In the given equations (Note : equations are not balanced)

\(Zn + \text{ conc. }HNO_3 \longrightarrow Zn(NO_3)_2 + x + H_2O\) -------(1)

\(Zn + \text{ Dil. }HNO_3 \longrightarrow Zn(NO_3)_2 + y + H_2O\) ---------(2)

The compound \(x\) and \(y\) respectively are:

\(NO_2 \text{ and } N_2O\)

The correct answer is \((4) \, (\text{NO}_2 \text{ and } \text{N}_2\text{O})\).
When zinc reacts with concentrated nitric acid, the products are zinc nitrate, nitrogen dioxide, and water. The reaction equation is as follows:
\[\text{Zn} + 4\text{HNO}_3(\text{conc.}) \rightarrow \text{Zn(NO}_3\text{)}_2 + 2\text{NO}_2 + 2\text{H}_2\text{O}\]
When zinc reacts with dilute nitric acid, the products are zinc nitrate, nitrous oxide, and water. The reaction equation is as follows:
\[4\text{Zn} + 10\text{HNO}_3(\text{dil.}) \rightarrow 4\text{Zn(NO}_3\text{)}_2 + 5\text{H}_2\text{O} + \text{N}_2\text{O}\]
Therefore, the compounds \(x\) and \(y\) respectively are \(\text{NO}_2\) and \(\text{N}_2\text{O}\).

The other options are incorrect:
\(1) \, (\text{NO}_2 \text{ and } \text{NO})\): NO is not a product of the reaction of zinc with concentrated nitric acid.
\(2) \, (\text{NO}_2 \text{ and } \text{NO}_2)\): \(\text{NO}_2\) is a product of the reaction of zinc with both concentrated and dilute nitric acid.
\(3) \, (\text{N}_2\text{O} \text{ and } \text{NO}_2)\): \(\text{N}_2\text{O}\) is a product of the reaction of zinc with dilute nitric acid, but NO is not a product of the reaction of zinc with concentrated nitric acid.