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

$\left|\begin{array}{ccc} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{array}\right| = $

$\left|\begin{array}{ccc}1 & bc+ad & b^2c^2+a^2d^2 \\ 1 & ca+bd & c^2a^2+b^2d^2 \\ 1 & ab+cd & a^2b^2+c^2d^2\end{array}\right|=$

If $x, y, z$ are distinct and $\Delta=\left|\begin{array}{lll}x & x^{2} & 1+x^{3} \\ y & y^{2} & 1+y^{3} \\ z & z^{2} & 1+z^{3}\end{array}\right|=0,$ then show that $1+x y z=0$.

If $a, b, c$ are all different from zero and $\left| \begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{array} \right| = 0$,then the value of $a^{-1} + b^{-1} + c^{-1}$ is

The value of $\left| \begin{matrix} \sin \alpha & \cos \alpha & \sin(\alpha + \gamma) \\ \sin \beta & \cos \beta & \sin(\beta + \gamma) \\ \sin \delta & \cos \delta & \sin(\delta + \gamma) \end{matrix} \right|$ is