Evaluate the determinants : $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

The following system of linear equations  $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$  ;$x-y+4 z=8$

The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

If the system of equations

$ x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 $

$ x+(\cos \alpha) y+(\sin \alpha) z=0 $

$ x+(\sin \alpha) y-(\cos \alpha) z=0$

has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to :

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations

$x+y+z=5$    ;    $x+2 y+3 z=\mu$   ;     $x+3 y+\lambda z=1$

is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :