If $\left| \begin{array}{ccc} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{array} \right| = k(x + y + z)(x - z)^2$,then $k = $

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 $.....$

If $a=1+2+4+\cdots$ up to $n$ terms,$b=1+3+9+\cdots$ up to $n$ terms and $c=1+5+25+\cdots$ up to $n$ terms,then $\Delta=\left|\begin{array}{ccc}a & 2b & 4c \\ 2 & 2 & 2 \\ 2^n & 3^n & 5^n\end{array}\right|=$

Prove that $\left|\begin{array}{ccc}a & a+b & a+b+c \\ 2a & 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c\end{array}\right|=a^{3}$

If $A$ is any square matrix of order $3 \times 3$,then $|3A|$ is equal to:

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}, \dots$ are in $G.P.$,then the value of the determinant $\left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} & \log {a_{n+2}} \\ \log {a_{n+3}} & \log {a_{n+4}} & \log {a_{n+5}} \\ \log {a_{n+6}} & \log {a_{n+7}} & \log {a_{n+8}} \end{array} \right|$ is