The number of elements in the set $\left\{A=\left(\begin{array}{ll}a & b \\ 0 & d\end{array}\right): a, b, d \in\{-1,0,1\}\right.$ and $\left.(I-A)^{3}=I-A^{3}\right\}$ where $I$ is $2 \times 2$ identity matrix, is :

Let $P = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\ 
  3&1&0 \\ 
  9&3&1 
\end{array}} \right]$ and $Q = [q_{ij}]$ be two $3\times3$ matrices such that $Q -P^5 = I_3$. Then $\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}$ is equal to

Let $p$ , $q$ , $r$ are three real numbers satisfying $\left[ {p\,\,q\,\,r} \right]\left[ {\begin{array}{*{20}{c}}
  2&p&q \\ 
  { – 3}&q&{ – p + r} \\ 
  {12}&r&{ – q + 3r} 
\end{array}} \right] = \left[ {5\,\,\,b\,\,c} \right]$ , then minimum value of $(b + c)$ is 

$\cos \theta \left[ {\begin{array}{*{20}{c}}{\cos \theta }&{\sin \theta }\\{ – \sin \theta }&{\cos \theta }\end{array}} \right] + \sin \theta \left[ {\begin{array}{*{20}{c}}{\sin \theta }&{ – \cos \theta }\\{\cos \theta }&{\sin \theta }\end{array}} \right] = $

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.