Maths Test

If \(\omega\) is a non-real cube root of unity, then what is one root of the equation
\[
\left|\begin{array}{ccc}
x+1 & \omega & \omega^{2} \\
\omega & x+\omega^{2} & 1 \\
\omega^{2} & 1 & x+\omega
\end{array}\right|=0
\]?

If \(x^{2}-x+1=0\) then
\[
\left(x-\frac{1}{x}\right)^{2}+\left(x-\frac{1}{x}\right)^{4}+\left(x-\frac{1}{x}\right)^{8}
\]
is equal to what?

If \[\left|\begin{array}{ccc}2 & 3+i & -1 \\3-i & 0 & i-1 \\-1 & -1-i & 1\end{array}\right| = A + i\,B\]where \(i=\sqrt{-1}\), then what is the value of \(A+B\)?

If ω is a non-real cube root of unity, then what is a root of the following equation?
\[
\left|\begin{array}{ccc}
x+1 & ω & ω^{2} \\
ω & x+ω^{2} & 1 \\
ω^{2} & 1 & x+ω
\end{array}\right|=0
\]

If \(A^{2}+B^{2}+C^{2}=0\), then what is the value of the following?
\[
\left|\begin{array}{ccc}
1 & \cos C & \cos B \\
\cos C & 1 & \cos A \\
\cos B & \cos A & 1
\end{array}\right|
\]

What is \( \left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{3} \) equal to?