#### Answer

The degree of unsaturation in the molecules is given below:
(a)$C_{6}H_{5}N$ = 5
(b)$C_{6}H_{5}NO_{2}$ = 5
(c)$C_{8}H_{9}Cl_{3}$ = 3
(d)$C_{9}H_{16}Br_{2}$ = 1
(e)$C_{10}H_{12}N_{2}O_{3}$ = 6
(f)$C_{20}H_{32}ClN$ = 5

#### Work Step by Step

The degree of unsaturation in a compound is the number of multiple bonds and/or rings present in the molecule of that compound.
To calculate the degree of unsaturation, subtract the number of hydrogen atoms present in the compound from the number of hydrogen atoms that its corresponding saturated hydrocarbon would possess.
For Example, If a compound has a formula $C_{4}H_{8}$, its corresponding saturated hydrocarbon would be, $C_{4}H_{10}$. Here, the degree of unsaturation is given by - ($H_{10} - H_{8}$ = $H_{2}$) and thus the degree of unsaturation is 1; i.e. the compound has one fewer pair of hydrogen.
For the compounds containing elements other than C and H, the degree of unsaturation is calculated by keeping in mind the following rules:
1. If a halogen substituent is present in the compound, we add the number of halogens and hydrogens to obtain the equivalent hydrocarbon formula to calculate the degree of unsaturation.
2. If oxygen is present as a substituent in the compound, there is no change in the formula of the equivalent hydrocarbon, and thus is ignored while calculating the degree of unsaturation.
3. If a nitrogen atom is present as a substituent in the compound, we subtract the number of Nitrogen atoms from the number of hydrogen atoms to arrive at the formula of the corresponding hydrocarbon.
(a) In $C_{6}H_{5}N$, the substituent is a N atom.
Subtracting the number of N atoms from the number of H atoms, we get the formula of the hydrocarbon - $C_{6}H_{4}$.
The saturated counterpart of this hydrocarbon is $C_{6}H_{14}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{14} - H_{4}$ = $H_{10} = 5H_{2}$).
Therefore the degree of unsaturation is 5.
(b) In $C_{6}H_{5}NO_{2}$,
the substituents are a N atom and 2 O atoms.
Subtracting the number of N atoms from the number of H atoms and ignoring the number of O atoms we get the formula of the hydrocarbon - $C_{6}H_{4}$.
The saturated counterpart of this hydrocarbon is $C_{6}H_{14}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{14} - H_{4}$ = $H_{10} = 5H_{2}$).
Therefore the degree of unsaturation is 5.
(c) In $C_{8}H_{9}Cl_{3}$, the substituents are 3 Cl atoms.
Adding the number of Cl atoms to the number of H atoms, we get the formula of the hydrocarbon - $C_{8}H_{12}$.
The saturated counterpart of this hydrocarbon is $C_{8}H_{18}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{18} - H_{12}$ = $H_{6} = 3H_{2}$).
Therefore the degree of unsaturation is 3.
(d) In $C_{9}H_{16}Br_{2}$, the substituents are 2 Br atoms.
Adding the number of Br atoms to the number of H atoms, we get the formula of the hydrocarbon - $C_{9}H_{18}$.
The saturated counterpart of this hydrocarbon is $C_{9}H_{20}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{20} - H_{18}$ = $H_{2}$).
Therefore the degree of unsaturation is 1.
(e) In $C_{10}H_{12}N_{2}O_{3}$, the substituents are 2 N atoms and 3 O atoms.
Subtracting the number of N atoms from the number of H atoms and ignoring the number of O atoms, we get the formula of the hydrocarbon - $C_{10}H_{10}$.
The saturated counterpart of this hydrocarbon is $C_{10}H_{22}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{22} - H_{10}$ = $H_{12} = 6H_{2}$).
Therefore the degree of unsaturation is 6.
(f) In $C_{20}H_{32}ClN$, the substituents are one Cl and one N atoms.
Subtracting the number of N atoms from the number of H atoms and adding the number of Cl atoms to the number of H atoms, we get the formula of the hydrocarbon - $C_{20}H_{32}$.
The saturated counterpart of this hydrocarbon is $C_{20}H_{42}$ and thus the degree of unsaturation is determined by calculating the number of missing hydrogen pairs, i.e. ($H_{42} - H_{32}$ = $H_{10} = 5H_{2}$).
Therefore the degree of unsaturation is 5.