If the system of equations $x + ay = 0,$ $az + y = 0$ and $ax + z = 0$ has infinite solutions, then the value of $a$ is

If the system of linear equations $x + ky + 3z = 0;3x + ky – 2z = 0$ ; $2x + 4y – 3z = 0$  has a non-zero solution $\left( {x,y,z} \right)$ then $\frac{{xz}}{{{y^2}}} = $. . . . .

For $\alpha, \beta \in \mathrm{R}$ and a natural number $\mathrm{n}$, let

$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$. Then $2 A_{10}-A_8$

If $\alpha ,\beta \ne 0$ and $f\left( n \right) = {\alpha ^n} + {\beta ^n}$ and $\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 – \alpha } \right)^2}$ ${\left( {1 – \beta } \right)^2}{\left( {\alpha – \beta } \right)^2}$ ,then $K=$ . . . . . .

Evaluate $\Delta=\left|\begin{array}{lll}3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3\end{array}\right|$