Answer

Correct Option: (i) recursive (Turing decidable = recursive)

Explanation

Turing decidable ≡ Recursive language ≡ Decided by a TM that always halts (decider).

Definitions:

• Recursively Enumerable (RE): TM recognizes $L$ (accepts if $w\in L$, may loop if $w
  otin L$). Options (ii) “TM recognizes” and (iii) “TM accepts” describe RE, not decidable.
• Recursive: TM decides $L$ : for every $w$, TM halts and says accept/reject.

So $L$ is Turing decidable iff $L$ is recursive.

Decider vs Recognizer distinction is key: recognizer may not halt on rejection.

Final Answer

Decidable = recursive → Option (i).