Option 1 states that both the assertion (A) and the reason (R) are individually true, and that R is the correct explanation of A. In this case, the assertion is that Ranjit Singh signed the Treaty of Amritsar in 1809, and the reason given is that he was defeated by the East India Company. However, option 1 is incorrect because Ranjit Singh was not defeated by the East India Company. He was a powerful ruler who successfully halted the expansion of the British in Punjab. Therefore, option 1 is ruled out as the correct answer.

Option 2 states that both A and R are individually true, but R is not the correct explanation of A. This option is also incorrect, as the reason provided does not explain why Ranjit Singh signed the Treaty of Amritsar.

Option 3 is the correct answer. It states that A is true, meaning that Ranjit Singh signed the Treaty of Amritsar in 1809. However, it states that R is false, meaning that he was not defeated by the East India Company. The treaty was actually signed between the British and Ranjit Singh, recognizing the boundaries of their respective territories.

Option 4 states that A is false, but R is true. This is