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

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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Bunuel Please provide the solution.

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

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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Will 1 hold if x is a big number, for instance, 10?

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.

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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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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