x2 + y2 + z2 = 2(x + z - 1), then the value of x3 + y3 + z3 = ?

Answer: Option B

$$\begin{array}{rl}& \text{Given,}\\ & x^2+y^2+z^2=2\left(x+z-1\right)\\ & \text{Find, }{x}^3+y^3+z^3=?\\ & \Rightarrow x^2+y^2+z^2=2\left(x+z-1\right)\\ & \Rightarrow x^2+y^2+z^2=2x+2z-2\\ & \Rightarrow x^2+y^2+z^2=2x+2z-1-1\\ & \Rightarrow \left(x^2+1-2x\right)+y^2+\left(z^2+1-2z\right)=0\\ & \Rightarrow {\left(x-1\right)}^2+y^2+{\left(z-1\right)}^2=0\\ & \Rightarrow {\left(x-1\right)}^2=0\\ & \Rightarrow x=1\\ & \Rightarrow y^2=0\\ & \Rightarrow y=0\\ & \Rightarrow {\left(z-1\right)}^2=0\\ & \Rightarrow z=1\\ & \text{Value substituted in question,}\\ & \Rightarrow x^3+y^3+z^3\\ & \Rightarrow 1^3+0+1^3\\ & \Rightarrow 2\end{array}$$