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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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Director
Joined: 08 Jun 2013
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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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23 Jan 2019, 07:48
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75% (01:14) correct 25% (01:31) wrong based on 83 sessions

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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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Updated on: 24 Jan 2019, 23:21
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

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Originally posted by KanishkM on 23 Jan 2019, 08:15.
Last edited by KanishkM on 24 Jan 2019, 23:21, edited 2 times in total.
Intern
Joined: 15 Nov 2018
Posts: 25
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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23 Jan 2019, 08:35
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

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Manager
Joined: 02 Aug 2015
Posts: 152
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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23 Jan 2019, 09:31
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!
Intern
Joined: 24 Jun 2018
Posts: 12
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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23 Jan 2019, 12:56
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.
Re: If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true   [#permalink] 23 Jan 2019, 12:56
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