An organic compound having the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{8} \mathrm{O}\) is insoluble in \(\mathrm{NaHCO}_{3}\) solution but dissolves in aqueous \(\mathrm{NaOH}\). When treated with bromine water the compound rapidly forms a precipitate having the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{OBr}_{3} .\) The organic compound is (a) o-cresol (b) \(\mathrm{m}\) -cresol (c) \(\mathrm{p}\) -cresol (d) anisole

To analyze the solubility behavior of the compound, we need to consider the acidity of the given options. Compounds that are insoluble in \(\mathrm{NaHCO}_{3}\) but soluble in \(\mathrm{NaOH}\) are generally less acidic than carboxylic acids but still acidic enough to react with a strong base such as \(\mathrm{NaOH}\).

Now, let's analyze each option based on the solubility behavior and molecular formulas: (a) o-cresol: This is a phenol, which is less acidic than carboxylic acid but acidic enough to react with a strong base like \(\mathrm{NaOH}\). It has the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{8}\mathrm{O}\). (b) m-cresol: Same as o-cresol, this is also a phenol, and it has the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{8}\mathrm{O}\). (c) p-cresol: Like o-cresol and m-cresol, p-cresol is also a phenol, and it has the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{8} \mathrm{O}\). (d) Anisole: This is an ether compound, not a phenol, and has the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{8} \mathrm{O}\). Among these options, anisole is a weakly acidic compound (essentially non-acidic) and would not react with either \(\mathrm{NaHCO}_{3}\) or \(\mathrm{NaOH}\). Others are phenols, which have acidic properties as described earlier.

When treated with bromine water, the compound forms a precipitate having the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{OBr}_{3}\). Now, let's check which phenols can undergo such reaction: (a) o-cresol: When treated with bromine water, it can form a compound with the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{OBr}_{3}\) by substituting three hydrogen atoms with bromine atoms. (b) m-cresol: The ortho and para position of the -OH group in m-cresol is blocked by the methyl group, so bromination will occur at the meta position. The resulting compound would have a molecular formula \(\mathrm{C}_{7}\mathrm{H}_{7}\mathrm{OBr}\). (c) p-cresol: When treated with bromine water, it can form a compound with the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{OBr}_{3}\) by substituting three hydrogen atoms with bromine atoms. So, o-cresol and p-cresol can both react with bromine water and form precipitates as described.

Based on the given information and the analysis of all available options, the correct answer is (a) o-cresol, as it fulfills all the required conditions - insoluble in \(\mathrm{NaHCO}_{3}\), soluble in \(\mathrm{NaOH}\), and forming a precipitate of molecular formula \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{OBr}_{3}\) when treated with bromine water.