If x + a + b + c = 0, then what is the value of $$\left| {\matrix{ {x + a} & b & c \cr a & {x + b} & c \cr a & b & {x + c} \cr } } \right|$$ ?

What are the values of x that satisfy the equation $$\left| {\matrix{ x & 0 & 2 \cr {2x} & 2 & 1 \cr 1 & 1 & 1 \cr } } \right| + \left| {\matrix{ {3x} & 0 & 2 \cr {{x^2}} & 2 & 1 \cr 0 & 1 & 1 \cr } } \right| = 0$$ ?

What is the value of the determinant $$\left| {\matrix{ {1!} & {2!} & {3!} \cr {2!} & {3!} & {4!} \cr {3!} & {4!} & {5!} \cr } } \right|$$ ?

For what value of $n$ is the determinant

$$ \left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) & C(12, n) \\ C(m, 7) & C(m, 6) & C(m+1, n+1) \end{array}\right|=0 $$

for every $m>n$ ?