GRE Exam Numerical Ability Geometry

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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GRE Other Question

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 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).