If $a \ne p,b \ne q,c \ne r$ and $\left| {\,\begin{array}{*{20}{c}}p&b&c\\{p + a}&{q + b}&{2c}\\a&b&r\end{array}\,} \right|$ =$ 0$, then $\frac{p}{{p - a}} + \frac{q}{{q - b}} + \frac{r}{{r - c}} = $

Let the system of linear equations  $x+y+k z=2$ ;  $2 x+3 y-z=1$ ; $3 x+4 y+2 z=k$ , have infinitely many solutions. Then the system $( k +1) x +(2 k -1) y =7$ ; $(2 k +1) x +( k +5) y =10 \text { has : }$

The roots of the determinant equation (in $x$) $\left| {\,\begin{array}{*{20}{c}}a&a&x\\m&m&m\\b&x&b\end{array}\,} \right| = 0$

The system of linear equations  $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if

Let $\alpha $ and $\beta $ be the roots of the equation $x^2 + x + 1 = 0.$ Then for $y \ne 0$ in $R,$ $\left| {\begin{array}{*{20}{c}}
{y\, + \,1}&\alpha &\beta \\
\alpha &{y\, + \,\beta }&1\\
\beta &1&{y\, + \,\alpha }
\end{array}} \right|$ is equal to

If $px^4 + qx^3 + rx^2 + sx + t$ $\equiv$ $\left| {\begin{array}{*{20}{c}}{{x^2}\, + \,\,3x}&{x\, - \,1}&{x\, + \,3}\\{x\, + \,1}&{2\, - \,x}&{x\, - \,3}\\{x\, - \,3}&{x\, + \,4}&{3x}\end{array}} \right|$ then $t =$