Complex Numbers Questions (654)

Let $z_1$ and $z_2$ be the non-real roots of the equation $3z^2+3z+b=0$. If the origin together with the points represented by $z_1$ and $z_2$ form an equilateral triangle, then the value of $b$ is :

Given \(x^2 - x + 1 = 0\) with roots \(\alpha\) and \(\beta\), the value of \(\alpha^{101} + \beta^{107}\) is:

Let $A(z)$ and $B(z_1)$ be two variable points such that $z z_1 = |z|^2$. If the area enclosed by $|z - \bar{z}| + |z_1 + \bar{z}_1| = 10$ is $A$ then the value of $A/8$ is ____.

The least positive integer \(n\) for which \(\left(\dfrac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)^n = 1\), is