The IUPAC name of  is:                                  [IOE-2068]a. 1,1-dimethyl - 1 - hydroxyl-2-propanoneb. 2-hydroxy-2-methyl-2-butanonec. 3-hydroxy-3-methyl-3-hydroxy butaned. 2-carboxy-3-methyl-3-hydroxy butane

I doubt that the question is wrong as for ( C*HC(CH3)-CO-CH3  ) compound,

                                                                                 |

                                                                                 OH

 the starred Carbon(C*) doesn't complete 4 bonds.

It should rather be CH3-C(CH3)-CO-CH3

                                           |

                                          OH

Take the longest change, it has functional group of ketone so set priority order accordingly.

4 Carbon in the longest change and methyl group and OH in 3rd place so;

3-hydroxy 3-Methyl Butan-2-one