A double star system consists of two stars having masses M and 2M. The distance between their centers is r. They revolve under their mutual gravitational attraction. Then the incorrect options are____

(i) they revolve about their C.O.M with same ‘$\omega$’

$$\begin{gathered} r_1=\frac{2M}{M+2M}r=\frac{2r}{3}=radius of small star \\ {r}_2=\frac{M}{M+2M}r=\frac{r}{3}=radius of big star \\ \left(iv\right) L_1=M{\left(\frac{2r}{3}\right)}^2\omega \\ {L}_2=2M{\left(\frac{r}{3}\right)}^2\omega \\ \therefore L_1\ne L_2 \\ \left(ii\right) M.\left(\frac{2r}{3}\right).{\omega }^2=\frac{G.M.2M}{r^2} \\ \Rightarrow {\omega }^2=\frac{2GM}{r^2.\frac{2r}{3}} \\ \Rightarrow {\omega }^2=\frac{3GM}{r^3} \\ \therefore T=2\pi \sqrt{\frac{r^3}{3GM}} \\ \left(iii\right) K.E_{small}=\frac{1}{2}\left(M{\left(\frac{2r}{3}\right)}^2\right){\omega }^2=\frac{2M{r}^2{\omega }^2}{9} \\ K.E_{big}=\frac{1}{2}\left(2M{\left(\frac{r}{3}\right)}^2\right){\omega }^2=\frac{M{r}^2{\omega }^2}{9} \end{gathered}$$