Suppose $D = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ and $D' = \left| {\,\begin{array}{*{20}{c}}{{a_1} + p{b_1}}&{{b_1} + q{c_1}}&{{c_1} + r{a_1}}\\{{a_2} + p{b_2}}&{{b_2} + q{c_2}}&{{c_2} + r{a_2}}\\{{a_3} + p{b_3}}&{{b_3} + q{c_3}}&{{c_3} + r{a_3}}\end{array}\,} \right|$, then

If $x = cy + bz,\,\,y = az + cx,\,\,z = bx + ay$ (where $x, y, z $ are not all zero) have a solution other than $x = 0$, $y = 0$, $z = 0$ then $a, b$  and $ c $ are connected by the relation

$2x + 3y + 4z = 9$,$4x + 9y + 3z = 10,$$5x + 10y + 5z = 11$ then the value of $ x$ is

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 $t =$