Let $a, b, c, d$ be in arithmetic progression with common difference $\lambda$. If
$\left|\begin{array}{lll} x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c \end{array}\right|=2$
then the value of $\lambda^{2}$ is equal to $.....$

By using properties of determinants,show that:
$\left|\begin{array}{ccc}a^{2}+1 & a b & a c \\ a b & b^{2}+1 & b c \\ c a & c b & c^{2}+1\end{array}\right|=1+a^{2}+b^{2}+c^{2}$

If $x, y$ and $z$ are greater than $1$,then the value of $\left|\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|$ is

If $\Delta = \begin{vmatrix} x & y & z \\ p & q & r \\ a & b & c \end{vmatrix}$,then $\begin{vmatrix} x & 2y & z \\ 2p & 4q & 2r \\ a & 2b & c \end{vmatrix}$ equals

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}, \dots$ are in $G$.$P$. and ${a_i} > 0$ for each $i$,then the value of the determinant $\Delta = \begin{vmatrix} \log {a_n} & \log {a_{n+2}} & \log {a_{n+4}} \\ \log {a_{n+6}} & \log {a_{n+8}} & \log {a_{n+10}} \\ \log {a_{n+12}} & \log {a_{n+14}} & \log {a_{n+16}} \end{vmatrix}$ is equal to

If $\left|\begin{array}{ccc}a+b+2c & a & b \\ c & 2a+b+c & b \\ c & a & a+2b+c\end{array}\right|=2$,then $a^3+b^3+c^3-3abc=$