c
By the law of conservation of angular momentum  \({I_1}{\omega _1} + {I_2}{\omega _2} = \left( {{I_1} + {I_2}} \right)\,\omega \)

Angular velocity of system \(\omega  = \frac{{{I_1}{\omega _1} + {I_2}{\omega _2}}}{{{I_1} + {I_2}}}\)

Rotational kinetic energy \( = \frac{1}{2}\left( {{I_1} + {I_2}} \right)\;{\omega ^2}\) \( = \frac{1}{2}\left( {{I_1} + {I_2}} \right)\,{\left( {\frac{{{I_1}{\omega _1} + {I_2}{\omega _2}}}{{{I_1} + {I_2}}}} \right)^2}\)\( = \frac{{{{({I_1}{\omega _1} + {I_2}{\omega _2})}^2}}}{{2({I_1} + {I_2})}}\).