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Frequently Asked GRE Latest Exam Pattern - GRE Sample Question with Solutions Page 39

(#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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(#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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(#M40151758) GRE QUESTION Binary operation denoted by + and * Keep an EYE

Suppose that two binary operations, denoted by + and * are defined on a nonempty set S, and that the
following conditions are satisfied for all x, y, and z in S:
(1) x+y and x * y are in S
(2) x+(y+z) = (x+y)+z and x*(y*z) = (x*y)*z
(3) x+y = y+x
Also, for each x in S and for each positive integer n, the elements nx and n x are defined recursively as x^n are defined recursively as follos:
1x = x^1 = x and
If kx = x^k have been defined, then (k+1)x = kx + x and x^(k+1) = x^k * x.
Which of the following must be true?
I. (x*y)^n = x^n * y^n for all x and y in S and for each positive integers n.
II. n(x+y) = nx ny for all x and y in S and for each positive integer n.
III. x^m * x^n = x^(m+n) for each x in S and for all positive integers m and n.
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

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(#M40151743) GRE QUESTION xy-plance Keep an EYE

Let f(x, y) = x^2 – 2xy + y^3 for all real x and y. Which of the following is true.
(A) f has all of its relative extrema on the line x y.
(B) f has all of its relative extrema on the parabola x = y^2
(C) f has a relative minimum at (0, 0)
(D) f has an absolute minimum at (2/3, 2/3)
(E) f has an absolute minimum at (1, 1)

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(#M40151742) GRE QUESTION xy-plance Keep an EYE

For what value of b is the line y = 10 x tangent to the curve y = e^(bx) at some point in the xy-plance?
(A) 10/e
(B) 10
(C) 10 e
(D) e^10
(E) e

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