I have managed to get solutions for almost all of the problems on the GRE math subject practice test posted on the ETS site with the exception of problems 51,58,64,65. I'm also not sure about 47. I would greatly appreciate solutions to these questions (posted below) and am willing to provide solutions to the other problems in gratitude. the practice test gives answers but no explanations (thanks guys). Here we go:

47) Let x and y be uniformly distributed, independent random variables on [0,1]. The probability that the distance between x and y is less than 1/2 is:

ans: 3/4

51) Let D be the region in the xy plane in which the series sum (k=1 to infinity) ((x + 2y)^k)/k converges. Then the interior of D is:

ans: an open region between two parallel lines

I tried using the ratio test and got the half plane y<1/2(1-x) . why is this wrong?

58) Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S={f(c): 0<c<1}

I. S is a connected subset of R

II. S is an open subset of R

III. S is a bounded subset of R

ans: I and III only.

This problem confuses me the most I think. since the preimage 0<c<1 is an open set and since f is cts, shouldn't S also be open? Isn't that a significant theorem, that continuous functions map open sets to open sets?

64) Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true?

I. There is a constant C>0 such that |f(x) - f(y)| <=C for all x,y in [0,1]

II. There is a constant D>0 s.t. |f(x) - f(y)| <=1 for all x,y in [0,1] that satisfy |x-y| <= D

III. There is a constant E>0 s.t. |f(x) - f(y)| <= E|x-y| for all x,y in [0,1]

ans: I and II only

II follows directly from the epsilon-delta definition of continuity, doesn't it? And III is def of lipschitz, which is stronger than just continuity. I follows from the extreme value theorem I believe, although what about tan(X+pi/2)? Isn't this is a cts unbounded function on [0,1] in the extended real number system?

65) Let p(x) be the polynomial x^3 + ax^2 +bx +c, where a,b, and c are real constants. If p(-3) = p(2) = 0 and p'(-3)<0, which of the following is a possible value of c?

ans: -27

I did not know where to even start on this one. Whoever solves this one gets extra genius points.

Thank you so much for helping if you can! I'm pretty nervous about the test....