Let $\overline{\mathrm{a}}\mathrm{and}\overline{\mathrm{b}}$ be two vectors such that $\left|\overline{\mathrm{a}}\right|=2022,\left|\overline{\mathrm{b}}\right|=2022\mathrm{and}\overline{\mathrm{a}}.\overline{\mathrm{b}}=0$ for some $\mathrm{\alpha },\mathrm{\beta }\in \mathrm{R},\overline{\mathrm{c}}=\mathrm{\alpha }\overline{\mathrm{a}}+\mathrm{\beta }\overline{\mathrm{b}}+\frac{\overline{\mathrm{a}}\times \overline{\mathrm{b}}}{2022}$ if $\left|\overline{\mathrm{c}}\right|=2022\sqrt{2}$ and the vector $\overline{\mathrm{c}}$ makes an angle $\text{'}\mathrm{\theta }\text{'}$ with both $\overline{\mathrm{a}}\mathrm{and}\overline{\mathrm{b}},$ then $\left[2024{\cos }^2\mathrm{\theta }\right]$ is….. (where [.] represents G.I.F)

$\begin{array}{l}\overline{\mathrm{c}}=\mathrm{\alpha }\overline{\mathrm{a}}+\mathrm{\beta }\overline{\mathrm{b}}+\frac{\overline{\mathrm{a}}\times \overline{\mathrm{b}}}{2022}\Rightarrow \overline{\mathrm{c}}\cdot \overline{\mathrm{a}}={\mathrm{\alpha a}}^2\Rightarrow \mathrm{\alpha }=\sqrt{2}\cos \mathrm{\theta }\\ \text{ and }\overline{\mathrm{c}}\cdot \overline{\mathrm{b}}={\mathrm{\beta b}}^2\Rightarrow \mathrm{\beta }=\sqrt{2}\cos \mathrm{\theta }|\overline{\mathrm{a}}\times \overline{\mathrm{b}}{|}^2=|\overline{\mathrm{a}}{|}^2|\overline{\mathrm{b}}{|}^2-(\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}{)}^2\\ |\overline{\mathrm{a}}\times \overline{\mathrm{b}}{|}^2=(2022{)}^4\Rightarrow |\overline{\mathrm{a}}\times \overline{\mathrm{b}}{|}^2=(2022{)}^2\\ \text{ and now }\overline{\mathrm{c}}\cdot \overline{\mathrm{c}}={\mathrm{\alpha }}^2{\mathrm{a}}^2+{\mathrm{\beta }}^2{\mathrm{b}}^2+(2022{)}^2\\ \Rightarrow \left(2022{)}^2\right(2)=(2022{)}^2\cdot \left[2{\cos }^2\mathrm{\theta }+2{\cos }^2\mathrm{\theta }+1\right]\\ \Rightarrow 2=4{\cos }^2\mathrm{\theta }+1\Rightarrow 4{\cos }^2\mathrm{\theta }=1\\ \therefore \left[2024{\cos }^2\mathrm{\theta }\right]=\left[506\left(4{\cos }^2\mathrm{\theta }\right)\right]=506\end{array}$