Suppose \( \omega \) is a cube root of unity with \( \omega \neq 1 \). Suppose P and Q are the points on the complex plane defined by \( \omega \) and \( \omega^{2} \). If O is the origin, then what is the angle between OP and OQ?
Suppose there is a relation * between the positive numbers x and y given by x * y if and only if \( x \leq y^{2} \). Then which one of the following is correct?
A.
* is reflexive but not transitive and symmetric
B.
* is transitive but not reflexive and symmetric
C.
* is symmetric and reflexive but not transitive
D.
* is neither reflexive, nor symmetric, nor transitive
If \( z = x + iy = \left(\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right)^{-25} \), where \( i = \sqrt{-1} \), then what is the fundamental amplitude of \( \frac{z - \sqrt{2}}{z - i\sqrt{2}} \)?
