" x="2+2^1/3+2^2/3" 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 <> 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, < y =" log10" y =" x-1">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 [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 - 3x+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˚."> 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 < a =" 2" c =" 0" ab =" 5" ad =" 1" df =" 1" def =" 3cm" abc =" 6" ab =" 2" ac =" 2" ab =" 2km" ac =" 2"> 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 < t =" 0" 2 =" 120t"> 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 < a =" (b+c+d)/3?" dc =" EA/CE" 4y =" 0" z =" 5/2" a =" <" b =" <" d =" 120Ëš," c =" <" bc =" 1" cd =" √3."> 120 km (b) < ac =" 3AB." 2 =" 0.">= 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˚," acm =" 30˚,"> 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 < x=" p/11"> 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