Let's analyze each statement given in the question:
- Statement I: All the pairs of molecules \((\mathrm{PbO}, \mathrm{PbO_2}); (\mathrm{SnO}, \mathrm{SnO_2})\) and \((\mathrm{GeO}, \mathrm{GeO_2})\) contain amphoteric oxides.
- Amphoteric oxides can react with both acids and bases to form salt and water. Hence, they have the ability to exhibit both acidic and basic properties.
- \(\mathrm{PbO}\) and \(\mathrm{SnO}\) are indeed amphoteric. However, \(\mathrm{PB0_2}\) and \(\mathrm{SnO_2}\) are generally considered as more acidic oxides rather than amphoteric. Similarly, \(\mathrm{GeO}\) is basic, while \(\mathrm{GeO_2}\) is typically acidic.
- Thus, the statement that all specified oxides in each pair are amphoteric is incorrect.
- Statement II: \(\mathrm{AlCl_3}, \mathrm{BH_3}, \mathrm{BeH_2}\) and \(\mathrm{NO_2}\) all have incomplete octet.
- The term "incomplete octet" refers to molecules in which the central atom possesses fewer than eight electrons in its valence shell.
- \(\mathrm{AlCl_3}\), \(\mathrm{BH_3}\), and \(\mathrm{BeH_2}\) are classic examples of compounds with an incomplete octet, as they have six, six, and four electrons respectively in their outermost shell.
- \(\mathrm{NO_2}\) also manifests an odd number of valence electrons, leading to an incomplete octet for the nitrogen atom. Hence, \(\mathrm{NO_2}\) can be considered as having an incomplete octet as well.
- Therefore, Statement II is correct.
Conclusion: Statement I is incorrect because not all mentioned oxides are amphoteric. On the other hand, Statement II is correct, as all the stipulated compounds possess an incomplete octet.
Thus, the correct answer is: Statement I is incorrect but Statement II is correct.