Maths Test -1

Summary \[\Delta=\left|\begin{array}{ccc}k(k+2) & 2 k+1 & 1 \\2 k+1 & k+2 & 1 \\3 & 3 & 1\end{array}\right|\]Consider the following statements regarding \(\Delta\): I. \(\Delta\) is positive if \(k > 0\). II. \(\Delta\) is negative if \(k < 0\). III. \(\Delta\) is zero if \(k = 0\). How many of the above statements are correct?

If \[\Delta = \left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|\] and \(A, B, C, D, G\) are the cofactors of the elements \(a, b, c, d, g\) respectively, then what is the value of \(bB + cC - dD - gG\)?

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

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 \(x^{2}-x+1=0\), 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?

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