The IUPAC name of is: [IOE-2068]
a. 1,1-dimethyl - 1 - hydroxyl-2-propanone
c. 3-hydroxy-3-methyl-3-hydroxy butane
d. 2-carboxy-3-methyl-3-hydroxy butane
I doubt that the question is wrong as for ( C*HC(CH3)-CO-CH3 ) compound,
the starred Carbon(C*) doesn't complete 4 bonds.
It should rather be CH3-C(CH3)-CO-CH3
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