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First off, I'm not sure whether this should be in the homework section since it's not homework, but this seems the best place to get help with it. If it should be moved, please feel free to do so.

These are a few problems I had with one of the GRE practice exams (from one of the Practicing to Take the GRE Mathematics Test books).

20. Suppose that f(1+x)=f(x) for all real x. If f is a polynomial and f(5)=11, then f(15/2) is...

a) -11

b) 0

c) 11

d) 33/2

e) Not uniquely determined

My first reaction was that it couldn't be determined since you can't reduce f(15/2) to f(5). But as I type this, I realize that, if it's a polynomial, it can't keep being 11 at every integer - it must eventually go to infinity or negative infinity - unless it's the constant function f(x)=11. This is the right answer, but is my reasoning correct?

25. Let x and y be positive integers such that 3x+7y is divisible by 11. Which of the following must also be divisible by 11?

a) 4x+6y

b) x+y+5

c) 9x+4y

d) 4x-9y

e) x+y-1

Not sure where to go with this one...:(

48. In the xy-plane, the graph of [itex]x^{\log y}=y^{\log x}[/itex] is:

a) Empty

b) A single point

c) A ray in the open first quadrant

d) a closed curve

e) the open first quadrant

My first realization was that there was at least one solution, x=y=1, so it wasn't empty. Then I realized x=y is a solution, so it's not b. I wasn't sure about the others, so I marked c since I had at least a ray, but the answer turns out to be e. Can anybody explain why?

49. If the finite group G contains a subgroup of order 7 but no element other than the identity is its own inverse, then the order of G could be:

a) 27

b) 28

c) 35

d) 37

e) 42

I know it's order must be divisible by 7, but I don't know what the second part about no element being its own inverse implies. The correct answer is c.

EDIT: If there is an element that is its own inverse, then a^2=e. But this implies a subgroup of order 2, so the order of G is not divisible by 2 since there is no a^2=e. The only answer that is divisible by 7 and not by 2 is c, which is the right answer. Is my reasoning sound here?

58. If f(z) an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped to:

a) the entire real axis

b) a point

c) a ray

d) an open finite interval

e) the empty set

My gut reaction was b (which turns out to be the correct answer) but I don't really have proof of it.

Thanks in advance!

These are a few problems I had with one of the GRE practice exams (from one of the Practicing to Take the GRE Mathematics Test books).

20. Suppose that f(1+x)=f(x) for all real x. If f is a polynomial and f(5)=11, then f(15/2) is...

a) -11

b) 0

c) 11

d) 33/2

e) Not uniquely determined

My first reaction was that it couldn't be determined since you can't reduce f(15/2) to f(5). But as I type this, I realize that, if it's a polynomial, it can't keep being 11 at every integer - it must eventually go to infinity or negative infinity - unless it's the constant function f(x)=11. This is the right answer, but is my reasoning correct?

25. Let x and y be positive integers such that 3x+7y is divisible by 11. Which of the following must also be divisible by 11?

a) 4x+6y

b) x+y+5

c) 9x+4y

d) 4x-9y

e) x+y-1

Not sure where to go with this one...:(

48. In the xy-plane, the graph of [itex]x^{\log y}=y^{\log x}[/itex] is:

a) Empty

b) A single point

c) A ray in the open first quadrant

d) a closed curve

e) the open first quadrant

My first realization was that there was at least one solution, x=y=1, so it wasn't empty. Then I realized x=y is a solution, so it's not b. I wasn't sure about the others, so I marked c since I had at least a ray, but the answer turns out to be e. Can anybody explain why?

49. If the finite group G contains a subgroup of order 7 but no element other than the identity is its own inverse, then the order of G could be:

a) 27

b) 28

c) 35

d) 37

e) 42

I know it's order must be divisible by 7, but I don't know what the second part about no element being its own inverse implies. The correct answer is c.

EDIT: If there is an element that is its own inverse, then a^2=e. But this implies a subgroup of order 2, so the order of G is not divisible by 2 since there is no a^2=e. The only answer that is divisible by 7 and not by 2 is c, which is the right answer. Is my reasoning sound here?

58. If f(z) an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped to:

a) the entire real axis

b) a point

c) a ray

d) an open finite interval

e) the empty set

My gut reaction was b (which turns out to be the correct answer) but I don't really have proof of it.

Thanks in advance!

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