If matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$,then the value of $a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33}$ is

The equation obtained by eliminating $a, b, c$ from the equations $x = \frac{a}{b-c}$,$y = \frac{b}{c-a}$,and $z = \frac{c}{a-b}$ is

If $a, b, c$ are in $A.P.$,then the value of $\left| \begin{array}{ccc} x+2 & x+3 & x+a \\ x+4 & x+5 & x+b \\ x+6 & x+7 & x+c \end{array} \right|$ is

If $a, b, c$ are sides of $\triangle ABC$ and $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$,then $\sin^2 A + \sin^2 B + \sin^2 C = $ . . . . . . .

For real numbers $x, y$ and $z$,if $x \neq y \neq z$,$\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$ and $\left|\begin{array}{ccc}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{array}\right| \neq 0$,then $xyz = $ . . . . . . .

If the area of a triangle is $3$ sq. units whose vertices are $A(1, 3)$,$B(0, 0)$,and $C(k, 0)$,then $k$ is equal to: