Step 1: Understanding the Question:
The question tests concepts of isomerism, specifically metamerism and tautomerism in cyclic ketones.
Step 2: Detailed Explanation:
Statement-I: Metamers are isomers that differ in the distribution of carbon atoms around a polyvalent functional group. In cyclic ketones like cyclohexanone, the ring is treated as the frame and substituents change position. In 2,4-Diethylcyclohexanone vs 2-methyl-6-propylcyclohexanone, the nature and arrangement of alkyl groups on the ring relative to the carbonyl group change, while the molecular formula remains the same. Hence they are considered metamers. Statement-I is correct.
Statement-II: Tautomerism (Keto-Enol) requires at least one acidic \(\alpha\)-hydrogen. In 2,2,6,6-tetraethylcyclohexanone, both \(\alpha\)-carbons (positions 2 and 6) are fully substituted with two ethyl groups each. There are no \(\alpha\)-hydrogens available on the ring. Therefore, it cannot show tautomerism. Statement-II is incorrect.
Step 3: Final Answer:
Statement-I is correct and Statement-II is incorrect.