If $A = \left[ {\begin{array}{*{20}{c}}
1&1\\
1&1
\end{array}} \right]$ and $\det ({A^n} - I) = 1 - {\lambda ^n}\,,\,n \in N$ then $\lambda $ is

If $D = \left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}\,} \right|$ for $x \ne 0,y \ne 0$ then $D$ is

If $S$ is the set of distinct values of $'b'$ for which the following system of linear equations $x + y + z = 1;x + ay + z = 1;ax + by + z = 0$ has no solution , then $S$ is :

If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $\mathrm{p}$ is $.........$

If $A, B, C$ are the angles of triangle then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {\sin \,2A}&{\sin \,C}&{\sin \,B} \\ 
  {\sin \,C}&{\sin \,2B}&{\sin A} \\ 
  {\sin \,B}&{\sin \,A}&{\sin \,2C} 
\end{array}} \right|$ is

Consider the system of linear equations

$-x+y+2 z=0$

$3 x-a y+5 z=1$

$2 x-2 y-a z=7$

Let $S_{1}$ be the set of all $\mathrm{a} \in {R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in {R}$ for which the system has infinitely many solutions. If $n\left(S_{1}\right)$ and $n\left(S_{2}\right)$ denote the number of elements in $S_{1}$ and $\mathrm{S}_{2}$ respectively, then