Which one of the following function(s) is/are homogeneous?

(a) $\mathrm{f}(\mathrm{\lambda x},\mathrm{\lambda y})=\frac{\mathrm{\lambda }(\mathrm{x}-\mathrm{y})}{{\mathrm{\lambda }}^2\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}={\mathrm{\lambda }}^{-1}\mathrm{f}(\mathrm{x},\mathrm{y})$
Thus, it is homogeneous of degree -1.
(b) $\mathrm{f}(\mathrm{\lambda x},\mathrm{\lambda y})=\left(\mathrm{\lambda x}{)}^{1/3}\right(\mathrm{\lambda y}{)}^{-2/3}{\tan }^{-1} \frac{\mathrm{x}}{\mathrm{y}}$
$\begin{array}{l}={\mathrm{\lambda }}^{-1/3}{\mathrm{x}}^{1/3}{\mathrm{y}}^{-2/3}{\tan }^{-1} \frac{\mathrm{x}}{\mathrm{y}}\\ ={\mathrm{\lambda }}^{-\frac{1}{3}}\mathrm{f}(\mathrm{x},\mathrm{y})\end{array}$
Thus, it is homogeneous.
(c) $\mathrm{f}(\mathrm{\lambda x},\mathrm{\lambda y})=\mathrm{\lambda x}\left(\ln \sqrt{{\mathrm{\lambda }}^2\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}-\ln \mathrm{\lambda y}\right)+{\mathrm{\lambda ye}}^{\mathrm{x}/\mathrm{y}}$
$\begin{array}{l}=\mathrm{\lambda x}\left[\ln \left(\frac{\mathrm{\lambda }\sqrt{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}}{\mathrm{\lambda y}}\right)\right]+{\mathrm{\lambda ye}}^{\mathrm{x}/\mathrm{y}}\\ =\mathrm{\lambda }\left[\mathrm{x}\left(\ln \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}-\ln \mathrm{y}\right)+{\mathrm{ye}}^{\mathrm{x}/\mathrm{y}}\right]=\mathrm{\lambda f}(\mathrm{x},\mathrm{y})\end{array}$
Thus, it is homogeneous.
(d) $\mathrm{f}(\mathrm{\lambda x},\mathrm{\lambda y})=\mathrm{\lambda x}\left[\ln \frac{2{\mathrm{\lambda }}^2{\mathrm{x}}^2+{\mathrm{\lambda }}^2{\mathrm{y}}^2}{\mathrm{\lambda x\lambda }(\mathrm{x}+\mathrm{y})}\right]+{\mathrm{\lambda }}^2{\mathrm{x}}^2\tan \frac{\mathrm{x}+2\mathrm{y}}{3\mathrm{x}-\mathrm{y}}$
$=\mathrm{\lambda x}\left[\ln \frac{2{\mathrm{x}}^2+{\mathrm{y}}^2}{\mathrm{x}(\mathrm{x}+\mathrm{y})}\right]+{\mathrm{\lambda }}^2{\mathrm{x}}^2\tan \frac{\mathrm{x}+2\mathrm{y}}{3\mathrm{x}-\mathrm{y}}$
Thus, it is non-homogeneous.