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12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which

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12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 07:00
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12 Days of Christmas GMAT Competition with Lots of Fun

If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
II. \(z^4 > y^4 > x^4\)
III. \(y^4 > x^4 > z^4\)

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


 


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C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 07:25
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Asked: If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
If x = 3 > y=2 > z = 1; x^4 = 81 > y^4 = 16 > z^4 = 1
MAY BE TRUE

II. \(z^4 > y^4 > x^4\)
If x=-1 > y=-2 > z>=-3; z^4 = 81 > y^4 = 16 > x^4 = 1
MAY BE TRUE

III. \(y^4 > x^4 > z^4\)
Since x>z & x^4 > z^4; x & z are positive; x>y>z>0
But It is not possible that y^4 > x^4 when x > y > 0
NOT POSSIBLE

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

IMO C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 07:41
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1. For any positive no's x>y>z this will be true. eg. x=2, y=1,z=0 so b out

Try negative no's for 2 and 3

2. x= -0.1 y= -0.2 z= -0.3 this will be true so a d out

3. Not possible. Try different set of positive and negative no's including 0. e out

Ans: C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 07:56
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If x>y>z, which of the following may be true?

I. \(x^4>y^4>z^4\)

II. \(z^4>y^4>x^4\)

III. \(y^4>x^4>z^4\)

Let's see... powers of 4. This means it will always be positive (even powers). Let's notice the combination of signs (+/-).

1. x(+) > y(+) > z(+)
--> \(x^4>y^4>z^4\)

2. x(+) > y(-) > z(-)
--> \(z^4>y^4>x^4\) (x = 2, y = -3, z = -10), or
--> \(x^4>z^4>y^4\) (x = 12, y = -3, z = -10)

3. x(+) > y(+) > z(-)
--> \(x^4>y^4>z^4\) (x = 4, y = 3, z = -1)
--> \(z^4>x^4>y^4\) (x = 4, y = 3, z = -10)

4. x(-) > y(-) > z(-)
--> Order reverses. \(z^4>y^4>x^4\)

I and II are possible. Answer.
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:11
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The value of x,y,z could be either integer or decimal. Since, stem is not saying anything..

First Condition
X=3
Y=2
Z=1
STEM: 3>2>1 (Satisfied)

I. 3^4>2^4>1^4
TRUE

II. 1^4>2^4>3^4
FALSE

III. 2^4>3^4>1^4
FALSE

Second Condition
X=-2
Y=-3
Z=-4
STEM: -2>-3>-4 (Satisfied)

II. (-2)^4>(-3)^4>(-2)^4
TRUE

III. (-3)^4>(-2)^4>(-4)^4
FALSE

Third Condition
X=1/2
Y=1/3
Z=1/4

STEM: 1/2>1/3>1/4 (Satisfied)

III. (1/3)^4>(1/2)^4>(1/4)^4
FALSE


Only I and II may be true

Answer: C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:13
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If we assume all value is positive the x,y,z will satisfy the 1st equation

If we assume all value negative then x,y,z will satisfy the 2nd equation

Here is no group value of x,y,z which can satisfy 3rd equation

So Ans-C

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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:14
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Bunuel wrote:
12 Days of Christmas GMAT Competition with Lots of Fun

If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
II. \(z^4 > y^4 > x^4\)
III. \(y^4 > x^4 > z^4\)

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


 


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for all positive integers
if x > y > z
then x^4 > y^4 > z^4

For all negative integers
if x > y > z
then z^4 > y^4 > x^4

For all numbers between 0 and 1 statement I will apply.

So answer is option C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:15
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given
If \(x > y > z\), which of the following may be true
I. \(x^4 > y^4 > z^4\)
2>1>0 ; true
II. \(z^4 > y^4 > x^4\)
(-8 )^4>(-6)^4>(-2)^4 ; true

III. \(y^4 > x^4 > z^4\)
this relation cannot be true because for any -ve value of z and +ve value of x , z>x when raised to power of 4
also when +ve x>y so y^4>x^4 is not possible

OPTION C is correct I and II only


Bunuel wrote:
12 Days of Christmas GMAT Competition with Lots of Fun

If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
II. \(z^4 > y^4 > x^4\)
III. \(y^4 > x^4 > z^4\)

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


 


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for the 12 Days of Christmas Competition

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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:39
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Bunuel wrote:
12 Days of Christmas GMAT Competition with Lots of Fun

If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
II. \(z^4 > y^4 > x^4\)
III. \(y^4 > x^4 > z^4\)

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


 


This question was provided by Experts'Global
for the 12 Days of Christmas Competition

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Solution:
Given, \(x > y > z\).

If x=4, y=3, z=2
or x=1/2, y=1/3, z=1/4
then \(x^4 > y^4 > z^4\). I is possible.

If x=-2, y=-3, z=-4
or x=-1/4, y=-1/3, z=-1/2
then \(z^4 > y^4 > x^4\). II is possible.

No other case is possible even after taking either of them as 0.

ANSWER C.
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:51
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If \(x > y > z\), which of the following may be true?

I. \(x^4 > y^4 > z^4\)
II. \(z^4 > y^4 > x^4\)
III. \(y^4 > x^4 > z^4\)

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

IMO C

I. Possible. For x=4, y=3, z=2

II. Possible. For x= -2, y= -3, z= -4

III. Not possible
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 08:51
taking fraction, negative values and positive values
only 1 & 2 can be true
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 09:23
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If x>y>zx>y>z, which of the following may be true?

I. x4>y4>z4x4>y4>z4
9^4>8^4>7^4 Correct
II. z4>y4>x4
(-10)^4>(-6)^4>5^4 Correct

IMO C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 09:46
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I) take x=3,y=2,z=1 , True

II) take x=1/5, y=1/3 z=1/2, True

II) take x=-1, y=-2, z=-3; False,
take x=-1/2, y=-1/3, z=-1/5 ; False,
take x=-1/3, y=-1/2 (we have x>y and y^4>x^4)
for y^4>x^4>z^4, we need z lesser than -1/2.2, which makes x^4>y^4
It's always False

Hence C)
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 09:54
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If all are +ve, say x=3, y=2, z=1 then I will hold true
If all are -ve, say x=-1, y=-2, z=-3 then II will hold true
III will not hold true in any scenario even when we consider any variable as 0 or fraction/decimal.

Hence, answer is C.
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 09:58
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I. x4>y4>z4x4>y4>z4; x=3, y=2, z=1 yes
II. z4>y4>x4z4>y4>x4; x=-1/4, y=-1/3, z=-1/2
III. y4>x4>z4--Not possible because z^4 will always be more

Ans C
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 10:30
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Answer is C
If x>y>z, which of the following may be true?

I. x^4>y^4>z^4 when x > y > z > 0. this is true
II. z^4>y^4>x^4 When 0 >x > y > z this is true
III. y^4>x^4>z^4 this cannot be true
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 10:34
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If x>y>z, which of the following may be true?

I. x4>y4>z4
II. z4>y4>x4
III. y4>x4>z4

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

Given x>y>z and all options have even power 4 for all varaiables
which means sign will change if there is a negative number
If all x,y,z are positive 1 will hold true

if z is negative , sign will change for z and will retain it magnitude
for eg if x=2 y=1 z= -5 and given
x>y>z
z^4 =625 which is highest so 2 will hold true

If y is negative, it cannot have the highest magnitude
as x>y>z has always has to be true
for eg x=8 y=-5 and z=-6
so 3 cannot be true


So C is the correct answer
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 10:38
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For any positive number greater than 1,
x^4>y^4>z^4 is true.

For example- z=2, y=3, x=4.

Now,
if z=-5, y=3 and x=4,
z^4>y^4>x^4 is true.

Can't think of any combination where statement III would be true. So, statement I and II are true.

Hence, IMO C.
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 11:35
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\(x > y > z\)
3 > 2> 1

So, 81 > 16> 1

\(x > y > z\) is true

\(x > y > z\)

-1> -2> -3

but, 1< 16< 81

II. \(z^4 > y^4 > x^4\) is true

Option c
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink] New post   23 Dec 2022, 12:35
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If x>y>z which of the following may be true?

I. x4>y4>z4x4>y4>z4
suppose x=3,y=2,z=1 hence 81>16>1 therefore I is true
II. z4>y4>x4
suppose x=3,y=-4,z=-5 hence 625>256>81 Therefore II is also true

hence only I and II are correct
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Re: 12 Days of Christmas GMAT Competition - Day 10: If x > y > z, which [#permalink]
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