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Bunuel

Fresh GMAT Club Tests' Challenge Question:



If \(xy ≠ 0\) and \(x^{(y + 1)} = 2x^y - 1\), what is the value of x?

(1) \(x^y = x\)

(2) \(x^2=1\)

(1) (x^y) * x = 2x^y - 1
x^2 -2x +1 = 0
(x-1)^2 = 0
x=1
Sufficient

(2)
x=+1 or -1
if x =1 then x^{(y + 1)} = 2x^y - 1 holds for all cases of y (odd,even)
if x= -1 then x^{(y + 1)} = 2x^y - 1 does not hold
hence x can only be 1
sufficient
D
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Bunuel

Fresh GMAT Club Tests' Challenge Question:



If \(xy ≠ 0\) and \(x^{(y + 1)} = 2x^y - 1\), what is the value of x?

(1) \(x^y = x\)

(2) \(x^2=1\)

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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Bunuel

Fresh GMAT Club Tests' Challenge Question:



If \(xy ≠ 0\) and \(x^{(y + 1)} = 2x^y - 1\), what is the value of x?

(1) \(x^y = x\)

(2) \(x^2=1\)


\(x^{(y + 1)} = 2x^y - 1.............x*x^y=2x^y-1....\)

(1) \(x^y = x\)
Substitute this in equation above...
\(x*x=2x-1\)......
\(x^2-2x+1=0\).....
\((x-1)^2=0\)
Therefore, x=1...
Sufficient

(2) \(x^2=1\)
so x=1 or -1..
Now \(x*x^y=2x^y-1.............(2-x)x^y=1\)..
Substitute x as 1..
\((2-x)x^y=1\)......
\((2-1)1^y=1\).. Possible, so X can be 1
Substitute x as -1
\((2-x)x^y=1\)......
\((2-(-1))*(-1)^y=1\).......
\(3*(-1)^y=1\).. not possible
So X is 1
Sufficient

D
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Bunuel

If \(xy ≠ 0\) and \(x^{(y + 1)} = 2x^y - 1\), what is the value of x?

(1) \(x^y = x\)

(2) \(x^2=1\)

\(xy ≠ 0…(x,y)≠0\)
\(x^{(y + 1)} = 2x^y - 1…(x^y)(x)=2(x^y)-1…x^y(x-2)=-1\)

(1) \(x^y = x\) sufic.
\(x^y(x-2)=-1…x(x-2)=-1…x^2-2x+1=0…(x-1)^2=0…x=1\)

(2) \(x^2=1\) sufic.
\(x^2=1…|x|=(1,-1)\)
\(x=-1:x^y(x-2)=-1…[-1]([-1]-2)=-1…-1(-3)≠-1\)
\(x=1:x^y(x-2)=-1…[1]([1]-2)=-1…(-1)=-1\)

Answer (D)
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Bunuel

Fresh GMAT Club Tests' Challenge Question:



If \(xy ≠ 0\) and \(x^{(y + 1)} = 2x^y - 1\), what is the value of x?

(1) \(x^y = x\)

(2) \(x^2=1\)

My solution is presented in the image attached.
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108174253_579052036306770_8866468411894112518_n.jpg
108174253_579052036306770_8866468411894112518_n.jpg [ 251.05 KiB | Viewed 4737 times ]

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