1. Write the structure clearly

   The given compound is: $\mathrm{HO-CH_2-(CH_2)_3-CH_2-COCH_3}$

   Expanding it: $\mathrm{HO-CH_2-CH_2-CH_2-CH_2-CH_2-CO-CH_3}$

   So the longest chain containing the carbonyl carbon has 7 carbon atoms.

2. Identify the principal functional group

   The compound contains:

   • an alcohol group $(-OH)$
   • a ketone group $(>C=O)$

   In IUPAC priority, ketone has higher priority than alcohol.

   Therefore, the parent name must use the suffix -one, and the $-OH$ group will be named as the prefix hydroxy.

3. Number the chain correctly

   Since ketone is the principal functional group, numbering must give the carbonyl carbon the lowest possible locant.

   Numbering from the methyl ketone end:

   $\mathrm{CH_3-CO-CH_2-CH_2-CH_2-CH_2-CH_2OH}$

   This gives:

   • ketone at carbon 2
   • hydroxy at carbon 7

   Hence the correct IUPAC name is: $\boxed{7\text{-hydroxyheptan-2-one}}$

4. Check Statement I

   Statement I says: IUPAC name is 7-hydroxyheptan-2-one.

   This is correct.

5. Check Statement II

   Statement II says: 2-oxoheptan-7-ol is the correct IUPAC name.

   This is incorrect, because:

   • using suffix -ol makes alcohol the principal functional group,
   • but ketone has higher priority than alcohol.

   So the compound should not be named as an alcohol with an oxo substituent.

6. Evaluate the options

   • A: Statement I incorrect, Statement II correct $\rightarrow$ false
   • B: Statement I correct, Statement II incorrect $\rightarrow$ true
   • C: Both correct $\rightarrow$ false
   • D: Both incorrect $\rightarrow$ false

Therefore, the correct option is: $\boxed{\text{B}}$