\(\mathrm{V}_2 \mathrm{O}_5, \mathrm{Cr}_2 \mathrm{O}_3\) is amphoteric oxide.
As we know that amphoteric oxides react with both acid as well as a base. So, from the above-given compounds, only vanadium pentoxide (\(\mathrm{V}_2 \mathrm{O}_5)\) and chromium oxide \((\mathrm{Cr}_2 \mathrm{O}_3)\) are amphoteric oxides.
This is so because vanadium pentoxide (\(\mathrm{V}_2 \mathrm{O}_5)\) can react with both acid and base. The vanadium pentoxide (\(\mathrm{V}_2 \mathrm{O}_5)\) reacts with base and acid to yield \(\mathrm{VO}_4^{-3}\) and \(\mathrm{VO}_4^{+}\)respectively.
Similarly, chromium oxide \((\mathrm{Cr}_2 \mathrm{O}_3)\) can also react with both acid and base.
It gives both acid as well as a base to give
\(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{3+}\) and \(\left[\mathrm{Cr}(\mathrm{OH})_6\right]^{3-}\) respectively.
Manganese heptoxide \( \mathrm{Mn}_2 \mathrm{O}_7\) is an acid in which manganese has the oxidation state of \(+7\) it means that manganese tends to accept the pair of electrons that's why it cannot react with a base to give a pair of electrons and is considered Lewis acid.
Chromium oxide \(\mathrm{CrO}_2\), is present which is basic and because it has an oxidation state of \(+2\) and we know that the compound having a low oxidation state is basic.
In \(\mathrm{V}_2 \mathrm{O}_4\), the oxidation state of vanadium is \(+4\) which shows that it tends to donate a pair of the electron that's why it is as basic only and not amphoteric.