If $z=x+i y$ and if the point $P$ in the argand diagram represents $z$, then the locus of the point $P$ satisfying the equation $2|z-2-3 i|=3|z+i-2|$ is a circle with centre

If $z$ is a non-real root of $x^7=1$, then $1+3 z+5 z^2+7 z^3+9 z^4+11 z^5+13 z^6=$

If $\cosh 2 x=199$, then $\cot h x=$

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$