Let's analyze both statements to determine their validity.
Statement I: Elements \(X\) and \(Y\) are the most and least electronegative elements, respectively, among \(N\), \(As\), \(Sb\), and \(P\). The nature of the oxides \(X_2O_3\) and \(Y_2O_3\) is acidic and amphoteric, respectively.
- The elements given are \(N\), \(As\), \(Sb\), and \(P\). The order of electronegativity is \(N > P > As > Sb\).
- Thus, \(N\) (Nitrogen) is the most electronegative, and \(Sb\) (Antimony) is the least electronegative.
- \(N_2O_3\) is acidic in nature because non-metal oxides like those of nitrogen are typically acidic.
- \(Sb_2O_3\) is amphoteric, meaning it can react with both acids and bases, which is true for certain metalloids and heavier non-metallic oxides.
Therefore, Statement I is true.
Statement II: \(BCl_3\) is covalent in nature and gets hydrolyzed in water. It produces \([B(OH)_4]^-\) and \([B(H_2O)_6]^{3+}\) in the aqueous medium.
- \(BCl_3\) is indeed covalent and susceptible to hydrolysis in water.
- During hydrolysis, \(BCl_3\) forms boric acid \((B(OH)_3)\) and hydrochloric acid \((HCl)\).
- \([B(OH)_4]^-\) can be formed when \(B(OH)_3\) further reacts with OH\(^-\), but \([B(H_2O)_6]^{3+}\) does not form because boron does not typically coordinate six water molecules in such a manner.
Therefore, Statement II is false because it incorrectly states the products of the hydrolysis of \(BCl_3\).
Hence, the correct answer is: Statement I is true but Statement II is false.