Step 1: Recall why the Act came about.
Before this law, abortion in India was covered under the Indian Penal Code and was allowed only to save the mother's life. Unsafe illegal abortions were a major cause of maternal death, so the government set up the Shantilal Shah Committee to study the problem and recommend a change.
Step 2: Recall when Parliament passed the Act.
Based on that committee's report, the Parliament of India passed the Medical Termination of Pregnancy Act in 1971. The Act came into force from April 1, 1972, and it laid down the medical and legal conditions under which a pregnancy could be ended safely by a registered doctor.
Step 3: Check the other years.
In 1950 India had no separate abortion law at all, so this year is too early. In 1965 the Act had not yet been drafted or discussed in Parliament. In 1981 the Act was already ten years old and in force, so this year does not mark its passing.
Step 4: Final answer.
The Medical Termination of Pregnancy Act was passed by the Parliament of India in 1971.
\[ \boxed{1971} \]