The area of a triangle with vertices $A(k, 1)$,$B(2, 4)$,and $C(1, 1)$ is $6$ sq. units. Find the value of $k$.

The value of $\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|$ is

Let $A = \begin{bmatrix} -2 & x & 1 \\ x & 1 & 1 \\ 2 & 3 & -1 \end{bmatrix}$. If the roots of the equation $\operatorname{det}(A) = 0$ are $l$ and $m$,then find the value of $l^3 - m^3$.

If $a, b, c$ are non-zero real numbers and if the equations $(a-1) x=y+z, (b-1) y=z+x, (c-1) z=x+y$ have a non-trivial solution,then $ab+bc+ca=$

Let $\left|\begin{array}{ccc}x^2+x+1 & x+1 & 2x-3 \\ 3x^2-1 & x+2 & x-1 \\ x^2+5x+1 & 2x+3 & x+4\end{array}\right| = ax^4+bx^3+cx^2+dx+e$ be an identity in $x$. If $a, b, c, d$ are known, then the value of $e$ is

If $\omega \neq 1$ is a cube root of unity, then the value of the determinant $\left|\begin{array}{ccc}\omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\omega^{27} \\ \omega^{30}+\omega^{41} & \omega^{41}+\omega^{19} & \omega^{19}+\omega^{30}\end{array}\right|$ is: