If $a, b, c$ and $d \in \mathbb{R}$ such that $a^2+b^2=4$ and $c^2+d^2=2$ and if $(a+ib)^2=(c+id)^2(x+iy)$,then $x^2+y^2$ is equal to

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to:

If $x = -5 + 2\sqrt{-4}$,then the value of the expression $x^4 + 9x^3 + 35x^2 - x + 4$ is

If $x_n = \cos \left(\frac{\pi}{4^n}\right) + i \sin \left(\frac{\pi}{4^n}\right)$,then the product $x_1 x_2 x_3 \ldots \infty$ is equal to

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is:

Given that the equation $z^2 + (p + iq)z + r + is = 0$,where $p, q, r, s$ are real and non-zero,has a real root,then: