Step 1: State the concept.
This is a factual public health question about when the World Health Organization (WHO) officially removed India from the list of polio-endemic countries.
Step 2: Recall the milestone.
After massive pulse polio immunisation campaigns, India completed a full year without a new wild polio case, and the WHO acknowledged this achievement in early $2012$.
Step 3: Pin down the exact date.
The official date on which India was struck off the polio-endemic list was $25$ February, $2012$.
Step 4: Eliminate the December date.
Option 1 ($25$ December, $2012$) is incorrect; the recognition came earlier in the year, not in December.
Step 5: Eliminate the other February and January dates.
Option 2 ($15$ February, $2012$) and option 3 ($25$ January, $2012$) are close but do not match the recorded date, so both are rejected.
Step 6: Conclude.
The correct date is $25$ February, $2012$, which is option 4.
\[ \boxed{\text{Option 4: 25 February, 2012}} \]