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

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

If $z = x + iy$ with $xy \neq 0$ satisfies the equation $z^2 + i\bar{z} = 0$,then $|z^2|$ is equal to:

If $x-iy = \sqrt{\frac{a-ib}{c-id}}$,prove that $(x^2+y^2)^2 = \frac{a^2+b^2}{c^2+d^2}$.

If the conjugate of $(x+iy)(1-2i)$ is $(1+i)$,then

If $|a_k| < 1, \lambda_k \ge 0$ for $k = 1, 2, ..., n$ and $\lambda_1 + \lambda_2 + ... + \lambda_n = 1$,then the value of $|\lambda_1 a_1 + \lambda_2 a_2 + ... + \lambda_n a_n|$ is