If $\mathrm{x}=\mathrm{a}+\mathrm{b}, \mathrm{y}=\mathrm{a\omega }+{\mathrm{b\omega }}^2, \mathrm{z}={\mathrm{a\omega }}^2+\mathrm{b\omega }$ then which of the following is true

$\mathrm{I} : \mathrm{x}+\mathrm{y}+\mathrm{z}\ne 0 \mathrm{II} : {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=6\mathrm{ab}$

$\mathrm{III} : {\mathrm{x}}^3+{\mathrm{y}}^3+{\mathrm{z}}^3=3({\mathrm{a}}^3+{\mathrm{b}}^3)$

$\begin{array}{rcl}\mathrm{x}& =& \mathrm{a}+\mathrm{b}, \mathrm{y}=\mathrm{aw}+{\mathrm{bw}}^2, \mathrm{z}={\mathrm{aw}}^2+\mathrm{bw}\end{array}$

$\begin{array}{rcl}\mathrm{x}+\mathrm{y}+\mathrm{z}& =& \mathrm{a}(1+\mathrm{w}+{\mathrm{w}}^2)+\mathrm{b}(1+{\mathrm{w}}^2+\mathrm{w})=0\end{array}$

$\begin{array}{rcl}{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2& =& {(\mathrm{a}+\mathrm{b})}^2+{(\mathrm{aw}+{\mathrm{bw}}^2)}^2+{({\mathrm{aw}}^2+\mathrm{bw})}^2 \end{array}$

$\begin{array}{rcl} & =& {\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}+{\mathrm{a}}^2{\mathrm{w}}^2+{\mathrm{b}}^2{\mathrm{w}}^4+2{\mathrm{abw}}^3+{\mathrm{a}}^2{\mathrm{w}}^4+{\mathrm{b}}^2{\mathrm{w}}^2+2{\mathrm{abw}}^3\\ & =& {\mathrm{a}}^2(1+{\mathrm{w}}^2+{\mathrm{w}}^4)+{\mathrm{b}}^2(1+{\mathrm{w}}^4+{\mathrm{w}}^2)+6\mathrm{ab}\\ & =& 6\mathrm{ab}\end{array}$$\begin{array}{rcl}\mathrm{x}+\mathrm{y}+\mathrm{z}& =& 0 \Rightarrow {\mathrm{x}}^3+{\mathrm{y}}^3+{\mathrm{z}}^3=3\mathrm{xyz}\\ & =& 3(\mathrm{a}+\mathrm{b}) (\mathrm{aw}+{\mathrm{bw}}^2) ({\mathrm{aw}}^2+\mathrm{bw})\\ & =& 3(\mathrm{a}+\mathrm{b}) ({\mathrm{a}}^2{\mathrm{w}}^3+{\mathrm{b}}^2{\mathrm{w}}^3+\mathrm{ab}({\mathrm{w}}^2+{\mathrm{w}}^4\left)\right)\\ & =& 3(\mathrm{a}+\mathrm{b}) ({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2)\\ & =& 3({\mathrm{a}}^3+{\mathrm{b}}^3)\end{array}$

only II, III are true.