If \( A + B + C = \pi \), then what is the value of the following?
\( \begin{vmatrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{vmatrix} \)
If \( \omega \) is a non-real cube root of unity, then what is a root of the following equation?
\( \begin{vmatrix} x+1 & \omega & \omega^{2} \\ \omega & x+\omega^{2} & 1 \\ \omega^{2} & 1 & x+\omega \end{vmatrix} = 0 \)
How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all consonants come together in each word?
