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GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x

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Bunuel
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GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink] GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x   Updated on: 22 Jul 2022, 08:17
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95% (hard)

Question Stats:

39% (02:09) correct 61% (01:58) wrong based on 197 sessions
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If a data set A consists of all x such that \((x^2 - 5x + 5)^{(x^2 + 8x)} = 1\), then what is the median of A ?

A. -4
B. 0.5
C. 1
D. 1.5
E. 2.5


 


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C

Originally posted by Bunuel on 22 Jul 2022, 08:00.
Last edited by Bunuel on 22 Jul 2022, 08:17, edited 1 time in total.
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ThatDudeKnows
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Re: GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink] GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x   Updated on: 29 May 2023, 05:11
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Bunuel wrote:
If a data set A consists of all x such that \((x^2 - 5x + 5)^{(x^2 + 8x)} = 1\), then what is the median of A ?

A. -4
B. 0.5
C. 1
D. 1.5
E. 2.5


 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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There are three ways to make this work:

1) Set \((x^2+8x) = 0\)
x = {-8,0}

2) Set \(x^2-5x+5 = 1\)
\(x^2-5x+4 = 0\)
\((x-4)(x-1) = 0\)
x = {1,4}

3) Set \(x^2-5x+5 = -1\) and confirm that \((x^2+8x)\) is even.
\(x^2-5x+6 = 0\)
\((x-3)(x-2) = 0\)
x = {2,3}
Confirm \((x^2+8x)\) is even for 2: \((2^2+8*2) = 4+16 = 20\). Okay.
Confirm \((x^2+8x)\) is even for 3: \((3^2+8*3) = 9+24 = 33\). That's odd. Eliminate 3.
x = {2}

x = {-8,0,1,2,4}

Median = 1

Answer choice C.


**Edited to conform to the revised question with an 8 in the exponent rather than a 9.

Originally posted by ThatDudeKnows on 22 Jul 2022, 08:12.
Last edited by ThatDudeKnows on 29 May 2023, 05:11, edited 2 times in total.
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Re: GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink] GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x   22 Jul 2022, 08:40
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We see that the equation becomes 1 , this occurs in the following three conditions -

1) The power is 0

2) The base is 1 and the power can be anything

3) The base is -1 and the power is even

Case 1: When the power is 0

\(x^2 \)+ 8x = 0

x(x + 8) = 0

x = -8
or
x = 0

Case 2: The base is 1 and the power can be anything

\(x^2 - 5x + 5 = 1\)

\(x^2 - 5x + 4 = 0\)

\((x-4)(x-1) = 0 \)

x = 4 or x = 1

Case 3: The base is -1 and the power is even

\(x^2 - 5x + 5 = -1\)

\(x^2 - 5x + 6 = 0\)

\((x - 3)(x-2) = 0\)

x = 3
or
x = 2

But if we have x = 3, the power becomes odd, hence we have to eliminate this.

A = {-8, 0, 1, 2, 4}

Median = 1

IMO C
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Re: GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink] GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x   22 Feb 2023, 06:43
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Bunuel wrote:
If a data set A consists of all x such that \((x^2 - 5x + 5)^{(x^2 + 8x)} = 1\), then what is the median of A ?

A. -4
B. 0.5
C. 1
D. 1.5
E. 2.5


 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

Compete, Get Better, Win prizes and more

 



Official Solution:

If a data set A consists of all x such that \((x^2 - 5x + 5)^{(x^2 + 8x)} = 1\), then what is the median of A ?

A. \(-4\)
B. \(0.5\)
C. \(1\)
D. \(1.5\)
E. \(2.5\)


Three cases are possible for given equation to hold true:

CASE 1: the base is 1, because \(1^m=1\), for all \(m\)

\(x^2 - 5x + 5=1\);

\(x^2 - 5x + 4=0\);

\(x=1\) or \(x=4\).

CASE 2: the exponent is 0, because \(n^0=1\), for any nonzero \(n\).

\(x^2 + 8x=0\);

\(x(x + 8)=0\);

\(x=0\) or \(x=-8\).

CASE 3: the base is -1 and the exponent is even, because \((-1)^k=1\), for any even \(k\).

\(x^2 - 5x + 5=-1\);

\(x^2 - 5x + 6=0\);

\(x=2\) or \(x=3\).

We need to check for which of these values of \(x\), is the exponent even:

When \(x=2\), then the exponent, \(x^2 + 8x=20=even\). Valid.

When \(x=3\), then the exponent, \(x^2 + 8x=33=odd\). Not valid.

Therefore, data set A is {-8, 0, 1, 2, 4 } and its median is 1.


Answer: C
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Re: GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink] GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x   22 Jul 2022, 18:21
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If a data set A consists of all x such that \((x^2 - 5x + 5)^{(x^2 + 8x)} = 1\), then what is the median of A ?

A. -4
B. 0.5
C. 1
D. 1.5
E. 2.5

(x^2 - 5x + 5)^{(x^2 + 8x)} = 1
L.H.S will be 1 if either (x^2 - 5x + 5)=1 or x^2+8x=0
(x^2 - 5x + 5)=1 --- equation 1
x^2+8x=0 ----equation 2
Possible solution for equation 1
(x^2-5x+5)=1
=>x^2-5x+4=0
=>x^2-4x-x+4=0
=>x(x-4)-1(x-4)=0
=>(x-4)(x-1)=0
Therefore x=1,4
Again if (x^2-5x+5)=-1 and (x^2 + 8x) is even then also the value of the expression would be 1
(x^2-5x+5)=-1
=>x^2-5x+6=0
=>x^2-3x-2x+6=0
=>x(x-3)-2(x-3)=0
=>(x-3)(x-2)=0
Therefor x=2,3 For x=2 (x^2+8x) is even but for x=3 (x^2+8x) is odd
Thus the only solution here is x=2
Now x^2+8x=0
=>x(x+8)=0
There x=-8,0
All the values in set A= {-8,0,1,2,4}
Therefore the median is 1
Answer is C
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Re: GMAT Club World Cup 2022 (DAY 10): If a data set A consists of all x [#permalink]
22 Jul 2022, 18:21
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Bunuel
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