If $A$ is a $2 \times 2$ matrix such that $\operatorname{det} A=-21$ and trace of $A^3$ is 2024 , then the trace of $A$ is

If $\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ is a skew-symmetric matrix and $b, c$ and $f$ are non-zero real numbers, then $\frac{b}{c}=$

In the matrix $\left[\begin{array}{ccc}-1 & x & 3 \\ -4 & -5 & -6 \\ -7 & y & 9\end{array}\right]$, if the cofactors of -6 and -7 are respectively 22 and 27 , then $5 x+y=$

Consider the simultaneous linear equations $\beta x+\alpha y-z=-1,3 x-\beta y+\alpha z=0 \alpha x+\beta y+z=1$, In the usual notation used in Crammer's rule, given that $\frac{\Delta_1}{\Delta}=-1, \frac{\Delta_2}{\Delta}=1, \frac{\Delta_3}{\Delta}=2$, then $(\alpha, \beta)=$