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The following sets are given, where is an integer greater than 1: 30 Sep 2021, 04:30
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64% (02:35) correct 36% (02:22) wrong based on 159 sessions

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The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
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E
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The following sets are given, where is an integer greater than 1: 11 Oct 2021, 06:32
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Bunuel
The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Show more

Let \(x \) be \(2\), then:

A \(= ( -1+2+4),(1-2+4),(1+2+4) = ( 5,3,7)\)

B\(= (-1+4+8), (1-4+8),(1+4+8) = (11,5,13)\)

I. The greatest element of A is greater than the least element of B-> \(7 > 5 \) True

II. The greatest element of A is less than the greatest element of B. -> \(7 < 13 \) True

III. The average of the elements of A is smaller than the average of the elements of B:
\(15 < 29 \) ( If sum is less avg. will be less as both sets contain equal no. of elements ) True

Ans - E

Hope it's clear.
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The following sets are given, where is an integer greater than 1: 22 Oct 2021, 02:48
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Bunuel Please provide the solution.
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The following sets are given, where is an integer greater than 1: 25 Oct 2021, 11:56
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stne
Bunuel
The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Let \(x \) be \(2\), then:

A \(= ( -1+2+4),(1-2+4),(1+2+4) = ( 5,3,7)\)

B\(= (-1+4+8), (1-4+8),(1+4+8) = (11,5,13)\)

I. The greatest element of A is greater than the least element of B-> \(7 > 5 \) True

II. The greatest element of A is less than the greatest element of B. -> \(7 < 13 \) True

III. The average of the elements of A is smaller than the average of the elements of B:
\(15 < 29 \) ( If sum is less avg. will be less as both sets contain equal no. of elements ) True

Ans - E

Hope it's clear.
Show more

Thanks very much for your explanations. A quick question on Statement I. It appears it doesn't hold in some instances: for instace X=3 . In that case, can we conclude Statement I is true.

Posted from my mobile device
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The following sets are given, where is an integer greater than 1: 26 Oct 2021, 05:50
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StacyArko
stne
Bunuel
The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Let \(x \) be \(2\), then:

A \(= ( -1+2+4),(1-2+4),(1+2+4) = ( 5,3,7)\)

B\(= (-1+4+8), (1-4+8),(1+4+8) = (11,5,13)\)

I. The greatest element of A is greater than the least element of B-> \(7 > 5 \) True

II. The greatest element of A is less than the greatest element of B. -> \(7 < 13 \) True

III. The average of the elements of A is smaller than the average of the elements of B:
\(15 < 29 \) ( If sum is less avg. will be less as both sets contain equal no. of elements ) True

Ans - E

Hope it's clear.

Thanks very much for your explanations. A quick question on Statement I. It appears it doesn't hold in some instances: for instace X=3 . In that case, can we conclude Statement I is true.

Posted from my mobile device
Show more

Dear StacyArko,

You are correct, in case of \(x= 3 \) statement \(1\) does not satisfy.

However please note: Question does not say " Which of the following statements MUST be true".

Hence we cannot disqualify statement 1 .

Although I agree with you, it could have been better if question read " can be true ".

Hope it helps.
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The following sets are given, where is an integer greater than 1: 28 Oct 2021, 20:26
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stne
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The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Let \(x \) be \(2\), then:

A \(= ( -1+2+4),(1-2+4),(1+2+4) = ( 5,3,7)\)

B\(= (-1+4+8), (1-4+8),(1+4+8) = (11,5,13)\)

I. The greatest element of A is greater than the least element of B-> \(7 > 5 \) True

II. The greatest element of A is less than the greatest element of B. -> \(7 < 13 \) True

III. The average of the elements of A is smaller than the average of the elements of B:
\(15 < 29 \) ( If sum is less avg. will be less as both sets contain equal no. of elements ) True

Ans - E

Hope it's clear.
Show more

Will 1 hold if x is a big number, for instance, 10?
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The following sets are given, where is an integer greater than 1: 28 Oct 2021, 21:39
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Bunuel
The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Show more

A couple of assumptions: The question says "... where x is an integer greater than 1".
"Which statements are true" means "which statements are true for all possible values of x"

Since x is an integer greater than 1, we know x^3 > x^2 > x > 1

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}......... Greatest: 1 + x + x^2, Least: 1 - x + x^2\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}.......... Greatest: 1 + x^2 + x^3, Least: 1 - x^2 + x^3 \)


I. The greatest element of A is greater than the least element of B.

\(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
\(Is \ 2x^2 > x^3?\)
If x = 2, it holds.
If x = 10, it doesn't hold.

Not always True.

II. The greatest element of A is less than the greatest element of B.

\(Is \ (1 + x + x^2) < (1 + x^2 + x^3)?\)
\(Is \ x < x^3\)
Yes, this is always true if x is greater than 1.

True.

III. The average of the elements of A is smaller than the average of the elements of B.

\(Avg of A = [(-1+x+x^2) + (1-x+x^2) + (1+x+x^2)]/3 = [3x^2 + x + 1]/3\)
\(Avg of B = [(-1+x^2+x^3) + (1-x^2+x^3) + (1+x^2+x^3)]/3 = [3x^3 + x^2 + 1]/3\)
x^3 is greater than x^2 and x^2 is greater than x. So avg of B will be greater than avg of A.

True.
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The following sets are given, where is an integer greater than 1: 28 Oct 2021, 21:43
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VeritasKarishma
Bunuel
The following sets are given, where is an integer greater than 1:

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}\)

Which statements are true?

I. The greatest element of A is greater than the least element of B.

II. The greatest element of A is less than the greatest element of B.

III. The average of the elements of A is smaller than the average of the elements of B

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

A couple of assumptions: The question says "... where x is an integer greater than 1".
"Which statements are true" means "which statements are true for all possible values of x"

Since x is an integer greater than 1, we know x^3 > x^2 > x > 1

\(A=\{(-1+x+x^2), \ (1-x+x^2), \ (1+x+x^2)\}......... Greatest: 1 + x + x^2, Least: 1 - x + x^2\)

\(B=\{(-1+x^2+x^3), \ (1-x^2+x^3), \ (1+x^2+x^3)\}.......... Greatest: 1 + x^2 + x^3, Least: 1 - x^2 + x^3 \)


I. The greatest element of A is greater than the least element of B.

\(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
\(Is \ 2x^2 > x^3?\)
If x = 2, it holds.
If x = 10, it doesn't hold.

Not always True.

II. The greatest element of A is less than the greatest element of B.

\(Is \ (1 + x + x^2) < (1 + x^2 + x^3)?\)
\(Is \ x < x^3\)
Yes, this is always true if x is greater than 1.

True.

III. The average of the elements of A is smaller than the average of the elements of B.

\(Avg of A = [(-1+x+x^2) + (1-x+x^2) + (1+x+x^2)]/3 = [3x^2 + x + 1]/3\)
\(Avg of B = [(-1+x^2+x^3) + (1-x^2+x^3) + (1+x^2+x^3)]/3 = [3x^3 + x^2 + 1]/3\)
x^3 is greater than x^2 and x^2 is greater than x. So avg of B will be greater than avg of A.

True.
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Hi Karishma,

Thanks for the explanation. I followed the same approach but didn't find this option in the answer choice.
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The following sets are given, where is an integer greater than 1: 28 Oct 2021, 22:05
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VeritasKarishma

I. The greatest element of A is greater than the least element of B.

\(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
\(Is \ 2x^2 > x^3?\)
If x = 2, it holds.
If x = 10, it doesn't hold.
Show more
VeritasKarishma How are you able to reach to \(Is \ 2x^2 > x^3?\)
When you solve \(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
You get
\(2x^2 + x > x^3?\) not \(2x^2 > x^3?\)
Did you intentionally miss \((+x)\) from the LHS? Or was it a misstype?
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The following sets are given, where is an integer greater than 1: 01 Nov 2021, 04:01
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PyjamaScientist
VeritasKarishma

I. The greatest element of A is greater than the least element of B.

\(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
\(Is \ 2x^2 > x^3?\)
If x = 2, it holds.
If x = 10, it doesn't hold.
VeritasKarishma How are you able to reach to \(Is \ 2x^2 > x^3?\)
When you solve \(Is \ (1 + x + x^2) > (1 - x^2 + x^3)?\)
You get
\(2x^2 + x > x^3?\) not \(2x^2 > x^3?\)
Did you intentionally miss \((+x)\) from the LHS? Or was it a misstype?
Show more

I missed the x.

It should be
\(Is \ x + 2x^2 > x^3?\)

But the point is that when x is a small value, x, x^2 and x^3 are comparable. But when x is a greater value, x^3 will become far greater than x and x^2.
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The following sets are given, where is an integer greater than 1: 05 Jun 2024, 13:51
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