Problems:
1) 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
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2) Given a dart board divided in two regions, one red, and 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
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3) The following sequence
1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17...
has one odd number followed by two evens, then three odds, four evens, and so on. What number is the 2003rd term?
(a) 3942 (b) 3943 (c) 3944 (d) 3945 (e) None of the foregoing
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4) In a quadrilateral ABCD sides AB and CD are equal with <A = 150˚, <B = 44˚, and <C = 72˚. Perpendicular bisector of the segment AD meets the sides BC at point X. Then m(<AXD) is
(1) 42˚ (2) 58˚ (3) 64˚ (4) 78˚ (5) none of these
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5) 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
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6) 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 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
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7) In a class there are 100 students. A division of the students in n sections is good if:
1) The sections have different numbers of students
2) For any partition of one of the sections in 2 smaller sections, among the (n+1) sections you get 2 with the same number of students (any section has at least 1 student).
The positive difference between the maximal and minimal possible value of n such that the division is good is
(a) 2 (b) 3 (c) 7 (d) 8 (e) none of the foregoing
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8) 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
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9) Consider a set P= {1, 2, 3..., 11, 12} of natural numbers. We define another set Q such that it contains no more than one out of any three consecutive natural numbers. How many subsets Q of P including the empty set is possible?
(a) 114 (b) 117 (c) 129 (d) 136 (e) 130
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10) 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
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11) What is the smallest positive integer k for which there are at least 11 even and 11 odd positive integers m so that (m^3 +k)/(m+2) is an integer?
(a) 268 (b) 448 (c) 638 (d) 858 (e) none of the foregoing
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12) ABCD is a square, point E is inside triangle ACD and point F is inside triangle ACB. < EAF = <ECF = 45˚. If DE = 3, and BF = 4, then EF equals
(1) 5 (2) 2√3 (3) 7/2 (4) 24/7 (5) none of these
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13) 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
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14) Given p <= 4 is a positive real. Let A be the area of the bounded region enclosed by the curves y = 1 - |1-x| and y = |2x-p|. Then which among the following best describes A?
(1)0<A<= ½ (2) 1/6<=A<=¼ (3) 0<A<=1/3 (4) 0<A< 2/3 (5) 1/6 <= A <= 1/3
---------------------------------------------------------------------------------------------------------------------------------------15) (Part A)
Three articles are sold at the profit of 20%, 25% and 40% and the net profit comes out to be 30%. Had the article sold at 25% been sold at 15% profit and the article sold at 40% been sold at its cost price then the profit on the sale of three articles would have been
(a) 5% (b) 10% (c) 15% (d) can not be determined (e) none of the foregoing
(Part B)
Let the cost price of an article A is as much more than another article B as much that of B is more than that of another article C whose cost price is more than that of fourth article D by the same amount. If A, B, C, D are sold at 10%, 15%, 30% and 5% profit respectively, then the net profit percent on the sale of four articles together is
(a) 5% (b) 10% (c) 15% (d) can not be determined (e) none of the foregoing
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16) 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
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17) Let a(n) = (n+9)!/(n-1)! for each positive integer n. If k is the smallest integer with the rightmost nonzero integer a(k) is odd. The right most nonzero digit of a(k) is
(1) 1 (2) 3 (3) 5 (4) 7 (5) 9