Given, mass of flywheel, $M=1 \mathrm{~kg}$, radius vectors $\mathbf{R}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \mathrm{m}$, $\mathbf{F}=(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}) \mathrm{N}$ and time, $t=4.5 \mathrm{~s}$ Magnitude of radius
$$
(R)=\sqrt{(2)^2+(1)^2+(2)^2}=\sqrt{9}=3 \mathrm{~m}
$$

Similarly, $F=\sqrt{29} \mathrm{~N}$
Torque on the flywheel,
$$
\begin{gathered}
\tau=I \alpha=F \cdot R=\frac{M R^2}{2} \alpha \\
\alpha=\frac{2 F}{M R}=\frac{2 \sqrt{29}}{1 \times 3}=\frac{2}{3} \sqrt{29}
\end{gathered}
$$

Now, the angular velocity,
$$
\begin{aligned}
\omega & =\omega_0+\alpha t \Rightarrow \omega=0+\frac{2}{3} \sqrt{29} \times 4.5\left(\because \omega_0=0\right) \\
\Rightarrow \quad \omega & =\sqrt{261} \mathrm{rad} \mathrm{s}^{-1}
\end{aligned}
$$

Hence, the correct option is (c).