Option 1 states that the Doctrine of Lapse did not allow Indian rulers to adopt any heir. This statement is incorrect. The Doctrine of Lapse did not prevent Indian rulers from adopting heirs, but it regulated the adoption process and required approval from the British authorities.

Option 2 states that the Doctrine of Lapse did not allow an adopted heir to rule a state after the death of the ruler. This statement is incorrect. The Doctrine of Lapse did allow an adopted heir to rule a state, as long as the adoption was approved by the British authorities.

Option 3 states that the Doctrine of Lapse made the annexation of Indian states compulsory after the death of a ruler. This statement is incorrect. While the Doctrine of Lapse did lead to several annexations of Indian states, it was not an automatic or compulsory process. Annexation was decided on a case-by-case basis, depending on factors such as the ruler`s ability to govern and the British East India Company`s interest in expanding their territories.

Option 4 correctly states that the Doctrine of Lapse made the annexation of Indian states compulsory if the adoption of an heir had not been approved by the British authorities. This means that if a ruler died without a legitimate natural or adopted heir that was approved by the British