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The starting compound is 2-methylbut-1-ene.
$$CH_2=C(CH_3)CH_2CH_3$$
Both reactions convert the alkene into an alcohol by adding $$H$$ and $$OH$$ across the double bond. The difference lies in the position at which the $$OH$$ group is introduced.
For reaction A, the reagents are $$BH_3/THF$$ followed by $$H_2O_2/OH^-$$.
This is the hydroboration-oxidation reaction.
Hydroboration-oxidation follows anti-Markovnikov addition. Therefore, the $$OH$$ group is introduced at the less substituted carbon of the double bond.
The product formed is
$$HOCH_2CH(CH_3)CH_2CH_3$$
which is 2-methylbutan-1-ol.
Thus, product A is a primary alcohol.
For reaction B, the reagents are $$Hg(OAc)_2,\ H_2O$$ followed by $$NaBH_4$$.
This is the oxymercuration-demercuration reaction.
Oxymercuration-demercuration follows Markovnikov addition without carbocation rearrangement. Therefore, the $$OH$$ group is introduced at the more substituted carbon of the double bond.
The product formed is
$$CH_3C(OH)(CH_3)CH_2CH_3$$
which is 2-methylbutan-2-ol.
Thus, product B is a tertiary alcohol.
Therefore,
$$A=\text{2-methylbutan-1-ol}$$
and
$$B=\text{2-methylbutan-2-ol}$$
Hence, the correct answer is
$$\text{Option A}$$
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