The values of $x$ for which the given matrix $\left[\begin{array}{ccc}-x & x & 2 \\ 2 & x & -x \\ x & -2 & -2\end{array}\right]$ will be non-singular are

If $a_i, b_i, c_i \in \mathbb{R}$ for $i=1, 2, 3$ and $x \in \mathbb{R}$ and $\begin{vmatrix} a_1+b_1 x & a_1 x+b_1 & c_1 \\ a_2+b_2 x & a_2 x+b_2 & c_2 \\ a_3+b_3 x & a_3 x+b_3 & c_3 \end{vmatrix} = 0$, then:

Which of the following is correct?

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:

Evaluate the determinant: $\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$

If ${a^2} + {b^2} + {c^2} + ab + bc + ca \leq 0$ for all $a, b, c \in R$,then find the value of the determinant $\left| {\begin{array}{*{20}{c}} {{(a + b + c)}^2} & {{a^2} + {b^2}} & 1 \\ 1 & {{(b + c + 2)}^2} & {{b^2} + {c^2}} \\ {{c^2} + {a^2}} & 1 & {{(c + a + 2)}^2} \end{array}} \right|$.