Question : If ${x^2+y^2+z^2=2(x+z-1)}$, then the value of $x^3+y^3+z^3$ is equal to:
Option 1: 6
Option 2: 1
Option 3: 2
Option 4: 8
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Correct Answer: 2
Solution :
Given, ${x^2+y^2+z^2=2(x+z-1)}$
$x^2+y^2+z^2=2x+2z−2$
⇒ $x^2−2x+1+y^2+z^2−2z+1=0$
⇒ $(x−1)^2+y^2+(z−1)^2=0$
⇒ $(x-1)^2=0,$ $y^2=0$, and $(z-1)^2=0$
⇒ $x-1=0,$ $y=0$, and $z-1=0$
⇒ $x=1,$ $y=0$, and $z=1$
Now, $x^3+y^3+z^3=1^3+0+1^3=1+1=2$
Hence, the correct answer is 2.
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