$R_f$ value for $2$-methylpropene in a solvent system (Ethyl acetate + ether) is $0.42$. $2$-methylpropene is treated with dilute $\text{H}_2\text{SO}_4$ to give major organic product (X). $R_f$ value for (X) in the same solvent system under identical condition will be:

$2$-methylpropene reacts with dilute $H_2SO_4$ to undergo hydration, forming tert-butyl alcohol as the major product (X).

The reaction is:

$CH_2=C(CH_3)_2 \xrightarrow{\text{dil. } H_2SO_4} (CH_3)_3C-OH$

Product (X) is an alcohol, which is significantly more polar than the starting alkene due to the presence of the $-OH$ group. In normal phase chromatography, the stationary phase (such as silica gel) is highly polar. The polar $-OH$ group forms strong hydrogen bonds with the stationary phase, causing the alcohol to be adsorbed more strongly than the non-polar alkene.

Consequently, the alcohol travels a shorter distance along the chromatography plate, resulting in a lower $R_f$ value compared to the alkene.

Since the $R_f$ value of $2$-methylpropene is $0.42$, the $R_f$ value of product (X) must be less than $0.42$. Among the given choices, only $0.12$ satisfies this condition.

Answer: $0.12$