Complex Numbers · Mathematics · IAT (IISER)

Let $z_1, z_2$, and $z_3$ be complex numbers satisfying the following conditions

$$ 2=\left|2 z_1\right|=\left|z_2-1\right|=\left|z_3+1\right|=\left|\frac{1}{z_1}+\frac{1}{z_2-1}+\frac{1}{z_3+1}\right| . $$

What is the value of $\left|4 z_1+z_2+z_3\right|$ ?

Consider the following subset of the $X Y$-plane.

$$ S=\left\{\left(|z-\mathrm{i} z|,|z|^2\right): \quad z \text { is a complex number }\right\} $$

Which one of the following statements is correct?

Let $\omega$ be a complex root of the quadratic polynomial $x^2+x+1$. The value of

$$ \left(\omega+\frac{1}{\omega}\right) \cdot\left(\omega^2+\frac{1}{\omega^2}\right) \cdots\left(\omega^{100}+\frac{1}{\omega^{100}}\right) $$

is