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gre Latest Exam Pattern - gre Sample Question with Solutions Page 21

(#M40151779) GRE QUESTION Analytic function Keep an EYE

Suppose f is an analytic function of the complex variable z = x + i y given by
f(z) = (2x + 3y) + i g (x, y)
where g(x, y) is a real-valued function of the real variables x and y. If g(2, 3) = 1, then g(7, 3) = ?
(A) -14
(B)-9
(C) 0
(D) 11
(E) 18

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(#M40151778) GRE QUESTION Matrices Keep an EYE

Suppose A and B are n*n invertible matrices, where n >1, and I is the n*n identity matrix. If A and B are similar matrices, which of the following statements must be true?
I. A - 2I and B – 2I are similar matrices.
II. A and B have the same trace.
III. A^-1 and B^-1 are similar matrices.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

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(#M40151777) GRE QUESTION Decimal representation Keep an EYE

For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeros?
(A) None
(B) One
(C) Four
(D) Five
(E) Twenty-four

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(#M40151776) GRE QUESTION Minimum value of the expression Keep an EYE

What is the minimum value of the expression x + 4z as a function defined onR^3 , subject to the constraint x^2 + y^2 + z^2 =< 2 ?
(A) 0
(B) 2
(C) – Sqrt(34)
(D) – Sqrt(35)
(E) -5 Sqrt(2)

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(#M40151775) GRE QUESTION Subset of the real line Keep an EYE

If A is a subset of the real line R and A contains each rational number, which of the following must be true?
(A) If A is open, then A = R.
(B) If A is closed, then A = R.
(C) If A is uncountable, then A = R .
(D) If A is uncountable, then A is open.
(E) If A is countable, then A is closed.

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(#M40151774) GRE QUESTION Matrix Keep an EYE

Let A be a real 2*2 matrix. Which of the following statements must be true?
I. All of the entries of A^2 are nonnegative.
II. The determinant of A^2 is nonnegative.
III. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

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(#M40151773) GRE QUESTION Isomorphism Keep an EYE

Up to isomorphism, how many additive abelian groups G of order 16 have the property that x+x+x+x = 0
for each x in G ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

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(#M40151772) GRE QUESTION Consider the theorem Keep an EYE

Consider the theorem: If f and f’ are both strictly increasing real-valued functions on the interval (0, infinite) then lim f(x) = infinite (x -> infinite). The following argument is suggested as a proof of this theorem.
(1) By the Mean Value Theorem, there is a c1 in the interval (1, 2) such that
f’(c1) = {f(2)-f(1)}/ (2-1) = f(2) – f(1) > 0.
(2) For each x > 2, there is a c(x) in (2, x) such that {f(x) – f(2)}/(x-2) = f’{c(x)}
(3) For each x > 2, {f(x)-f(2)}/(x-2) = f’(c(x)) > f’(c1) since f’ strictly incrasing
(4) For each x > 2, f(x) > f(2) + (x-2) f’(c1)
(5) lim f(x) = infinite (x -> infinite)
Which of the following statements is true?
(A) The argument is valid.
(B) The argument is not valid since the hypotheses of the Mean Value Theorem are not satisfied in (1) and (2).
(C) The argument is not valid since (3) is not valid.
(D) The argument is not valid since (4) cannot be deduced from the previous steps.
(E) The argument is not valid since (4) does not imply (5).

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(#M40151771) GRE QUESTION xyz-space Keep an EYE

Let F be a constant unit force that is parallel to the vector (-1, 0, 1) in xyz-space. What is the work done by F on a particle that moves along the path given by (t, t^2, t^3) between time t = 0 and time t = 1?
(A) – 1/4
(B) – 1/4 Sqrt(2)
(C) 0
(D) Sqrt(2)
(E) 3 Sqrt(2)

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(#M40151770) GRE QUESTION Complex number Keep an EYE

Let G be the group of complex numbers {1, i, -1, -i} under multiplication. Which of the following statements are true about the homomorphism of G into itself?
I. z -> z defines one such homomorphism, where z denotes the complex conjugate of z,
II. z -> z^2 defines one such homomorphism.
III. For every such homomorphism, there is an integer k such that the homomorphism has the form z -> z^k .
(A) None
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

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