The correct answer is option 4, Mysore. The Doctrine of Lapse was a policy introduced by the British East India Company, which stated that if a ruler of a princely state died without a male heir, the state would be annexed by the British. This policy was used to expand British control over Indian states.

Option 1, Satara, was annexed by applying the Doctrine of Lapse. Option 2, Sambalpur, was also annexed using this policy. Option 3, Jhansi, was annexed after the death of Rani Lakshmibai, who fought against the British during the Indian Rebellion of 1857. However, option 4, Mysore, was not annexed through the Doctrine of Lapse. Mysore was an independent princely state until it was incorporated into British India after the death of the ruler, Tipu Sultan, in the Fourth Anglo-Mysore War.