Answer
Please see the work below.
Work Step by Step
a) In $H_{3}C-CH_{3}$, there are only single bonds around carbon atoms and hence the hybridization of carbon atoms in ethane is $sp^{3}$.
b) In $H_{3}C-CH꓿CH_{2}$, there are only single bonds around the first carbon atom $H_{3}C-$ and hence the hybridization of this carbon atom is $sp^{3}$. The other two carbon atoms $-CH꓿CH_{2}$, have one double bond and two single bonds and hence they are in $sp^{2}$ hybridized state.
c) In $H_{3}C-C≡C-CH_{2}OH$, there are only single bonds around the first carbon atom $H_{3}C-$ and last carbon atom $-CH_{2}OH$ hence the hybridization of these two carbon atoms are $sp^{3}$. The other two carbon atoms $-C≡C-$ have one triple bond and one single bond and hence they are in a $sp$ hybridized state.
d) In $H_{3}C-CH꓿O$, there are only single bonds around the first carbon atom $H_{3}C-$ and hence the hybridization of this carbon atom is $sp^{3}$. The other carbon atom $-CH꓿O$, has one double bond and two single bonds, and hence the hybridization is $sp^{2}$.
e) In $H_{3}C-COOH$, there are only single bonds around the first carbon atom $H_{3}C-$ and hence the hybridization of this carbon atom is $sp^{3}$. The other carbon atom $-COOH$, has one double bond and two single bonds, and hence the hybridization is $sp^{2}$.
