Find the real numbers $x$ and $y$ if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

If $(1 + i) \cdot (1 + 2i) \dots \dots \dots (1 + ni) = x + iy$ (where $i = \sqrt{-1}$), then find the value of $(2) \cdot (5) \cdot (10) \dots \dots \dots (1 + n^2)$.

The modulus of the square root of the complex number $6+8i$ (where $i=\sqrt{-1}$) is

Let $z = a - \frac{i}{2}$,where $a \in R$. Then $|i + z|^2 - |i - z|^2$ is equal to

For any two complex numbers $z_1$ and $z_2$ and any real numbers $a$ and $b$,what is the value of $|az_1 - bz_2|^2 + |bz_1 + az_2|^2$?

Let $Z$ be a complex number such that $|Z|+Z=2+i$ (where $i=\sqrt{-1}$),then $|Z|$ is equal to