Step 1: Understanding the reaction.
The given reaction involves the hydration of an alkene (but-1-ene) to form an alcohol (butan-2-ol). This reaction typically proceeds through the acid-catalyzed hydration of alkenes.
Step 2: Analyzing the options.
(A) H$_2$O$_2$: Hydrogen peroxide is not suitable for this transformation as it does not catalyze the hydration of alkenes.
(B) KOH aqueous: Potassium hydroxide in aqueous solution typically induces elimination reactions, not hydration.
(C) (i) cold Conc. H$_2$SO$_4$, (ii) H$_2$O: This is the correct answer. Cold concentrated sulfuric acid adds to the alkene, and hydrolysis with water results in the formation of butan-2-ol.
(D) B$_2$H$_6$: Diborane is used in hydroboration-oxidation reactions, but this is not the correct reagent for this specific transformation.
Step 3: Conclusion.
The correct answer is (C) (i) cold Conc. H$_2$SO$_4$, (ii) H$_2$O.