Answer: Option B
$$\eqalign{
& {\text{Given,}} \cr
& {x^2} + {y^2} + {z^2} = 2\left( {x + z - 1} \right) \cr
& {\text{Find, }}{x^3} + {y^3} + {z^3} = ? \cr
& \Rightarrow {x^2} + {y^2} + {z^2} = 2\left( {x + z - 1} \right) \cr
& \Rightarrow {x^2} + {y^2} + {z^2} = 2x + 2z - 2 \cr
& \Rightarrow {x^2} + {y^2} + {z^2} = 2x + 2z - 1 - 1 \cr
& \Rightarrow \left( {{x^2} + 1 - 2x} \right) + {y^2} + \left( {{z^2} + 1 - 2z} \right) = 0 \cr
& \Rightarrow {\left( {x - 1} \right)^2} + {y^2} + {\left( {z - 1} \right)^2} = 0 \cr
& \Rightarrow {\left( {x - 1} \right)^2} = 0 \cr
& \Rightarrow x = 1 \cr
& \Rightarrow {y^2} = 0 \cr
& \Rightarrow y = 0 \cr
& \Rightarrow {\left( {z - 1} \right)^2} = 0 \cr
& \Rightarrow z = 1 \cr
& {\text{Value substituted in question,}} \cr
& \Rightarrow {x^3} + {y^3} + {z^3} \cr
& \Rightarrow {1^3} + 0 + {1^3} \cr
& \Rightarrow 2 \cr} $$