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If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true

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If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true  [#permalink]

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New post 23 Jan 2019, 08:48
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If \(XY ≠ 0\) and \(X^3Y = XY^3\), then which of the following must be true?

I) \(|X| = -|Y|\)
II) \(|X| = |Y|\)
III) \(X = Y = 1\)

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III

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If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true  [#permalink]

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New post Updated on: 25 Jan 2019, 00:21
1
Helium wrote:
If \(XY ≠ 0\) and \(X^3Y = XY^3\), then which of the following must be true?

I) \(|X| = -|Y|\)
II) \(|X| = |Y|\)
III) \(X = Y = 1\)

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III


\(X^3Y = XY^3\)
\(X^3Y - XY^3 = 0\)
\(XY (X^2 - Y^2) = 0\)
\(XY (X - Y) (X+Y) = 0\)

Now XY != 0(given) so X^2 = Y^2

I) \(|X| = -|Y|\)
Now this inequality can show 4 cases, We can't say anything about XY(This can be +ive or -ive), i wont go with this. Not sufficient
X= - Y, X= -(-Y), -X = -Y and -X = - (-Y)

II) \(|X| = |Y|\)
This is always true, square root of X^2 = Y^2 => |X| = |Y|

III) \(X = Y = 1\)
Not necessary, what if the value is different

Answer B
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Originally posted by KanishkM on 23 Jan 2019, 09:15.
Last edited by KanishkM on 25 Jan 2019, 00:21, edited 2 times in total.
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Re: If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true  [#permalink]

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New post 23 Jan 2019, 09:35
If XY≠0XY≠0 and X3Y=XY3X3Y=XY3, then which of the following must be true?

I) |X|=−|Y||X|=−|Y|
II) |X|=|Y||X|=|Y|
III) X=Y=1X=Y=1

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III

IMO B
1- |X| = -|Y|
Not sufficient coz modulus never gives a negative value ,

2- |X| = |Y|
4 cases but it will always end up in 2 saying x=Y or x= -Y
Upon simplifying the original equation we will get
Xy= 0 - not possible
X= Y or x= -y
Sufficient

3- x=Y=1
Not sufficient coz x and u could 2 and 2 or 3 and 3 or 2 and -2
Not sufficient

So answer is B

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Re: If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true  [#permalink]

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New post 23 Jan 2019, 10:31
Helium wrote:
If \(XY ≠ 0\) and \(X^3Y = XY^3\), then which of the following must be true?

I) \(|X| = -|Y|\)
II) \(|X| = |Y|\)
III) \(X = Y = 1\)

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III


\(X^3Y = XY^3\)

\(X^3Y - XY^3\) = 0

XY(\(X^2 -Y^2\)) = 0

Given \(XY ≠ 0\)

So \(X^2 -Y^2\) = 0

\(X^2 = Y^2\)

Options II and III satisfy the above condition.

Also in I) \(|X| = -|Y|\),

|X| is always positive, how can it be equal to a negative value? The case is only possible when X=Y=0 but questions says \(XY ≠ 0\).

Are the given options correct or am I missing something?

Cheers!
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Re: If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true  [#permalink]

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New post 23 Jan 2019, 13:56
KanishkM

If C must be true ie X=Y=1 that means X & Y cant hold any other value, other than 1.

However, X=Y=2 or X=Y=3 also achieves the condition mentioned in the question.
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Re: If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true   [#permalink] 23 Jan 2019, 13:56
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