$x=-5\text{i}y=2$  

$\text{a)}{x}^{-2}+y^2$

${\left(-5\right)}^{-2}+2^2={\left(-\frac{1}{5}\right)}^2+4=\frac{1}{25}+4=4\frac{1}{25}$  

$\underline{}$  

$\text{b)}{\left(-x{y}^2\right)}^2$  

${\left(-\left(-5\right)\cdot 2^2\right)}^2={\left(-\left(-5\right)\cdot 4\right)}^2={20}^2=400$  

$\underline{}$  

$\text{c)}{\left(x-2y^{-2}\right)}^2$  

${\left(\left(-5\right)-2\cdot 2^{-2}\right)}^2={\left(\left(-5\right)-2\cdot {\left(\frac{1}{2}\right)}^2\right)}^2={\left(\left(-5\right)-{\sout{2}}^1\cdot \frac{1}{{\sout{4}}^2}\right)}^2=$  

$={\left(-5\frac{1}{2}\right)}^2={\left(-\frac{11}{2}\right)}^2=\frac{121}{4}=30\frac{1}{4}$  

$\underline{}$  

$\text{d)}{\left(3x^{-2}y\right)}^2$  

${\left(3\cdot {\left(-5\right)}^{-2}\cdot 2\right)}^2={\left(3\cdot {\left(-\frac{1}{5}\right)}^2\cdot 2\right)}^2={\left(3\cdot \frac{1}{25}\cdot 2\right)}^2={\left(\frac{6}{25}\right)}^2=\frac{36}{625}$