Let $a, b, c, d \in \mathbf{R}$ be such that $a d-b c \neq 0$ and $e$ be a positive number other than 1 .

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|, \Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$, then the values of $x$ and $y$ are respectively.

For a square matrix $B$ of order 3 , if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

For $\alpha, \beta \in[0,2 \pi]$ and $\gamma \in[0, \pi)$ consider the system of equations

$$ \begin{aligned} & 2 \sin \alpha-\cos \beta+3 \tan \gamma=3 \\ & 4 \sin \alpha+2 \cos \beta-2 \tan \gamma=2 \\ & 6 \sin \alpha-3 \cos \beta+\tan \gamma=9 \end{aligned} $$

Then, which one of the following is true?

$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$