- #1
island-boy
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Hello, I'm preparing for the GRE Math Subject Test, as such, I am using ETS's free practice book which can be downloaded from the gre.com website. The answers are given for the questions, but the solutions are not. I am having difficulty getting the solutions to the following questions. Help and input is appreciated:
I gathered all the questions into one thread so as not to spam the forums:
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24) Which of the ff sets of vectors is a basis for the subspace of Euclidean 4-space consisting of all vectors that are orthogonal to both (0,1,1,1) and (1,1,1,0)?
A) {(0,-1,1,0)}
B) {(1,0,0,0), (0,0,0,1)}
C) {(-2,1,1,-2), ((0,1,-1,0)}
D) {(1,-1,0,1), (-1,1,0,-1), (0,1,-1,0)}
E) {(0,0,0,0),(-1,1,0,-1),(0,1,-1,0)}
answer is C.
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32)When 20 children in a classroom line up for lunch, Pat insists on being somewhere ahead of Lynn, If Pat's demand is to be satisfied, in how many ways can the children line up?
answer is 20!/2
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37) Summation of (k^2)/k! from k = 1 to infinity is ____
answer is 2e
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44) Let f denote the function defined for all x>0 by f(x) = x^(x/2), which of the folowing statement is false?
answer is the derivative f'(x) is positive for all x>0.
I was able to get why the other choices are true, but I can't understand why this is false. Isn't the derivative [x^(x/2) x lnx]/2? which is postive for all x>0?
----------------
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 __
answer is 3/4
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50)How many continuous real-valued functions f are there with domain [-1,1] such that (f(x))^2 = x^2 for each x in [-1,1]
answer is 4
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54) The inside of a certain water tank is a cube measuring 10 ft on each edge and having vertical sides and no top. Let h(t) denote the water levele in feet, above the floor of the tank at time t seconds. Starting at t=0, water pours into the tank at a constan rate of 1 cubic foot per second and simultaneously, water is removed from the tank at a rate of 0.25 h(t) cubic ft per second as t -> infinity, what is the limit of the volume of the tank?
answr is 400 cubic ft.
--------------
56)For every set S and every metric d on S, which of the ff is a metric on S?
a) 4 + d
b) e^d - 1
c)d - |d|
d)d^2
e)d^1/2
answer is e
--------------
57) Let R be the field of real numbers and R(x) the ring of polynomials in x with coefficent in R. Which of ff subsets of R(x) is a subring of R(x)?
answer: I was able to get why the others are subrings, but I can't get why: All polynomials whoose degree is an even integer, together with the zero polynomial, is not a subring.
------------------
59)A cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly 2 elements. THe number of elements in the set {x^13n: n a positive integer} is:
answer is 3
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60) If S is a ring with the property that s = s^2, which of the ff must be true?
I. s + s = 0 for all s in S
II (s+t)^2 = s^2 + t^2 for all s, t in S
II S is commutative.
answer is all 3 are true.
------------
61) What is the greatest integer that divides p^4 -1 for every prime number p greater than 5?
answer is 240.
--------------
63)At how many points in the xy plane do the graphs of y= x^12 and y =2^x intersect?
answer is 3. Shouldn't it be 2 since y=x^12 is just a parabola (am not sure of this) and y = 2^x is just similar to the graph of y = e^x?
----------
64) Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the ff must be true?
I. There is a constant C>0 st |f(x) - f(y)| <= C for all x and y in [0,1]
II. There is a constant D>0 st |f(x) - f(y)| <= 1 for all x and y in [0,1] that satisfy |x-y\<=D
III. There is a constant E>0 st |f(x) - f(y)| <= E|x-y| for all x, y in [0,1]
answer is I and II only. I know this has something to do with the mean value theorem.
_________
that's it. sorry for the long post and thanks for the help!
I gathered all the questions into one thread so as not to spam the forums:
----------
24) Which of the ff sets of vectors is a basis for the subspace of Euclidean 4-space consisting of all vectors that are orthogonal to both (0,1,1,1) and (1,1,1,0)?
A) {(0,-1,1,0)}
B) {(1,0,0,0), (0,0,0,1)}
C) {(-2,1,1,-2), ((0,1,-1,0)}
D) {(1,-1,0,1), (-1,1,0,-1), (0,1,-1,0)}
E) {(0,0,0,0),(-1,1,0,-1),(0,1,-1,0)}
answer is C.
-------------
32)When 20 children in a classroom line up for lunch, Pat insists on being somewhere ahead of Lynn, If Pat's demand is to be satisfied, in how many ways can the children line up?
answer is 20!/2
-------------
37) Summation of (k^2)/k! from k = 1 to infinity is ____
answer is 2e
-------------
44) Let f denote the function defined for all x>0 by f(x) = x^(x/2), which of the folowing statement is false?
answer is the derivative f'(x) is positive for all x>0.
I was able to get why the other choices are true, but I can't understand why this is false. Isn't the derivative [x^(x/2) x lnx]/2? which is postive for all x>0?
----------------
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 __
answer is 3/4
----------
50)How many continuous real-valued functions f are there with domain [-1,1] such that (f(x))^2 = x^2 for each x in [-1,1]
answer is 4
--------------
54) The inside of a certain water tank is a cube measuring 10 ft on each edge and having vertical sides and no top. Let h(t) denote the water levele in feet, above the floor of the tank at time t seconds. Starting at t=0, water pours into the tank at a constan rate of 1 cubic foot per second and simultaneously, water is removed from the tank at a rate of 0.25 h(t) cubic ft per second as t -> infinity, what is the limit of the volume of the tank?
answr is 400 cubic ft.
--------------
56)For every set S and every metric d on S, which of the ff is a metric on S?
a) 4 + d
b) e^d - 1
c)d - |d|
d)d^2
e)d^1/2
answer is e
--------------
57) Let R be the field of real numbers and R(x) the ring of polynomials in x with coefficent in R. Which of ff subsets of R(x) is a subring of R(x)?
answer: I was able to get why the others are subrings, but I can't get why: All polynomials whoose degree is an even integer, together with the zero polynomial, is not a subring.
------------------
59)A cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly 2 elements. THe number of elements in the set {x^13n: n a positive integer} is:
answer is 3
------------
60) If S is a ring with the property that s = s^2, which of the ff must be true?
I. s + s = 0 for all s in S
II (s+t)^2 = s^2 + t^2 for all s, t in S
II S is commutative.
answer is all 3 are true.
------------
61) What is the greatest integer that divides p^4 -1 for every prime number p greater than 5?
answer is 240.
--------------
63)At how many points in the xy plane do the graphs of y= x^12 and y =2^x intersect?
answer is 3. Shouldn't it be 2 since y=x^12 is just a parabola (am not sure of this) and y = 2^x is just similar to the graph of y = e^x?
----------
64) Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the ff must be true?
I. There is a constant C>0 st |f(x) - f(y)| <= C for all x and y in [0,1]
II. There is a constant D>0 st |f(x) - f(y)| <= 1 for all x and y in [0,1] that satisfy |x-y\<=D
III. There is a constant E>0 st |f(x) - f(y)| <= E|x-y| for all x, y in [0,1]
answer is I and II only. I know this has something to do with the mean value theorem.
_________
that's it. sorry for the long post and thanks for the help!