Explanation:

To determine the number of signals in the 13C-NMR spectrum, we need to consider the different types of carbon atoms present in each compound. Each unique chemical environment of carbon atoms will result in a distinct signal in the spectrum.

The compounds given are:
1. O-xylene
2. p-xylene
3. 1,4-dimethoxybenzene
4. trans-1,2-dimethylcyclopropane
5. 4-bromoanisole
6. cis-1,2-dimethylcyclopropane
7. Resorcinol

O-xylene:
- O-xylene has three different types of carbon atoms.
- It contains a methyl group (-CH3) attached to an aromatic ring, which produces one signal.
- It also has two aromatic carbon atoms, which are chemically equivalent and produce another signal.
- Therefore, O-xylene shows a total of two signals in the 13C-NMR spectrum.

p-xylene:
- P-xylene also has three different types of carbon atoms.
- It contains a methyl group (-CH3) attached to an aromatic ring, which produces one signal.
- It also has two aromatic carbon atoms, which are chemically equivalent and produce another signal.
- Therefore, p-xylene shows a total of two signals in the 13C-NMR spectrum.

1,4-dimethoxybenzene:
- 1,4-dimethoxybenzene has three different types of carbon atoms.
- It contains two methoxy (-OCH3) groups attached to an aromatic ring, which produce one signal.
- It also has one aromatic carbon atom, which produces another signal.
- Therefore, 1,4-dimethoxybenzene shows a total of two signals in the 13C-NMR spectrum.

trans-1,2-dimethylcyclopropane:
- Trans-1,2-dimethylcyclopropane has four different types of carbon atoms.
- It contains two methyl groups (-CH3), which are chemically equivalent and produce one signal.
- It also has two different types of methylene (-CH2-) groups, which produce another two signals.
- Therefore, trans-1,2-dimethylcyclopropane shows a total of three signals in the 13C-NMR spectrum.

4-bromoanisole:
- 4-bromoanisole has four different types of carbon atoms.
- It contains a bromine atom (-Br) attached to an aromatic ring, which produces one signal.
- It also has two methoxy (-OCH3) groups attached to an aromatic ring, which produce another one signal.
- It has one aromatic carbon atom, which produces another signal.
- Therefore, 4-bromoanisole shows a total of three signals in the 13C-NMR spectrum.

cis-1,2-dimethylcyclopropane:
- Cis-1,2-dimethylcyclopropane has four different types of carbon atoms.
- It contains two methyl groups (-CH3), which are chemically equivalent and produce one signal.
- It also has two different types of methylene (-CH2-) groups, which produce another two