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(#M40151769) GRE QUESTION Circular region divide 5 sector Keep an EYE

A circular region is divided by 5 radii into sectors. Twenty-one points are chosen in the circular region, none of which is on any of the 5 radii. Which of the following statements must be true?
I. Some sector contains at least 5 of the points.
II. Some sector contains at most 3 of the points.
III. Some pair of adjacent sectors contains a total of at least 9 of the points.
(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

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(#M40151768) GRE QUESTION Greatest probability Keep an EYE

A fair coin is to be tossed 100 times, with each toss resulting in a head or a tail. If H is the total number of heads and T is the total number of tails, which of the following events has the greatest probability?
(A) H = 50
(B) T >= 60
(C) 51 =< H =< 55
(D) H >= 48 and T >= 48
(E) H =< 5 or H >= 95

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(#M40151767) GRE QUESTION Solve operation Keep an EYE

If z = e^(2*Pi*i/5), then 1 + z + z^2 + z^3 + 5z^4 + 4z^5 ++4z^6 + 4z^7 + 4z^8 + 5z^9 = ?
(A) 0
(B) 4e^(3*Pi*i/5)
(C) 5e^(4*Pi*i/5)
(D) – 4e^(2*Pi*i/5)
(E) – 5e^(3*Pi*i)/5

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(#M40151766) GRE QUESTION Probability Keep an EYE

Suppose X is a discrete random variable on the set of positive integers such that for each positive integer n, the probability that X = n is 1/2^n. If Y is a random variable with the same probability distribution and X and Y are independent, what is the probability that the value of at least one of the variables X and Y is greater than 3 ?
(A) 1/64
(B) 15/64
(C) 1/4
(D) 3/8
(E) 4/9

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(#M40151765) GRE QUESTION Non zero elements Keep an EYE

For which of the following rings is it possible for the product of two nonzero elements to be zero?
(A) The ring of complex numbers
(B) The ring of integers modulo 11
(C) The ring of continuous real-valued functions on [0, 1]
(D) The ring {a + b Sqrt 2 : a and b are rational numbers}
(E) The ring of polynomials in x with real coefficients

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(#M40151764) GRE QUESTION Complex z-plane Keep an EYE

In the complex z-plane, the set of points satisfying the equation z^2 = |z|^2 is a
(A) pair of points
(B) circle
(C) half-line
(D) line
(E) union of infinitely many different lines

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(#M40151763) GRE QUESTION Matrix Keep an EYE

Let M be a 5*5 real matrix. Exactly four of the following five conditions on M are equivalent to each other. Which of the five conditions is equivalent to NONE of the other four?
(A) For any two distinct column vectors u and v of M, the set {u, v} is linearly independent.
(B) The homogeneous system Mx = 0 has only the trivial solution.
(C) The system of equations M x = b has a unique solution for each real 5*1 column vector b.
(D) The determinant of M is nonzero.
(E) There exists a 5*5 real matrix N such that NM is the 5*5 identity matrix.

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(#M40151762) GRE QUESTION Probability Keep an EYE

At a banquet, 9 women and 6 men are to be seated in a row of 15 chairs. If the entire seating arrangement is to be chosen at random, what is the probability that all of the men will be seated next to each other in 6 consecutive positions?
(A) 1/(15/6)
(B) 6!/(15/6)
(C) 10!/15!
(D) 6! 9! /14!
(E) 6! 10! / 15!

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(#M40151761) GRE QUESTION Point distance on the sphere Keep an EYE

The minimal distance between any point on the sphere (x-2)^2 + (y-1)^2 + (z-3)^2 = 1 and any point on the sphere (x+3)^2 + (y-2)^2 + (z-4)^2 = 4 is
(A) 0
(B) 4
(C) Sqrt(27)
(D) 2(Sqrt2 + 1)
(E)3(Sqrt3 – 1)

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(#M40151760) GRE QUESTION Greatest common divisor (gcd) Keep an EYE

The Euclidean algorithm is used to find the greatest common divisor (gcd) of two positive integers a and b .
input(a)
input(b)
while b>0
begin
r := a mod b
a := b
b := r
end
gcd := a
output(gcd)
When the algorithm is used to find the greatest common divisor of a = 273 and b = 110, which of the
following is the sequence of computed values for r ?
(A) 2, 26, 1, 0
(B) 2, 53, 1, 0
(C) 53, 2, 1, 0
(D) 53, 4, 1, 0
(E) 53, 5, 1, 0

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