If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}?$

Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ vanish?

1. a + b + c = 0

2. a³ + b³ + c³ = 3abc

3. a² + b² + c² - ab - bc - ca = 0

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For what value of k is the matrix $\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}$ singular?

If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?