Find the value of $(x+y)$ ,if ${(x+\frac{y^3}{x^2})}^{-1}-{(\frac{x^2}{y}+\frac{y^2}{x})}^{-1}+{(\frac{x^3}{y^2}+y)}^{-1}=\frac{1}{3}$

Find the values of $x\&y$ when $\frac{2x}{3}+\frac{y}{2}=3$ and $\frac{x}{2}-\frac{2y}{3}=\frac{1}{6}$

Find the value of $(\frac{1}{x}+\frac{1}{y})(\frac{1}{x}-\frac{1}{y})(\frac{1}{x^2}+\frac{1}{y^2})$

Show that:${\left(a^{x-y}\right)}^{x+y}.{\left(a^{y-z}\right)}^{y+z}.{\left(x^{z-x}\right)}^{z+x}=1$

If $xandy$ are of same sign, then the value of $\frac{x^3}{2}\cos e{c}^2\left(\frac{1}{2}\frac{{\tan }^{-1}x}{y}\right)+\frac{y^3}{2}{\sec }^2\left(\frac{1}{2}\frac{{\tan }^{-1}y}{x}\right)$ is equal to $(x-y)(x^2+y^2)$ (b) $(x+y)(x^2-y^2)$ $(x+y)(x^2+y^2)\stackrel{.}{}$ (d) $(x-y)(x^2-y^2)$

Find the value of $x$ , if $\left[\begin{array}{cc}3x+y& -y\\ 2y-x& 3\end{array}\right]=\left[\begin{array}{cc}1& 2\\ -5& 3\end{array}\right]$ .

x-y relation is given by $y=x^n$ calculate the value of n in the following :-
(i)$y=\frac{x^2}{x^{2/3}}$
(ii)$y=\frac{x^{3/2}}{x^1}$
(iii)$y=\frac{x^{1/2}}{x^{1/2}}$
(iv)$y=\frac{x^{3/2}}{\sqrt{x}}$

Find the following products and verify the result for x $=-1,y=-2:$ $(3x-5y)(x+y)$