Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$

Given below are two statements:

Statement I: $f(-x)$ is the inverse of the matrix $f(x)$.

Statement II: $f(x) f(y)=f(x+y)$.

In the light of the above statements, choose the correct answer from the options given below

Let $A =\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$ and $B =\left[\begin{array}{ll}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in R$. Let $\alpha_{1}$ be the value of $\alpha$ which satisfies $( A + B )^{2}= A ^{2}+\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ and $\alpha_{2}$ be the value of $\alpha$ which satisfies $( A + B )^{2}= B ^{2}$. Then $\left|\alpha_{1}-\alpha_{2}\right|$ is equal to.

If $R(t) = \left[ {\begin{array}{*{20}{c}}{\cos t}&{\sin t}\\{ – \sin t}&{\cos t}\end{array}} \right],$then $R(s).\,R(t) = $

$A$ and $B$ be $3 \times 3$ matrices such that $AB + A + B = 0$ , then